Marek Blok, Maciej Sac: Variable fractional delay filter design using a symmetric window, Circuits, Systems and Signal Processing, 2014

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1 Marek Blok, Maciej Sac: Variable fractional delay filter design using a symmetric window, Circuits, Systems and Signal Processing, 2014 Gdansk University of Technology, Department of Teleinformation Networks 11/12 Narutowicza, Gdansk, Poland Corresponding author: Marek Blok, Marek.Blok@eti.pg.gda.pl Online Resource 1: Comments for the demonstration code accompanying paper 1. Gain correction test Farrow structure coefficients calculation Optimal filter implementation using the proposed structure Coefficients computation for the proposed window method Sample rate conversion (SRC) using the proposed structure... 9 References Gain correction test script: test_gain_correction.m The script compares peak error of the minimax filter given as an example in [3] (N = 16, f a = 0.35 and total delay = 7.8 (net delay equal to 0.3)), which is reported in [3] as 80 db and only 0.1 db worse from optimal solution. For the same parameters we obtain the following results: for optimal filter PE opt = db and for our implementation with directly designed reference window and approximate gain correction factor computed using formula (39) [2] PE wind = db (after gain correction factor optimization the error can be decreased to PE wind,opt = db). Since we do not know the exact implementation used in paper [3] we have decided that we would not present additional comparisons in our paper. 2. Farrow structure coefficients calculation: comparison of Tseng concept [4] based on differentiator bank (red lines) with the approach used in [2] based on piecewise polynomial approximation of overall filter [1] (black lines) script: test_sinc_approx.m The script tests how well Farrow structures of different orders (q parameter) approximate truncated impulse response of the ideal FD filter (sampled sinc function) (here for ). The figure below presents squared error (SE) (for ) computed based on the following complex approximation error ( ) ( ) (1)

2 ( ) ( ) ( ) (2) where ( ) is the frequency response of the FD filter implemented using Farrow structure and ( ) is the frequency response of the ideal FD filter impulse response. As we can see, even a low order structure offers performance similar to that of truncated sinc. Nevertheless, the presented figure does not give a proper assessment of the structure usefulness for implementation of optimal filters based on the window method proposed in [2]. The next figure presents SE (for ) calculated based on the following error where ( ) ( ) ( ) (3) ( ) is the frequency response of truncated impulse response of the ideal FD filter (truncated sinc) of length. This measure illustrates how well a structure of given order approximates truncated sinc. As we will see further, this error is directly related to how well optimal filter will be approximated when the structure is used in the window method [2]. The increase of Farrow structure order by one results in improvement by about 15 db for both structure coefficients calculation methods ([1] and [4]). Comparing to [1], the approach based on differentiators bank [4] performs better around but is significantly worse for around 0.5.

3 3. Optimal filter implementation using the structure proposed in [2] script: test_opt_approx.m q Farrow structure order. The script tests the Farrow structure modification proposed in [2] (Fig. 13b). This approach uses the Farrow structure implementing truncated sinc with reference window and additional gain correction. The next figure presents SE (for LS filter of and ) calculated based on error (2). As already mentioned, this error has properties similar to the error presented in the previous figure Additionally, we can see here that the proposed window method based on direct design of the reference window ( ) allows us to obtain filters better by about 20 db than using direct optimal filter design method.

4 4. Coefficients computation for the proposed window method script: compute_coefs.m function [wind_e, wind_a, h_farrow] = compute_coefs(n_fd, fa, filter_type, window_design_mode, alpha_poly_ord, Farrow_mode, p_farrow, do_save) % Input: % N_FD - impulse response length of designed FD filter % fa - upper frequency of the approximation band % filter_type - FD filter type % 'MF' - filter with maximally flat error at f = 0 % 'LS' - filter with least squared error in [0, fa] % 'minimax' - filter with minimum maximum error in [0, fa] % window_design_mode - % 'design' - directly designed symmetric window % 'extract' - even part of the extracted window % alpha_poly_ord - order of polynomial approximating gain correction factor % Farrow_mode - method for computation of coefficients of Farrow structure % implementing truncated sinc impulse response: % 'diff' - bank of differentiators, 'poly' - polynomial approximation % p_farrow - Farrow structure order % do_save save coefs to files, 0 - do not save % Output: % wind_e - symmetric window for VFD filter implementation % wind_a - coefficients of polynomial approximating gain correction factor % h_farrow - coefficients of the Farrow structure implementing impulse % response % % Output files: wind_e.txt, wind_a.txt, h_farrow.txt % % Without output variables specified figures are generated The script which head lines are listed above computes coefficients of the modified Farrow structure proposed in [2]. User obtains reference window coefficients, coefficients of the polynomial approximating gain correction factor and coefficients of the Farrow structure approximating truncated sinc. All these coefficients can be saved to text files.

5 SE(d) [db] SE(d) [db] script: test_compute_coefs.m Examples of compute_coefs uses. In the next part of the comments several filter designs are presented. Figures on the left show SE versus fractional delay for FD filter with and figures on the right demonstrate magnitude of the complex approximation error (2) versus normalized frequency for several filters with different fractional delays. In each case we can see that the use of gain correction is necessary to obtain high performance VFD filters using the proposed window method. Input parameters: N_FD=251, fa=0.480, minimax, design, alpha_poly_ord=3, Farrow_mode= poly, p_farrow=10 Above: with gain correction results are virtually optimal; additionally, use of the window method gives better SE results for low fractional delays. Input parameters: N_FD=251, fa=0.480, minimax, design, alpha_poly_ord=3, Farrow_mode= diff, p_farrow=10 Above: use of differentiators bank [4] (all other parameters are the same as in the previous example) results in poor performance for filters with.

6 Input parameters: N_FD=25, fa=0.400, LS, design, alpha_poly_ord=4, Farrow_mode= poly, p_farrow=6 SE(d) [db] SE(d) [db] SE(d) [db] Above: another example with virtually optimal results obtained using the proposed method. Input parameters: N_FD=25, fa=0.400, minimax, design, alpha_poly_ord=4, Farrow_mode= diff, p_farrow=8 Above: in this example differentiators bank concept [4] is used; as a result, Farrow structure order has to be increased by 3, comparing to the solution from [1], to obtain virtualy optimal results. Input parameters: N_FD=25, fa=0.400, minimax, design, alpha_poly_ord=2, Farrow_mode= poly, p_farrow=6 Above: in the right figure we can see performance degradation for filters with resulting from low order of polynomial approximating gain correction factor ( ). Nevertheless, since for approximation error decreases significantly selection of seems acceptable (left figure).

7 SE(d) [db] SE(d) [db] Input parameters: N_FD=25, fa=0.400, minimax, design, alpha_poly_ord=4, Farrow_mode= poly, p_farrow=5 Above: here Farrow structure order is too low, which leads to increase in approximation error in frequency range ; this degradation increases with frequency. In the next part we present examples for which the window method with direct reference window design performs better (because of lower computational errors) in comparison to direct optimal filter design of LS FD filters. Input parameters: N_FD=89, fa=0.450, LS, design, alpha_poly_ord=4, Farrow_mode= poly, p_farrow=8 Above: although the window method does not lead here to optimal solution, the achieved performance is significantly better in comparison to direct use of optimal filter design

8 Input parameters: N_FD=89, fa=0.450, LS, design, alpha_poly_ord=2, Farrow_mode= poly, p_farrow=8 SE(d) [db] SE(d) [db] SE(d) [db] Above: decrease in gain correction polynomial order from 4 to 2 results in worse performance for but since those filters are not worse than the filter with, selection of seems acceptable Input parameters: N_FD=230, fa=0.480, LS, design, alpha_poly_ord=4, Farrow_mode= poly, p_farrow=11 Above: again, window method does not lead to optimal solution but we can observe significant improvement in performance in comparison to direct use of optimal filter design Input parameters: N_FD=230, fa=0.480, LS, design, alpha_poly_ord=2, Farrow_mode= poly, p_farrow=10 Above: gain correction polynomial order can be decreased from 4 to 2 and Farrow structure order from 11 to 10 with still acceptable performance of VFD filter

9 5. Sample rate conversion (SRC) using the proposed structure In the proposed examples a band limited noise with linearly increasing instantaneous bandwidth is resampled with incommensurate resampling ratio r = 2π/3. The results are compared with ideally resampled signal (both input signal and reference resampled signal are generated using the get_signal.m script). The SRC process is based on VFD (variable fractional delay) filters with coefficients changed based on instantaneous signal bandwidth. Below input signal and its spectrogram are presented. In the following figures resampling errors and all spectrograms are presented in decibels. script: SRC_demo.m SRC implemented with the proposed structure based on the window method. Filter switching is equivalent to window and gain correction polynomial coefficients change. Input parameters: windows_set_no (window set number), p_farrow (Farrow structure order). script: SRC_demo_opt.m SRC based on the standard Farrow structure implementing optimal filters designed using the proposed window method since direct design of the optimal filter in some cases results in higher computational errors. Filter switching is equivalent to change of all structure coefficients. Input parameters are the same as in the case of the SRC_demo script. script: run_src_demo.m examples of using SRC_demo and SRC_demo_opt

10 E(f) [db] Example 1. SRC_demo.m, windows_set_no = 6; p_farrow = 7; constant filter length Magnitude of complex approximation errors [db] versus normalized frequency [1/Sa] for the filters in the used set. f [1/Sa] resampled signal and resampling error for VFD filter no 1

11 resampled signal and resampling error for VFD filter no 2 resampled signal and resampling error for VFD filter no 3

12 resampled signal and resampling error for VFD filter no 4 resampled signal and resampling error for VFD filter no 5

13 E(f) [db] resampled signal and resampling error for resampling different signal segments using different VFD filters filters are selected based on the instantaneous bandwidth of the input signal Example 2a. SRC_demo.m, windows_set_no = 1.1; p_farrow = 8; variable filter length with approximately constant resampling error N_FD = {9, 17, 29, 71, 255} f [1/Sa]

14 signal resampled using VFD filter no 1 signal resampled using VFD filter no 2

15 signal resampled using VFD filter no 3 signal resampled using VFD filter no 4

16 signal resampled using VFD filter no 5 signal resampled using all VFD filters switching windows

17 The difference in filter length ( { 9, 17, 29, 71, 255}) results in the difference in computation time for particular filters (when one filter is used for the whole SRC process without filter switching). Using our computer we measured [s]. It is worth noting that in the performed experiment with filter switching resampling error of the longest filter is at similar level as in the case of filters switching. However, such an approach leads to significant decrease in computation time, which for filter switching was measured as t = [s]. Thus, the implementation with filters switching allows for decrease of computational costs (example 2a) or decrease of resampling error without the need for increase in computational costs (example 1). Example 2b. SRC_demo_opt.m, windows_set_no = 1.1; p_farrow = 8; variable filter length with approximately constant resampling error N_FD = {9, 17, 29, 71, 255} signal resampled using all VFD filters switching all Farrow structure coefficients As we can see, the performance of both structures (from example 2a and 2b) is similar. The main differences are: 1. Only slightly smaller computation time of the standard Farrow structure: on our computer we measured [s] for single filter and [s] for filters switching. 2. For switching filters standard Farrow structure requires much more coefficients, which have to be stored and switched when needed, comparing to the modified structure.

18 References: [1] Blok, M.: Farrow structure implementation of fractional delay filter optimal in Chebyshev sense. In: Proc. SPIE (2005), vol. 6159, 61594K [2] Blok, M., Sac, M.: Variable fractional delay filter design using a symmetric window. Circuits Syst. Signal Process (2014) [3] Hermanowicz, E.: A nearly optimal variable fractional delay filter with extracted Chebyshev window. In: Proc. ICECS 98, Lisboa, Portugal (1998), pp [4] Tseng, Ch.-Ch.: Design of variable fractional delay FIR filter using differentiator bank. In: Proc. ISCAS 2002 (2002), vol. 4, pp