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STFT with Adaptive Window Width Based on the Chirp Rate

Soo-Chang Pei, Shih-Gu Huang

arXiv:1705.08795v1cs.ITmath.IT

TL;DR

Adaptive TFRs face a trade-off between energy concentration and complexity, and existing approaches may be limited across signal types. This paper modifies a chirp-rate ASTFT with low-complexity CM-based IF estimation, PCA chirp-rate estimation, and a time-frequency-varying Gaussian window. It reports improved comparisons with other ASTFTs and stronger IF estimation than many adaptive TFRs at low SNR, while noting a high-SNR comparison boundary.

  • Problem

    Adaptive TFRs seek concentrated energy at manageable complexity, while no single TFR is best for all signals and CM optimization can be costly.

    Source: §I · p. 1

  • Method

    The method combines low-complexity CM-based IF estimation, PCA-based chirp-rate estimation, and a Gaussian window with time-frequency-varying width.

    Source: §I · p. 2

  • Results

    The proposed method outperforms CM-based and chirp-rate-based ASTFTs in noisy and noiseless settings; its IF estimation is superior to many adaptive TFRs at low SNR but inferior to adaptive bilinear TFRs at high SNR.

    Source: §I · p. 2

  • Takeaways & Limitations

    The authors suggest the method may be more useful for signal analysis and synthesis at both low and high SNR because it is a linear transform.

    Source: §I · p. 2

    At high SNR, the method's IF estimation is inferior to adaptive bilinear TFRs.

    Source: §I · p. 2

Abstract

from arXiv · show

An adaptive time-frequency representation (TFR) with higher energy concentration usually requires higher complexity. Recently, a low-complexity adaptive short-time Fourier transform (ASTFT) based on the chirp rate has been proposed. To enhance the performance, this method is substantially modified in this paper: i) because the wavelet transform used for instantaneous frequency (IF) estimation is not signal-dependent, a low-complexity ASTFT based on a novel concentration measure is addressed; ii) in order to increase robustness to IF estimation error, the principal component analysis (PCA) replaces the difference operator for calculating the chirp rate; and iii) a more robust Gaussian kernel with time-frequency-varying window width is proposed. Simulation results show that our method has higher energy concentration than the other ASTFTs, especially for multicomponent signals and nonlinear FM signals. Also, for IF estimation, our method is superior to many other adaptive TFRs in low signal-to-noise ratio (SNR) environments.

I. INTRODUCTION

The paper modifies chirp-rate-based adaptive STFTs to balance energy concentration and computational complexity. It introduces a low-complexity concentration-measure approach and a time-frequency-varying Gaussian window, reporting advantages over other adaptive STFTs.

I. INTRODUCTION

  • No single TFR is considered best for every signal; CM optimization can be computationally costly, while reassignment is expensive and noise-sensitive.CMs quantify concentration and can be maximized to choose TFR parameters.Source: §I · p. 1
  • The earlier chirp-rate ASTFT adapts window width to IF variation, but its wavelet-based IF estimate is not signal-dependent and differencing is sensitive to IF error.Its width–chirp-rate relationship is also described as insufficiently accurate, with no guidance for choosing its threshold.Source: §I · pp. 1–2
  • The method combines a low-complexity CM-based ASTFT for IF estimation, PCA-based chirp-rate estimation, and a Gaussian kernel with time-frequency-varying width.The width is designed using an approximate optimal-width/chirp-rate relationship and 2D interpolation.Source: §I · p. 2
  • The proposed ASTFT outperforms CM-based and chirp-rate-based ASTFTs in noiseless and noisy environments; its IF estimation excels at low SNR but trails adaptive bilinear TFRs at high SNR.For signal analysis and synthesis, the authors suggest its linearity may make it more useful at both low and high SNR.Source: §I · p. 2

II. ADAPTIVE SHORT-TIME FOURIER TRANSFORMS AND ADAPTIVE S-TRANSFORMS

  • The paper focuses on linear TFRs, including STFTs and S-transforms, as background for its method.The cited CM-based S-transform variants use frequency-varying widths, including an unconstrained positive width optimized with CM2.Source: §II · p. 2, A. Adaptive STFT and Adaptive… · p. 2

A. Adaptive STFT and Adaptive S-Transform Based on Concentration Measures

  • CM-based TFR optimization is computationally intensive, and a width optimized for total concentration along time may not suit every component of a multicomponent signal.CM1 and CM2 are described as criteria for optimizing the parameter or width.Source: A. Adaptive STFT and Adaptive… · p. 2

B. Adaptive STFT Based on the Chirp Rate

The chirp-rate ASTFT adapts a Gaussian window to keep the signal locally quasi-stationary. Its window width is derived from IF variation, with threshold and multicomponent limitations.

B. Adaptive STFT Based on the Chirp Rate

  • The method estimates IF from a wavelet-transform ridge and adapts window width to chirp rate: smoother IF permits a wider window, while sharper variation calls for a narrower one.The width is tuned by a threshold to keep the signal within the window quasi-stationary.Source: B. Adaptive STFT Based on the Chirp Rate · p. 3
  • For discrete signals, the method derives chirp rate from sampled IF and uses the resulting quasi-stationary width at each time sample.The discrete formulation depends on sampling interval and threshold.Source: B. Adaptive STFT Based on the Chirp Rate · p. 3
  • The wavelet transform may not suit all signals, differencing is sensitive to IF error, and threshold- and sampling-dependent widths may be suboptimal for multicomponent signals.The optimal threshold is not derived, and averaging component chirp rates may fail to give each component its own optimal width.Source: B. Adaptive STFT Based on the Chirp Rate · p. 3

III. ADAPTIVE STFT BASED ON THE CONCENTRATION MEASURES AND THE CHIRP RATE

The method generalizes the ASTFT window width to vary over time and frequency, while a low-complexity concentration-measure ASTFT estimates instantaneous frequency. Its CM5 criterion balances the different behavior of CM3 and CM4, and the resulting method offers adjustable complexity.

III. ADAPTIVE STFT BASED ON THE CONCENTRATION MEASURES AND THE CHIRP RATE

  • The generalized ASTFT uses a time-frequency-varying window width suited to monocomponent nonlinear-FM and multicomponent signals.The paper contrasts this with using a frequency-varying or time-varying width alone.Source: §III · p. 3
  • ASTFT-tf reduces to ASTFT-f when σ(t, f)=σ(f), and to ASTFT-t when σ(t, f)=σ(t).The paper uses these names for the concentration-measure and chirp-rate-based methods, respectively.Source: §III · p. 3

A. Instantaneous Frequency Estimation Using a Low-Complexity CM-Based ASTFT

  • The concentration-measure approach is chosen for IF estimation because it is less sensitive to noise than reassignment methods.The ASTFT is preferred over other concentration-measure TFRs to reduce hardware cost, with low complexity prioritized over exact IF estimates.Source: A. Instantaneous Frequency Estimation… · pp. 3, 5
  • Complexity is reduced by optimizing one window width over the whole time-frequency plane, using only part of the observations and a limited candidate set.Larger p and smaller L further reduce complexity at the cost of performance.Source: A. Instantaneous Frequency Estimation… · p. 5
  • CM3 induces lower total energy and suits the Fig. 1 signal, whereas CM4 induces higher total energy and suits Fig. 2; CM5 is introduced as a compromise between them.The CM5-based ASTFT has performance between the CM3-based and CM4-based ASTFTs in those examples.Source: A. Instantaneous Frequency Estimation… · p. 5

B. Chirp Rate Estimation Using Principal Component Analysis

PCA estimates chirp rate from local instantaneous-frequency measurements to reduce sensitivity to IF-estimation error. The paper relates Gaussian window width to chirp rate and uses an approximate relationship for highly nonlinear FM signals.

B. Chirp Rate Estimation Using Principal Component Analysis

  • Unlike the two-measurement difference operator, PCA can use more IF measurements, with larger K recommended for lower-SNR signals to improve robustness to IF-estimation error.The difference operator is described as a special case of PCA using only two IF measurements.Source: B. Chirp Rate Estimation Using… · p. 5
  • PCA estimates chirp rate from the slope of the first principal component of nearby time-frequency IF measurements.The first principal component is the eigenvector associated with the largest eigenvalue of the measurements’ covariance matrix.Source: B. Chirp Rate Estimation Using… · p. 5

C. Optimal Standard Deviation of the Gaussian Window Versus the Chirp Rate

  • For a linear-FM signal, the optimal Gaussian-window width is obtained by minimizing the variance of its time-frequency envelope.The paper states that this variance is strictly convex for positive σ^2.Source: C. Optimal Standard Deviation of the… · p. 6
  • For highly nonlinear FM with time-varying chirp rate, the exact optimal σ(t) involves phase derivatives of arbitrarily high order, so the paper uses an approximate relationship.Source: C. Optimal Standard Deviation of the… · p. 6

D. Time-Frequency-Varying Standard Deviation Using 2D Interpolation

The method chooses a Gaussian kernel’s standard deviation from its two-dimensional mask shape, adapting it across time and frequency using chirp rates. It uses interpolation to estimate off-ridge values, trading some energy concentration for low complexity.

D. Time-Frequency-Varying Standard Deviation Using 2D Interpolation

  • For nonlinear FM and multicomponent signals, time-frequency-varying window width is preferred over time-varying width for higher energy concentration.Different components or locations can have different chirp rates, so a single width across a time instant or frequency band may be suboptimal.Source: D. Time-Frequency-Varying Standard… · pp. 6, 8
  • The Gaussian kernel’s standard deviation σ is determined by the height-to-width ratio γ of its two-dimensional mask.The TFR is treated as the convolution of the ideal TFR with that mask; tuning its shape at each time-frequency point aims to concentrate energy on the ridges.Source: D. Time-Frequency-Varying Standard… · pp. 6–7
  • On the ridge of a nonlinear FM signal, the mask’s height-to-width ratio γ is set to the absolute chirp rate.For a linear FM signal, γ = |a| makes the highest-envelope region equivalent to the ideal TFR, and the normalized convolution envelope is most concentrated at that ratio.Source: D. Time-Frequency-Varying Standard… · pp. 7–8, Figure 3 · p. 7, Figure 4 · p. 7
  • Off-ridge, γ is interpolated in two dimensions: match the adjacent ridge mask at q1 and choose an intermediate value at q3 to avoid overlap.The resulting chirp-rate estimates are used to obtain approximate optimal standard deviations at all time-frequency points.Source: D. Time-Frequency-Varying Standard… · p. 8, Figure 5 · p. 8
  • The chosen interpolation is not necessarily optimal, but triangle-based linear interpolation achieved higher energy concentration than several other tested interpolation methods.The comparisons included nearest-neighbor, triangle-based cubic, and MATLAB 4 griddata interpolation.Source: D. Time-Frequency-Varying Standard… · p. 8
  • The proposed technique prioritizes low complexity over the highest possible energy concentration, since greater concentration can require impractical computation.The paper describes this as a tradeoff in designing an adaptive TFR.Source: D. Time-Frequency-Varying Standard… · p. 8

IV. SIMULATION RESULTS

The simulations compare three ASTFT variants and evaluate IF estimators based on adaptive TFRs in noisy environments. The proposed method’s chirp rate is also applied to the original chirp-rate ASTFT in the reported simulations.

IV. SIMULATION RESULTS

  • The simulations compare ASTFT-f, ASTFT-t, and ASTFT-tf, representing the CM-based, chirp-rate-based, and proposed methods, respectively.The comparisons examine energy-concentration performance and IF estimators based on the ASTFT-tf and other adaptive TFRs, including adaptive bilinear TFRs.Source: §IV · p. 8
  • The IF-estimation experiments evaluate the adaptive TFR methods in noisy environments.The comparison includes popular adaptive bilinear TFRs.Source: §IV · p. 8
  • The simulations use α = 0.1, β = 5, p = 4, and L = 64 in the ASTFT-tf’s CM5.The ASTFT-tf chirp rate is used for ASTFT-t because its wavelet-based IF estimation is not signal-dependent and its difference operator is sensitive to IF error.Source: §IV · p. 8

A. Effect of IF Estimation Error on the Performance of the ASTFT-tf

The experiments test how IF-estimation errors affect ASTFT-tf energy concentration. Higher estimation error reduces concentration, while the loss from the low-complexity CM5 estimator is reported as tolerable.

A. Effect of IF Estimation Error on the Performance of the ASTFT-tf

B. Energy Concentration Analysis of the ASTFT-f, the ASTFT-t and the ASTFT-tf

Across the reported multicomponent and nonlinear-FM examples, ASTFT-tf generally has higher energy concentration than ASTFT-f and ASTFT-t. The comparisons attribute this advantage to a time-frequency-varying window width that adapts to different chirp rates.

B. Energy Concentration Analysis of the ASTFT-f, the ASTFT-t and the ASTFT-tf

  • For a multicomponent signal with two linear FM components, ASTFT-tf has higher energy concentration than ASTFT-f and ASTFT-t.The components have different chirp rates, so a time-frequency-varying standard deviation can adapt to each, unlike widths that vary only in time or frequency.Source: B. Energy Concentration Analysis of… · p. 9, Figure 7 · p. 10
  • For the nonlinear FM example, ASTFT-tf achieves much higher energy concentration than ASTFT-t, even with the same estimated chirp rate.ASTFT-tf also somewhat outperforms ASTFT-f; ASTFT-t’s threshold values were selected by trial and error.Source: B. Energy Concentration Analysis of… · p. 9
  • For a signal with a linear-FM component and a nonlinear-FM component, ASTFT-tf is somewhat better than ASTFT-f and ASTFT-t, especially in the marked time-frequency regions.The comparison uses a signal with a sinusoidal envelope on one component and a positive random envelope on the other.Source: B. Energy Concentration Analysis of… · p. 9

C. Energy Concentration Analysis of the FFT-based ASTFT-tf

The FFT-based ASTFT-tf improves energy concentration over ASTFT-f and ASTFT-t in the illustrated nonlinear-FM and multicomponent cases. Averaging chirp rates enables FFT-based implementation but sacrifices a fully time-frequency-varying window.

C. Energy Concentration Analysis of the FFT-based ASTFT-tf

  • Averaging chirp rates along frequency yields σ[m], while averaging along time yields σ[n], enabling FFT-based ASTFT-tf implementations.The corresponding standard deviation is no longer time-frequency-varying, and chirp-rate interpolation is unnecessary.Source: C. Energy Concentration Analysis of… · p. 10
  • For the illustrated nonlinear FM signal, ASTFT-tf has the highest energy concentration among ASTFT-f, ASTFT-t, and ASTFT-tf.The comparison is shown in Fig. 8.Source: Figure 8 · p. 10
  • Choose frequency-axis averaging when chirp-rate magnitudes are similar across frequencies at each time, and time-axis averaging when they are similar across times at each frequency.The paper notes that averaging incurs a performance loss because the window width is no longer time-frequency-varying.Source: C. Energy Concentration Analysis of… · p. 10
  • For the multicomponent signal, FFT-based ASTFT-tf is more concentrated than ASTFT-f near (t = 0.3, f = −32) and than ASTFT-t near two illustrated regions.Averaging chirp rates makes concentration more even across components than ASTFT-f’s total-energy CM2 optimization; ASTFT-t uses a window-width relationship aimed at quasistationarity rather than concentration.Source: C. Energy Concentration Analysis of… · p. 11, Figure 10 · p. 12

ASTFT-tf

The paper compares ASTFT-tf with other adaptive TFRs and reports advantages in concentration and low-SNR IF estimation, alongside limits at high SNR. It concludes that the linear transform may suit signal analysis and synthesis across SNR conditions.

ASTFT-tf

  • The original ASTFT-t uses a non-adaptive wavelet transform and difference operator, trading much lower energy concentration for lower complexity.The ASTFT-t discussed in the simulations instead uses the CM-based ASTFT and PCA estimators shared with ASTFT-tf.Source: ASTFT-tf · p. 12, D. Comparison of ASTFT-tf with other… · p. 12

D. Comparison of ASTFT-tf with other Adaptive TF Representations in Noisy Environments

  • In noiseless tests, reassigned SM and SPWVD have the highest concentration; in noisy tests, reassigned methods enhance both noise-only and signal-plus-noise components, while chirp-rate methods are affected through chirp-rate estimation error.The reported comparison includes seven adaptive TFRs, and the chirp-rate estimator is described as somewhat robust to that error.Source: D. Comparison of ASTFT-tf with other… · pp. 12–13, Figure 12 · p. 14
  • ASTFT-tf outperforms all other adaptive TFRs for IF estimation at low SNR but is inferior to adaptive bilinear TFRs at high SNR.The authors identify its linearity as a reason it may be more useful for signal analysis and synthesis across both SNR regimes.Source: D. Comparison of ASTFT-tf with other… · p. 13

V. CONCLUSION

  • ASTFT-tf has higher energy concentration than CM-based and chirp-rate-based ASTFTs, while its IF estimation is superior to many adaptive TFRs at low SNR but inferior to adaptive bilinear TFRs at high SNR.The authors note that its linear-transform form may make it more useful for signal analysis and synthesis at both low and high SNR.Source: §V · p. 13

APPENDIX A ENVELOPE OF THE ASTFT-TF OF A LINEAR FM SIGNAL

The appendix derives the ASTFT-tf envelope for a linear FM signal, stating a condition and the operation used to obtain the envelope.

APPENDIX A ENVELOPE OF THE ASTFT-TF OF A LINEAR FM SIGNAL

  • The appendix considers a linear FM signal and gives an expression for its ASTFT-tf.The displayed transform expression follows the stated signal form.Source: Appendix A · p. 14
  • The derivation proceeds under the condition Re{µ2} > 0.Source: Appendix A · p. 15
  • The envelope is determined by evaluating the real part of c1, after which the appendix states the resulting envelope.Source: Appendix A · p. 15

APPENDIX B FFT-BASED IMPLEMENTATION OF THE ASTFT-TF

The appendix presents FFT-based ASTFT-tf implementations and gives conditions under which the relevant standard deviation permits FFT use. It also describes truncating the sums when their tails tend to zero.

APPENDIX B FFT-BASED IMPLEMENTATION OF THE ASTFT-TF

  • The appendix truncates the discrete ASTFT-tf sum to 2Q1 + 1 terms when the specified right-hand side tends to zero for |m| > Q1.The passage defines σmax as an upper bound on all σ[m,n] values.Source: Appendix B · p. 15
  • FFT use in the first implementation requires σ[m,n] to be independent of n.Source: Appendix B · p. 15
  • A second implementation expresses the transform using X[k], the DFT of x[l], and truncates its sum to 2Q2 + 1 terms when the stated tail tends to zero.This form is equivalent to the discrete S-transform when σ[m,n] = 1/(n∆f), and FFT use requires σ[m,n] to be independent of m.Source: Appendix B · pp. 15–16
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