Rethinking Time Series Tokenization from a Frequency Perspective

# Rethinking Time Series Tokenization from a Frequency Perspective
Source: [https://arxiv.org/html/2610.28997](https://arxiv.org/html/2610.28997)
###### Abstract

Tokenization that partitions time series into subsequences has become a foundational paradigm in modern time series modeling\. However, we find that current tokenization strategies inherently introduce spectral distortions, forcing the learned representations to diverge from the true underlying patterns such as trend, periodicity, and seasonality, thereby impairing model performance\. Through theoretical analysis from a frequency perspective, we derive the boundary conditions under which such distortions occur\. To overcome this distortion, we proposeTimeFT, a nearly distortion\-free frequency\-based tokenizer for time series data\. TimeFT obtains tokens through frequency\-domain partitioning, frequency shifting, and Nyquist sampling\. The method is parameter\-free, incurs negligible computational complexity, and can serve as a drop\-in replacement for existing tokenizers\. Our extensive experiments on diverse tasks such as forecasting, classification, and anomaly detection demonstrate that TimeFT consistently and significantly improves performance\. The code has been made available at[https://anonymous\.4open\.science/r/TimeFT\-75E3/](https://anonymous.4open.science/r/TimeFT-75E3/)\.

## 1Introduction

Figure 1:*Left:*The processes of patch and phase tokenization in the time domain\.*Right:*Spectral shifting moves the frequency spectrum, thereby preventing high frequency components from aliasing into low frequencies\.Models\[[22](https://arxiv.org/html/2610.28997#bib.bib18)\]such as PatchTST\[[30](https://arxiv.org/html/2610.28997#bib.bib5)\], PatchMLP\[[36](https://arxiv.org/html/2610.28997#bib.bib4)\]and PhaseFormer\[[31](https://arxiv.org/html/2610.28997#bib.bib3)\]have established a tokenization paradigm\[[11](https://arxiv.org/html/2610.28997#bib.bib16),[18](https://arxiv.org/html/2610.28997#bib.bib17)\]for time series data by partitioning the raw sequence into a set of non\-overlapping \(or overlapping\) segments:

𝐙patch​i:=X\[∑j=1i−1Sj:∑j=1i−1Sj\+Li\]i∈\{1,2,…,I\}\\begin\{split\}\\mathbf\{Z\}\_\{\\mathrm\{patch\}\\ i\}:=&X\\\!\\left\[\\,\\sum\_\{j=1\}^\{i\-1\}S\_\{j\}\\,:\\,\\sum\_\{j=1\}^\{i\-1\}S\_\{j\}\+L\_\{i\}\\,\\right\]\\\\ &i\\in\\\{1,2,\\dots,I\\\}\\end\{split\}\(1\)
𝐙phase​i:=\[X⁡\[j\]​for​j=i,i\+S,…\]j≤N−1,i∈\{0,1,2,…,I\}\\begin\{split\}\\mathbf\{Z\}\_\{\\mathrm\{phase\}\\ i\}&:=\\left\[\\,X\[\\;j\\;\]\\;\\text\{ for \}\\;j=i,i\+S,\\dots\\right\]\\\\ &j\\leq N\-1,\\ i\\in\\\{0,1,2,\\dots,I\\\}\\end\{split\}\(2\)
where𝐙patch​i\\mathbf\{Z\}\_\{\\mathrm\{patch\}\\ i\}is the patch token\[[30](https://arxiv.org/html/2610.28997#bib.bib5)\]and𝐙phase​i\\mathbf\{Z\}\_\{\\mathrm\{phase\}\\ i\}is the phase token\[[31](https://arxiv.org/html/2610.28997#bib.bib3)\], which represent two main classes of tokenization methods\.XXdenotes the original input sequence of lengthNN,LiL\_\{i\}is the length of each patch,SiS\_\{i\}orSSis the stride, andIIis the number of tokens\. As shown in Figure[1](https://arxiv.org/html/2610.28997#S1.F1), the patch tokenizer can be seen as segmenting the sequence in the time domain, and the phase tokenizer can be regarded as downsampling in the time domain\. Extensive existing work shows that the tokenization paradigm has become a fundamental component of modern time series modeling \(§[5](https://arxiv.org/html/2610.28997#S5)\)\. However, the frequency domain effects of tokenization remain largely unexplored\.

For patch tokens, rectangular window truncation in the time domain leads to issues such as information loss and spectral leakage in the frequency domain \(§[A\.1](https://arxiv.org/html/2610.28997#A1.SS1)\)\. For phase tokens, downsampling in the time domain leads to frequency aliasing in the frequency domain \(§[A\.2](https://arxiv.org/html/2610.28997#A1.SS2)\)\. These distortions in tokenization cause the patterns of tokens, such as trend, periodicity, and seasonality, to diverge significantly from those of the original time series, thereby leading the model to learn erroneous patterns and degrading model performance\. The fundamental reason for this phenomenon lies in distortions caused by tokenization methods and the constraints that the sequence length imposes on the frequency\. Towards this, we proposeTimeFT, a nearly distortion\-free frequency\-based tokenizer for time series data in a model\-agnostic and parameter\-free manner\.

𝐙TimeFT:=Reshape⁡\(Nyquist​SamplingS⁡\(ℱℝ−1​\(Shift⁡\(BlockK⁡\(ℱℝ​\(X\)\)\)\)\)\)\\mathbf\{Z\}\_\{\\mathrm\{TimeFT\}\}:=\\operatorname\{Reshape\}\\Big\(\\operatorname\{Nyquist\\ Sampling\}\_\{S\}\\Big\(\\mathcal\{F\}\_\{\\mathbb\{R\}\}^\{\-1\}\\big\(\\operatorname\{Shift\}\\big\(\\operatorname\{Block\}\_\{K\}\\big\(\\mathcal\{F\}\_\{\\mathbb\{R\}\}\(X\)\\big\)\\big\)\\big\)\\Big\)\\Big\)\(3\)
where𝐙TimeFT\\mathbf\{Z\}\_\{\\mathrm\{TimeFT\}\}is the set of TimeFT tokens,SSis the sampling stride, andKKis the number of blocks\.BlockK⁡\(⋅\)\\operatorname\{Block\}\_\{K\}\(\\cdot\),Shift⁡\(⋅\)\\operatorname\{Shift\}\(\\cdot\),Nyquist​SamplingS⁡\(⋅\)\\operatorname\{Nyquist\\ Sampling\}\_\{S\}\(\\cdot\)andReshape⁡\(⋅\)\\operatorname\{Reshape\}\(\\cdot\)denote the operations[7](https://arxiv.org/html/2610.28997#S3.E7),[9](https://arxiv.org/html/2610.28997#S3.E9),[12](https://arxiv.org/html/2610.28997#S3.E12)and[13](https://arxiv.org/html/2610.28997#S3.E13)of TimeFT, which split the time series spectrum into multiple sub\-bands that satisfy the alias\-free condition, shift all of them to baseband, and then apply Nyquist sampling\.ℱℝ\\mathcal\{F\}\_\{\\mathbb\{R\}\}andℱℝ−1\\mathcal\{F\}\_\{\\mathbb\{R\}\}^\{\-1\}denote the real\-valued fast Fourier transform \(RFFT\) and its inverse transform\. The comparison of three tokenization methods is illustrated in Figure[2](https://arxiv.org/html/2610.28997#S1.F2)\.

Figure 2:The comparison of tokenization methods across the time domain, frequency domain and reconstruction versus origin\. \(a\) Patch tokens lead to a completely different frequency spectrum and cause the loss of all information outside the rectangular window\. \(b\) Phase tokens cause frequency components above the Nyquist frequency to completely alias into the low frequencies, thereby exhibiting erroneous patterns\. \(c\) Our TimeFT effectively solves the above issues and achieves almost distortion\-free tokenization\. The details of reconstruction methods are provided in the Appendix[B](https://arxiv.org/html/2610.28997#A2)\.As mentioned, TimeFT is a parameter\-free method that can serve as a fundamental time series tokenization module for a wide range of models, and its time complexity is negligible \(§[4\.3](https://arxiv.org/html/2610.28997#S4.SS3)\)\. We prove that our method solves the above data distortion issue \(§[A\.4](https://arxiv.org/html/2610.28997#A1.SS4)\)\. As a result, TimeFT can consistently and effectively improve model performance in various tasks \(§[4\.2](https://arxiv.org/html/2610.28997#S4.SS2)\)\. The main contributions of this work are as follows\.

- •We find that the existing time series tokenization paradigm causes distortions in the frequency domain, which leads the data to exhibit erroneous temporal patterns\. We then derive the transformation relationships in the time domain and the frequency domain during tokenization and further establish the boundary conditions of such distortions\.
- •We propose TimeFT based on the above boundary conditions, which generates tokens while preserving frequency information by employing frequency\-domain partitioning, spectrum shifting, and Nyquist sampling\. Our method effectively solves the frequency distortion issue during tokenization\.
- •Extensive experiments demonstrate that TimeFT achieves state\-of\-the\-art performance compared to existing tokenization methods in forecasting, classification, and anomaly detection tasks\. As a plug\-in module, TimeFT results in an improvement of nearly 10% in many cases, and the time cost is consistently less than 1\.4% of the training time per epoch\.

## 2Preliminaries

In this section, we first define the tokenization problem\. Then, from a frequency\-domain perspective, we theoretically analyze the causes of the distortions shown in Figure[2](https://arxiv.org/html/2610.28997#S1.F2)for patch and phase tokens\.

##### Problem Definition\.

Consider a multivariate time series input denoted as𝐗∈ℝV×N\\mathbf\{X\}\\in\\mathbb\{R\}^\{V\\times N\}, whereVVis the number of variates \(channels\) andNNis the length of the temporal sequence\. The fundamental objective of our tokenization strategy is to discretize time series𝐗∈ℝV×N\\mathbf\{X\}\\in\\mathbb\{R\}^\{V\\times N\}into a finite collection of tokens𝐙∈ℝV×S×∗\\mathbf\{Z\}\\in\\mathbb\{R\}^\{V\\times S\\times\*\}that can be learned by models, whereSSand∗\*are the shapes of the tokens\.

##### Patch Tokenization in Frequency Domain\.

Letx⁡\[n\]x\[n\],n∈\{0,1,…,N−1\}n\\in\\\{0,1,\\dots,N\-1\\\}, be a discrete sequence of finite lengthNN\. The patch tokenization in Equation[1](https://arxiv.org/html/2610.28997#S1.E1)can be equivalent to the operation of rectangular windowing\. The subsequence of lengthLLis obtained by extracting the segment in the rectangular window and re\-indexing it from zero, i\.e\.,y⁡\[m\]=x⁡\[n0\+m\]y\[m\]=x\[n\_\{0\}\+m\]form=0,…,L−1m=0,\\dots,L\-1\. This is equivalent to convolving with a shifted Dirichlet kernel in the frequency domain\.

Intuitively, it is easy to assume that patch tokens distort high frequency information\[[45](https://arxiv.org/html/2610.28997#bib.bib44),[15](https://arxiv.org/html/2610.28997#bib.bib35),[50](https://arxiv.org/html/2610.28997#bib.bib36)\]due to reduced data length, but the actual situation is much more complex\. Applying a rectangular window of lengthLLto a sequence of lengthNNinduces three main distortions\. First, the kernel’s slowly decaying sidelobes producespectral leakage, spreading energy from each frequency component across the entire spectrum and potentially obscuring weak signals\. Second, the width of the mainlobe establishes an absoluteresolution limit: components spaced closer than half this interval merge into a single broadened peak, irrespective of the original sequence lengthNN\. Moreover, the windowed sequence is further truncated to lengthLL, so itdiscards informationbeyond lengthLLwhile inheriting identical leakage and resolution penalties\.

##### Phase Tokenization in Frequency Domain\.

The phase tokenization in Equation[2](https://arxiv.org/html/2610.28997#S1.E2)is equivalent to the operation of downsampling, i\.e\.,y⁡\[k\]=x⁡\[k​S\]y\[k\]=x\[kS\]fork=0,1,…,L−1k=0,1,\\dots,L\-1, whereL=⌈N/S⌉L=\\lceil N/S\\rceil\. If the original frequency spectrum contains nonzero energy for frequencies higher than the boundBB, the adjacent replicas will overlap when downsampling, causing the higher frequency components to fold into lower frequencies\. This phenomenon is known asspectral aliasingand constitutes an irreversible distortion\. To avoid aliasing, downsampling must satisfy theNyquist sampling theorem\[[32](https://arxiv.org/html/2610.28997#bib.bib37)\]\.

2​π​Bfs≤πS⟹B≤fs2​S2\\pi\\frac\{B\}\{f\_\{s\}\}\\leq\\frac\{\\pi\}\{S\}\\quad\\Longrightarrow\\quad B\\leq\\frac\{f\_\{s\}\}\{2S\}\(4\)
wherefsf\_\{s\}represents the sampling frequency of the original discrete time sequence\. The downsampled signal will be free from aliasing if and only if the original continuous\-time signal is bandlimited to less than or equal to the newNyquist frequencyfs/2​Sf\_\{s\}/2S\. Sampling that satisfies this condition is referred to asNyquist sampling\. Detailed theoretical analysis and illustrations are provided in the Appendix[A](https://arxiv.org/html/2610.28997#A1)\.

## 3Methodology

As shown in Figure[3](https://arxiv.org/html/2610.28997#S3.F3), our TimeFT consists of three principal stages: \(1\) frequency\-domain decomposition based on boundary conditions that extracts different sub\-bands, \(2\) frequency shifting translates high\-frequency components into the low\-frequency band, and \(3\) information\-preserving token extraction that transforms these representations into final tokens\.

### 3\.1Boundary Conditions for Distortion\-Free Tokenization

Section[2](https://arxiv.org/html/2610.28997#S2)shows that for any finite discrete sequence𝐗\\mathbf\{X\}, there exists a simple method \(such as Digital\-to\-Analog Conversion\) that can reconstruct it into a continuous process𝒳\\mathcal\{X\}\(an infinite sequence\) over a period of time\.𝐗\\mathbf\{X\}can be regarded as being obtained from𝒳\\mathcal\{X\}\. When the boundary conditions are satisfied, both the sampling and reconstruction processes are invertible and lossless, and in this case𝐗\\mathbf\{X\}is equivalent to𝒳\\mathcal\{X\}\. However, due to the finite length constraint on the information contained in𝐗\\mathbf\{X\}, an incorrect𝒳\\mathcal\{X\}is often reconstructed, leading to tokens of erroneous patterns\. We next present the boundary conditions for this conversion\.

##### Boundary Condition for Patch Tokenization\.

Patch tokenization multipliesx⁡\[n\]x\[n\]by a rectangular window of lengthLLand extracts the non\-zero segment\. The necessary and sufficient condition for the token to retain all the information required for an exact reconstruction of the original signal is that the entire valid sequence ofx⁡\[n\]x\[n\]is contained within the windowed interval\. For a finite\-duration sequence of lengthNN, this condition is satisfied if and only if the window covers all non\-zero values\. Ifx⁡\[n\]x\[n\]is periodic with periodTT, the condition becomesL=m​TL=mT\(m∈ℕ\+m\\in\\mathbb\{N\}^\{\+\}\), which means that any contiguousLLsample block then suffices for perfect periodic extension, completely eliminating information loss\. However, these situations rarely occur for general time series\.

##### Boundary Condition for Phase Tokenization\.

Compared to patch tokenization, the boundary condition of phase tokenization is much simpler\. Due to the Nyquist sampling theorem, the alias\-free condition requiresx⁡\[n\]x\[n\]to satisfy Equation[4](https://arxiv.org/html/2610.28997#S2.E4)\. Therefore, for any discretex⁡\[n\]x\[n\], we define:

BN=⌊N2​S⌋B\_\{N\}=\\lfloor\\frac\{N\}\{2S\}\\rfloor\(5\)
As long as the maximum frequency component ofx⁡\[n\]x\[n\]is less than or equal toBNB\_\{N\}, it satisfies the boundary condition for phase tokenization\. In this case, the original sequence can be perfectly recovered from the tokens, making the tokenization free from information loss\.

### 3\.2Frequency Domain Decomposition Based on Boundary Conditions

Figure 3:The pipeline of TimeFT is implemented by \(1\) frequency domain decomposition, \(2\) spectral shifting and \(3\) information\-preserving token extraction\. Different colors and markers are used to denote the processes of operations\. Based on the frequency thresholdBNB\_\{N\}derived from downsampling without information loss, TimeFT shifts all spectra of bandwidthBNB\_\{N\}to the baseband, and then obtains almost distortion\-free tokens by Nyquist sampling\.In order to achieve lossless tokenization in the frequency domain, we first project the input time series into the frequency domain via the real\-valued fast Fourier transform applied along the temporal axis, as shown in Stage 1 in Figure[3](https://arxiv.org/html/2610.28997#S3.F3)\. Letℱℝ\\mathcal\{F\}\_\{\\mathbb\{R\}\}denote the RFFT operator\. The resulting complex\-valued tensor is as follows:

𝐗freq=ℱℝ​\(𝐗\)∈ℂV×Nf\\mathbf\{X\}\_\{\\mathrm\{freq\}\}=\\mathcal\{F\}\_\{\\mathbb\{R\}\}\(\\mathbf\{X\}\)\\in\\mathbb\{C\}^\{V\\times N\_\{f\}\}\(6\)
whereNf=⌊N/2⌋\+1N\_\{f\}=\\lfloor N/2\\rfloor\+1is the number of frequency bins corresponding to the non\-negative frequencies of a real signal of lengthNN\. We adopt a channel\-independent approach for different variables to avoid frequency domain mixing\. For a univariate real sequence𝐱=\(x0,…,xN−1\)\\mathbf\{x\}=\(x\_\{0\},\\dots,x\_\{N\-1\}\), itskk\-th RFFT coefficient is given by\[ℱℝ\(𝐱\)\]k=∑n=0N−1xn⋅e−i2πkn/N,k=0,1,…,Nf−1\[\\mathcal\{F\}\_\{\\mathbb\{R\}\}\(\\mathbf\{x\}\)\]\_\{k\}=\\sum\_\{n=0\}^\{N\-1\}x\_\{n\}\\cdot e^\{\-i2\\pi kn/N\},\\ k=0,1,\\dots,N\_\{f\}\-1\. To retain multi\-frequency information, we partition the frequency representation along the spectral dimension intoKKcontiguous blocks, each of sizeBNB\_\{N\}\. This makes the spectral bandwidth satisfy Equation[5](https://arxiv.org/html/2610.28997#S3.E5)and the boundary condition in §[3\.1](https://arxiv.org/html/2610.28997#S3.SS1)\. For thekk\-th band \(k=0,…,K−1k=0,\\dots,K\-1\), we define a full extraction that retains only the frequency components within the interval\[k​BN,k​BN\+BN−1\]\[kB\_\{N\},kB\_\{N\}\+B\_\{N\}\-1\]and zeros out the rest:

\[𝐗block\(k\)\]v,f=\{\[𝐗freq\]v,f,if​k​BN≤f<\(k\+1\)​BN,0,otherwise,f=0,…,Nf−1\\bigl\[\\mathbf\{X\}\_\{\\mathrm\{block\}\}^\{\(k\)\}\\bigr\]\_\{v,f\}=\\begin\{cases\}\[\\mathbf\{X\}\_\{\\mathrm\{freq\}\}\]\_\{v,f\},&\\text\{if \}\\ kB\_\{N\}\\leq f<\(k\+1\)B\_\{N\},\\\\\[4\.0pt\] 0,&\\text\{otherwise\},\\end\{cases\}\\quad f=0,\\dots,N\_\{f\}\-1\(7\)
𝐗blocks=\[𝐗block\(0\);𝐗block\(1\);…;𝐗block\(K−1\)\]∈ℂV×K×Nf\\mathbf\{X\}\_\{\\mathrm\{blocks\}\}=\\bigl\[\\mathbf\{X\}\_\{\\mathrm\{block\}\}^\{\(0\)\};\\mathbf\{X\}\_\{\\mathrm\{block\}\}^\{\(1\)\};\\dots;\\mathbf\{X\}\_\{\\mathrm\{block\}\}^\{\(K\-1\)\}\\bigr\]\\in\\mathbb\{C\}^\{V\\times K\\times N\_\{f\}\}\(8\)
whereK=⌈Nf/BN⌉K=\\lceil N\_\{f\}/B\_\{N\}\\rceildenotes the total number of spectral blocks, and the last portion of𝐗freq\\mathbf\{X\}\_\{\\mathrm\{freq\}\}is filled by zero\-padding\. In this step, we separate the frequency components that correspond to characteristics such as trend, periodicity, and seasonality\. Through this partitioning, we obtain a series of sequences in the frequency\-domain that satisfy the required frequency band length under the boundary conditions, which is the first step in avoiding aliasing\.

### 3\.3Spectral Shifting

However, existing effective frequency bands are located at high frequencies, causing high frequency components to irreversibly alias into low frequencies during downsampling\. Therefore, we need to shift the high frequency information down to the low frequency range\. As shown in Stage 2 in Figure[3](https://arxiv.org/html/2610.28997#S3.F3), we align all bands to the baseband and multiply by a factor that compensates for the band starting frequency:

\[𝐗shift\]v,k,b=\[𝐗blocks\]v,k,k​BN\+b⋅e−i2π\(kBN\)b/N,b=0,1,…,BN−1\\bigl\[\\mathbf\{X\}\_\{\\mathrm\{shift\}\}\\bigr\]\_\{v,k,b\}=\\bigl\[\\mathbf\{X\}\_\{\\mathrm\{blocks\}\}\\bigr\]\_\{v,k,\\,kB\_\{N\}\+b\}\\cdot e^\{\-i2\\pi\(kB\_\{N\}\)b/N\},\\quad b=0,1,\\dots,B\_\{N\}\-1\(9\)
The frequency portion of𝐗shift\\mathbf\{X\}\_\{\\mathrm\{shift\}\}fromBNB\_\{N\}toNfN\_\{f\}is zero\-padded\. Equivalently, one may view this as applying the shifting operator𝒮\\mathcal\{S\}to each band, which is𝐗shift=𝒮⁡\(𝐗blocks\)∈ℂV×K×Nf\\mathbf\{X\}\_\{\\mathrm\{shift\}\}=\\mathcal\{S\}\\bigl\(\\mathbf\{X\}\_\{\\mathrm\{blocks\}\}\\bigr\)\\in\\mathbb\{C\}^\{V\\times K\\times N\_\{f\}\}\. The resulting sequences𝐗shift\\mathbf\{X\}\_\{\\mathrm\{shift\}\}satisfy the Nyquist sampling condition when sampled with a step size ofSS, ensuring that no aliasing occurs\. In Equation[9](https://arxiv.org/html/2610.28997#S3.E9), the factor is applied to align the phase reference with the time origin after shifting the frequency band\. It compensates for the phase shift introduced by the translation without altering the magnitude of the complex signal in the frequency domain\. Since TimeFT does not compress the temporal dimension, the phase factor in Equation[9](https://arxiv.org/html/2610.28997#S3.E9)is ignored for models \(§[A\.4\.2](https://arxiv.org/html/2610.28997#A1.SS4.SSS2)\)\. Therefore, Equations[7](https://arxiv.org/html/2610.28997#S3.E7)\-[9](https://arxiv.org/html/2610.28997#S3.E9)can be simplified to the following form to avoid computing the above complex steps:

𝐗shift=ZeroPadBN→Nf⁡\(Reshape⁡\(𝐗freq,\(V,K,BN\)\)\)∈ℂV×K×Nf\\mathbf\{X\}\_\{\\mathrm\{shift\}\}=\\operatorname\{ZeroPad\}\_\{B\_\{N\}\\to N\_\{f\}\}\\Bigl\(\\operatorname\{Reshape\}\\bigl\(\\mathbf\{X\}\_\{\\mathrm\{freq\}\},\(V,K,B\_\{N\}\)\\bigr\)\\Bigr\)\\in\\mathbb\{C\}^\{V\\times K\\times N\_\{f\}\}\(10\)
where theZeroPadBN→Nf\\operatorname\{ZeroPad\}\_\{B\_\{N\}\\to N\_\{f\}\}operation denotes appending zeros to the end of each frequency\-domain sequence of lengthBNB\_\{N\}to extend its dimension toNfN\_\{f\}\. TheReshape⁡\(⋅\)\\operatorname\{Reshape\(\\cdot\)\}operation zero\-pads at the end if the dimension is insufficient\. Here, the spectrum will slightly lose the imaginary part at zero frequency\. In this way, we obtain a frequency\-domain sequence that satisfies the Nyquist sampling theorem\.

### 3\.4Information\-Preserving Token Extraction

However, the tokenization extraction operation should be performed in the time domain\. As shown in Stage 3 in Figure[3](https://arxiv.org/html/2610.28997#S3.F3), to recover the corresponding time domain sequence, we apply the inverse real fast Fourier transform \(IRFFT\)ℱℝ−1\\mathcal\{F\}\_\{\\mathbb\{R\}\}^\{\-1\}to every frequency block independently\.

𝐗time=ℱℝ−1​\(𝐗shift\)∈ℝV×K×N\\mathbf\{X\}\_\{\\mathrm\{time\}\}=\\mathcal\{F\}\_\{\\mathbb\{R\}\}^\{\-1\}\\bigl\(\\mathbf\{X\}\_\{\\mathrm\{shift\}\}\\bigr\)\\in\\mathbb\{R\}^\{V\\times K\\times N\}\(11\)
where𝐗time\\mathbf\{X\}\_\{\\mathrm\{time\}\}denotes the time\-domain representation of the sequences ofKKdifferent frequency bands of widthBNB\_\{N\}for each variableVV\. Then we perform*dense sampling*: for every possible offsetλ∈\{0,1,…,S−1\}\\lambda\\in\\\{0,1,\\dots,S\-1\\\}, we sample the signal starting at the indexλ\\lambdawith stepSS\. Formally, for thekk\-th band of the variablevv, let𝐱v,k=𝐗time\[v,k,⋅\]∈ℝN\\mathbf\{x\}\_\{v,k\}=\\mathbf\{X\}\_\{\\mathrm\{time\}\}\[v,k,\\cdot\]\\in\\mathbb\{R\}^\{N\}\. The dense sampling operator𝒟S\\mathcal\{D\}\_\{S\}produces a matrix of sizeS×LS\\times L, whereL=⌈N/S⌉L=\\lceil N/S\\rceil:

\[𝒟S\(𝐱v,k\)\]λ,ℓ=𝐱v,k\[λ\+ℓ⋅S\],λ=0,…,S−1,ℓ=0,…,L−1\\bigl\[\\mathcal\{D\}\_\{S\}\(\\mathbf\{x\}\_\{v,k\}\)\\bigr\]\_\{\\lambda,\\ell\}=\\mathbf\{x\}\_\{v,k\}\[\\lambda\+\\ell\\cdot S\],\\quad\\lambda=0,\\dots,S\-1,\\;\\;\\ell=0,\\dots,L\-1\(12\)
Applying this operator independently to every sub\-band yields the final token tensor, which is𝐗sampling=𝒟S​\(𝐗time\)∈ℝV×K×S×L\\mathbf\{X\}\_\{\\mathrm\{sampling\}\}=\\mathcal\{D\}\_\{S\}\\bigl\(\\mathbf\{X\}\_\{\\mathrm\{time\}\}\\bigr\)\\in\\mathbb\{R\}^\{V\\times K\\times S\\times L\}\. Thus, each variate is represented byK×SK\\times Svectors of lengthLL, whereKKindexes the frequency bands andSScaptures different temporal phases\. In our framework, before sampling, thekk\-th band has been shifted to the baseband and truncated to theBBfrequency bins\. Due to Equation[5](https://arxiv.org/html/2610.28997#S3.E5)the sampling strideSSsatisfiesS≤N/\(2​BN\)S\\leq\{N\}/\(2B\_\{N\}\)\. Hence, the operator𝒟S​\(⋅\)\\mathcal\{D\}\_\{S\}\(\\cdot\)corresponds to theNyquist samplingoperation\. Consequently, the dense sampling procedure preserves information and is lossless with respect to the sub\-band representation\. Finally, to obtain a unified token, we first permute the dimension to group the frequency and sampling dimensions, then flatten the frequency bands into the feature dimension\.

𝐙TimeFT=reshape⁡\(permute⁡\(𝐗sampling\),\(V,S,K⋅L\)\)∈ℝV×S×\(K⋅L\)\\mathbf\{Z\}\_\{\\mathrm\{TimeFT\}\}=\\operatorname\{reshape\}\\Bigl\(\\operatorname\{permute\}\\bigl\(\\mathbf\{X\}\_\{\\mathrm\{sampling\}\}\\bigr\),\\,\(V,S,K\\cdot L\)\\Bigr\)\\in\\mathbb\{R\}^\{V\\times S\\times\(K\\cdot L\)\}\(13\)
wherepermute⁡\(⋅\)\\operatorname\{permute\}\(\\cdot\)reorders the dimension from\(V,K,S,L\)\(V,K,S,L\)to\(V,S,K,L\)\(V,S,K,L\), and the subsequent reshape collapses the last two dimensions into a single feature dimension of sizeK⋅LK\\cdot L\. Since models typically apply embedding to tokens, theK⋅LK\\cdot Ldimension here does not lead to the curse of dimensionality\. Each step of TimeFT is designed to be approximately invertible\. The simplified spectral shifting only introduces a negligible loss at the zero\-frequency component, and we can almost perfectly reconstruct the original spectrum from each token \(§[B\.3](https://arxiv.org/html/2610.28997#A2.SS3)\)\.

## 4Experiments

In this section, we conduct a comprehensive evaluation of TimeFT, including three major time series tasks \(§[4\.2](https://arxiv.org/html/2610.28997#S4.SS2)\), the computational efficiency \(§[4\.3](https://arxiv.org/html/2610.28997#S4.SS3)\), the plug\-in experiment \(§[4\.4](https://arxiv.org/html/2610.28997#S4.SS4)\), the ablation study \(§[4\.5](https://arxiv.org/html/2610.28997#S4.SS5)\) and the hyperparameter sensitivity \(§[4\.7](https://arxiv.org/html/2610.28997#S4.SS7)\)\. Detailed experiments are presented in Appendix[C](https://arxiv.org/html/2610.28997#A3), and visualizations of TimeFT are provided in Appendix[D](https://arxiv.org/html/2610.28997#A4)\.

### 4\.1Experimental Setup

##### Datasets\.

We evaluate TimeFT on a comprehensive set of benchmarks spanning three core time\-series tasks, and we select commonly used datasets for each task\.Long\-term forecastingis performed on seven real\-world datasets: ETTh1, ETTh2, ETTm1, ETTm2, Weather, Electricity and Traffic\[[42](https://arxiv.org/html/2610.28997#bib.bib38),[41](https://arxiv.org/html/2610.28997#bib.bib10),[27](https://arxiv.org/html/2610.28997#bib.bib8)\]\.Classificationuses 10 UEA datasets\[[1](https://arxiv.org/html/2610.28997#bib.bib1),[20](https://arxiv.org/html/2610.28997#bib.bib41),[7](https://arxiv.org/html/2610.28997#bib.bib39)\]: EthanolConcentration, FaceDetection, Handwriting, Heartbeat, JapaneseVowels, PEMS\-SF, SelfRegulationSCP1, SelfRegulationSCP2, SpokenArabicDigits, and UWaveGestureLibrary\.Anomaly detectionis conducted on five widely used benchmarks\[[1](https://arxiv.org/html/2610.28997#bib.bib1),[47](https://arxiv.org/html/2610.28997#bib.bib42),[17](https://arxiv.org/html/2610.28997#bib.bib40)\]: SMD, MSL, SMAP, SWaT, and PSM\.

##### Baselines\.

We compare TimeFT with state\-of\-the\-art tokenization methods and classic time series models\. We consider EntroPE\[[1](https://arxiv.org/html/2610.28997#bib.bib1)\], TimeMosaic\[[13](https://arxiv.org/html/2610.28997#bib.bib2)\], PhaseFormer\[[31](https://arxiv.org/html/2610.28997#bib.bib3)\], PatchMLP\[[36](https://arxiv.org/html/2610.28997#bib.bib4)\]and PatchTST\[[30](https://arxiv.org/html/2610.28997#bib.bib5)\]as standard baselines\. In addition, we also use TimeFilter\[[19](https://arxiv.org/html/2610.28997#bib.bib6)\], xPatch\[[34](https://arxiv.org/html/2610.28997#bib.bib7)\], iTransformer\[[27](https://arxiv.org/html/2610.28997#bib.bib8)\], TimeMixer\[[37](https://arxiv.org/html/2610.28997#bib.bib9)\], TimesNet\[[41](https://arxiv.org/html/2610.28997#bib.bib10)\], LightTS\[[9](https://arxiv.org/html/2610.28997#bib.bib11)\], DLinear\[[48](https://arxiv.org/html/2610.28997#bib.bib15)\], GPT4TS\[[52](https://arxiv.org/html/2610.28997#bib.bib14)\], ETSformer\[[40](https://arxiv.org/html/2610.28997#bib.bib13)\], FEDformer\[[51](https://arxiv.org/html/2610.28997#bib.bib12)\]as baselines\. The specific baselines differ slightly between tasks, due to the choice of baselines in previous papers\. Since TimeFT is a general tokenization method rather than a model and due to expensive pre\-train costs, we do not use time series foundation models \(TSFMs\) as baselines or plug\-in models\.We use PatchTST as the backbone model for TimeFT in all experiments except the plug\-in experiment\.

### 4\.2Main Results on Various Tasks

For the long\-term forecasting task, we evaluate the performance of TimeFT against state\-of\-the\-art tokenization baselines across diverse datasets\. The experiments adopt the settings in TimeMosaic\[[13](https://arxiv.org/html/2610.28997#bib.bib2)\]\. As demonstrated in Table[1](https://arxiv.org/html/2610.28997#S4.T1), TimeFT achieves the best performance in the majority of datasets and metrics\. By outperforming both classic and recent baselines, TimeFT establishes an excellent paradigm for obtaining tokens of time series\. The detailed performance is provided in Table[8](https://arxiv.org/html/2610.28997#A3.T8)\.

Table 1:For long\-term forecasting task, we evaluate average MSE and MAE across prediction horizonsH∈\{96,192,336,720\}H\\in\\\{96,192,336,720\\\}\. All baselines use a fixed input sequence length of 320\. Best results are inboldand second best areunderlined\. The detailed performance is provided in Table[8](https://arxiv.org/html/2610.28997#A3.T8)\.ModelsTimeFTEntroPETimeMosaicPhaseFormerTimeFilterxPatchPatchMLPiTransformerTimeMixerPatchTST\(Ours\)\(2026\)\(2026\)\(2026\)\(2025\)\(2025\)\(2025\)\(2024\)\(2024\)\(2023\)MetricMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEETTm10\.3480\.3730\.3690\.3830\.3600\.3770\.3640\.3920\.3990\.4080\.3540\.3860\.3740\.3960\.3860\.4060\.3560\.3850\.3720\.399ETTm20\.2550\.3070\.2640\.3200\.2560\.3100\.2680\.3240\.2780\.3300\.2620\.3220\.2690\.3240\.2930\.3380\.2650\.3230\.2780\.332ETTh10\.4170\.4230\.4240\.4330\.4120\.4230\.4890\.4750\.5110\.4820\.4460\.4450\.4620\.4600\.4870\.4750\.4300\.4380\.5030\.485ETTh20\.3580\.3900\.3740\.4010\.3630\.3930\.3840\.4170\.4020\.4210\.3540\.3970\.3800\.4160\.4110\.4250\.3580\.3960\.3980\.427Weather0\.2320\.2610\.2370\.2640\.2280\.2560\.2330\.2680\.2310\.2700\.2240\.2620\.2300\.2670\.2370\.2720\.2270\.2660\.2360\.271Electricity0\.1650\.2580\.1770\.2710\.1700\.2630\.1770\.2730\.1670\.2600\.1700\.2640\.2000\.3020\.1800\.2770\.1730\.2670\.1730\.272Traffic0\.4200\.2900\.4440\.3000\.4340\.2870\.4410\.3050\.4200\.2840\.4300\.3000\.5250\.3820\.4630\.3350\.4440\.3160\.4280\.297

Table 2:For classification task, we evaluate accuracy \(%\) across 10 UEA datasets\. Best results are inboldand second best areunderlined\. The detailed performance is provided in Table[9](https://arxiv.org/html/2610.28997#A3.T9)\.ModelsTimeFTEntroPETimeMosaicPhaseFormerPatchMLPGPT4TSPatchTSTTimesNetMLP\(Ours\)\(2026\)\(2026\)\(2026\)\(2025\)\(2024\)\(2023\)\(2023\)\-Average72\.2971\.3370\.1468\.9368\.9164\.3771\.1469\.3668\.97

Table 3:For anomaly detection task, we evaluate F1\-score \(%\) on five benchmark datasets\. Best results are inboldand second best areunderlined\. The detailed performance is provided in Table[11](https://arxiv.org/html/2610.28997#A3.T11)\.ModelsTimeFTEntroPETimeMosaicPhaseFormerLightTSTimesNetPatchTSTDLinearETSformerFEDformer\(Ours\)\(2026\)\(2026\)\(2026\)\(2024\)\(2023\)\(2023\)\(2023\)\(2023\)\(2022\)Average86\.1285\.3882\.5083\.0184\.2385\.2482\.7982\.4682\.8784\.97

For the classification task, we evaluate the performance on 10 commonly used UEA datasets\. As shown in Tables[2](https://arxiv.org/html/2610.28997#S4.T2)and[9](https://arxiv.org/html/2610.28997#A3.T9), TimeFT achieves the highest average accuracy and achieves the best or second\-best result in a majority of datasets\. This strong performance improvement is due to TimeFT’s almost distortion\-free tokenization ability, which faithfully preserves discriminative temporal patterns that are often corrupted by tokenizers\.

For the anomaly detection task, we evaluate the capability on five standard datasets\. TimeFT preserves the integrity of anomalous patterns and shows excellent detection performance\. As shown in Tables[3](https://arxiv.org/html/2610.28997#S4.T3)and[11](https://arxiv.org/html/2610.28997#A3.T11), experiments on extensive datasets demonstrate that TimeFT exhibits superior performance and achieves the highest average F1\-score of 86\.12%\.

### 4\.3Time Complexity Analysis

The computational complexity of the proposed TimeFT tokenization is𝒪⁡\(V​N​log⁡N\+V​K​N​log⁡N\+V​K​N\)\\mathcal\{O\}\(VN\\log N\+VKN\\log N\+VKN\)\. Each of theV×KV\\times Kfrequency blocks is transformed back to the time domain via an IRFFT \(Equation[11](https://arxiv.org/html/2610.28997#S3.E11)\), contributing the dominant term of𝒪⁡\(V​K​N​log⁡N\)\\mathcal\{O\}\(VKN\\log N\)\. As shown in Table[4](https://arxiv.org/html/2610.28997#S4.T4), the one\-time overhead introduced by TimeFT is negligible relative to the aggregate cost of model training and evaluation\. Moreover, tokenization can be efficiently batched and parallelized, further amortizing its practical runtime\. For multiple epochs, TimeFT produces identical tokens as long as the input sequence is the same, thus eliminating the need for repeated execution\.

Table 4:Comparison of TimeFT tokenization time and training time per epoch across different datasets\. The percentage represents the proportion of TimeFT tokenization time relative to the training time per epoch\.DatasetsETTm1ETTm2ETTh1ETTh2WeatherElectricityTrafficTimeFT\(s\)1\.4921\.3760\.3520\.3623\.79928\.5050\.27Training \(s\)299\.5177\.580\.55127\.8833\.721573629Percentage0\.50%0\.78%0\.44%0\.28%0\.46%1\.32%1\.38%

Table 5:Plug\-in performance on long\-term forecasting task\. Evaluation of TimeFT as a plug\-in tokenization module on prediction horizonH=96H=96and input length of 320\.DatasetsETTm1ETTm2ETTh1ETTh2WeatherElectricityTrafficMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEPatchTST0\.3210\.3610\.1810\.2700\.4030\.4230\.3240\.3740\.1550\.2040\.1420\.2450\.4030\.285\+TimeFT0\.2900\.3390\.1620\.2440\.3760\.3920\.2970\.3390\.1470\.1940\.1390\.2340\.3950\.279Δ%\\Delta\\%Improvement9\.7%6\.1%10\.5%9\.6%6\.7%7\.3%8\.3%9\.4%5\.2%4\.9%2\.1%4\.5%2\.0%2\.1%PhaseFormer0\.3120\.3520\.1740\.2610\.3890\.4110\.3090\.3610\.1520\.2010\.1450\.2480\.4180\.289\+TimeFT0\.2930\.3410\.1610\.2490\.3790\.3970\.2990\.3430\.1490\.1930\.1430\.2420\.4060\.278Δ%\\Delta\\%Improvement6\.1%3\.1%7\.5%4\.6%2\.6%3\.4%3\.2%5\.0%1\.9%3\.9%1\.4%2\.4%2\.9%3\.8%

Table 6:Ablation study on long\-term forecasting task on prediction horizonH=96H=96and input length of 320\. The “w/o” indicates that the operations of the corresponding stage in Figure[3](https://arxiv.org/html/2610.28997#S3.F3)are skipped in the pipeline, while the rest remains unchanged\. Stage 1 is frequency domain decomposition, Stage 2 is spectral shifting and Stage 3 is information\-preserving token extraction\.DatasetsETTm1ETTm2ETTh1ETTh2WeatherElectricityTrafficMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAETimeFT0\.2900\.3390\.1620\.2440\.3760\.3920\.2970\.3390\.1470\.1940\.1390\.2340\.3950\.279w/o Stage 10\.3280\.3640\.1750\.2610\.4120\.4210\.3150\.3580\.1620\.2110\.1510\.2550\.4080\.288w/o Stage 20\.3140\.3520\.1940\.2820\.3980\.4150\.3420\.3840\.1580\.2070\.1460\.2490\.4230\.301w/o Stage 30\.3750\.4120\.2280\.3150\.4820\.4850\.3960\.4320\.1890\.2420\.1780\.2940\.4750\.342

### 4\.4Plug\-in Experiment

We evaluate the universality of TimeFT using it as a plug\-in tokenization module for PatchTST and PhaseFormer\. As shown in Table[5](https://arxiv.org/html/2610.28997#S4.T5), integrating TimeFT consistently improves performance in all seven datasets\. For PatchTST, performance in MSE and MAE improves from nearly10%10\\%in many cases\. For PhaseFormer, the corresponding improvements are more than5%5\\%in many cases\. These results demonstrate that TimeFT’s near\-lossless tokenization is readily transferable to different architectures, delivering immediate gains without modifying the base model design\.

### 4\.5Ablation Study

We conduct the ablation study to demonstrate the contribution of each part in TimeFT\. Removing Stage 1 means that only a single block exists, and the subsequent operations artificially introduce aliasing\. Removing Stage 2 means that the high\-frequency components of each block will individually alias into the low frequencies, and removing Stage 3 is equivalent to using different frequency components directly as tokens\. As shown in Table[6](https://arxiv.org/html/2610.28997#S4.T6), removing any individual stage significantly degrades the performance, showing that every component is essential to the model\.

### 4\.6Distortion Comparison of Different Tokenizations

To intuitively illustrate the degree of distortion for different types of tokens, we compare the mean absolute error between the reconstructed sequences and the original sequences across different datasets in the time domain\. This experiment can be viewed as a statistical summary of Figure[2](https://arxiv.org/html/2610.28997#S1.F2)and the visualization section[D](https://arxiv.org/html/2610.28997#A4)\. Due to Appendices[A](https://arxiv.org/html/2610.28997#A1)and[B](https://arxiv.org/html/2610.28997#A2), the reconstructions from tokens to the reconstructed signals here are theoretically an invertible and lossless transformation\. We ensure that all hyperparameters remain identical in this experiment\. As shown in Figure[5](https://arxiv.org/html/2610.28997#S4.F5), TimeFT reduces more than 80% reconstruction error in many datasets, which is shown by the percentage of “Reduce” on the right side of each bar\.\. This phenomenon arises because patch and phase tokenization introduce data distortion, causing the reconstructed sequence to differ substantially from the original sequence\. In contrast, TimeFT achieves nearly distortion\-free tokenization\. The details are provided in Table[13](https://arxiv.org/html/2610.28997#A3.T13)and the reconstruction methods are provided in Appendix[B](https://arxiv.org/html/2610.28997#A2)\.

Figure 4:Reconstruction performance of different tokens types across six commonly used datasets\. The details are provided in Table[13](https://arxiv.org/html/2610.28997#A3.T13)\.Figure 5:Hyperparameter sensitivity analysis of tokenization strideSSfor different input lengthsNN\. Metric:*Left:*MSE,*Right:*MAE\.

### 4\.7Sensitivity Analysis

We further examine the impact of the tokenization stride S on model performance across various input lengths in the ETTm1 dataset\. As shown in Figure[5](https://arxiv.org/html/2610.28997#S4.F5), the results reveal a significant synergy between the stride and the input scale, where the optimal configuration changes according to the input length\. For shorter input sequences such asN=96N=96, smaller strides ensure sufficient coverage of temporal details\. However, as the input length increases toN=720N=720, the model demonstrates a clear preference for larger strides\. Since the stride determines the Nyquist frequency, a smaller stride results in a higher Nyquist frequency and longer tokens, while also leading to fewer tokens \(downsampling\)\. This trade\-off affects the model’s performance, making it necessary to choose an appropriate stride for the input\. Moreover, longer input sequences generally result in better performance\.

## 5Related Work

Existing time series tokenizers can be classified into two fundamental categories: patch tokens and phase tokens\. We consider only the partitioning operation here without the embedding step\. Quantization methods such as "bins"\[[5](https://arxiv.org/html/2610.28997#bib.bib43),[46](https://arxiv.org/html/2610.28997#bib.bib45)\]and some frequency methods\[[2](https://arxiv.org/html/2610.28997#bib.bib46),[24](https://arxiv.org/html/2610.28997#bib.bib47),[33](https://arxiv.org/html/2610.28997#bib.bib51),[8](https://arxiv.org/html/2610.28997#bib.bib52)\]are not discussed because they are irrelevant to our topic \(§[A\.5](https://arxiv.org/html/2610.28997#A1.SS5)\)\. According to the modeling paradigm, token\-based methods fall into the following three categories\.

For deep learning models, PatchTST\[[30](https://arxiv.org/html/2610.28997#bib.bib5)\]introduces a patch\-based approach to time series analysis, establishing patching as a key component of time series modeling\. Subsequently, PITS\[[23](https://arxiv.org/html/2610.28997#bib.bib19)\], PatchMLP\[[36](https://arxiv.org/html/2610.28997#bib.bib4)\]and xPatch\[[34](https://arxiv.org/html/2610.28997#bib.bib7)\]demonstrate that patch tokens can improve performance\. Beyond static patching\[[21](https://arxiv.org/html/2610.28997#bib.bib20),[14](https://arxiv.org/html/2610.28997#bib.bib21),[10](https://arxiv.org/html/2610.28997#bib.bib22)\], some dynamic patching methods\[[19](https://arxiv.org/html/2610.28997#bib.bib6),[3](https://arxiv.org/html/2610.28997#bib.bib24),[26](https://arxiv.org/html/2610.28997#bib.bib26),[44](https://arxiv.org/html/2610.28997#bib.bib27),[29](https://arxiv.org/html/2610.28997#bib.bib28)\]have also been proposed for time series tokenization\. SRSNet\[[43](https://arxiv.org/html/2610.28997#bib.bib23)\]leverages information from contextual time series to improve the forecasting performance of patch\-based models\. TimeMosaic\[[13](https://arxiv.org/html/2610.28997#bib.bib2)\]employs a lightweight classifier to determine the length of patches\. Additionally, PhaseFormer\[[31](https://arxiv.org/html/2610.28997#bib.bib3)\]introduces time series phase tokens, which can be viewed as an approximate transpose of patch tokenization\.

For pre\-trained tokenizers, TOTEM\[[35](https://arxiv.org/html/2610.28997#bib.bib25)\]divides the time series into patches and maps them to a discrete embedding using a pre\-trained model\. EntroPE\[[1](https://arxiv.org/html/2610.28997#bib.bib1)\]employs a pre\-trained entropy model to determine optimal patch boundaries\. These methods still belong to the patch token class\.

For pre\-trained foundation models, recent state\-of\-the\-art TSFMs\[[25](https://arxiv.org/html/2610.28997#bib.bib34)\], including Chronos\-2\[[4](https://arxiv.org/html/2610.28997#bib.bib29)\], Sundial\[[28](https://arxiv.org/html/2610.28997#bib.bib30)\], TiRex\[[6](https://arxiv.org/html/2610.28997#bib.bib31)\], Moirai\[[39](https://arxiv.org/html/2610.28997#bib.bib33)\], and TimeFM\[[12](https://arxiv.org/html/2610.28997#bib.bib32)\], have adopted patch tokenization as a preprocessing step\. Tokenization has become a fundamental component of TSFMs\.

## 6Conclusion

We identify the inherent spectral distortion in the dominant time series tokenization paradigm and derive its boundary conditions for distortion\-free tokenization\. We propose TimeFT, an approach obtains time series tokens in a nearly distortion\-free manner through frequency\-domain partitioning, shifting, and Nyquist sampling\. TimeFT is parameter\-free, lightweight, and plug\-and\-play tokenizer for time series data\. Extensive experiments on various tasks showcase that TimeFT outperforms existing practices and yields nearly 10% improvement as a plug\-in module in many cases\.

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## Appendix AFrequency Domain Theoretical Analysis and Boundary Conditions

### A\.1Patch Tokenization

Letx⁡\[n\]x\[n\],n∈\{0,1,…,N−1\}n\\in\\\{0,1,\\dots,N\-1\\\}, be a discrete time sequence of finite lengthNN\. The patch tokenization in Equation[1](https://arxiv.org/html/2610.28997#S1.E1)can be equivalent to the operation of rectangular windowing\. A rectangular windowwL​\[n;n0\]w\_\{L\}\[n;n\_\{0\}\]of lengthLLstarting at the indexn0n\_\{0\}is defined aswL​\[n;n0\]=1w\_\{L\}\[n;n\_\{0\}\]=1forn0≤n≤n0\+L−1n\_\{0\}\\leq n\\leq n\_\{0\}\+L\-1and00otherwise\. The windowed sequence isxw​\[n\]=x⁡\[n\]⋅wL​\[n;n0\]x\_\{w\}\[n\]=x\[n\]\\cdot w\_\{L\}\[n;n\_\{0\}\], and the sliced sequence of lengthLLis obtained by extracting the segment in the window and re\-indexing it from zero, i\.e\.,y⁡\[m\]=xw​\[n0\+m\]y\[m\]=x\_\{w\}\[n\_\{0\}\+m\]form=0,…,L−1m=0,\\dots,L\-1\. We analyze the spectral consequences of these operations within the continuous frequency domain provided by the discrete time Fourier transform \(DTFT\)\. The DTFT ofx⁡\[n\]x\[n\]is

X⁡\(ej​ω\)=∑n=−∞∞x⁡\[n\]​e−j​ω​n=∑n=0N−1x⁡\[n\]​e−j​ω​nX\(e^\{j\\omega\}\)=\\sum\_\{n=\-\\infty\}^\{\\infty\}x\[n\]e^\{\-j\\omega n\}=\\sum\_\{n=0\}^\{N\-1\}x\[n\]e^\{\-j\\omega n\}\(14\)
whereω∈\[−π,π\)\\omega\\in\[\-\\pi,\\pi\)denotes the normalized angular frequency in radians\. Note thatDTFT\\operatorname\{DTFT\}of a sequencex⁡\[n\]x\[n\]of lengthNNis well\-defined, since by treating the values at undefinednnas00, the sequence naturally extends to an infinite\-length sequence, and the infinite summation inDTFT\\operatorname\{DTFT\}reduces to a finite sum\. The DTFT of the rectangular windowwL​\[n;n0\]w\_\{L\}\[n;n\_\{0\}\]is derived as follows:

WL​\(ej​ω,n0\)\\displaystyle W\_\{L\}\(e^\{j\\omega\};n\_\{0\}\)=∑n=n0n0\+L−1e−j​ω​n\\displaystyle=\\sum\_\{n=n\_\{0\}\}^\{n\_\{0\}\+L\-1\}e^\{\-j\\omega n\}\(15\)=e−j​ω​n0​1−e−j​ω​L1−e−j​ω\\displaystyle=e^\{\-j\\omega n\_\{0\}\}\\frac\{1\-e^\{\-j\\omega L\}\}\{1\-e^\{\-j\\omega\}\}=e−j​ω​n0⋅2je−jωL/2sin\(ωL/2\)2je−jω/2sin\(ω/2\)\\displaystyle=e^\{\-j\\omega n\_\{0\}\}\\cdot\\frac\{2je^\{\-j\\omega L/2\}\\sin\(\\omega L/2\)\}\{2je^\{\-j\\omega/2\}\\sin\(\\omega/2\)\}=e−j​ω​n0e−jω\(L−1\)/2sin⁡\(ω​L/2\)sin⁡\(ω/2\)\\displaystyle=e^\{\-j\\omega n\_\{0\}\}\\,e^\{\-j\\omega\(L\-1\)/2\}\\,\\frac\{\\sin\(\\omega L/2\)\}\{\\sin\(\\omega/2\)\}=e−j​ω​\(n0\+L−12\)⋅sin⁡\(ω​L/2\)sin⁡\(ω/2\)\\displaystyle=e^\{\-j\\omega\\left\(n\_\{0\}\+\\frac\{L\-1\}\{2\}\\right\)\}\\cdot\\frac\{\\sin\(\\omega L/2\)\}\{\\sin\(\\omega/2\)\}
where the first step expresses the windowed sum as a finite geometric series\. Factor out the term corresponding to the starting index by lettingn=n0\+k,k=0,1,…,L−1n=n\_\{0\}\+k,\\ k=0,1,\\dots,L\-1:

∑k=0L−1e−j​ω​\(n0\+k\)=e−j​ω​n0​∑k=0L−1\(e−j​ω\)k\\sum\_\{k=0\}^\{L\-1\}e^\{\-j\\omega\(n\_\{0\}\+k\)\}=e^\{\-j\\omega n\_\{0\}\}\\sum\_\{k=0\}^\{L\-1\}\\left\(e^\{\-j\\omega\}\\right\)^\{k\}\(16\)
The sum on the right is a finite geometric series with the first term11, the common ratior=e−j​ωr=e^\{\-j\\omega\}, andLLterms\. Using the geometric sum formula∑k=0L−1rk=1−rL1−r\\sum\_\{k=0\}^\{L\-1\}r^\{k\}=\\frac\{1\-r^\{L\}\}\{1\-r\}\(forr≠1r\\neq 1\), we obtain

∑k=0L−1\(e−j​ω\)k=1−\(e−j​ω\)L1−e−j​ω=1−e−j​ω​L1−e−j​ω\\sum\_\{k=0\}^\{L\-1\}\\left\(e^\{\-j\\omega\}\\right\)^\{k\}=\\frac\{1\-\\left\(e^\{\-j\\omega\}\\right\)^\{L\}\}\{1\-e^\{\-j\\omega\}\}=\\frac\{1\-e^\{\-j\\omega L\}\}\{1\-e^\{\-j\\omega\}\}\(17\)
Substituting this back gives the closed form

WL​\(ej​ω,n0\)=e−j​ω​n0​1−e−j​ω​L1−e−j​ωW\_\{L\}\(e^\{j\\omega\};n\_\{0\}\)=e^\{\-j\\omega n\_\{0\}\}\\frac\{1\-e^\{\-j\\omega L\}\}\{1\-e^\{\-j\\omega\}\}\(18\)
In Equation[15](https://arxiv.org/html/2610.28997#A1.E15), the factorDL​\(ω\)≜sin⁡\(ω​L/2\)sin⁡\(ω/2\)D\_\{L\}\(\\omega\)\\triangleq\\frac\{\\sin\(\\omega L/2\)\}\{\\sin\(\\omega/2\)\}is the Dirichlet kernel, a periodic sinc\-like function whose main lobe has a null\-to\-null width of4​π/L4\\pi/Land whose highest side lobe is approximately−13\-13dB below the main peak\. The complex exponential factore−j​ω​\(n0\+\(L−1\)/2\)e^\{\-j\\omega\(n\_\{0\}\+\(L\-1\)/2\)\}imparts a linear phase shift, but since we are concerned with magnitude spectral characteristics, this phase term does not affect the analysis of problems\. The windowed sequencexw​\[n\]=x⁡\[n\]⋅wL​\[n;n0\]x\_\{w\}\[n\]=x\[n\]\\cdot w\_\{L\}\[n;n\_\{0\}\]is the pointwise product of the original sequence and the rectangular window\. By the DTFT multiplication property, multiplication in the time domain corresponds to*circular convolution*of their DTFTs over a period of length2​π2\\pi:

Xw​\(ej​ω\)=12​π​∫−ππX⁡\(ej​θ\)​WL​\(ej⁡\(ω−θ\),n0\)​𝑑θX\_\{w\}\(e^\{j\\omega\}\)=\\frac\{1\}\{2\\pi\}\\int\_\{\-\\pi\}^\{\\pi\}X\(e^\{j\\theta\}\)\\,W\_\{L\}\(e^\{j\(\\omega\-\\theta\)\};n\_\{0\}\)\\,d\\theta\(19\)
whereXw​\(ej​ω\)X\_\{w\}\(e^\{j\\omega\}\)is the DTFT of sequencexw​\[n\]x\_\{w\}\[n\]\. We then yield the following:

Xw​\(ej​ω\)\\displaystyle X\_\{w\}\(e^\{j\\omega\}\)=12​π​∫−ππX⁡\(ej​θ\)​e−j⁡\(ω−θ\)​\(n0\+L−12\)​sin⁡\(\(ω−θ\)​L/2\)sin⁡\(\(ω−θ\)/2\)​𝑑θ\\displaystyle=\\frac\{1\}\{2\\pi\}\\int\_\{\-\\pi\}^\{\\pi\}X\(e^\{j\\theta\}\)\\,e^\{\-j\(\\omega\-\\theta\)\\left\(n\_\{0\}\+\\frac\{L\-1\}\{2\}\\right\)\}\\frac\{\\sin\\big\(\(\\omega\-\\theta\)L/2\\big\)\}\{\\sin\\big\(\(\\omega\-\\theta\)/2\\big\)\}\\,d\\theta\(20\)=e−j​ω​\(n0\+L−12\)⋅12​π∫−ππ\[X\(ej​θ\)ej​θ​\(n0\+L−12\)\]DL\(ω−θ\)dθ\\displaystyle=e^\{\-j\\omega\\left\(n\_\{0\}\+\\frac\{L\-1\}\{2\}\\right\)\}\\cdot\\frac\{1\}\{2\\pi\}\\int\_\{\-\\pi\}^\{\\pi\}\\Big\[X\(e^\{j\\theta\}\)\\,e^\{j\\theta\\left\(n\_\{0\}\+\\frac\{L\-1\}\{2\}\\right\)\}\\Big\]\\,D\_\{L\}\(\\omega\-\\theta\)\\,d\\theta
whereDL​\(ω−θ\)=sin⁡\(\(ω−θ\)​L/2\)sin⁡\(\(ω−θ\)/2\)D\_\{L\}\(\\omega\-\\theta\)=\\frac\{\\sin\(\(\\omega\-\\theta\)L/2\)\}\{\\sin\(\(\\omega\-\\theta\)/2\)\}is the Dirichlet kernel shifted toθ\\theta\. Equation[20](https://arxiv.org/html/2610.28997#A1.E20)reveals that, up to a global linear phase shift,Xw​\(ej​ω\)X\_\{w\}\(e^\{j\\omega\}\)is the convolution of a phase\-rotated version ofX⁡\(ej​ω\)X\(e^\{j\\omega\}\)with the Dirichlet kernelDL​\(⋅\)D\_\{L\}\(\\cdot\)\. The distortion mainly takes the following three forms: \(1\)spectral leakage:Slowly decaying sidelobes ofDL​\(⋅\)D\_\{L\}\(\\cdot\)spread the energy from each frequency component across the entire spectrum, potentially masking weaker components\. \(2\)resolution limit:The main lobe ofDL​\(⋅\)D\_\{L\}\(\\cdot\)has a width of4​π/L4\\pi/L, which limits the ability to distinguish adjacent spectral peaks\. Components closer than half this interval merge into a single broadened lobe, irrespective of the original sequence lengthNN\. Moreover, the windowed sequence is further truncated to length L, so itdiscards informationbeyond length L while inheriting the identical leakage and resolution penalties\. Since discarding information outside the window is intuitive in the time domain, we next analyze the underlying causes of spectral leakage and resolution limit\.

##### Spectral Leakage\.

ForXw​\(ej​ω\)X\_\{w\}\(e^\{j\\omega\}\)in Equation[20](https://arxiv.org/html/2610.28997#A1.E20), the linear phase factore−j​ω​\(n0\+\(L−1\)/2\)e^\{\-j\\omega\(n\_\{0\}\+\(L\-1\)/2\)\}does not affect the magnitude spectrum, so the magnitude distortion is entirely determined by the convolution with the Dirichlet kernel\. This convolution is the origin of*spectral leakage*: each frequency component of the original signal is not confined to its ideal location but spreads to neighboring frequencies due to the side lobes ofDL​\(ω\)D\_\{L\}\(\\omega\), as shown in Figure[6](https://arxiv.org/html/2610.28997#A1.F6):

Figure 6:Illustration of spectral leakage and weak\-signal masking caused by rectangular window truncation\. \(a\) Rectangular window truncation of a time\-domain signal\. \(b\) Spectral characteristics of the Dirichlet kernel corresponding to the rectangular window\. \(c\) Original spectrum containing adjacent strong and weak signals\. \(d\) Masking of weak signals caused by sidelobe leakage after windowing\. The decibel \(dB\) is a logarithmic unit, so a small difference can represent a large change\.To illustrate the effect concretely with a typical signal from practical time series, consider a real sinusoid infinite\-duration sequencex⁡\[n\]=cos⁡\(ω0​n\)x\[n\]=\\cos\(\\omega\_\{0\}n\)\. Using Euler’s formula,x⁡\[n\]=12​ej​ω0​n\+12​e−j​ω0​nx\[n\]=\\frac\{1\}\{2\}e^\{j\\omega\_\{0\}n\}\+\\frac\{1\}\{2\}e^\{\-j\\omega\_\{0\}n\}, whose DTFT consists of two impulses at±ω0\\pm\\omega\_\{0\}\. Since the DTFT of the window sequencexw​\[n\]x\_\{w\}\[n\]is the convolution ofX⁡\(ej​ω\)X\(e^\{j\\omega\}\)withWL​\(ej​ω,n0\)W\_\{L\}\(e^\{j\\omega\};n\_\{0\}\), linearity allows us to treat each complex exponential\. Substituting the rectangular window DTFT from Equation[19](https://arxiv.org/html/2610.28997#A1.E19)yields the following:

Xw​\(ej​ω\)\\displaystyle X\_\{w\}\(e^\{j\\omega\}\)=12​WL​\(ej⁡\(ω−ω0\),n0\)\+12​WL​\(ej⁡\(ω\+ω0\),n0\)\\displaystyle=\\frac\{1\}\{2\}W\_\{L\}\(e^\{j\(\\omega\-\\omega\_\{0\}\)\};n\_\{0\}\)\+\\frac\{1\}\{2\}W\_\{L\}\(e^\{j\(\\omega\+\\omega\_\{0\}\)\};n\_\{0\}\)\(21\)=12​e−j⁡\(ω−ω0\)​\(n0\+L−12\)​DL​\(ω−ω0\)\+12​e−j⁡\(ω\+ω0\)​\(n0\+L−12\)​DL​\(ω\+ω0\)\\displaystyle=\\frac\{1\}\{2\}e^\{\-j\(\\omega\-\\omega\_\{0\}\)\\left\(n\_\{0\}\+\\frac\{L\-1\}\{2\}\\right\)\}D\_\{L\}\(\\omega\-\\omega\_\{0\}\)\+\\frac\{1\}\{2\}e^\{\-j\(\\omega\+\\omega\_\{0\}\)\\left\(n\_\{0\}\+\\frac\{L\-1\}\{2\}\\right\)\}D\_\{L\}\(\\omega\+\\omega\_\{0\}\)
Whenω0\\omega\_\{0\}is sufficiently far from00andπ\\pi, the main lobes centered at±ω0\\pm\\omega\_\{0\}are well separated, and the cross terms between them become negligible\. In this case, the magnitude spectrum is approximately:

\|Xw​\(ej​ω\)\|≈12​\|DL​\(ω−ω0\)\|\+12​\|DL​\(ω\+ω0\)\|\|X\_\{w\}\(e^\{j\\omega\}\)\|\\approx\\frac\{1\}\{2\}\\big\|D\_\{L\}\(\\omega\-\\omega\_\{0\}\)\\big\|\+\\frac\{1\}\{2\}\\big\|D\_\{L\}\(\\omega\+\\omega\_\{0\}\)\\big\|\(22\)
which explicitly shows that each ideal spectral line is replaced by a broadened main lobe accompanied by oscillatory side lobes\. The factor1/21/2accounts for the splitting of the sinusoid’s energy between positive and negative frequencies\. The side lobes of\|DL​\(⋅\)\|\|D\_\{L\}\(\\cdot\)\|slowly decay, with the highest side lobe only about1313dB below the main peak\. Consequently, a weak sinusoidal component located near a strong one may be completely masked by the side\-lobe energy of the stronger component\.

##### Resolution Limit\.

To understand the resolution limit imposed by the rectangular window, consider two complex exponentialsej​ω1​ne^\{j\\omega\_\{1\}n\}andej​ω2​ne^\{j\\omega\_\{2\}n\}withω1<ω2\\omega\_\{1\}<\\omega\_\{2\}\. After windowing withwL​\[n;n0\]w\_\{L\}\[n;n\_\{0\}\], the DTFT of each component becomes a shifted Dirichlet kernelDL​\(ω−ωi\)D\_\{L\}\(\\omega\-\\omega\_\{i\}\)\(up to a phase factor\)\. The magnitude spectrum therefore contains two main lobes centered atω1\\omega\_\{1\}andω2\\omega\_\{2\}\. According to theRayleigh criterioncommonly adopted in spectral analysis, two peaks are resolved when their separation is at least half the null\-to\-null width of the main lobe\. Since the main lobe ofDL​\(ω\)D\_\{L\}\(\\omega\)has a null\-to\-null width of4​π/L4\\pi/L, the critical separation is2​π/L2\\pi/L\. Hence, two sinusoidal components can be distinguished in the DTFT magnitude spectrum only if

\|ω1−ω2\|≥2​πL\|\\omega\_\{1\}\-\\omega\_\{2\}\|\\geq\\frac\{2\\pi\}\{L\}\(23\)
If the separation is smaller, the two main lobes merge into a single broadened peak, and the components become indistinguishable regardless of the continuous nature of the DTFT\. This is a trade\-off relationship, reducing the window lengthLLimproves temporal localization but inevitably degrades frequency resolution and exacerbates spectral leakage\.

First, we analyze whether the transformation fromxw​\[n\]x\_\{w\}\[n\]toy⁡\[n\]y\[n\]introduces the resolution limit problem\. The patch tokenization extracts the windowed segment and re\-indexes it to start at zero, yieldingy⁡\[m\]=xw​\[n0\+m\]y\[m\]=x\_\{w\}\[n\_\{0\}\+m\]form=0,1,…,L−1m=0,1,\\dots,L\-1\. We now derive its DTFT:

Y⁡\(ej​ω\)\\displaystyle Y\(e^\{j\\omega\}\)=∑m=0L−1y⁡\[m\]​e−j​ω​m\\displaystyle=\\sum\_\{m=0\}^\{L\-1\}y\[m\]\\,e^\{\-j\\omega m\}=∑n=n0n0\+L−1xw​\[n\]​e−j​ω​\(n−n0\)\\displaystyle=\\sum\_\{n=n\_\{0\}\}^\{n\_\{0\}\+L\-1\}x\_\{w\}\[n\]\\,e^\{\-j\\omega\(n\-n\_\{0\}\)\}\(24\)=ej​ω​n0​∑n=n0n0\+L−1xw​\[n\]​e−j​ω​n\\displaystyle=e^\{j\\omega n\_\{0\}\}\\sum\_\{n=n\_\{0\}\}^\{n\_\{0\}\+L\-1\}x\_\{w\}\[n\]\\,e^\{\-j\\omega n\}=ej​ω​n0​Xw​\(ej​ω\)\\displaystyle=e^\{j\\omega n\_\{0\}\}\\,X\_\{w\}\(e^\{j\\omega\}\)\(25\)
Equation[25](https://arxiv.org/html/2610.28997#A1.E25)shows that slicing multiplies the DTFT of the windowed sequence byej​ω​n0e^\{j\\omega n\_\{0\}\}, which corresponds to a time advance ofn0n\_\{0\}samples\. This shows that cutting merely applies a pure time advance, leaving the magnitude spectrum unchanged\. Substituting the expression forXw​\(ej​ω\)X\_\{w\}\(e^\{j\\omega\}\)from Equation[20](https://arxiv.org/html/2610.28997#A1.E20):

Xw​\(ej​ω\)=e−j​ω​\(n0\+L−12\)​12​π​∫−ππX⁡\(ej​θ\)​ej​θ​\(n0\+L−12\)​DL​\(ω−θ\)​𝑑θX\_\{w\}\(e^\{j\\omega\}\)=e^\{\-j\\omega\\left\(n\_\{0\}\+\\frac\{L\-1\}\{2\}\\right\)\}\\frac\{1\}\{2\\pi\}\\int\_\{\-\\pi\}^\{\\pi\}X\(e^\{j\\theta\}\)\\,e^\{j\\theta\\left\(n\_\{0\}\+\\frac\{L\-1\}\{2\}\\right\)\}D\_\{L\}\(\\omega\-\\theta\)\\,d\\theta\(26\)
We obtain

Y⁡\(ej​ω\)\\displaystyle Y\(e^\{j\\omega\}\)=ej​ω​n0​e−j​ω​\(n0\+L−12\)​12​π​∫−ππX⁡\(ej​θ\)​ej​θ​\(n0\+L−12\)​DL​\(ω−θ\)​𝑑θ\\displaystyle=e^\{j\\omega n\_\{0\}\}\\,e^\{\-j\\omega\\left\(n\_\{0\}\+\\frac\{L\-1\}\{2\}\\right\)\}\\frac\{1\}\{2\\pi\}\\int\_\{\-\\pi\}^\{\\pi\}X\(e^\{j\\theta\}\)\\,e^\{j\\theta\\left\(n\_\{0\}\+\\frac\{L\-1\}\{2\}\\right\)\}D\_\{L\}\(\\omega\-\\theta\)\\,d\\theta=12​π​∫−ππX⁡\(ej​θ\)​ej​θ​n0​e−j⁡\(ω−θ\)​L−12​DL​\(ω−θ\)​𝑑θ\\displaystyle=\\frac\{1\}\{2\\pi\}\\int\_\{\-\\pi\}^\{\\pi\}X\(e^\{j\\theta\}\)\\,e^\{j\\theta n\_\{0\}\}\\,e^\{\-j\(\\omega\-\\theta\)\\frac\{L\-1\}\{2\}\}D\_\{L\}\(\\omega\-\\theta\)\\,d\\theta\(27\)
From Equation[25](https://arxiv.org/html/2610.28997#A1.E25)it immediately follows that the magnitude spectra are identical:

\|Y⁡\(ej​ω\)\|=\|Xw​\(ej​ω\)\|\|Y\(e^\{j\\omega\}\)\|=\|X\_\{w\}\(e^\{j\\omega\}\)\|\(28\)
Thus, the stepy⁡\[m\]=xw​\[n0\+m\]y\[m\]=x\_\{w\}\[n\_\{0\}\+m\]form=0,…,L−1m=0,\\dots,L\-1does not alter the leak pattern or the frequency resolution; it simply realigns the temporal origin\. All subsequent discussion of spectral leakage and resolution applies equally to the sliced sequencey⁡\[m\]y\[m\]that constitutes the patch token\.

Figure 7:Illustration of the picket\-fence effect\. \(a\) Picket\-fence effect at linear magnitude\. \(b\) Picket\-fence effect at dB magnitude\.Although DTFTY⁡\(ej​ω\)Y\(e^\{j\\omega\}\)is a continuous function ofω\\omega, any practical spectral analysis must evaluate it at a finite set of discrete frequencies\. When we sample the DTFT atMMuniformly spaced frequenciesωk=2​π​k/M\\omega\_\{k\}=2\\pi k/M,k=0,…,M−1k=0,\\dots,M\-1, we obtain a discrete representation that is subject to thepicket\-fence effect\. When viewing a landscape through a picket fence, one can see only the portions of the scene that align with the gaps between the slats, while the rest remains hidden\. In the spectral context, the continuous DTFT is the “landscape,” and theMMfrequency samplesωk\\omega\_\{k\}are the “gaps\.” If a signal component’s true frequency lies exactly on one of these sample points, its full amplitude is observed; if it falls between two points, the observed value at the nearest sample is only an attenuated version of the actual spectral peak\. The severity of this attenuation depends on how far the true frequency deviates from the nearest sampling point\.

To quantify this effect, consider a single complex sinusoid of frequencyω0\\omega\_\{0\}after rectangular windowing\. Its continuous DTFT is a shifted Dirichlet kernel, i\.e\.,Y⁡\(ej​ω\)=ej​ϕ​\(ω\)​DL​\(ω−ω0\)Y\(e^\{j\\omega\}\)=e^\{j\\phi\(\\omega\)\}D\_\{L\}\(\\omega\-\\omega\_\{0\}\), whereϕ⁡\(ω\)\\phi\(\\omega\)is a linear phase term\. The true magnitude peak occurs exactly atω=ω0\\omega=\\omega\_\{0\}and is equal to\|DL​\(0\)\|=L\|D\_\{L\}\(0\)\|=L\. Suppose that the frequencyω0\\omega\_\{0\}falls between two discrete frequency grid points, written asω0=2​π​\(k0\+δ\)/M\\omega\_\{0\}=2\\pi\(k\_\{0\}\+\\delta\)/M, wherek0k\_\{0\}is the index of the nearest grid point and0<\|δ\|<0\.50<\|\\delta\|<0\.5measures the fractional bin offset\. The discrete spectrum is evaluated only on the fixed gridωk=2​π​k/M\\omega\_\{k\}=2\\pi k/M\. Hence, the maximum amplitude that we can observe among the discrete bins is

\|Y⁡\(ej​ωk0\)\|=\|DL​\(2​π​δ/M\)\|=\|sin⁡\(π​δ​L/M\)sin⁡\(π​δ/M\)\|\|Y\(e^\{j\\omega\_\{k\_\{0\}\}\}\)\|=\|D\_\{L\}\(2\\pi\\delta/M\)\|=\\left\|\\frac\{\\sin\(\\pi\\delta L/M\)\}\{\\sin\(\\pi\\delta/M\)\}\\right\|\(29\)
which is strictly smaller thanLLwheneverδ≠0\\delta\\neq 0\. The ratio of the observed peak to the true peak,

\|Y⁡\(ej​ωk0\)\|\|DL​\(0\)\|=1L​\|sin⁡\(π​δ​L/M\)sin⁡\(π​δ/M\)\|\\frac\{\|Y\(e^\{j\\omega\_\{k\_\{0\}\}\}\)\|\}\{\|D\_\{L\}\(0\)\|\}=\\frac\{1\}\{L\}\\left\|\\frac\{\\sin\(\\pi\\delta L/M\)\}\{\\sin\(\\pi\\delta/M\)\}\\right\|\(30\)
This is known as the scalloping loss or picket\-fence loss\. For example, as shown in Figure[7](https://arxiv.org/html/2610.28997#A1.F7), in the worst\-case offsetδ=0\.5\\delta=0\.5withM=LM=L\(no zero\-padding\), this ratio is about\|sin⁡\(π/2\)/\(π/2\)\|≈0\.637\|\\sin\(\\pi/2\)/\(\\pi/2\)\|\\approx 0\.637, implying an amplitude reduction of more than36%36\\%and nearly 4 dB compared to the true peak\. The effect can be mitigated by increasingMM, the number of frequency samples over the same interval, which is usually achieved by zero padding the time\-domain sequencey⁡\[m\]y\[m\]before computing the DFT\. Zero\-padding appends zeros toy⁡\[m\]y\[m\], resulting in a denser set of DTFT evaluation pointsωk\\omega\_\{k\}without altering the underlying continuous spectrumY⁡\(ej​ω\)Y\(e^\{j\\omega\}\)\.

### A\.2Phase Tokenization

Letx⁡\[n\]x\[n\]be a finite\-length discrete\-time sequence of lengthNN, and letS≥2S\\geq 2be an integer downsampling factor\. The downsampled sequence isy\[m\]=x\[mS\],m=0,1,…,⌈N/S⌉−1y\[m\]=x\[mS\],\\ m=0,1,\\dots,\\lceil N/S\\rceil\-1\. We decompose this operation into two stages: \(i\) zero insertion, and \(ii\) time compression\. To analyze the spectral consequences, it is convenient to introduce an auxiliary zero\-filled sequence:

w⁡\[n\]=\{x⁡\[n\],n=kS,k∈ℤ,0,otherwise\.w\[n\]=\\begin\{cases\}x\[n\],&n=kS,\\;k\\in\\mathbb\{Z\},\\\\ 0,&\\text\{otherwise\}\.\\end\{cases\}\(31\)
which is simplyx⁡\[n\]x\[n\]sampled on everySS\-th index and padded with zeros elsewhere\. The relation betweenw⁡\[n\]w\[n\]and the final downsampled signaly⁡\[m\]y\[m\]is

y⁡\[m\]=w⁡\[m​S\],m∈ℤy\[m\]=w\[mS\],\\ m\\in\\mathbb\{Z\}\(32\)
The sequencew⁡\[n\]w\[n\]can be expressed as the product ofx⁡\[n\]x\[n\]with a periodic impulse sequence

p⁡\[n\]=∑k=−∞∞δ⁡\[n−k​S\]p\[n\]=\\sum\_\{k=\-\\infty\}^\{\\infty\}\\delta\[n\-kS\]\(33\)
whereδ⁡\[n\]\\delta\[n\]denotes the unit sample \(δ⁡\[0\]=1\\delta\[0\]=1andδ⁡\[n\]=0\\delta\[n\]=0forn≠0n\\neq 0\)\. That is,

w⁡\[n\]=x⁡\[n\]​p​\[n\]w\[n\]=x\[n\]\\,p\[n\]\(34\)
The DTFT of the impulse sequencep⁡\[n\]p\[n\]is obtained via the Poisson summation formula\. The sequencep⁡\[n\]=∑k=−∞∞δ⁡\[n−k​S\]p\[n\]=\\sum\_\{k=\-\\infty\}^\{\\infty\}\\delta\[n\-kS\]is periodic with periodSS\. Within one periodn=0,1,…,S−1n=0,1,\\dots,S\-1, we havep⁡\[0\]=1p\[0\]=1andp⁡\[n\]=0p\[n\]=0forn≠0n\\neq 0\. Its discrete Fourier series coefficients are therefore

Pc\[k\]=1S∑n=0S−1p\[n\]e−j2πkn/S=1S,k=0,1,…,S−1P\_\{c\}\[k\]=\\frac\{1\}\{S\}\\sum\_\{n=0\}^\{S\-1\}p\[n\]\\,e^\{\-j2\\pi kn/S\}=\\frac\{1\}\{S\},\\ k=0,1,\\dots,S\-1\(35\)
Therefore,

p⁡\[n\]=∑k=0S−1Pc​\[k\]​ej​2​π​k​n/S=1S​∑k=0S−1ej​2​π​k​n/Sp\[n\]=\\sum\_\{k=0\}^\{S\-1\}P\_\{c\}\[k\]\\,e^\{j2\\pi kn/S\}=\\frac\{1\}\{S\}\\sum\_\{k=0\}^\{S\-1\}e^\{j2\\pi kn/S\}\(36\)
Taking the DTFT of both sides,

P⁡\(ej​ω\)\\displaystyle P\(e^\{j\\omega\}\)=∑n=−∞∞p⁡\[n\]​e−j​ω​n=1S​∑k=0S−1∑n=−∞∞e−j⁡\(ω−2​π​k/S\)​n\\displaystyle=\\sum\_\{n=\-\\infty\}^\{\\infty\}p\[n\]\\,e^\{\-j\\omega n\}=\\frac\{1\}\{S\}\\sum\_\{k=0\}^\{S\-1\}\\;\\sum\_\{n=\-\\infty\}^\{\\infty\}e^\{\-j\(\\omega\-2\\pi k/S\)n\}=2​πS​∑k=0S−1∑r=−∞∞δ⁡\(ω−2​π​\(k\+r​S\)S\)\\displaystyle=\\frac\{2\\pi\}\{S\}\\sum\_\{k=0\}^\{S\-1\}\\sum\_\{r=\-\\infty\}^\{\\infty\}\\delta\\\!\\left\(\\omega\-\\frac\{2\\pi\(k\+rS\)\}\{S\}\\right\)\(37\)
where we have used∑n=−∞∞e−j​α​n=2​π​∑r=−∞∞δ⁡\(α−2​π​r\)\\sum\_\{n=\-\\infty\}^\{\\infty\}e^\{\-j\\alpha n\}=2\\pi\\sum\_\{r=\-\\infty\}^\{\\infty\}\\delta\(\\alpha\-2\\pi r\)\. Lettingm=k\+r​Sm=k\+rS, the double sum collapses to a single sum over all integersmm:

P⁡\(ej​ω\)=2​πS​∑m=−∞∞δ⁡\(ω−2​π​mS\)P\(e^\{j\\omega\}\)=\\frac\{2\\pi\}\{S\}\\sum\_\{m=\-\\infty\}^\{\\infty\}\\delta\\\!\\left\(\\omega\-\\frac\{2\\pi m\}\{S\}\\right\)\(38\)
Due to the2​π2\\pi\-periodicity of the DTFT, onlySSdistinct impulses fall within any interval of length2​π2\\pi\. For the principal interval\[−π,π\)\[\-\\pi,\\pi\)one may restrict the index set tom=0,1,…,S−1m=0,1,\\dots,S\-1when operating inside the convolution integral\. Multiplication in the time domain corresponds to circular convolution in the frequency domain:

W⁡\(ej​ω\)=12​π​∫−ππX⁡\(ej​θ\)​P​\(ej⁡\(ω−θ\)\)​𝑑θW\(e^\{j\\omega\}\)=\\frac\{1\}\{2\\pi\}\\int\_\{\-\\pi\}^\{\\pi\}X\(e^\{j\\theta\}\)\\,P\\bigl\(e^\{j\(\\omega\-\\theta\)\}\\bigr\)\\,d\\theta\(39\)
Equations[38](https://arxiv.org/html/2610.28997#A1.E38)and[39](https://arxiv.org/html/2610.28997#A1.E39)give

W⁡\(ej​ω\)\\displaystyle W\(e^\{j\\omega\}\)=12​π​∫−ππX⁡\(ej​θ\)​2​πS​∑m=−∞∞δ⁡\(ω−θ−2​π​mS\)​𝑑θ\\displaystyle=\\frac\{1\}\{2\\pi\}\\int\_\{\-\\pi\}^\{\\pi\}X\(e^\{j\\theta\}\)\\,\\frac\{2\\pi\}\{S\}\\sum\_\{m=\-\\infty\}^\{\\infty\}\\delta\\\!\\left\(\\omega\-\\theta\-\\frac\{2\\pi m\}\{S\}\\right\)d\\theta=1S​∑m=−∞∞∫−ππX⁡\(ej​θ\)​δ​\(ω−θ−2​π​mS\)​𝑑θ\\displaystyle=\\frac\{1\}\{S\}\\sum\_\{m=\-\\infty\}^\{\\infty\}\\int\_\{\-\\pi\}^\{\\pi\}X\(e^\{j\\theta\}\)\\,\\delta\\\!\\left\(\\omega\-\\theta\-\\frac\{2\\pi m\}\{S\}\\right\)d\\theta\(40\)
For a fixedω∈\(−π,π\]\\omega\\in\(\-\\pi,\\pi\], the argument of the delta function is zero whenθ=ω−2​π​m/S\\theta=\\omega\-2\\pi m/S\. The integral is not zero only if thisθ\\thetais within the integration limits\[−π,π\)\[\-\\pi,\\pi\)\. BecauseX⁡\(ej​θ\)X\(e^\{j\\theta\}\)is2​π2\\pi\-periodic, exactlySSdistinct values ofmmcontribute \(conventionallym=0,1,…,S−1m=0,1,\\dots,S\-1after reindexing\)\. Exploiting the sifting property of the Dirac delta, we obtain:

W⁡\(ej​ω\)=1S​∑k=0S−1X⁡\(ej⁡\(ω−2​π​k/S\)\)W\(e^\{j\\omega\}\)=\\frac\{1\}\{S\}\\sum\_\{k=0\}^\{S\-1\}X\\\!\\left\(e^\{j\(\\omega\-2\\pi k/S\)\}\\right\)\(41\)
Equation[41](https://arxiv.org/html/2610.28997#A1.E41)shows that the zero\-filled sequence possesses a spectrum consisting ofSSshifted replicas of the original spectrumX⁡\(ej​ω\)X\(e^\{j\\omega\}\)\(each scaled by1/S1/S\), packed into the interval\[−π,π\)\[\-\\pi,\\pi\)\. From Equation[32](https://arxiv.org/html/2610.28997#A1.E32)we havey⁡\[m\]=w⁡\[m​S\]y\[m\]=w\[mS\]\. Its DTFT is

Y⁡\(ej​ω\)=∑m=−∞∞y⁡\[m\]​e−j​ω​m=∑m=−∞∞w⁡\[m​S\]​e−j​ω​mY\(e^\{j\\omega\}\)=\\sum\_\{m=\-\\infty\}^\{\\infty\}y\[m\]\\,e^\{\-j\\omega m\}=\\sum\_\{m=\-\\infty\}^\{\\infty\}w\[mS\]\\,e^\{\-j\\omega m\}\(42\)
Settingn=m​Sn=mS, we see that the sum runs over all integer multiples ofSS\. Becausew⁡\[n\]=0w\[n\]=0fornnnot a multiple ofSS, we may sum over all integersnn:

Y⁡\(ej​ω\)=∑n=−∞∞w⁡\[n\]​e−j⁡\(ω/S\)​n=W⁡\(ej​ω/S\)Y\(e^\{j\\omega\}\)=\\sum\_\{n=\-\\infty\}^\{\\infty\}w\[n\]\\,e^\{\-j\(\\omega/S\)n\}=W\\\!\\left\(e^\{j\\omega/S\}\\right\)\(43\)
Thus, the time\-compressionSSin the time domain expands the frequency axis by a factorSS; i\.e\., the DTFT ofy⁡\[m\]y\[m\]is the same as that ofw⁡\[n\]w\[n\]but evaluated atω/S\\omega/S\. Substituting Equation[41](https://arxiv.org/html/2610.28997#A1.E41)into Equation[43](https://arxiv.org/html/2610.28997#A1.E43)yields the DTFT of the downsampled sequence:

Y⁡\(ej​ω\)=1S​∑k=0S−1X⁡\(ej⁡\(ω/S−2​π​k/S\)\)Y\(e^\{j\\omega\}\)=\\frac\{1\}\{S\}\\sum\_\{k=0\}^\{S\-1\}X\\\!\\left\(e^\{j\(\\omega/S\-2\\pi k/S\)\}\\right\)\(44\)
This is a fundamental description of downsampling in the frequency domain\. The original spectrumX⁡\(ej​ω\)X\(e^\{j\\omega\}\)is first compressed by the factorSS\(argumentω/S\\omega/S\) and then shifted by multiples of2​π/S2\\pi/S\. IfX⁡\(ej​ω\)X\(e^\{j\\omega\}\)contains non\-zero energy for frequencies\|ω\|≥π/S\|\\omega\|\\geq\\pi/S, the shifted replicas will overlap, causing high frequency components to irreversibly fold into lower frequencies, which isspectral aliasing\. To avoid aliasing, the replicas must remain distinct\.

Equation[44](https://arxiv.org/html/2610.28997#A1.E44)shows both spectral stretching and the potential for aliasing\. Consider first the term corresponding tok=0k=0, namely1S​X​\(ej​ω/S\)\\frac\{1\}\{S\}X\(e^\{j\\omega/S\}\)\. Here, the original spectrumX⁡\(ej​ω\)X\(e^\{j\\omega\}\)undergoes a scaling of the frequency axis by the factorSS\. A frequency component originally located atω0\\omega\_\{0\}inx⁡\[n\]x\[n\]will appear atS​ω0S\\omega\_\{0\}inY⁡\(ej​ω\)Y\(e^\{j\\omega\}\)\. This expansion of the frequency axis is called spectral stretching\. Consequently, a narrowband signal inx⁡\[n\]x\[n\]spreads over a proportionally wider bandwidth in the downsampled sequence, and its amplitude is reduced by the factorSS\.

The remainingS−1S\-1terms ofk=1,…,S−1k=1,\\dots,S\-1, have the form1S​X​\(ej⁡\(ω/S−2​π​k/S\)\)\\frac\{1\}\{S\}X\\bigl\(e^\{j\(\\omega/S\-2\\pi k/S\)\}\\bigr\)\. These arise from the shifted replicas created during the intermediate zero\-insertion step\. In the spectrum of the zero\-filled sequenceW⁡\(ej​ω\)W\(e^\{j\\omega\}\)in Equation[41](https://arxiv.org/html/2610.28997#A1.E41), these replicas are centred atω=2​π​k/S\\omega=2\\pi k/S\. The time\-compression stepy⁡\[m\]=w⁡\[m​S\]y\[m\]=w\[mS\]corresponds to the frequency mappingY⁡\(ej​ω\)=W⁡\(ej​ω/S\)Y\(e^\{j\\omega\}\)=W\(e^\{j\\omega/S\}\), which expands the frequency axis bySS\. Hence, a replica originally centered at2​π​k/S2\\pi k/Sis mapped to the new center:

S⋅2​π​kS=2​π​kS\\cdot\\frac\{2\\pi k\}\{S\}=2\\pi k\(45\)
Because the DTFT is2​π2\\pi\-periodic, all these centers fall at integer multiples of2​π2\\pi, which are exactly the boundaries of the principal interval\[−π,π\)\[\-\\pi,\\pi\)\. Thus, after the downsampling operation, every one of theSSreplicas is assigned to the same baseband interval\.

Figure 8:Illustration of spectral aliasing in the frequency domain\. When the original spectrum is not confined to\(−π/S,π/S\)\(\-\\pi/S,\\pi/S\), high frequency components appear as spurious low frequency components\.Whether these replicas overlap destructively depends on the bandwidth ofX⁡\(ej​ω\)X\(e^\{j\\omega\}\)\. IfX⁡\(ej​ω\)X\(e^\{j\\omega\}\)contains non\-zero energy for\|ω\|≥π/S\|\\omega\|\\geq\\pi/S, then thek=0k=0baseband replica already extends beyond\|ω\|≤π\|\\omega\|\\leq\\piafter stretching, and thek≠0k\\neq 0terms bring additional copies of the spectrum that intrude into the same region\. More precisely, a frequencyω\\omegainY⁡\(ej​ω\)Y\(e^\{j\\omega\}\)receives contributions from allkkfor whichω/S−2​π​k/S\\omega/S\-2\\pi k/Sfalls within the support ofX⁡\(ej​ω\)X\(e^\{j\\omega\}\)\. As shown in Figure[8](https://arxiv.org/html/2610.28997#A1.F8), when the original spectrum is not confined to\(−π/S,π/S\)\(\-\\pi/S,\\pi/S\), there exist frequencies at which two or more replicas contribute, causing high frequency components to appear as spurious low frequency components\. This irreversible superposition is spectral aliasing\. Because the overlapping terms all share the same frequency inY⁡\(ej​ω\)Y\(e^\{j\\omega\}\), the aliased components cannot be separated after downsampling, leading to permanent distortion\.

To avoid aliasing entirely, the support ofX⁡\(ej​ω\)X\(e^\{j\\omega\}\)must be a subset of\(−π/S,π/S\)\(\-\\pi/S,\\pi/S\)\. In other words, the maximum normalized frequencyωmax\\omega\_\{\\max\}present inx⁡\[n\]x\[n\]must satisfy

ωmax≤πS\\omega\_\{\\max\}\\leq\\frac\{\\pi\}\{S\}\(46\)
When this condition is satisfied, each frequencyω\\omegain the principal interval receives a contribution from exactly one value ofkk\(the one for whichω/S−2​π​k/S\\omega/S\-2\\pi k/Slies within the support ofXX\)\. The replicas are then placed side\-by\-side without overlapping, and the original signal can in principle be recovered from the downsampled version\.

The condition can be expressed in terms of physical frequencies by relatingω\\omegato the underlying continuous\-time signal from whichx⁡\[n\]x\[n\]was obtained\. Assumex​\[n\]=xc​\(n​T\)x\[n\]=x\_\{c\}\(nT\), with the sampling periodT=1/fsT=1/f\_\{s\}andfsf\_\{s\}the original sampling rate\. The normalized frequency isω=2​π​f/fs\\omega=2\\pi f/f\_\{s\}\. If the analog signalxc​\(t\)x\_\{c\}\(t\)is bandlimited, say\|f\|≤B\|f\|\\leq B, thenX⁡\(ej​ω\)X\(e^\{j\\omega\}\)is non\-zero only for\|ω\|≤2​π​B/fs\|\\omega\|\\leq 2\\pi B/f\_\{s\}\. Imposing the no\-aliasing requirement Equation[46](https://arxiv.org/html/2610.28997#A1.E46)gives

2​π​Bfs≤πS⟹B≤fs2​S\\frac\{2\\pi B\}\{f\_\{s\}\}\\leq\\frac\{\\pi\}\{S\}\\quad\\Longrightarrow\\quad B\\leq\\frac\{f\_\{s\}\}\{2S\}\(47\)
After downsampling, the effective sampling rate becomesfs′=fs/Sf\_\{s\}^\{\\prime\}=f\_\{s\}/S, and its associated Nyquist frequency isfs′/2=fs/\(2​S\)f\_\{s\}^\{\\prime\}/2=f\_\{s\}/\(2S\)\. Therefore, alias\-free downsampling is possible if and only if the original continuous\-time signal is bandlimited to less than or equal to the new Nyquist frequency\.

### A\.3Boundary Conditions for Distortion\-Free Tokenization

The preceding analyses establish that patch tokenization and phase tokenization are governed by distinct mathematical constraints\. We now formalize the precise boundary conditions to avoid irreversible distortion\.

#### A\.3\.1Boundary Condition for Patch Tokenization

Patch tokenization, as defined in Equation[1](https://arxiv.org/html/2610.28997#S1.E1), is mathematically equivalent to multiplying the original sequencex⁡\[n\]x\[n\]by a rectangular window of lengthLLstarting at indexn0n\_\{0\}, and then re\-indexing the non\-zero segment from zero\. The window sequence isxw​\[n\]=x⁡\[n\]​wL​\[n;n0\]x\_\{w\}\[n\]=x\[n\]\\,w\_\{L\}\[n;n\_\{0\}\], and the final patch token isy⁡\[m\]=xw​\[n0\+m\]y\[m\]=x\_\{w\}\[n\_\{0\}\+m\]form=0,…,L−1m=0,\\dots,L\-1\.

For an exact reconstruction of the original finite length sequencex⁡\[n\]x\[n\]of lengthNNfrom its patch token, it is necessary and sufficient that the window covers all non\-zero samples ofx⁡\[n\]x\[n\]\. This requiresn0≤0n\_\{0\}\\leq 0andn0\+L−1≥N−1n\_\{0\}\+L\-1\\geq N\-1\. When the window starts atn0=0n\_\{0\}=0, the condition reduces toL≥NL\\geq N\. Under this condition, the tokeny⁡\[m\]=x⁡\[m\]y\[m\]=x\[m\]form=0,…,N−1m=0,\\dots,N\-1\(and zero form≥Nm\\geq N\), so the originalx⁡\[n\]x\[n\]can be recovered losslessly by discarding the zero\-padded tail\.

Ifx⁡\[n\]x\[n\]is periodic with a fundamental periodT∈ℕ\+T\\in\\mathbb\{N\}^\{\+\}, i\.e\.,x⁡\[n\+T\]=x⁡\[n\]x\[n\+T\]=x\[n\]for all integersnn, then the entire signal is fully determined by any contiguous block ofTTsamples\. Hence, choosing the window length as an integer multiple of the period,L=m​TL=mT\(m∈ℕ\+m\\in\\mathbb\{N\}^\{\+\}\), guarantees that the token contains exactly the information needed to reconstruct the whole periodic sequence by periodic extension, irrespective of the starting indexn0n\_\{0\}\.

Although time domain preservation is straightforward, the windowing operation inherently alters the frequency\-domain representation\. As derived in Section[A\.1](https://arxiv.org/html/2610.28997#A1.SS1), multiplication by the rectangular window convolves the true spectrum with a Dirichlet kernel, whose main\-lobe width \(4​π/L4\\pi/L\) limits frequency resolution and whose sidelobes cause spectral leakage\.

For periodic signals withL=m​TL=mT, an important special case arises: the discrete Fourier transform \(or the DTFT evaluated at the harmonic frequencies\) samples the Dirichlet kernel exactly at its zero crossings between harmonics\. This coherent sampling completely eliminates spectral leakage at those discrete frequency bins because the contribution from any harmonic to a different harmonic’s bin is identically zero\. However, this cancellation occurs only at the sampled frequencies; the continuous DTFT still exhibits sidelobes between the bins, and the fundamental resolution limit imposed by the main\-lobe width persists\. For non\-periodic signals or whenL≠m​TL\\neq mT, the Dirichlet kernel’s nulls no longer align with the analysis frequencies, and spectral leakage becomes inevitable\.

The resolution limit states that two sinusoids can be separated in the magnitude spectrum only if their frequency separation exceeds half the main\-lobe width,\|ω1−ω2\|≥2​πL\|\\omega\_\{1\}\-\\omega\_\{2\}\|\\geq\\frac\{2\\pi\}\{L\}\. This limit is entirely independent of whether the lossless condition in the time domain is satisfied\. This follows only from the finite observation lengthLL\.

In summary, the time\-domain lossless condition for patch tokenization isL≥NL\\geq Nfor finite signals, orL=m​TL=mTfor periodic signals\. In the frequency domain, spectral leakage is an unavoidable consequence of the finite rectangular window, except that for coherently sampled periodic signals it is null at the DFT bins\.

#### A\.3\.2Boundary Condition for Phase Tokenization

Phase tokenization, equivalent to downsampling the original sequence by an integer factorS≥2S\\geq 2, imposes a fundamentally different and more restrictive boundary condition\. Compared to patch tokenization, the boundary condition of phase tokenization is much simpler\. As derived from Equation[44](https://arxiv.org/html/2610.28997#A1.E44), the alias\-free condition for the discrete time signal is given by Equation[47](https://arxiv.org/html/2610.28997#A1.E47)\. This constitutes the necessary and sufficient condition for distortion\-free phase tokenization\. If the original signal violates this bandlimit, high\-frequency energy residing outside the interval\|ω\|≤π/S\|\\omega\|\\leq\\pi/Swill be folded into the baseband, causing irreversible distortion\.

### A\.4Analysis of TimeFT Methodology

#### A\.4\.1Analysis of Distortion

Consider a real sequence𝐗∈ℝN\\mathbf\{X\}\\in\\mathbb\{R\}^\{N\}\. Its RFFT𝐗freq∈ℂNf\\mathbf\{X\}\_\{\\mathrm\{freq\}\}\\in\\mathbb\{C\}^\{N\_\{f\}\}collects the non\-negative frequency components, with theff\-th bin corresponding to the angular frequencyωf=2​π​f/N\\omega\_\{f\}=2\\pi f/N,f=0,…,Nf−1f=0,\\dots,N\_\{f\}\-1\. The block size is chosen asBN=⌊N/\(2​S\)⌋B\_\{N\}=\\bigl\\lfloor N/\(2S\)\\bigr\\rfloor, satisfying the boundary condition[5](https://arxiv.org/html/2610.28997#S3.E5)\.

Thek−t​hk\-thspectral block𝐗block\(k\)\\mathbf\{X\}\_\{\\mathrm\{block\}\}^\{\(k\)\}retains only the frequency binsf∈\[k​BN,k​BN\+BN−1\]f\\in\[kB\_\{N\},\\,kB\_\{N\}\+B\_\{N\}\-1\]and zeros out the remaining ones\. After the shifting step, i\.e\., reshaping into blocks of sizeBNB\_\{N\}and zero\-padding back to lengthNfN\_\{f\}, each block becomes a spectrum𝐗~\(k\)\\widetilde\{\\mathbf\{X\}\}^\{\(k\)\}concentrated in the low\-frequency intervalf=0,…,BN−1f=0,\\dots,B\_\{N\}\-1:

\[𝐗~\(k\)\]b=\{\[𝐗freq\]k​BN\+b,0≤b≤BN−1,0,otherwise\.\\bigl\[\\widetilde\{\\mathbf\{X\}\}^\{\(k\)\}\\bigr\]\_\{b\}=\\begin\{cases\}\[\\mathbf\{X\}\_\{\\mathrm\{freq\}\}\]\_\{kB\_\{N\}\+b\},&0\\leq b\\leq B\_\{N\}\-1,\\\\ 0,&\\text\{otherwise\}\.\\end\{cases\}\(48\)
The inverse RFFT of𝐗~\(k\)\\widetilde\{\\mathbf\{X\}\}^\{\(k\)\}yields a real\-valued time\-domain sequence𝐱time\(k\)∈ℝN\\mathbf\{x\}\_\{\\mathrm\{time\}\}^\{\(k\)\}\\in\\mathbb\{R\}^\{N\}\. By the properties of the DFT, the spectral support of𝐱time\(k\)\\mathbf\{x\}\_\{\\mathrm\{time\}\}^\{\(k\)\}is confined to \(normalized\) frequencies

\|ω\|≤2​π​\(BN−1\)N<2​π​BNN\|\\omega\|\\leq\\frac\{2\\pi\(B\_\{N\}\-1\)\}\{N\}<\\frac\{2\\pi B\_\{N\}\}\{N\}\(49\)
We then obtain the strict band limit:

\|ω\|<2​πN⋅N2​S=πS\|\\omega\|<\\frac\{2\\pi\}\{N\}\\cdot\\frac\{N\}\{2S\}=\\frac\{\\pi\}\{S\}\(50\)
Equation[50](https://arxiv.org/html/2610.28997#A1.E50)is exactly the alias\-free condition for downsampling by a factorSS\(§[A\.2](https://arxiv.org/html/2610.28997#A1.SS2)\)\. Hence, every sub\-band signal𝐱time\(k\)\\mathbf\{x\}\_\{\\mathrm\{time\}\}^\{\(k\)\}obeys the Nyquist criterion for the subsequent decimation step\. The operator𝒟S\\mathcal\{D\}\_\{S\}defined in Equation[12](https://arxiv.org/html/2610.28997#S3.E12)extracts the polyphase components of𝐱time\(k\)\\mathbf\{x\}\_\{\\mathrm\{time\}\}^\{\(k\)\}with offsetsλ=0,…,S−1\\lambda=0,\\dots,S\-1\. Because the signal is bandlimited to\|ω\|<π/S\|\\omega\|<\\pi/S, each decimated polyphase stream is free from spectral aliasing\. Therefore, the entire information contained in thekk\-th frequency sub\-band is almost perfectly transferred to the tokens\.

The simplified spectral shifting of Equation[10](https://arxiv.org/html/2610.28997#S3.E10)moves a complex Fourier coefficient originally located at frequency bink​BN\+bkB\_\{N\}\+bdirectly to positionbb\. Whenb=0b=0, this coefficient becomes the DC term of the shifted block\. For a real signal, the DC component of the RFFT must be purely real\. The IRFFT therefore discards any imaginary part at that point\. Consequently, a single complex value per block loses its imaginary part, a negligible fraction of the total coefficients\. As shown in the visualization in Appendix[D](https://arxiv.org/html/2610.28997#A4), TimeFT tokens can be well reconstructed into the original sequence, and thus this discarding is negligible\. All other frequencies \(b≠0b\\neq 0\) are almost perfectly preserved\. Hence, TimeFT is almost distortion\-free\.

#### A\.4\.2Linear Phase Induced by Frequency Shifting

Letx⁡\[n\]x\[n\],n=0,…,N−1n=0,\\dots,N\-1, be a real sequence andXr​\[f\]X\_\{r\}\[f\],f=0,…,Nf−1f=0,\\dots,N\_\{f\}\-1, its RFFT coefficients, whereNf=⌊N/2⌋\+1N\_\{f\}=\\lfloor N/2\\rfloor\+1\. We isolate thekk\-th sub\-band of widthBNB\_\{N\}, starting at indexk​BNkB\_\{N\}\(BN=⌊N/\(2​S\)⌋B\_\{N\}=\\lfloor N/\(2S\)\\rfloor,SSis the downsampling factor\)ℐk=\{k​BN,k​BN\+1,…,k​BN\+BN−1\}\\mathcal\{I\}\_\{k\}=\\\{\\,kB\_\{N\},\\,kB\_\{N\}\+1,\\dots,\\,kB\_\{N\}\+B\_\{N\}\-1\\,\\\}\. The spectral shifting stage moves the content ofℐk\\mathcal\{I\}\_\{k\}to the baseband0,…,BN−10,\\dots,B\_\{N\}\-1\. A natural implementation copies the coefficients without modification:

X~r\[b\]=Xr\[kBN\+b\],b=0,…,BN−1\\widetilde\{X\}\_\{r\}\[b\]=X\_\{r\}\[kB\_\{N\}\+b\],\\quad b=0,\\dots,B\_\{N\}\-1\(51\)
and zeros the remaining bins\. We then perform aNN\-point inverse RFFT to obtain a real sequencex~\(k\)​\[n\]\\tilde\{x\}^\{\(k\)\}\[n\]of lengthNN\. We will show that this simple copy operation already produces the correct baseband signal, no extra phase rotation is required when the inverse transform retains the original lengthNN\.

##### Why Direct Copy Preserves the Phase Relations\.

TheNN\-point DFT of the original signal satisfiesX⁡\[ℓ\]=∑n=0N−1x⁡\[n\]​e−j​2​π​ℓ​nNX\[\\ell\]=\\sum\_\{n=0\}^\{N\-1\}x\[n\]\\,e^\{\-j\\frac\{2\\pi\\ell n\}\{N\}\}and for realx⁡\[n\]x\[n\]we haveXr​\[f\]=X​\[f\]X\_\{r\}\[f\]=X\[f\]for0≤f≤N/20\\leq f\\leq N/2\. Consider the contribution of thekk\-th sub\-band tox⁡\[n\]x\[n\]:

xband,k​\[n\]=2N​∑b=0BN−1Re⁡\(X⁡\[k​BN\+b\]​ej​2​π​\(k​BN\+b\)​nN\)x\_\{\\text\{band\},k\}\[n\]=\\frac\{2\}\{N\}\\sum\_\{b=0\}^\{B\_\{N\}\-1\}\\operatorname\{Re\}\\\!\\Bigl\(X\[kB\_\{N\}\+b\]\\,e^\{j\\frac\{2\\pi\(kB\_\{N\}\+b\)n\}\{N\}\}\\Bigr\)\(52\)
Now, take the DFT of the sequencex~\(k\)\\tilde\{x\}^\{\(k\)\}obtained from the copied spectrum[51](https://arxiv.org/html/2610.28997#A1.E51)\. By construction, its DFT is non\-zero only at indicesbbandN−bN\-bwith coefficientsX⁡\[k​BN\+b\]X\[kB\_\{N\}\+b\]andX∗​\[k​BN\+b\]X^\{\*\}\[kB\_\{N\}\+b\], respectively\. Therefore,

x~\(k\)​\[n\]\\displaystyle\\tilde\{x\}^\{\(k\)\}\[n\]=1N​∑b=0BN−1\(X⁡\[k​BN\+b\]​ej​2​π​b​nN\+X∗​\[k​BN\+b\]​e−j​2​π​b​nN\)\\displaystyle=\\frac\{1\}\{N\}\\sum\_\{b=0\}^\{B\_\{N\}\-1\}\\Bigl\(X\[kB\_\{N\}\+b\]\\,e^\{j\\frac\{2\\pi bn\}\{N\}\}\+X^\{\*\}\[kB\_\{N\}\+b\]\\,e^\{\-j\\frac\{2\\pi bn\}\{N\}\}\\Bigr\)=2N​∑b=0BN−1Re⁡\(X⁡\[k​BN\+b\]​ej​2​π​b​nN\)\\displaystyle=\\frac\{2\}\{N\}\\sum\_\{b=0\}^\{B\_\{N\}\-1\}\\operatorname\{Re\}\\\!\\Bigl\(X\[kB\_\{N\}\+b\]\\,e^\{j\\frac\{2\\pi bn\}\{N\}\}\\Bigr\)\(53\)
Comparing Equation[53](https://arxiv.org/html/2610.28997#A1.E53)with the demodulated version of[52](https://arxiv.org/html/2610.28997#A1.E52), i\.e\., multiplyingxband,k​\[n\]x\_\{\\text\{band\},k\}\[n\]bye−j​2​π​\(k​BN\)N​ne^\{\-j\\frac\{2\\pi\(kB\_\{N\}\)\}\{N\}n\}and taking the real part, reveals that the two expressions are identical after canceling the complex exponential factor\. Hencex~\(k\)​\[n\]\\tilde\{x\}^\{\(k\)\}\[n\]is exactly the real baseband signal obtained by demodulating thekk\-th band to zero frequency and low\-pass filtering\. All sinusoids maintain the same relative phases as they had in the original band\.

##### When Does a Phase Factor Become Necessary?

If we were to directly obtain a critically downsampled baseband signal of lengthL=N/SL=N/Swithout going through theNN\-point grid, we would need to perform an inverse transform of lengthLL\. In that case, the time\-axis compression changes the time origin: the new sample indexm=0m=0corresponds to the original timen=0n=0, but the frequency grid changes from2​π​b/N2\\pi b/Nto2​π​b/L2\\pi b/L\. To correctly align the phases of the complex exponentials, each coefficient must be multiplied byexp⁡\(−j​2​π​\(k​BN\)N​b\)\\exp\(\-j\\frac\{2\\pi\(kB\_\{N\}\)\}\{N\}b\), compensating for the shift of the frequency centerk​BNkB\_\{N\}relative to the new sampling grid\. Explicitly, if we form a length\-LLDFT as

Z\[b\]=1SX\[kBN\+b\]e−j​2​π​\(k​BN\)N​b,b=0,…,BN−1Z\[b\]=\\frac\{1\}\{S\}\\,X\[kB\_\{N\}\+b\]\\,e^\{\-j\\frac\{2\\pi\(kB\_\{N\}\)\}\{N\}b\},\\quad b=0,\\dots,B\_\{N\}\-1\(54\)
and apply a length\-MMinverse DFT, the resulting sequence is exactly the downsampled version ofx~\(k\)​\[n\]\\tilde\{x\}^\{\(k\)\}\[n\]without any aliasing\. This is the context in which the phase rotationexp\(−j2π\(kBN\)b/N\)\\exp\(\-j2\\pi\(kB\_\{N\}\)b/N\)appears; it accounts for the different time origin of the shortened sequence\.

##### TimeFT Avoids the Explicit Phase Factor\.

In the TimeFT pipeline we deliberately stay on the originalNN\-point grid\. After copying the coefficients, we apply anNN\-point inverse RFFT and then employ the dense sampling operator𝒟S\\mathcal\{D\}\_\{S\}to extract the tokens\. Because the downsampling is performed after recovering the full\-length baseband signal, the phase relations are automatically correct because the phase factor is implicitly absorbed by the subsequent decimation step\. Had we shortened the inverse transform toBNB\_\{N\}orLLpoints, the phase compensation would have been indispensable\.

#### A\.4\.3Analysis of Multiple Tokens

A natural question that arises is whether the distortions introduced by individual tokens can be mitigated when multiple tokens are processed jointly\. One might argue that since a collection of tokens together covers the entire original sequence, no information is lost in principle\. However, in the case of patch tokenization, this argument does not hold when tokens are processed independently, which is the standard step in models such as PatchTST\[[30](https://arxiv.org/html/2610.28997#bib.bib5)\]\. Consequently, the patterns that the model learns are inherently distorted\.

Several works have proposed cross\-token learning mechanisms that allow information to flow between tokens\[[49](https://arxiv.org/html/2610.28997#bib.bib48),[16](https://arxiv.org/html/2610.28997#bib.bib49)\]\. These methods can partially alleviate the distortion by enabling tokens to share contextual information, but their cross\-token capacity is inherently bounded, which cannot cover all tokens simultaneously and therefore cannot fully recover the missing structure\. Although these methods demonstrate the existence of the problem, they cannot fundamentally solve it\.

In contrast, our TimeFT tokenization solves this problem\. The token can almost losslessly reconstruct the original signal\. At first glance, this may appear to introduce redundant information\. However, this can be approximately viewed as a phase decomposition in the frequency domain\. Extensive experiments on forecasting, classification, and anomaly detection confirm that this representation consistently improves model performance rather than introducing harmful redundancy\. Furthermore, based on Appendix[A\.2](https://arxiv.org/html/2610.28997#A1.SS2), Phase Tokens also contain only the low\-frequency information of the original sequence, resulting in similar information redundancy and even worse because of frequency aliasing\. However, phase tokens have also been shown to be an effective tokenization method, such as PhaseFormer\[[31](https://arxiv.org/html/2610.28997#bib.bib3)\]\. Therefore, this phenomenon does not pose a risk\.

### A\.5Difference between Quantization Methods and Our Main Topic

Time series quantization methods originate from attempts to process time series in a text\-like fashion inspired by large language models\[[5](https://arxiv.org/html/2610.28997#bib.bib43)\]\. They discretize individual time points from continuous values into a finite set of bins, leaving the input shape unchanged\. This paradigm is fundamentally different from the tokenization framework that dominates current time series modeling\. A large number of current TSFMs\[[4](https://arxiv.org/html/2610.28997#bib.bib29),[28](https://arxiv.org/html/2610.28997#bib.bib30),[12](https://arxiv.org/html/2610.28997#bib.bib32)\]and deep learning time series models\[[30](https://arxiv.org/html/2610.28997#bib.bib5),[31](https://arxiv.org/html/2610.28997#bib.bib3),[43](https://arxiv.org/html/2610.28997#bib.bib23)\]adopt patch or phase tokenizations, whereas quantization remains a marginal approach\.

For instance, Apollo\[[46](https://arxiv.org/html/2610.28997#bib.bib45)\]views quantization as a sampling procedure and proposes to first remove high\-frequency content before discretizing the continuous data points, a scheme analogous to the “bins” used in Chronos\[[5](https://arxiv.org/html/2610.28997#bib.bib43)\]\. Because such methods do not involve partitioning the time series into subsequences, they fall outside the scope of our tokenization focused analysis\. We therefore do not discuss them further\.

Our work is the first to provide a rigorous frequency domain theoretical foundation for the distortions introduced by the prevailing tokenization paradigm, and to derive precise boundary conditions for distortion\-free tokenization\. Building on this novel perspective, we propose TimeFT, a general, parameter\-free, and highly effective tokenizer that consistently improves performance in diverse tasks and models\.

## Appendix BReconstruction Methods

### B\.1Reconstruction from Patch Tokens

A patch token is obtained by multiplying the original sequencex⁡\[n\]x\[n\]of lengthNNby a rectangular window of lengthLLstarting atn0n\_\{0\}, and then discarding the zero entries\. Reconstruction therefore amounts to padding the token with zeros to the original length and shifting it back to its original temporal position\.

x^​\[n\]=\{y⁡\[n−n0\],n0≤n≤n0\+L−1,0,otherwise\.\\hat\{x\}\[n\]=\\begin\{cases\}y\[n\-n\_\{0\}\],&n\_\{0\}\\leq n\\leq n\_\{0\}\+L\-1,\\\\ 0,&\\text\{otherwise\}\.\\end\{cases\}\(55\)
Equivalently, in the frequency domain,

X^​\(ej​ω\)=Y⁡\(ej​ω\)​e−j​ω​n0=Xw​\(ej​ω\)\\hat\{X\}\(e^\{j\\omega\}\)=Y\(e^\{j\\omega\}\)\\,e^\{\-j\\omega n\_\{0\}\}=X\_\{w\}\(e^\{j\\omega\}\)\(56\)
whereXwX\_\{w\}is the DTFT of the windowed sequence\. As shown in Appendix[A\.1](https://arxiv.org/html/2610.28997#A1.SS1), it inherits these distortions without any possibility of recovery, the discarded information is permanently lost\.

### B\.2Reconstruction from Phase Tokens

A phase token is obtained by downsampling the original sequencex⁡\[n\]x\[n\]of lengthNNby an integer factorSS, which yieldsy⁡\[m\]=x⁡\[m​S\]y\[m\]=x\[mS\]form=0,…,Mf−1m=0,\\dots,M\_\{f\}\-1, whereMf=⌊N/S⌋M\_\{f\}=\\lfloor N/S\\rfloor\. Reconstruction consists in upsamplingy⁡\[m\]y\[m\]back to the original lengthNN\. The practical method used in our experiments is Fourier\-based sinc interpolation\. In terms of RFFT, the algorithm computes theMfM\_\{f\}\-point RFFT of the real sequenceyy,

Yr\[k\]=∑m=0Mf−1y\[m\]e−j2πkm/Mf,k=0,…,⌊Mf/2⌋Y\_\{r\}\[k\]=\\sum\_\{m=0\}^\{M\_\{f\}\-1\}y\[m\]\\,e^\{\-j2\\pi km/M\_\{f\}\},\\qquad k=0,\\dots,\\lfloor M\_\{f\}/2\\rfloor\(57\)
and then zero\-pads it to the length required by the original signal\. The RFFT of the reconstructed sequencex^​\[n\]\\hat\{x\}\[n\]thus becomes

X^r​\[k\]=\{Yr​\[k\],0≤k≤⌊Mf/2⌋,0,⌊Mf/2⌋<k≤⌊Nf/2⌋\.\\hat\{X\}\_\{r\}\[k\]=\\begin\{cases\}Y\_\{r\}\[k\],&0\\leq k\\leq\\lfloor M\_\{f\}/2\\rfloor,\\\\\[6\.0pt\] 0,&\\lfloor M\_\{f\}/2\\rfloor<k\\leq\\lfloor N\_\{f\}/2\\rfloor\.\\end\{cases\}\(58\)
The time domain reconstruction is finally obtained by IRFFT,x^=ℱℝ−1​\(X^r\)\\hat\{x\}=\\mathcal\{F\}\_\{\\mathbb\{R\}\}^\{\-1\}\(\\hat\{X\}\_\{r\}\)\. We denote the upsampling operator here asℛ⁡\(⋅\)\\mathcal\{R\}\(\\cdot\)\. Since⌊Mf/2⌋=⌊N/\(2​S\)⌋\\lfloor M\_\{f\}/2\\rfloor=\\lfloor N/\(2S\)\\rfloor, zero\-padding in Equation[58](https://arxiv.org/html/2610.28997#A2.E58)is equivalent to applying an ideal low\-pass filter with cutoff frequencyπ/S\\pi/Sto the upsampled sequence\. The quality of reconstruction critically depends on whether the original signal is bandlimited to\|ω\|≤π/S\|\\omega\|\\leq\\pi/S\. Ifx⁡\[n\]x\[n\]contains energy at frequencies\|ω\|\>π/S\|\\omega\|\>\\pi/S, the downsampling step produces spectral aliasing that maps high\-frequency components to the low\-frequency region \(§[A\.2](https://arxiv.org/html/2610.28997#A1.SS2)\)\. The upsampling process cannot distinguish between a true low\-frequency component and an aliased high\-frequency one, so the reconstructed signal will still contain wrong patterns\.

### B\.3Reconstruction from TimeFT Tokens

The TimeFT tokenizer producesSStokens per variate\. Each token is a vector of lengthK⋅LK\\cdot L, whereK=⌈Nf/B⌉K=\\lceil N\_\{f\}/B\\rceilis the number of frequency blocks,B=⌊N/\(2​S\)⌋B=\\lfloor N/\(2S\)\\rflooris the bandwidth, andL=⌈N/S⌉L=\\lceil N/S\\rceilis an intermediate variable\. Reconstruction proceeds by inverting the two stages of the tokenizer: \(i\) polyphase recomposition and \(ii\) inverse spectral shifting\.

##### Polyphase Recomposition\.

For a fixed variate and frequency block indexkk, theSStokens with offsetsλ=0,…,S−1\\lambda=0,\\dots,S\-1represent theSSpolyphase components of the bandlimited time domain sequence𝐱time\(k\)\\mathbf\{x\}\_\{\\mathrm\{time\}\}^\{\(k\)\}obtained after spectral shifting and IRFFT[11](https://arxiv.org/html/2610.28997#S3.E11)\. The reconstruction first upsamples every sequence back to lengthNNusing the same Fourier\-based sinc interpolation described in the previous subsection:

𝐱^time\(k\)=ℛ⁡\(𝐱time\(k\)\)\\hat\{\\mathbf\{x\}\}\_\{\\mathrm\{time\}\}^\{\(k\)\}=\\mathcal\{R\}\(\\mathbf\{x\}\_\{\\mathrm\{time\}\}^\{\(k\)\}\)\(59\)
Equivalently, this can be expressed as the output of aSS\-channel synthesis filter bank with ideal interpolation filters\.

##### Inverse Spectral Shifting\.

The reconstructed bandlimited signal𝐱^time\(k\)\\hat\{\\mathbf\{x\}\}\_\{\\mathrm\{time\}\}^\{\(k\)\}occupies the low\-frequency region of thekk\-th spectral block\. To recover the contribution of this block to the full spectrum, we applyNN\-point RFFT, obtaining𝐗^block\(k\)∈ℂNf\\hat\{\\mathbf\{X\}\}\_\{\\mathrm\{block\}\}^\{\(k\)\}\\in\\mathbb\{C\}^\{N\_\{f\}\}\. Only the firstBBfrequency bins are retained and the remainder are set to zero\.

\[𝐗^block\(k\)\]f=\{\[RFFT⁡\(𝐱^time\(k\)\)\]f,0≤f<B,0,otherwise\.\\bigl\[\\hat\{\\mathbf\{X\}\}\_\{\\mathrm\{block\}\}^\{\(k\)\}\\bigr\]\_\{f\}=\\begin\{cases\}\[\\operatorname\{RFFT\}\(\\hat\{\\mathbf\{x\}\}\_\{\\mathrm\{time\}\}^\{\(k\)\}\)\]\_\{f\},&0\\leq f<B,\\\\ 0,&\\text\{otherwise\}\.\\end\{cases\}\(60\)
The original spectrum was partitioned into contiguous blocks of sizeBB[7](https://arxiv.org/html/2610.28997#S3.E7)\)\. The inverse operation simply places the retained frequency bins of thekk\-th block back into their original positions:

𝐗^freq\[kB\+b\]=\[𝐗^block\(k\)\]b,k=0,…,K−1,b=0,…,B−1\\hat\{\\mathbf\{X\}\}\_\{\\mathrm\{freq\}\}\[kB\+b\]=\\bigl\[\\hat\{\\mathbf\{X\}\}\_\{\\mathrm\{block\}\}^\{\(k\)\}\\bigr\]\_\{b\},\\qquad k=0,\\dots,K\-1,\\;\\;b=0,\\dots,B\-1\(61\)
All bins of𝐗^freq\\hat\{\\mathbf\{X\}\}\_\{\\mathrm\{freq\}\}that lie beyond the last block \(if any\) are obtained from the zero\-padding that was applied during the forward decomposition\. Finally, an IRFFT produces the reconstructed time domain sequence:

𝐱^=ℱℝ−1​\(𝐗^freq\)∈ℝN\\hat\{\\mathbf\{x\}\}=\\mathcal\{F\}\_\{\\mathbb\{R\}\}^\{\-1\}\\bigl\(\\hat\{\\mathbf\{X\}\}\_\{\\mathrm\{freq\}\}\\bigr\)\\in\\mathbb\{R\}^\{N\}\(62\)

##### Fidelity of the Reconstruction\.

The only deviation from perfect reconstruction comes from the simplified spectral shifting of Equation[10](https://arxiv.org/html/2610.28997#S3.E10), causing the DC coefficient of each block to lose its imaginary part\. Apart from this single complex value per block, the entire process is algebraically equivalent to passing the signal through a bank of ideal bandpass filters followed by perfect reconstruction synthesis\. Because the block bandwidthBBis chosen to respect the alias\-free conditionB≤N/\(2​S\)B\\leq N/\(2S\), no aliasing occurs during downsampling, and polyphase decomposition captures all samples without loss\. Consequently, the reconstructed signal𝐱^\\hat\{\\mathbf\{x\}\}differs from the original𝐱\\mathbf\{x\}only in the imaginary parts of the zero\-frequency bins of each spectral block\. This distortion is negligible in magnitude and that leaves the amplitude spectrum essentially identical to that of the original\. This explains why TimeFT tokens are almost distortion\-free, instead of information loss of patch tokens and aliasing artifacts from phase tokens\.

## Appendix CFull Experiment Results

In this section, we present detailed experiments of TimeFT, including three major tasks of long\-term forecasting[C\.1](https://arxiv.org/html/2610.28997#A3.SS1), classification[C\.2](https://arxiv.org/html/2610.28997#A3.SS2)and anomaly detection[C\.3](https://arxiv.org/html/2610.28997#A3.SS3)\. Moreover, we provide settings for other experiments[C\.4](https://arxiv.org/html/2610.28997#A3.SS4), variation in model parameters[C\.5](https://arxiv.org/html/2610.28997#A3.SS5)and reconstruction performance of different tokenizations[C\.6](https://arxiv.org/html/2610.28997#A3.SS6)\. The computational efficiency \(§[4\.3](https://arxiv.org/html/2610.28997#S4.SS3)\), the plug\-in experiment \(§[4\.4](https://arxiv.org/html/2610.28997#S4.SS4)\), the ablation study \(§[4\.5](https://arxiv.org/html/2610.28997#S4.SS5)\) and the hyperparameter sensitivity \(§[4\.7](https://arxiv.org/html/2610.28997#S4.SS7)\) please see Section[4](https://arxiv.org/html/2610.28997#S4)\. All experiments were performed using Intel Core i7\-13700H CPU and NVIDIA GeForce RTX 4060 GPU\. The training and testing procedures in the experiments follow the Time Series Library\[[38](https://arxiv.org/html/2610.28997#bib.bib50)\]\.

### C\.1Long\-Term Forecasting Task

We evaluate the long\-term forecasting performance of TimeFT against state\-of\-the\-art tokenization baselines across diverse benchmarks\. The experiments adopt the same settings as TimeMosaic\[[13](https://arxiv.org/html/2610.28997#bib.bib2)\]\. We evaluate our model on widely used real\-world time series forecasting datasets, including Electricity Transformer Temperature \(ETT\), Weather, Electricity, and Traffic\. The details of the datasets for this task are as follows:

- •ETT111[https://github\.com/zhouhaoyi/ETDataset](https://github.com/zhouhaoyi/ETDataset)comprises four subsets: ETTh1 and ETTh2 \(hourly\), and ETTm1 and ETTm2 \(every 15 minutes\), collected from two electricity transformers\.
- •Weather222[https://www\.bgc\-jena\.mpg\.de/wetter](https://www.bgc-jena.mpg.de/wetter)contains 21 meteorological variables recorded every 10 minutes in 2020 at the Max Planck Institute for Biogeochemistry\.
- •
- •Traffic444[https://pems\.dot\.ca\.gov](https://pems.dot.ca.gov/)consists of hourly road occupancy rates from 862 freeway sensors in the San Francisco Bay Area between January 2015 and December 2016\.

Table[7](https://arxiv.org/html/2610.28997#A3.T7)summarizes the key characteristics of the datasets used in our evaluation\. The collection encompasses various sampling rates, ranging from 10\-minute and 15\-minute intervals to hourly and daily recordings\. And they exhibits considerable diversity in dimensionality, spanning from the 7\-dimensional ETT subsets to the 321\-dimensional ECL dataset\. For each dataset, we report the partition sizes in the format \(Training, Validation, Test\) to clearly specify the data splitting strategy adopted across all experiments\.

Table 7:Summary of datasets for long\-term forecasting task\. Dim denotes dimension, which is the variate number of each dataset\. Dataset sizes represent the number of temporal observations in each partition \(Training, Validation, Test\)\. Frequency denotes the sampling interval of time points\.DatasetDimDataset Size\(Train, Val, Test\)FrequencyDomain InformationETTm17\(34465, 11521, 11521\)15minEnergy InfrastructureETTm27\(34465, 11521, 11521\)15minEnergy InfrastructureETTh17\(8545, 2881, 2881\)HourlyEnergy InfrastructureETTh27\(8545, 2881, 2881\)HourlyEnergy InfrastructureWeather21\(36792, 5271, 10540\)10minMeteorological MonitoringElectricity321\(18317, 2633, 5261\)HourlyElectricity ConsumptionTraffic862\(12185,1757,3509\)HourlyTraffic FlowAs demonstrated in Table[8](https://arxiv.org/html/2610.28997#A3.T8), TimeFT achieves the best performance in the majority of datasets and metrics\. By outperforming both classic and recent baselines, TimeFT establishes an excellent paradigm for obtaining tokens of time series\.

Table 8:For long\-term forecasting, we evaluate prediction horizonsH∈\{96,192,336,720\}H\\in\\\{96,192,336,720\\\}\. All baselines use a fixed input sequence length of320320\. The term “Avg\.” refers to the average metric over the four horizons\. The best and second\-best results are marked asbestandsecond, respectively\. The column “1s​t1^\{st\}Count” reports the number of times each method ranks first among all competitors\.ModelsTimeFTEntroPETimeMosaicPhaseFormerTimeFilterxPatchPatchMLPLightTSiTransformerTimeMixerPatchTSTDLinear\(Ours\)\(2026\)\(2026\)\(2026\)\(2025\)\(2025\)\(2025\)\(2024\)\(2024\)\(2024\)\(2023\)\(2023\)MetricMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEETTm1960\.2900\.3390\.3160\.3490\.3040\.3440\.3120\.3520\.3550\.3730\.2930\.3470\.3050\.3570\.3130\.3630\.3270\.3710\.3070\.3520\.3210\.3610\.2990\.3441920\.3250\.3570\.3440\.3710\.3390\.3640\.3390\.3810\.3760\.3960\.3320\.3710\.3510\.3840\.3500\.3870\.3570\.3880\.3320\.3730\.3420\.3850\.3370\.3703360\.3600\.3810\.3860\.3940\.3750\.3850\.3750\.3980\.4010\.4130\.3640\.3950\.3870\.4030\.3990\.4180\.3930\.4120\.3620\.3900\.3860\.4100\.3690\.3857200\.4180\.4150\.4310\.4190\.4220\.4130\.4280\.4370\.4650\.4490\.4280\.4320\.4540\.4410\.4890\.4790\.4670\.4520\.4220\.4260\.4390\.4410\.4240\.419Avg\.0\.3480\.3730\.3690\.3830\.3600\.3770\.3640\.3920\.3990\.4080\.3540\.3860\.3740\.3960\.3880\.4120\.3860\.4060\.3560\.3850\.3720\.3990\.3570\.380ETTm2960\.1620\.2440\.1730\.2560\.1640\.2470\.1740\.2610\.1740\.2610\.1680\.2590\.1790\.2650\.1870\.2860\.1870\.2710\.1690\.2580\.1810\.2700\.1720\.2701920\.2230\.2870\.2310\.3010\.2260\.2910\.2310\.3060\.2290\.2980\.2260\.2990\.2320\.3010\.3110\.3840\.2510\.3140\.2400\.3080\.2410\.3110\.2390\.3223360\.2750\.3210\.2880\.3340\.2790\.3240\.2910\.3420\.3120\.3510\.2810\.3370\.2850\.3340\.3990\.4180\.3190\.3550\.2820\.3340\.3030\.3480\.3050\.3677200\.3610\.3750\.3640\.3880\.3530\.3760\.3740\.3880\.3960\.4080\.3720\.3910\.3800\.3940\.4890\.4790\.4130\.4120\.3690\.3910\.3850\.3990\.4120\.432Avg\.0\.2550\.3070\.2640\.3200\.2560\.3100\.2680\.3240\.2780\.3300\.2620\.3220\.2690\.3240\.3470\.3920\.2930\.3380\.2650\.3230\.2780\.3320\.2820\.348ETTh1960\.3760\.3920\.3780\.3960\.3620\.3870\.3890\.4110\.4600\.4570\.3800\.4030\.4110\.4260\.4320\.4430\.4030\.4170\.3760\.3980\.4030\.4230\.3770\.4001920\.4170\.4160\.4090\.4180\.4000\.4110\.4610\.4720\.4390\.4430\.4250\.4300\.4430\.4450\.4740\.4720\.4430\.4440\.4120\.4200\.4790\.4780\.4100\.4203360\.4340\.4290\.4470\.4460\.4360\.4350\.4850\.4760\.4740\.4630\.4440\.4390\.4680\.4610\.5150\.5010\.4700\.4640\.4450\.4460\.4900\.4810\.4520\.4587200\.4460\.4580\.4620\.4740\.4510\.4600\.6190\.5420\.6700\.5640\.5340\.5090\.5250\.5080\.5920\.5670\.6320\.5730\.4880\.4890\.6410\.5590\.4770\.497Avg\.0\.4170\.4230\.4240\.4330\.4120\.4230\.4890\.4750\.5110\.4820\.4460\.4450\.4620\.4600\.5030\.4960\.4870\.4750\.4300\.4380\.5030\.4850\.4290\.444ETTh2960\.2970\.3390\.3080\.3540\.3000\.3420\.3090\.3610\.3670\.3890\.2800\.3430\.3110\.3710\.3240\.3830\.3230\.3690\.2850\.3460\.3240\.3740\.3050\.3681920\.3490\.3830\.3660\.3920\.3540\.3870\.3640\.4030\.3720\.4040\.3450\.3860\.3810\.4130\.4380\.4550\.4210\.4220\.3580\.3910\.3780\.4160\.4070\.4353360\.3750\.4010\.3870\.4110\.3780\.4030\.3920\.4250\.4020\.4260\.3820\.4150\.4010\.4270\.4960\.4900\.4400\.4410\.3800\.4090\.4050\.4320\.4930\.4907200\.4100\.4370\.4340\.4470\.4210\.4390\.4690\.4770\.4650\.4640\.4080\.4430\.4260\.4511\.1510\.7490\.4590\.4660\.4100\.4390\.4840\.4840\.7620\.620Avg\.0\.3580\.3900\.3740\.4010\.3630\.3930\.3840\.4170\.4020\.4210\.3540\.3970\.3800\.4160\.6020\.5200\.4110\.4250\.3580\.3960\.3980\.4270\.4920\.478Weather960\.1470\.1940\.1530\.1970\.1440\.1890\.1520\.2010\.1560\.2090\.1480\.1960\.1530\.2060\.1520\.2120\.1600\.2090\.1500\.1990\.1550\.2040\.1770\.2421920\.1990\.2360\.2010\.2390\.1970\.2320\.2010\.2490\.1950\.2460\.1900\.2380\.1970\.2450\.1970\.2550\.2070\.2510\.1930\.2440\.2010\.2490\.2170\.2763360\.2480\.2790\.2540\.2830\.2450\.2760\.2490\.2820\.2510\.2860\.2410\.2800\.2470\.2820\.2490\.2990\.2560\.2890\.2440\.2830\.2550\.2890\.2620\.3127200\.3320\.3350\.3390\.3360\.3270\.3270\.3310\.3400\.3220\.3390\.3180\.3330\.3240\.3340\.3230\.3570\.3250\.3380\.3190\.3360\.3330\.3400\.3260\.367Avg\.0\.2320\.2610\.2370\.2640\.2280\.2560\.2330\.2680\.2310\.2700\.2240\.2620\.2300\.2670\.2300\.2810\.2370\.2720\.2270\.2660\.2360\.2710\.2460\.299Electricity960\.1390\.2340\.1480\.2410\.1400\.2360\.1450\.2480\.1370\.2330\.1430\.2410\.1650\.2710\.1710\.2850\.1450\.2440\.1410\.2380\.1420\.2450\.1430\.2421920\.1510\.2450\.1600\.2580\.1560\.2500\.1600\.2610\.1530\.2460\.1570\.2520\.1830\.2880\.1920\.3050\.1650\.2630\.1580\.2540\.1590\.2590\.1570\.2553360\.1660\.2590\.1810\.2750\.1730\.2670\.1780\.2750\.1690\.2630\.1720\.2660\.2010\.3040\.2140\.3240\.1840\.2820\.1750\.2700\.1760\.2760\.1720\.2727200\.2060\.2930\.2190\.3090\.2120\.2980\.2240\.3090\.2080\.2960\.2090\.2970\.2500\.3430\.2510\.3490\.2270\.3170\.2170\.3070\.2150\.3080\.2070\.305Avg\.0\.1650\.2580\.1770\.2710\.1700\.2630\.1770\.2730\.1670\.2600\.1700\.2640\.2000\.3020\.2070\.3160\.1800\.2770\.1730\.2670\.1730\.2720\.1700\.269Traffic960\.3950\.2790\.4210\.3040\.4130\.2970\.4180\.2890\.3970\.2740\.4130\.2970\.4860\.3610\.4930\.3670\.4220\.3100\.4210\.2970\.4030\.2850\.4220\.2961920\.4120\.2810\.4330\.2880\.4240\.2810\.4260\.3010\.4110\.2780\.4250\.2980\.5150\.3780\.5110\.3730\.4490\.3270\.4410\.3110\.4190\.2920\.4350\.3023360\.4210\.2910\.4460\.2950\.4350\.2870\.4430\.3040\.4230\.2840\.4300\.2970\.5330\.3860\.5210\.3780\.4690\.3390\.4250\.3170\.4320\.2980\.4470\.3087200\.4520\.3090\.4770\.3140\.4640\.3020\.4780\.3240\.4490\.3000\.4500\.3060\.5640\.4010\.5040\.3600\.5120\.3640\.4890\.3400\.4590\.3130\.4760\.327Avg\.0\.4200\.2900\.4440\.3000\.4340\.2870\.4410\.3050\.4200\.2840\.4300\.3000\.5250\.3820\.5070\.3700\.4630\.3350\.4440\.3160\.4280\.2970\.4450\.3081s​t1^\{st\}Count400140108000000

### C\.2Classification

For the classification task, we employ 10 publicly available multivariate time series datasets from the University of East Anglia \(UEA555[https://www\.timeseriesclassification\.com/](https://www.timeseriesclassification.com/)\) Time Series Classification repository, which are widely recognized benchmarks in the field\. These datasets capture a diverse range of real\-world scenarios,including human activity recognition, sensor data interpretation, and complex system monitoring, and thus provide a multidimensional foundation for evaluating time series models\.

We evaluate the classification performance of TimeFT against state\-of\-the\-art tokenization methods on 10 commonly used UEA datasets\. We run and evaluate several recent tokenization methods for comparison\. As shown in Table[9](https://arxiv.org/html/2610.28997#A3.T9), TimeFT achieves the highest average accuracy and achieves the best or second\-best result in a majority of datasets\. This strong performance improvement is due to TimeFT’s almost distortion\-free tokenization ability, which faithfully preserves discriminative temporal patterns that are often corrupted by tokenizers\.

Table 9:For classification task, we evaluate accuracy \(%\) across 10 UEA datasets\. Best results are inboldand second best areunderlined\.ModelsTimeFTEntroPETimeMosaicPhaseFormerPatchMLPGPT4TSPatchTSTTimesNetMLP\(Ours\)\(2026\)\(2026\)\(2026\)\(2025\)\(2024\)\(2023\)\(2023\)\-EthanolConcentration31\.9429\.2728\.9026\.1329\.6525\.4828\.9027\.7328\.09FaceDetection69\.4869\.0367\.9564\.6567\.1465\.5868\.9667\.4767\.42Handwriting24\.7023\.3124\.8023\.1623\.873\.7626\.0026\.1822\.47Heartbeat77\.0676\.6473\.3476\.1073\.5436\.5976\.5974\.4873\.17JapaneseVowels98\.8798\.7697\.2395\.5797\.4998\.1198\.6597\.8397\.84PEMS\-SF88\.8687\.1287\.1586\.8078\.8587\.2888\.4488\.1382\.08SelfRegulationSCP189\.0588\.5889\.3586\.0389\.3691\.4789\.7677\.4388\.40SelfRegulationSCP259\.4457\.4252\.0756\.4653\.6251\.6754\.4452\.8451\.67SpokenArabicDigits99\.7198\.7598\.7397\.6096\.9999\.3699\.6898\.3696\.68UWaveGestureLibrary83\.7584\.3881\.8876\.8178\.6384\.3880\.0083\.1381\.88Average72\.2971\.3370\.1468\.9368\.9164\.3771\.1469\.3668\.97

### C\.3Anomaly Detection

For anomaly detection, we adopt five widely recognized datasets that span diverse application domains, allowing a thorough evaluation of detection capabilities under varied conditions\. The detailed characteristics of these datasets are presented in Table[10](https://arxiv.org/html/2610.28997#A3.T10)\.

- •SMD \(Server Machine Dataset\): Comprising multivariate server telemetry, SMD targets the identification of anomalous server behavior indicative of failures or security breaches\.
- •MSL \(Mars Science Laboratory\): Originating from the Mars rover, this telemetry dataset focuses on space exploration, where detecting anomalies is critical for diagnosing spacecraft subsystem faults\.
- •SMAP \(Soil Moisture Active Passive\): Containing satellite soil moisture measurements, SMAP supports environmental monitoring; detected irregularities can reveal critical changes in environmental and agricultural conditions\.
- •SWaT \(Secure Water Treatment\): Derived from a water treatment testbed, SWaT simulates plant operations and is essential for ensuring the safety and integrity of water treatment processes through anomaly detection\.
- •PSM \(Pump Sensor Monitoring\): Collected from industrial pump sensors, PSM enables anomaly detection to flag equipment malfunctions and maintenance needs, thereby helping to prevent industrial incidents\.

Table 10:Summary of datasets used for anomaly detection evaluation\. Dim denotes the variate number of each dataset\. Dataset sizes represent the number of temporal observations in each partition \(Training, Validation, Test\)\. Frequency denotes the sampling interval of time points\.DatasetDimDataset Size\(Train, Val, Test\)LengthDomain InformationSMD38\(566724, 141681, 708420\)100Server MachineMSL55\(44653, 11664, 73729\)100SpacecraftSMAP25\(108146, 27037, 427617\)100SpacecraftSWaT51\(396000, 99000, 449919\)100InfrastructurePSM25\(105984, 26497, 87841\)100Server MachineWe evaluate the anomaly detection capability of TimeFT against state\-of\-the\-art tokenization baselines and classic time series models on five standard benchmarks\. We run and evaluate several recent tokenization methods for comparison\. As shown in Table[11](https://arxiv.org/html/2610.28997#A3.T11), TimeFT achieves the highest average F1\-score of 86\.12%\. TimeFT preserves the integrity of anomalous patterns and shows excellent detection performance\. Experiments on diverse tasks and extensive benchmarks demonstrate that TimeFT exhibits superior performance compared to existing tokenization methods\.

Table 11:For anomaly detection task, we evaluate F1\-score \(%\) on five benchmark datasets\. Best results are inboldand second best areunderlined\.ModelsTimeFTEntroPETimeMosaicPhaseformerLightTSTimesNetPatchTSTDLinearETSformerFEDformer\(Ours\)\(2026\)\(2026\)\(2026\)\(2024\)\(2023\)\(2023\)\(2023\)\(2023\)\(2022\)SMD85\.5284\.7584\.2584\.8582\.5384\.6184\.6277\.1083\.1385\.08MSL82\.9581\.9578\.4279\.1578\.9581\.8478\.7084\.8885\.0378\.57SMAP71\.8069\.8068\.1068\.7569\.2169\.3968\.8269\.2669\.5070\.76SWaT93\.6593\.1085\.9086\.1293\.3393\.0285\.7287\.5284\.9193\.19PSM96\.6897\.2895\.8596\.2097\.1597\.3496\.0893\.5591\.7697\.23Average86\.1285\.3882\.5083\.0184\.2385\.2482\.7982\.4682\.8784\.97

### C\.4Settings for Other Experiments

##### Time Complexity Analysis\.

In the time complexity analysis experiment, we ensure that all experimental parameters are identical\. The test is conducted on the ETTm1 dataset, with an input length of 320 and a prediction length of 336\.

##### Plu\-in Experiments\.

In the plug\-in experiments, we replace the tokenization modules of different methods with TimeFT, leaving the other components unchanged\. For each method, we carefully tune the hyperparameters both before and after the substitution to ensure strong performance\.

##### Ablation Study\.

In the ablation study, removing a Stage means removing that module from the pipeline while keeping the rest unchanged\. The evaluation is conducted on the long\-term forecasting task, with an input length of 320 and a prediction length of 96\.

##### Hyperparameter Sensitive Analysis\.

Generally speaking, the only hyperparameter of TimeFT is the strideSS, and all other parameters are mathematically derived\. Moreover, as a tokenizer, TimeFT is closely related to input lengthNN\. Therefore, in the hyperparameter sensitivity experiment, we compare the effects of different stridesSSand input lengthsNNon model performance\. For each setting, we tune other hyperparameters because the optimal configuration changes accordingly\.

### C\.5Variation in Model Parameters

When analyzing the impact of TimeFT on the model parameters, we observed that hyperparameters significantly affect this metric\. Therefore, we keep all hyperparameters identical across experiments and compare only the variation in the embedding module parameters and the total parameter size, since the tokenizer mainly influences the number of embedding parameters\. "Total of Model" denotes the total parameter count of the model under identical hyperparameter settings\. As shown in Table[12](https://arxiv.org/html/2610.28997#A3.T12), although TimeFT increases the number of embedding parameters due to the change in input dimensionality, this change is negligible compared to the total number of parameters\.

Table 12:Variation in Model Parameter Count\.LayersTimeFT Embedding LayerPatchTST Embedding LayerTotal of ModelParams \(MB\)0\.6990\.03150\.337
### C\.6Reconstruction Performance of Different Token Types

To intuitively illustrate the degree of distortion for different types of tokens, we compare the mean absolute error between the reconstructed sequences and the original sequences across different datasets\. This experiment can be viewed as a statistical summary of Figure[2](https://arxiv.org/html/2610.28997#S1.F2)and the visualization section[D](https://arxiv.org/html/2610.28997#A4)\. We ensure that all hyperparameters remain identical in the experiments\. As shown in Table[13](https://arxiv.org/html/2610.28997#A3.T13), TimeFT tokens are almost distortion\-free and reduce more than 80% reconstruction error in many datasets\. The reconstruction methods are provided in Section[B](https://arxiv.org/html/2610.28997#A2)\.

Table 13:Reconstruction performance of different token types across different datasets\. The percentage represents the MAE reduction of TimeFT tokens relative to the best one of others\. Metric: MAE\.DatasetsETTm1ETTm2ETTh1ETTh2WeatherElectricityTrafficPatch0\.7680\.7470\.7550\.7220\.7260\.8210\.719Phase0\.4490\.5250\.8230\.7140\.2800\.9980\.986TimeFT0\.1020\.1150\.1440\.1330\.0790\.1220\.138Reduction77\.3%78\.1%80\.9%81\.4%71\.8%85\.1%80\.8%

## Appendix DVisualization of TimeFT

Since TimeFT is a tokenization method, we present visualizations comparing different classes of tokens in the time domain, frequency domain, and after equivalent reconstruction\. As shown in[Figures9](https://arxiv.org/html/2610.28997#A4.F9),[10](https://arxiv.org/html/2610.28997#A4.F10),[11](https://arxiv.org/html/2610.28997#A4.F11),[12](https://arxiv.org/html/2610.28997#A4.F12),[13](https://arxiv.org/html/2610.28997#A4.F13),[14](https://arxiv.org/html/2610.28997#A4.F14)and[15](https://arxiv.org/html/2610.28997#A4.F15), patch tokens lead to \(b\) a completely different frequency spectrum and \(c\) cause the loss of all information outside the rectangular window\. Phase tokens cause \(e\) frequency components above the Nyquist frequency to completely alias into the low frequency, thereby \(f\) exhibiting erroneous patterns\. Our TimeFT \(h\) effectively solves the above issues and \(i\) achieves almost distortion\-free tokenization\.

Figure 9:The visualization of tokenization methods across the time domain, frequency domain and reconstruction vs origin in ETTh1 dataset\.Figure 10:The visualization of tokenization methods across the time domain, frequency domain and reconstruction vs origin in ETTh2 dataset\.Figure 11:The visualization of tokenization methods across the time domain, frequency domain and reconstruction vs origin in ETTm1 dataset\.Figure 12:The visualization of tokenization methods across the time domain, frequency domain and reconstruction vs origin in ETTm2 dataset\.Figure 13:The visualization of tokenization methods across the time domain, frequency domain and reconstruction vs origin in Weather dataset\.Figure 14:The visualization of tokenization methods across the time domain, frequency domain and reconstruction vs origin in Electricity dataset\.Figure 15:The visualization of tokenization methods across the time domain, frequency domain and reconstruction vs origin in Traffic dataset\.
## Appendix ELimitations and Social Impacts

##### Limitations\.

Our work carries two limitations\. Owing to the substantial computational expense involved in pre\-training large\-scale time series foundation models, our experimental evaluation does not encompass pre\-trained TSFMs\. Consequently, the behavior and potential benefits of TimeFT in a pre\-training and fine\-tuning pipeline remain unverified, and investigating its effectiveness under TSFMs regimes is left to future work\. Furthermore, although tokenization has become a prevailing paradigm in modern time series analysis, our method intrinsically assumes a tokenization\-based processing flow and therefore yields no direct improvement for models that operate directly on raw sequences without any tokenization step\.

##### Social Impacts\.

As a fundamental component in time series modeling, our work can enhance the performance of a wide range of time series methods on various real\-world tasks, such as forecasting, anomaly detection, and classification\. It can serve as a foundational module in improving the application of time series models in fields such as healthcare, transportation, and economics\.

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3. Answer:\[Yes\]
4. Justification: The paper’s contributions and scope are well\-explained throughout the abstract and introduction, and the entire paper is written to provide a consistent and comprehensive conclusion\.
5. Guidelines: - •The answer\[N/A\]means that the abstract and introduction do not include the claims made in the paper\. - •The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations\. A\[No\]or\[N/A\]answer to this question will not be perceived well by the reviewers\. - •The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings\. - •It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper\.
6. 2\.Limitations
7. Question: Does the paper discuss the limitations of the work performed by the authors?
8. Answer:\[Yes\]
9. Justification: Our paper discusses the limitations in the Limitation section[E](https://arxiv.org/html/2610.28997#A5)\.
10. Guidelines: - •The answer\[N/A\]means that the paper has no limitation while the answer\[No\]means that the paper has limitations, but those are not discussed in the paper\. - •The authors are encouraged to create a separate “Limitations” section in their paper\. - •The paper should point out any strong assumptions and how robust the results are to violations of these assumptions \(e\.g\., independence assumptions, noiseless settings, model well\-specification, asymptotic approximations only holding locally\)\. The authors should reflect on how these assumptions might be violated in practice and what the implications would be\. - •The authors should reflect on the scope of the claims made, e\.g\., if the approach was only tested on a few datasets or with a few runs\. In general, empirical results often depend on implicit assumptions, which should be articulated\. - •The authors should reflect on the factors that influence the performance of the approach\. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting\. Or a speech\-to\-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon\. - •The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size\. - •If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness\. - •While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren’t acknowledged in the paper\. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community\. Reviewers will be specifically instructed to not penalize honesty concerning limitations\.
11. 3\.Theory assumptions and proofs
12. Question: For each theoretical result, does the paper provide the full set of assumptions and a complete \(and correct\) proof?
13. Answer:\[Yes\]
14. Justification: Our paper provides the full set of assumptions and a complete and correct proof in Section[A](https://arxiv.org/html/2610.28997#A1)\.
15. Guidelines: - •The answer\[N/A\]means that the paper does not include theoretical results\. - •All the theorems, formulas, and proofs in the paper should be numbered and cross\-referenced\. - •All assumptions should be clearly stated or referenced in the statement of any theorems\. - •The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition\. - •Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material\. - •Theorems and Lemmas that the proof relies upon should be properly referenced\.
16. 4\.Experimental result reproducibility
17. Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper \(regardless of whether the code and data are provided or not\)?
18. Answer:\[Yes\]
19. Justification: Our paper provides detailed information to reproduce all experiments and the complete experiment code\.
20. Guidelines: - •The answer\[N/A\]means that the paper does not include experiments\. - •If the paper includes experiments, a\[No\]answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not\. - •If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable\. - •Depending on the contribution, reproducibility can be accomplished in various ways\. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model\. In general\. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model \(e\.g\., in the case of a large language model\), releasing of a model checkpoint, or other means that are appropriate to the research performed\. - •While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution\. For example 1. \(a\)If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm\. 2. \(b\)If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully\. 3. \(c\)If the contribution is a new model \(e\.g\., a large language model\), then there should either be a way to access this model for reproducing the results or a way to reproduce the model \(e\.g\., with an open\-source dataset or instructions for how to construct the dataset\)\. 4. \(d\)We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility\. In the case of closed\-source models, it may be that access to the model is limited in some way \(e\.g\., to registered users\), but it should be possible for other researchers to have some path to reproducing or verifying the results\.
21. 5\.Open access to data and code
22. Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
23. Answer:\[Yes\]
24. Justification: Please see the code link in Abstract\.
25. Guidelines: - •The answer\[N/A\]means that paper does not include experiments requiring code\. - • - •While we encourage the release of code and data, we understand that this might not be possible, so\[No\]is an acceptable answer\. Papers cannot be rejected simply for not including code, unless this is central to the contribution \(e\.g\., for a new open\-source benchmark\)\. - •The instructions should contain the exact command and environment needed to run to reproduce the results\. See the NeurIPS code and data submission guidelines \([https://neurips\.cc/public/guides/CodeSubmissionPolicy](https://neurips.cc/public/guides/CodeSubmissionPolicy)\) for more details\. - •The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc\. - •The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines\. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why\. - •At submission time, to preserve anonymity, the authors should release anonymized versions \(if applicable\)\. - •Providing as much information as possible in supplemental material \(appended to the paper\) is recommended, but including URLs to data and code is permitted\.
26. 6\.Experimental setting/details
27. Question: Does the paper specify all the training and test details \(e\.g\., data splits, hyperparameters, how they were chosen, type of optimizer\) necessary to understand the results?
28. Answer:\[Yes\]
29. Justification: Our paper provides detailed information about the experiments, including data, hyperparameters, how they were chosen, and the type of optimizer, in the main text and the appendix\.
30. Guidelines: - •The answer\[N/A\]means that the paper does not include experiments\. - •The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them\. - •The full details can be provided either with the code, in appendix, or as supplemental material\.
31. 7\.Experiment statistical significance
32. Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
33. Answer:\[Yes\]
34. Justification: Experiments are repeated three times and the best results are reported in Tables\. We also evaluate the sensitivity of hyperparameter in §[4\.7](https://arxiv.org/html/2610.28997#S4.SS7)\.
35. Guidelines: - •The answer\[N/A\]means that the paper does not include experiments\. - •The authors should answer\[Yes\]if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper\. - •The factors of variability that the error bars are capturing should be clearly stated \(for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions\)\. - •The method for calculating the error bars should be explained \(closed form formula, call to a library function, bootstrap, etc\.\) - •The assumptions made should be given \(e\.g\., Normally distributed errors\)\. - •It should be clear whether the error bar is the standard deviation or the standard error of the mean\. - •It is OK to report 1\-sigma error bars, but one should state it\. The authors should preferably report a 2\-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified\. - •For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range \(e\.g\., negative error rates\)\. - •If error bars are reported in tables or plots, the authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text\.
36. 8\.Experiments compute resources
37. Question: For each experiment, does the paper provide sufficient information on the computer resources \(type of compute workers, memory, time of execution\) needed to reproduce the experiments?
38. Answer:\[Yes\]
39. Justification: We reported the computer resources used in the experiments in Appendix\.
40. Guidelines: - •The answer\[N/A\]means that the paper does not include experiments\. - •The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage\. - •The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute\. - •The paper should disclose whether the full research project required more compute than the experiments reported in the paper \(e\.g\., preliminary or failed experiments that didn’t make it into the paper\)\.
41. 9\.Code of ethics
43. Answer:\[Yes\]
44. Justification: The research conducted in the paper fully conforms to the NeurIPS Code of Ethics in every respect\.
45. Guidelines: - •The answer\[N/A\]means that the authors have not reviewed the NeurIPS Code of Ethics\. - •If the authors answer\[No\], they should explain the special circumstances that require a deviation from the Code of Ethics\. - •The authors should make sure to preserve anonymity \(e\.g\., if there is a special consideration due to laws or regulations in their jurisdiction\)\.
46. 10\.Broader impacts
47. Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
48. Answer:\[Yes\]
49. Justification: Our paper discusses the potential broader impacts of our research in the Appendix[E](https://arxiv.org/html/2610.28997#A5)\.
50. Guidelines: - •The answer\[N/A\]means that there is no societal impact of the work performed\. - •If the authors answer\[N/A\]or\[No\], they should explain why their work has no societal impact or why the paper does not address societal impact\. - •Examples of negative societal impacts include potential malicious or unintended uses \(e\.g\., disinformation, generating fake profiles, surveillance\), fairness considerations \(e\.g\., deployment of technologies that could make decisions that unfairly impact specific groups\), privacy considerations, and security considerations\. - •The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments\. However, if there is a direct path to any negative applications, the authors should point it out\. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate Deepfakes for disinformation\. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster\. - •The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from \(intentional or unintentional\) misuse of the technology\. - •If there are negative societal impacts, the authors could also discuss possible mitigation strategies \(e\.g\., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML\)\.
51. 11\.Safeguards
52. Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse \(e\.g\., pre\-trained language models, image generators, or scraped datasets\)?
53. Answer:\[N/A\]
54. Justification: Our model does not pose such risks\.
55. Guidelines: - •The answer\[N/A\]means that the paper poses no such risks\. - •Released models that have a high risk for misuse or dual\-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters\. - •Datasets that have been scraped from the Internet could pose safety risks\. The authors should describe how they avoided releasing unsafe images\. - •We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort\.
56. 12\.Licenses for existing assets
57. Question: Are the creators or original owners of assets \(e\.g\., code, data, models\), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
58. Answer:\[Yes\]
59. Justification: We have properly credited the creators or original owners of assets used in the paper\.
60. Guidelines: - •The answer\[N/A\]means that the paper does not use existing assets\. - •The authors should cite the original paper that produced the code package or dataset\. - •The authors should state which version of the asset is used and, if possible, include a URL\. - •The name of the license \(e\.g\., CC\-BY 4\.0\) should be included for each asset\. - •For scraped data from a particular source \(e\.g\., website\), the copyright and terms of service of that source should be provided\. - •If assets are released, the license, copyright information, and terms of use in the package should be provided\. For popular datasets,[paperswithcode\.com/datasets](https://paperswithcode.com/datasets)has curated licenses for some datasets\. Their licensing guide can help determine the license of a dataset\. - •For existing datasets that are re\-packaged, both the original license and the license of the derived asset \(if it has changed\) should be provided\. - •If this information is not available online, the authors are encouraged to reach out to the asset’s creators\.
61. 13\.New assets
62. Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
63. Answer:\[N/A\]
64. Justification: This paper does not include new assets\.
65. Guidelines: - •The answer\[N/A\]means that the paper does not release new assets\. - •Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates\. This includes details about training, license, limitations, etc\. - •The paper should discuss whether and how consent was obtained from people whose asset is used\. - •At submission time, remember to anonymize your assets \(if applicable\)\. You can either create an anonymized URL or include an anonymized zip file\.
66. 14\.Crowdsourcing and research with human subjects
67. Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation \(if any\)?
68. Answer:\[N/A\]
69. Justification: This paper does not involve crowdsourcing and research with human subjects\.
70. Guidelines: - •The answer\[N/A\]means that the paper does not involve crowdsourcing nor research with human subjects\. - •Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper\. - •According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector\.
71. 15\.Institutional review board \(IRB\) approvals or equivalent for research with human subjects
72. Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board \(IRB\) approvals \(or an equivalent approval/review based on the requirements of your country or institution\) were obtained?
73. Answer:\[N/A\]
74. Justification: This paper does not involve research with human subjects\.
75. Guidelines: - •The answer\[N/A\]means that the paper does not involve crowdsourcing nor research with human subjects\. - •Depending on the country in which research is conducted, IRB approval \(or equivalent\) may be required for any human subjects research\. If you obtained IRB approval, you should clearly state this in the paper\. - •We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution\. - •For initial submissions, do not include any information that would break anonymity \(if applicable\), such as the institution conducting the review\.
76. 16\.Declaration of LLM usage
77. Question: Does the paper describe the usage of LLMs if it is an important, original, or non\-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does*not*impact the core methodology, scientific rigor, or originality of the research, declaration is not required\.
78. Answer:\[N/A\]
79. Justification: LLM is not an important, original, or non\-standard component of the core methods in this research\.
80. Guidelines: - •The answer\[N/A\]means that the core method development in this research does not involve LLMs as any important, original, or non\-standard components\. - •Please refer to our LLM policy in the NeurIPS handbook for what should or should not be described\.