1. On Quality Criteria for Time- Varying Filters By: Lior Assouline Supervisor: Dr. Moshe Porat

2. Agenda Motivation (The need for Quality Criteria). The Time-Frequency (TF) filtering problem. Short theoretical introduction to TF Filter analysis. Existing solutions. Proposed solutions. Validation and Simulations.

3. Motivation In recent years there have been considerable strides made in developing a combined TF description of signals by the use of joint time- frequency distributions. This development offered the possibility of generalizing the concepts and methods of classical filtering theory to the combined TF domain.

4. The Problem In many applications (e.g. Seismic analysis, Evoked potentials, E.C.G/E.E.G. analysis etc.) it is desirable to filter a signal such that its components inside given time-frequency regions are specifically weighted. The TF Filtering Model M(t,f)

5. The Problem (cont ’ d) Time-Varying filtering presents unique difficulties that have not been fully overcome: The large variety of filtering methods is due in part to the multiple Time-Frequency representation methods, each with its own merits and drawbacks. There is still no accepted way to determine which filtering method to use for a specific filtering task, nor is there a way to find the optimal scheme suitable for a certain filtering task.

6. Time-Frequency Representations and Operator Symbols STFT Wigner Distribution (WD) We can generalize both representations as an inner product of the signal and a symmetric/asymmetric TF shifted version of the signal itself (WD) or a normalized window (STFT)

7. Time-Frequency Representations and Operator Symbols (cont ’ d) The input-output relation of any linear, time-varying (LTV) filter H is: where is the impulse response of H. Weyl Symbol: Generalization of Fourier using the Heisenberg Group Theory for LTV systems.

8. Time-Frequency Representations and Operator Symbols (cont ’ d) Spreading Function: The 2D Fourier of Weyl Symbol.  The Weyl Symbol can be interpreted as a TF transfer function of the operator H with certain limitations due to the uncertainty principle.  The Spreading Function can be interpreted as the amount of potential time and frequency shifts caused by the linear operator.

9. LTV Filter Design Methods Every linear filter can be represented as. Current LTV Filter design approaches are based primarily on three main methods: 1. Composition of timepass and bandpass filters. 2. STFT: Restriction of the reproducing formulas based on coherent state expansions. 3. Weyl: pseudodifferential operators with symbols of compact support.

10. STFT based methods Calculate the STFT of a signal : Multiply the STFT by the mask : Synthesize the output signal from the masked STFT : The resultant impulse response is STFT ANALYSIS using window l(t) STFT SYNTHESIS using window g(t) Multiplicative Modification M(t,f) x(t)... y(t)

11. Optimal Window STFT (Kozek-92) Matching the analysis and synthesis windows to the filtering model. Optimal diagonalization of the operator via a Weyl-Heisenberg matched signal set.

12. Weyl Correspondence based methods Based on the interpretation of the Weyl Symbol (WS) as a TF transfer function: We construct a filter H such that its WS is the Model, where,

13. TF Weighting filter Weyl filter has a substantial amount of TF energy displacements. A constraint on H to be a positive semi- definite operator leads to a minimization problem:

14. TF Weighting filter (cont ’ d) Algorithm (Hlawatsch 94) Calculate as in a Weyl filter. Calculate the eigenvalues and eigenfunctions of the filter. The TF Weighting filter (TFWF) is given by

15. Generalized LTV Filter The Weyl Symbol of the STFT filter is i.e., the Model smoothed by the Cross Wigner Distribution (CWD) of the windows used in the analysis and synthesis of the STFT.  This suggests a generalized filter where the STFT and the Weyl based methods are special cases.

16. Generalized LTV Filter (cont ’ d) Proposed algorithm 2D Convolve with an arbitrary (or part of a family of a) smoothing function to yield, Calculate from the modified as in a Weyl filter:

17. Generalized LTV Filter (cont ’ d) Properties: The proposed filter is related to the STFT and the Weyl filters: The STFT filter results from The Weyl filter results from is a 2D smoothing function.  Using smoothing functions along the continuum between the two extreme cases of a physical window (Heisenberg cell) and an unrealizable Dirac TF function ( ) yields a new family of filters.  This resultant family of filters has smooth transition between the extreme properties of STFT and Weyl filters, as will be shown.

18. Current LTV Filter Quality Criteria - SNR improvement (Kozek-92) SNR improvement : SNR_in = SNR_out = SNR_improvement is based on the difference between SNR_out and SNR_in. Problems: It is not clear if a small SNR improvement is due to low immunity to noise or high distortion of the internal signal. A (internal) test signal must be found: Signal synthesis is an as yet unsolved problem.

19. Current LTV Filters Quality Criteria (cont ’ d), Dubiner - 97 Dubiner ’ s Max/Mean criteria is based on the fact that an internal (to the passing area) signal should be passed undistorted, and an external signal should be blocked. Problems: This QC is suitable only for rectangular areas TF filters

20. Proposed Quality Criterion Mean Distortion Error (MDE-QC)

21. Eigenvalues analysis of a self- adjoint operator Self-Adjoint operators can be decomposed into real eigenvalues and their corresponding eigenfunctions: The unitary property of WD enables an interpretation of concentration for the eigenvalues

22. Positive and negative eigenvalues Positive eigenvalues correspond to eigenfunctions whose TF support is inside the model, defined as weighted functions: Negative eigenvalues correspond to eigenfunctions that are passed with inverted phase, defined as corrections:

23. Positive and negative eigenvalues (cont ’ d) Negative eigenvalues appear when trying to filter an area sharply localized or smaller than Heisenberg cell. This is a natural manifestation of the Heisenberg Uncertainty principle.

24. The proposed criterion The QC main idea:  Measure distortion i.e. difference between resulting weighting operator and the required TF model.  Penalty for corrections induced by the operator. We can see a filtering process as projecting a signal onto the eigenfunctions linear space. It is therefore of interest to determine the TF support of these eigenfunctions.

25. Definition (Hlawatsch-91) : The Wigner Space (WSP) of an operator H is defined as: where are the eigenvalues and eigenfunctions of the operator H.  The WSP is an approximate TF transfer function. Wigner space of an operator

26. MDE-QC The Mean Distortion Error QC (MDE-QC) is defined as where is the target TF filter model.

27. QC-MDE (cont ’ d) This criterion measures the fidelity of the resulting linear filter with respect to the TF weighting specification. This criterion is a natural generalization of the LTI case.

28. Quality Criteria - Analysis and Comparison The MDE-QC is a suitable quality criterion : It is signal independent (unlike existing quality criteria). It permits an arbitrarily shaped TF weighting filter analysis (unlike most of the existing quality criteria). It permits an arbitrary weight specification of the TF location error (unlike SNR criteria).

29. Quality Criteria - Analysis and Comparison (cont ’ d) It is independent of the filter implementation method (like most other methods) It accounts for the inherent tradeoff in LTV filtering: TF localization vs. TF ripple. It is a generalization of the LTI filter theory.

30. Quality Criteria - Analysis and Comparison (cont ’ d) The choice of the WD is optimal in the sense of resolution. Linear combination of weighted WD diminishes the effect of cross-terms. However, due to practical considerations (finite number of eigenfunctions) minimal smoothing is required for enhanced analysis.

31. Upper bounds on QC values For STFT based filters: For Weyl based filters:

32. Optimal filter We can search over a family of smoothing functions to obtain the lowest value for the MDE-QC. This family of filters is defined here as GTFF (Generalized TF filter): where is the model smoothed by.

33. Optimal filter (cont ’ d) Unlike previously proposed filters such as STFT (Daubechies-88) and Weyl (Hlawatch-92), which use extreme cases of a smoothing function, the solution here is superior to both by selecting a smoothing function that suits the model and the user ’ s choice of TF localization and ripple errors.

34. LTV filtering comparison The filtering model (a) and MDE-QC weighting part (b,c,d,e) for various filtering methods.

35. LTV filtering comparison (cont ’ d)

36. GTFF OGTFF synthesis and eigenvalue analysis

37. Signal Pass validation experiments Validation of MDE-QC with SPNS-QC (STFT representation)

38. Signal Pass validation experiments

39. Noise Stop validation experiments Validation of MDE-QC with SPNS-QC (STFT representation)

40. Noise Stop validation experiments

41. Signal Pass/Noise Stop validation experiments Validation experiments showing the relative quality of the filters using SPNS-QC.

42. Conclusions from experiments The MDE-QC is in agreement with the SPNS-QC criterion and can be used to predict filter performances. The optimal GTFF (OGTFF) has superior properties: TF localization and ripple, compared to both STFT and Weyl based filters.

43. Summary A QC based on eigenvalue analysis is proposed. The QC can predict the filter performances for a given filtering task. LTV filtering trade-off (TF localization and ripple) are accounted for.

44. Summary (cont ’ d) MDE-QC is in accordance with the unrelated SPNS-QC criterion. MDE-QC enables synthesis of an optimal LTV filter, which was found best also according to SPNS-QC.

45. Additional Quality Criteria Extended Dubiner Quality Criterion Signal Pass/Noise Stop (SPNS) QC

46. Extended Dubiner Quality Criterion (alpha) Internal/external signals are created using a (arbitrary area) TF signal synthesis technique by Hlawatsch & Krattenthaler. The filters are tested using these signals to yield the mean/max criterion. Problem: This technique relies on a TF (bilinear) signal synthesis technique that will bias the results to Weyl based filter techniques (since its basis functions are the eigenvalues of the filter operator). This QC is therefore suitable only for STFT based methods.

47. SPNS-QC based Quality Criterion Measure the passing of internal signals: Generate a TF model based on a representative signal. Check all filtering operators available using SPNS-QC with the same signal used for the synthesis of the TF model.

48. SPNS-QC based Quality Criterion (cont ’ d) Measure the blocking of external signals: Generate random noise distributed uniformly in the TF plane. Check the total amount of noise passed in each filtering method. The best filter passes the least amount of noise.

49. SPNS-QC based Quality Criterion (cont ’ d) This criterion will serve as a validation QC since it relies on a test signal (available only in synthesized cases). Current methods of TF signal synthesis bias the results of the QC towards the related filtering method.

50. Generalization of WD and STFT Wigner Distribution STFT Distribution

51. Weyl Symbol as a generalization of Fourier A general (time-varying) linear operator is defined by and its 2D Fourier transform Representing a filtering operation as a weighted superposition of TF shifted versions of the signal:

52. Weighting and Correcting effects in TF plane

53. WS as a weighted superposition of eigenfunctions WD Moyal formula:

54. The Heisenberg Uncertainty principle for WD Using the Weyl correspondence, the positive semi- definite operator H obeys Heisenberg uncertainty principle for Wigner Distributions (Folland-97)  A restriction on the minimum spread of the symbol

55. The Heisenberg Uncertainty principle for WD (cont ’ d)  The following inequality shows that the symbol cannot be too peaked locally (Janssen-89 ):

56. Filtering Experiments – 1

57. Filtering Experiments – 2