ESE 531, Introduction to Digital Signal Processing Final Project

Transcription

1 Department of Materials Science and Engineering, University of Pennsylvania ESE 531, Introduction to Digital Signal Processing Final Project Some of the designs and Matlab codes are referred to lecture note 9 and note 10. Part A Multi-Channel FIR Filter-Bank (1-D Application) A.1 Design of a 4-channel "octave band" scheme Yang Lu The 4-channel octave bank is shown below in Fig.A1, which uses the building element is 2- channel filter bank. As long as signal in each 2-channel filter bank can be reconstructed perfectly, the 4-channel octave bank is also a design for perfect reconstruction. As the question requires, the original full band signal (x[n]) is decomposed into: 2 low frequency bands (1/8 of full-band) as channel 0 and channel 1 respectively, one intermediate 1/4 of full-band (channel 2), and the upper 1/2 high-frequency band (channel 3). In order to add those decomposed signals in each band at the end and compare the reconstructed output signal with original x[n], the filtering delays in each band must be counted precisely. In Fig.A1, the filtered signal after decimation/interpolation are labelled based on the fact that each H(z) and G(z) filter produce a delay of N/2 with respect to n. (N is the order of FIR filter.) The filtering delay turns out to be N for channel 3, 3N for channel 2, and 7N for channel 1 and channel 0. These facts will be used in Matlab codes. 7N/2 3N/2 HL(z) 2 2 GL(z) 11N/2 13N/2 7N HL(z) 2 Channel GL(z) 2 GL(z) HH(z) 2 2 GH(z) 3N x[n] HL(z) HH(z) 2 2 HH(z) N/2 2 3N/2 Channel 1 Channel 2 Channel 3 2 GH(z) 2 GL(z) 5N/2 2 GH(z) + + y[n] N Fig.A1 4-channel octave bank for perfect reconstruction!1

2 A.2 Test of the 4-channel octave bank To realize the designed 4-channel filter bank in Matlab, Matlab routines "firpr2chfb", "mfilt.firdecim", and "mfilt.firinterp" are used to create the fundamental HL, HH, GL, and GH filters in a 2-channel filter bank. Then, filters in each channel can be treated equivalently as single Hx or Gx, which is certain cascade of HL, HH, GL, and GH filters. The frequency response of H0-3 and G0-4 are shown in Fig.A2a and A2b, respectively (where the filter length is 99 and band edge is 0.45). As indicated in Fig.A2, the equivalent H and G filters indeed separate the full band into 1/8, 1/8, 1/4, and 1/2 as designed. (a) 1/8 1/8 1/4 1/2 (b) 1/8 1/8 1/4 1/2 Fig.A2 Frequency response of filters (a) H0-3 and (b) G0-3 in 4-channel octave bank. Fig.A3 Input signal x[n] for simulation of perfect construction As for the input signal, I chose the same x[n] as the one in note 9 (Fig.A3). As explained earlier, each channel contains different amounts of delays. So signals in channel 2-3 are shifted!2

3 according to their delays (N or 3N) before the final output. Lastly, errors of the reconstructed signal from the original one are calculated after shifting the input x[n] by 7N. Under this test, N varies from 9 to 109. As shown in Fig.A4, the signal error of the reconstructed y[n] is very sensitive to the filter length N. However, the maximum error (max{e[n]}) is not inversely proportional to N. (e[n] is fluctuated with minimum in the order of 10-3 at N = 39, no general trend can be summarized.) Fig.A4 Relation between max{e[n]} and filter length N. max{e[n]} is minimized at N = 39. Fig. A5 compares y[n] and e[n] when N = 39 (a-b) and N = 99 (c-d). (a) N = 39!3

4 (b) N = 39 (c) N = 99 (d) N = 99 Fig.A5 Output y[n] and signal error e[n] for N = 39 (a,b) and N = 99 (c,d). Matlab codes in this section are attached below. %% Perfect Reconstruction 4 Channel Octave Bank % Define the order of filters, which must be an odd number N=39;!4

5 % Building element, i.e., 2 channel filter bank, construction [H0,H1,G0,G1]=firpr2chfb(N,.45); Hl=mfilt.firdecim(2,H0); Hh=mfilt.firdecim(2,H1); Gl=mfilt.firinterp(2,G0); Gh=mfilt.firinterp(2,G1); % Hx and Gx are cascaded from Hl, Hh, Gl, and Gh based on Fig.A1 H0=cascade(Hl,Hl,Hl); H1=cascade(Hl,Hl,Hh); H2=cascade(Hl,Hh); H3=Hh; G0=cascade(Gl,Gl,Gl); G1=cascade(Gh,Gl,Gl); G2=cascade(Gh,Gl); G3=Gh; % Check the analysis/synthesis filters hfv=fvtool(h0); set(hfv,'filters',{h0,h1,h2,h3}); legend(hfv,'channel 0 Decimator','Channel 1 Decimator','Channel 2 Decimator','Channel 3 Decimator'); gfv=fvtool(g0); set(gfv,'filters',{g0,g1,g2,g3}); legend(gfv,'channel 0 Interpolator','Channel 1 Interpolator','Channel 2 Interpolator','Channel 3 Interpolator'); % Input Signal x=zeros(1024,1); x(1:3)=1;x(8:10)=2;x(16:18)=3;x(24:26)=4;x(32:34)=3;x(40:42)=2;x(48:50)=1; stemplot(x,'input x[n]'); % Simulation of Perfect Reconstruction % Channel 0 and 1 both have the delay of 7N x0=filter(h0,x);x0=filter(g0,x0);!5

6 x1=filter(h1,x);x1=filter(g1,x1); % Channel 2 has the delay of 3N x2=filter(h2,x);x2=filter(g2,x2); delay4n=zeros(1,4*n+1);delay4n(4*n+1)=1; x2=filter(delay4n,1,x2); % Channel 3 has the delay of N x3=filter(h3,x);x3=filter(g3,x3); delay6n=zeros(1,6*n+1);delay6n(6*n+1)=1; x3=filter(delay6n,1,x3); xtilde=x0+x1+x2+x3; stemplot(xtilde,'output y[n]'); % Shift input x[n] by 7N to check the error delay7n=zeros(1,7*n+1);delay7n(7*n+1)=1; shiftedx=filter(delay7n,1,x); xerror=shiftedx-xtilde; stemplot(xerror,'error e[n]'); A.3 Audio Waveform Test To further test the designed 4-channel octave band on an audible waveform, a segment of music named "Super Mario.wav" is loaded as the input for the filter-bank. (Wave music typically has two inherent channels: left and right. Only one channel is under study here for simplicity.) Different gains (a0-3) are applied on the 4 subband signals (0-3) before reconstruction to enhance certain frequency band relative to others. For example, when a = (100,1,1,1) is applied, the bass boost output of this music can be clearly heard (check output file "Super Mario_bass boost.wav"). When a = (1,1,1,100) is applied, the high frequency part of this music is enhanced (check output file "Super Mario_hf boost.wav"). Moreover, if a is set as (1,0,0,0), only the low frequency part is output (check output file "Super Mario_bass only.wav") while only the high frequency part can be heard in the case of (0,0,0,1) (check output file "Super Mario_hf only.wav"). In addition, delays D0-3 before reconstruction are added to four subband signals. As the delay of channel 0 increases, my ear starts to tell some different when it becomes 50 samples (check!6

7 output file "Super Mario_delay 50.wav"), and an obvious beat mismatch occurs when delay is up to 100 samples (check output file "Super Mario_delay 100.wav"). Matlab codes in this section are attached below. (a0-3 and D0-3 will vary according to the testing conditions of frequency enhancement and delays.) %% 4 Channel Octave Bank on Audible Waveform % Define the order of filters, which must be an odd number N=39; % Define enhancement for each band a0=1;a1=1;a2=1;a3=1; % Delay for each band D0=0;D1=0;D2=0;D3=0; % Building element, i.e., 2 channel filter bank [H0,H1,G0,G1]=firpr2chfb(N,.45); Hl=mfilt.firdecim(2,H0); Hh=mfilt.firdecim(2,H1); Gl=mfilt.firinterp(2,G0); Gh=mfilt.firinterp(2,G1); % Hx and Gx are cascaded from Hl, Hh, Gl, and Gh based on Fig.A1 H0=cascade(Hl,Hl,Hl); H1=cascade(Hl,Hl,Hh); H2=cascade(Hl,Hh); H3=Hh; G0=cascade(Gl,Gl,Gl); G1=cascade(Gh,Gl,Gl); G2=cascade(Gh,Gl); G3=Gh; % Input signal from music sample Super Mario.wav which has two channels [y,fs,bits]=wavread('super Mario'); yleft=y(:,1); yright=y(:,2); % Perfect reconstruction for left channel!7

8 x0=filter(h0,yleft);x0=filter(g0,x0); x1=filter(h1,yleft);x1=filter(g1,x1); x2=filter(h2,yleft);x2=filter(g2,x2); delay4n=zeros(1,4*n+1);delay4n(4*n+1)=1; x2=filter(delay4n,1,x2); x3=filter(h3,yleft);x3=filter(g3,x3); delay6n=zeros(1,6*n+1);delay6n(6*n+1)=1; x3=filter(delay6n,1,x3); % Delay the subband signals according to D0-D3 delayd0=zeros(1,d0*n+1);delayd0(d0*n+1)=1; x0=filter(delayd0,1,x0); delayd1=zeros(1,d1*n+1);delayd1(d1*n+1)=1; x1=filter(delayd1,1,x1); delayd2=zeros(1,d2*n+1);delayd2(d2*n+1)=1; x2=filter(delayd2,1,x2); delayd3=zeros(1,d3*n+1);delayd3(d3*n+1)=1; x3=filter(delayd3,1,x3); % Enhance certain bands according to a0-a3 yleftout=a0*x0+a1*x1+a2*x2+a3*x3; % Final output for left channel only yout=yleftout; wavwrite(yout,fs,bits,'super Mario_new');!8

9 Part B Adaptive Notch Filter B.1 Adaptive notch filter to cancel sinusoidal interference The design of adaptive notch filter to reject a strong sinusoidal interference is illustrated in Fig.B1. The input signal can be treated as x[n]=s[n]+w[n], where s[n] is the desired clean signal and w[n] is the sinusoidal noise. If w[n] is much stronger than s[n], it will be rejected simply when E(y[n] 2 ) is minimized. Under the given assumption, a[n+1]=a[n]-µy[n]x[n-1] (a=0 at the beginning). Note that a = -2cos(ω0), so it has to be reset to 0 if a[n]-µy[n]x[n-1] is out of bounds. x[n] = s[n]+w[n] Notch Filter y[n] = x[n]+ax[n-1]+ x[n-2]-ray[n-1]-r 2 y[n-2] Update Coefficient a a[n+1] = a[n]-µy[n]x[n-1] Fig.B1 Schematic of designed adaptive notch filter In the simulation, w[n] is set to be c0 cos(2πnf0) and s[n] is set to be c1 cos(2πnf1) with c1 << c0. An example to demonstrate the results of designed adaptive notch filter is shown below in Fig.B2 for c0=1000, f0=0.1, c1=1, f1=0.5, r=0.98, and µ=10-9. It is clearly observed that a converges to (the corresponding ω0=2πnf0 being 0.2π) around 5000 steps and the output y[n] indeed only contains s[n] with much lower amplitude and higher frequency. (a)!9

10 (b) (c) (d) Fig.B2 Results of the designed adaptive notch filter (c0=1000, f0=0.1, c1=1, f1=0.5, r=0.98, and µ=10-9 ). (a) input x[n] contains high power sinusoidal interference (low-frequency), (b) output y[n] only contains low power single-highfrequency signal, (c) convergence of a vs. n steps, (d) frequency response of the notch filter after convergence. In the next sections, different values of parameters u, r, c0-1 and f0-1 are tested to check their effects on the results. Note that these parameters are actually correlated with each other. For simplicity, all the other parameters are fixed to study only one changing variable at one time. (i) u dependence!10

11 Effect of different values of u are examined in Fig.B3 after fixing c0=1000, f0=0.1, c1=1, f1=0.5, r=0.98. Generally speaking, u determines how fast a converges. When u is not small enough, e.g., 10-3, a always goes out of bound so it is preset to 0 in every step (Fig.B3a). When increases to 10-5, a starts to fluctuate from -2 to 2 but never converges (Fig.B3b-c). a begins to converge within 2000 steps when u= (Fig.B3d). As u further decreases, the convergence of a becomes slower (Fig.B2c and Fig.B3e) until u=10-11 when a fails to converge within steps (Fig.B3f). (The convergence of a is also related to r and c0/c1, which will be discussed later.) (a) µ=10-3 (b) µ=10-5 (c) µ=10-6!11

12 (d) µ= (e) µ=10-10 (f) µ=10-11 Fig.B3 Convergence of a at various µ values. (ii) r dependence Effect of r on designed adaptive notch filter is illustrated in Fig.B4-5 with constant c0=1000, f0=0.1, c1=1, f1=0.11, µ=10-9. According to the transfer equation of notch filter, r defines the!12

13 positions of poles in this system. As the poles get closer to zeroes, i.e., r is closer to 1, the notch at zero will become sharper (Fig.B4), which implies a better cancellation especially when f1 is near f0 (Fig.B5): when f1 equals to 0.11, the amplitude of output signal decreases when r decreases from 0.98 to (a) r=0.98 (b) r=0.915 (c) r=0.85 Fig.B4 Frequency response of converged notch filter for various r values.!13

14 (a) r=0.98 (b) r=0.915 (c) r=0.85 Fig.B5 Output y[n] with suppressed amplitude as r decreases. (iii) f0 (f1) dependence As discussed in (ii), effect of frequency f0 (f1) is sometimes correlated with r. For simplicity, c0=1000, c1=1, µ=10-9, and r=0.98 are fixed in this section. First of all, converged a value is!14

15 determined by a=-2cos(2πf0). Secondly, the resolution of this adaptive notch filter is limited: f1 cannot be too close to f0 as shown in Fig.B6. Lastly, µ has to decrease with smaller f0 as illustrated in Fig.B7: µ is decreased from 10-9 to in order to have the sensitivity to converge a (f0 =0.01). (a) f1=0.11 (b) f1=0.101 (c) f1= Fig.B6 Output y[n] with suppressed amplitude as f1 gets closer to f0=0.1.!15

16 (a) µ=10-9 (b) µ=10-10 Fig.B7 Smaller µ has to be applied for convergence at low frequency f0=0.01. (iv) c0 (c1) dependence Two effects of c0 (c1) has been revealed: 1) As shown in Fig.B8, c0 has to be larger than c1 by at least one order of magnitude, which is the fundamental assumption for the entire system. 2) As the decrease of c0, µ has to increase for convergence (Fig.B9). (a) c1=10!16

17 (b) c1=100 (c) c1=1000=c0 Fig.B8 Cancellation fails when c1 increases to c0 with fixed c0=1000, f0=0.1, f0=0.5. (a) c0=1000, µ=10-9!17

18 (b) c0=100, µ=10-9 (c) c0=100, µ=10-8 Fig.B9 As c0 increases, µ has to decrease for convergence (fixed c0=1000, f0=0.1, f0=0.5). Matlab codes in this section are attached below. %% Adaptive Notch Filter for sinusoidal interference cancellation % Set interference w[n]'s frequency and amplitude f0=0.1; c0=1000; % Set desired s[n]'s frequency and amplitude f1=0.5; c1=1; % Input length N = 50,000 N=50000; % Input signal!18

19 x=c0*cos(2*pi*f0*[1:n])+c1*cos(2*pi*f1*[1:n]); % Initialize a=zeros(1,n); y=zeros(1,n); mu= ; r=0.98; % Main loop for n=100:n y(n)=x(n)+a(n)*x(n-1)+x(n-2)-r*a(n)*y(n-1)-r*r*y(n-2); if abs(a(n)-mu*y(n)*x(n-1))<=2 a(n+1)=a(n)-mu*y(n)*x(n-1); else a(n+1)=0; end end % Plot x, y, a figure(1); plot(x(49000:49100)); figure(2); plot(y(49000:49100)); figure(3); plot(a); % Plot frequency response of the final converged notch filter b1=[1,a(n),1]; b2=[1,r*a(n),r*r]; [h,w]=freqz(b1,b2,'whole',2001); figure(4); plot(w/pi,20*log10(abs(h))) ax=gca; ax.ylim=[-60 10]; ax.xtick=0:.2:2;!19

20 xlabel('normalized Frequency (\times\pi rad/sample)') ylabel('magnitude (db)') B.2 Sinusoidal interference with slowly changing frequency In order to check the tracking ability of designed adaptive notch filter on a sinusoidal interference with slowly changing frequency, w[n] now becomes c0 cos[2πn(f0+dfn)], where df is defined as the changing rate of sinusoidal interference. To enable the fast frequency tracking of adaptive notch filter, one has to make sure the convergence of a is fast enough. According to Fig.B3, the convergence is dependent on µ. It will be also been verified here that the fast tracking is also extremely sensitive to µ. Fig.B10-13 illustrate the convergence of a and output y[n] with various df and µ while c0 (1000), f0(0.2), c1(1), f1(0.5), and r(0.98) are fixed. Note that N is set to be 1,000,000 in order to fully demonstrate the tracking ability. df is chosen to be 10-8 in Fig.B10 while µ=10-7, 10-9, in (a)-(b), (c)-(d), and (e)-(f) respectively. With such a small df, adaptive filters with all these three µ values are able to track the frequency change: a changes continuously and the output is purely the desired signal without any interference after convergence. Different µ only affect the starting point of convergence, consistent with the observation in previous section. (a) df=10-8, µ=10-7 (b) df=10-8, µ=10-7!20

21 (c) df=10-8, µ=10-9 (d) df=10-8, µ=10-9 (e) df=10-8, µ=10-10 (f) df=10-8, µ=10-10 Fig.B10 Convergence of a and output y[n] (with y[999000:999100] as inset) at df=10-8 and µ=10-7, 10-9, !21

22 (a) df=10-7, µ=10-7 (b) df=10-7, µ=10-7 (c) df=10-7, µ=10-9 (d) df=10-7, µ=10-9 Fig.B11 Convergence of a and output y[n] (with y[999000:999100] as inset) at df=10-7 and µ=10-7, 10-9.!22

23 (a) df=10-6, µ=10-7 (b) df=10-6, µ=10-7 (c) df=10-6, µ=10-9 (d) df=10-6, µ=10-9 Fig.B12 Convergence of a and output y[n] (with y[999000:999100] as inset) at df=10-6 and µ=10-7, 10-9.!23

24 df is increased to 10-7 in Fig.B11 (µ=10-7 and 10-9 ). As df increases, a can still follow the change of frequency so the output signal is still the desired one with small amplitude. But the exact shape of filtered output is not perfectly preserved (see inset of (b) and (d)). When df is 10-6 as shown in Fig.B12, a fails to converge thus doesn t follow the change of frequency at the upper or lower bounds (-2 and 2), which is reflected in the output with high power interference near those n values. For µ=10-9, the adaptive filter is even unable to reject the high power interference completely at all the n values (see high amplitude in the inset of (d)). It may be concluded that, the designed adaptive notch filter is able to track the slow change of interference frequency as long as a doesn t hit the upper/lower bound. Matlab codes in this section are attached below. %% Adaptive Notch Filter for Slowly Changing Sinusoidal interference % Set interference w[n]'s frequency f0, changing rate df, and amplitude f0=0.2; df=1e-8; c0=1000; % Set desired s[n]'s frequency and amplitude f1=0.5; c1=1; % Input length N = 1,000,000 N= ; % Input signal for n=1:n x(n)=c0*cos(2*pi*(f0+df*n)*n)+c1*cos(2*pi*f1*n); end % Initialize a=zeros(1,n); y=zeros(1,n); mu= ; r=0.98; % Main loop for n=100:n!24

25 y(n)=x(n)+a(n)*x(n-1)+x(n-2)-r*a(n)*y(n-1)-r*r*y(n-2); if abs(a(n)-mu*y(n)*x(n-1))<=2 a(n+1)=a(n)-mu*y(n)*x(n-1); else a(n+1)=0; end end % Plot x, y, a figure(1); plot(x); figure(2); plot(y); figure(3); plot(a); figure(4); plot(y(999000:999100)); B.3 Adaptive notch filter cascade to reject two sinusoidal interference x[n] = s[n]+w1[n]+w1[n] Notch Filter y1[n] Notch Filter y2[n] Update Coefficient a1 Update Coefficient a2 a1[n+1] = a1[n]-µy2[n]x[n-1] a2[n+1] = a2[n]-µy2[n]y1[n-1] Fig.B13 Design scheme of two-notch adaptive filter!25

26 The design scheme of the adaptive notch filter cascade for two sinusoidal interference cancellation is illustrated in Fig.B13, where the input sinusoidal interference now becomes c01 cos(2πnf01)+c02 cos(2πnf02). In order to reject two strong sinusoidal interference with different frequencies, two notch filters are cascaded together: one s output serves as the other s input. The key point of design is how to update coefficients a1 and a2 in these two filters. According to B1, a[n+1]=a[n]-µy[n]x[n-1]. The problem becomes what are the "y[n]" and "x[n-1]" in these two coefficient-update units. Obviously, the inputs should serve as "x[n-1]", which are x[n-1] and y1[n-1] for a1 and a2, respectively. As for "y[n]", which represents the signal that are targeted to be minimized, it is reasonable to use the final output y2[n] for both two coefficients. Using this design, the adaptive notch filter cascade works quite well to reject two strong frequency components as the interference. Fig.B14 shows an example of the input x[n], intermediate output y1[n], final output y2[n], convergence of a1 and a2, and frequency response of the converged two notch filters with c01=1000, f01=0.1, c02=1000, f02=0.3, c1=1, f1=0.5, r=0.98, and µ=10-8. It is interesting to observe that a1 and a2 initially go to the same direction to converge to the value corresponds to f01=0.1. Since a2 arrives first, a1 has to turn around and converge to another value associated with f02=0.3. (a) (b)!26

27 (c) (d) (e) (f)!27

28 (g) Fig.B14 Example of the results of adaptive notch filter cascade to cancel two sinusoidal interference with difference frequencies (0.1 and 0.3). (a) Input, (b) intermediate output, (c) final output, (d) convergence of a1, (e) convergence of a2, (f) frequency response of converged notch filter 1 (reject 0.3 frequency), and (g) frequency response of converged notch filter 1 (reject 0.1 frequency). (a) (b) Fig.B15 Convergence of when are close to each other (being 0.1 and 0.12, respectively). This design also works for different parts of f01 and f02 that are close to each other. Again, µ is very sensitive to different pairs of a1 and a2. And it is actually very interesting to check how a1 and!28

29 a2 converge. Fig.B15 shows another example with f01=0.1 and f02=0.12 (µ=10-9 this time). It seems a1 and a2 cannot converge at the same time. It must be one of them converge first, and then the other will converge very quickly. Matlab codes in this section are attached below. %% Adaptive Notch Filter Cascade for Two Sinusoidal interference Cancellation % Set interference w[n]'s frequency and amplitude f01=0.1; c01=1000; f02=0.3; c02=1000; % Set desired s[n]'s frequency and amplitude f1=0.5; c1=1; % Input length N = 50,000 N=50000; % Input signal x=c01*cos(2*pi*f01*[1:n])+c02*cos(2*pi*f02*[1:n])+c1*cos(2*pi*f1*[1:n]); % Initialize a1=zeros(1,n); a2=zeros(1,n); y1=zeros(1,n); y2=zeros(1,n); mu= ; r=0.98; % Main loop for n=100:n y1(n)=x(n)+a1(n)*x(n-1)+x(n-2)-r*a1(n)*y1(n-1)-r*r*y1(n-2); y2(n)=y1(n)+a2(n)*y1(n-1)+y1(n-2)-r*a2(n)*y2(n-1)-r*r*y2(n-2); if abs(a1(n)-mu*y2(n)*x(n-1))<=2 a1(n+1)=a1(n)-mu*y2(n)*x(n-1); else!29

30 a1(n+1)=0; end if abs(a2(n)-mu*y2(n)*y1(n-1))<=2 a2(n+1)=a2(n)-mu*y2(n)*y1(n-1); else a2(n+1)=0; end end % Plot x, y, a figure(1); plot(x(49000:49100)); figure(2); plot(y1(49000:49100)); figure(3); plot(y2(49000:49100)); figure(4); plot(a1); figure(5); plot(a2); % Plot frequency response of the final converged notch filter b1=[1,a1(n),1]; b2=[1,r*a1(n),r*r]; [h,w]=freqz(b1,b2,'whole',2001); figure(6); plot(w/pi,20*log10(abs(h))) ax=gca; ax.ylim=[-60 10]; ax.xtick=0:.2:2; xlabel('normalized Frequency (\times\pi rad/sample)') ylabel('magnitude (db)') b1=[1,a2(n),1];!30

31 b2=[1,r*a2(n),r*r]; [h,w]=freqz(b1,b2,'whole',2001); figure(7); plot(w/pi,20*log10(abs(h))) ax=gca; ax.ylim=[-60 10]; ax.xtick=0:.2:2; xlabel('normalized Frequency (\times\pi rad/sample)') ylabel('magnitude (db)') B.4 Sinusoidal interference with slowly changing frequency Strong sinusoidal interference is introduced at each channel of 4-channel octave-band filterbank with frequencies of 0.01( f00 in channel 0), 0.08 ( f01 in channel 1), 0.2 ( f02 in channel 2), and 0.3 ( f03 in channel 3) in addition to the desired signal with very small power with frequency f01 as 0.5. For the adaptive notch filters in each subband, a constant µ (10-10 ) is selected to make sure all of the a values will get converged within N=50,000 steps. As for the filter bank, in order to guarantee only one frequency exists in each channel, the order of FIR filters (M) has to be large enough so that there exists high enough ratio after LP/HP filtering (M=49 and 99 are used for comparison). Fig.B16 shows the input signal, signals in each channel before and after adaptive notch filters, the convergence of a values in each channel, and the final output with the desired signal. However, the efficiency of this method is not very high considering the total number of filters that are used (low order FIR filters in filter-bank is unable to make it work). It is better to use one multi-notch (adaptive) filter to reject all the interference without splitting the signal. (a)!31

32 (b) f00=0.02 in channel 0 (c) (d) After notch filter (e) f01=0.08 in channel 1!32

33 (f) (g) After notch filter (h) f02=0.2 in channel 2 (i)!33

34 (j) After notch filter (k) f03=0.3 in channel 3 (l) (m) After notch filter!34

35 (n) Final output, M=99 (o) Final output, M=49 Fig.B16 Results of applying adaptive notch filter in each subband of 4-channel octave-band filter-bank. (a) Input with four strong sinusoidal interference, (b)-(d): signals in channel 0, (e)-(g): signals in channel 1, (h)-(j): signals in channel 2, (k)-(m): signals in channel 3, (n) final output for filter length of 99, (o) final output for filter length of 49. Matlab codes in this section are attached below. %% 4-Channel Octave-Band Filter-Bank with Adaptive Notch Filter in Each Band % Define the order of filters, which must be an odd number M=49; % Building element, i.e., 2 channel filter bank, construction [H0,H1,G0,G1]=firpr2chfb(M,.45); Hl=mfilt.firdecim(2,H0); Hh=mfilt.firdecim(2,H1); Gl=mfilt.firinterp(2,G0); Gh=mfilt.firinterp(2,G1); % Hx and Gx are cascaded from Hl, Hh, Gl, and Gh based on Fig.A1 H0=cascade(Hl,Hl,Hl); H1=cascade(Hl,Hl,Hh);!35

36 H2=cascade(Hl,Hh); H3=Hh; G0=cascade(Gl,Gl,Gl); G1=cascade(Gh,Gl,Gl); G2=cascade(Gh,Gl); G3=Gh; % Set interference w[n]'s frequency and amplitude f00=0.01; f01=0.08; f02=0.2; f03=0.3; c0=1000; % Set desired s[n]'s frequency and amplitude f1=0.5; c1=1; % Input length N = 50,000 N=50000; % Input signal x=c0*cos(2*pi*f00*[1:n])+c0*cos(2*pi*f01*[1:n])+c0*cos(2*pi*f02*[1:n]) +c0*cos(2*pi*f03*[1:n])+c1*cos(2*pi*f1*[1:n]); figure(1); plot(x(n-100:n)); % Initialize a0=zeros(1,n); a1=zeros(1,n); a2=zeros(1,n); a3=zeros(1,n); y0=zeros(1,n); y1=zeros(1,n); y2=zeros(1,n); y3=zeros(1,n);!36

37 mu= ; r=0.98; % Notch Filtering and Perfect Reconstruction % Channel 0 with interference f00 x0=filter(h0,x); x0=filter(g0,x0); figure(2); plot(x0(n-100:n)); for n=100:n y0(n)=x0(n)+a0(n)*x0(n-1)+x0(n-2)-r*a0(n)*y0(n-1)-r*r*y0(n-2); if abs(a0(n)-mu*y0(n)*x0(n-1))<=2 a0(n+1)=a0(n)-mu*y0(n)*x0(n-1); else a0(n+1)=0; end end figure(3); plot(a0); figure(4); plot(y0(n-100:n)); % Channel 1 with interference f01 x1=filter(h1,x); x1=filter(g1,x1); figure(5); plot(x1(n-100:n)); for n=100:n y1(n)=x1(n)+a1(n)*x1(n-1)+x1(n-2)-r*a1(n)*y1(n-1)-r*r*y1(n-2); if abs(a1(n)-mu*y1(n)*x1(n-1))<=2 a1(n+1)=a1(n)-mu*y1(n)*x1(n-1); else a1(n+1)=0;!37

38 end end figure(6); plot(a1); figure(7); plot(y1(n-100:n)); % Channel 2 with interference f02 x2=filter(h2,x); x2=filter(g2,x2); delay4m=zeros(1,4*m+1);delay4m(4*m+1)=1; x2=filter(delay4m,1,x2); figure(8); plot(x2(n-100:n)); for n=100:n y2(n)=x2(n)+a2(n)*x2(n-1)+x2(n-2)-r*a2(n)*y2(n-1)-r*r*y2(n-2); if abs(a2(n)-mu*y2(n)*x2(n-1))<=2 a2(n+1)=a2(n)-mu*y2(n)*x2(n-1); else a2(n+1)=0; end end figure(9); plot(a2); figure(10); plot(y2(n-100:n)); % Channel 2 with interference f03 and desired signal f1 x3=filter(h3,x); x3=filter(g3,x3); delay6m=zeros(1,6*m+1);delay6m(6*m+1)=1; x3=filter(delay6m,1,x3); figure(11);!38

39 plot(x3(n-100:n)); for n=100:n y3(n)=x3(n)+a3(n)*x3(n-1)+x3(n-2)-r*a3(n)*y3(n-1)-r*r*y3(n-2); if abs(a3(n)-mu*y3(n)*x3(n-1))<=2 a3(n+1)=a3(n)-mu*y3(n)*x3(n-1); else a3(n+1)=0; end end figure(12); plot(a3); figure(13); plot(y3(n-100:n)); y=y0+y1+y2+y3; figure(14); plot(y(n-100:n));!39

40 Part C Adaptive Equalization C.1 Training Mode Equalization The design scheme of adaptive FIR equalizer with training sequence is shown in Fig.C1: a random sequence s of ±A amplitudes as input to a LTI channel with unit response h (order L); the channel s output x=h*s+n, where n is Gaussian noise with SNR mourned db; the FIR adaptive equalizer filter (order M) with input x, reference (training) sequence s with delays, and output as the delayed version of original s. Input s Channel h x=h*s+n FIR Filter Output y Update Coefficients - + Delay Fig.C1 Design scheme of adaptive equalizer filter Desired Output Fig.C2 shows an example of the results of designed equalizer for h=[0.3, 1, 0.7, 0.3, 0.2], training sequence of length ~ 5,000, A=100 (SNR~35dB), M=20, channel delay=5, step size µ =10-6, where (a) shows output signal after convergence (red) almost identical to the original delayed signal (blue), (b) shows the error signal converges around 2500 steps, (c-e) are the impulse response, frequency response, and pole-zero plot of channel h, (f-h) are those of converged FIR filter, (i-k) are those of their cascade, i.e., the equivalent system response. (a)!40

41 (b) (c) (d) (e)!41

42 (f) (g) (h) (i)!42

43 (j) Fig.C2 Results of adaptive equalizer filter with h=[0.3, 1, 0.7, 0.3, 0.2], training sequence of length ~ 5,000, A=100 (SNR~35dB), M=20, channel delay=5, step size µ=10-6. The dependence of equalizer filter order M, filter step size µ, adaptive filer initialization, SNR, channel impulse response response h are examined below. (i) Equalizer filter order M Fig.C3 shows the results (y[n] & s[n-d] and convergence of error) with the same parameters as Fig.C2 s except M. The converged error decreases when M is increased from 10 to 20, but doesn t change much when M is further increased to 30. (a) (b)!43

44 (c) (d) Fig.C3 Comparison between results with different M (all the other parameters are the same as C2). (ii) filter step size µ Similar to the adaptive notch filter, the number of steps required to converge and the final error are highly sensitive to the step size µ. Fig.C4 compares the convergence of error with different µ. Note that the choice of step size is also related to SNR, which will be discussed later. (a)!44

45 (b) (c) Fig.C4 Comparison between results with different µ (all the other parameters are the same as C2). (iii) adaptive filer initialization In Fig.C2, the initialization is done by setting all the hh components (the initial impulse response of FIR adaptive filter) zero except the one in the middle set as 1. Other initializations are also used (hh all set to the same value a) and shown for comparison in Fig.C5. The results donate change much for this particular way of initialization. (a)!45

46 (b) (c) (d) Fig.C5 Comparison between results with different a (the initial value of all the hh component in FIR adaptive filter) (all the other parameters are the same as C2). (iv) SNR The influence of SNR values is shown in Fig.C6. The smaller the SNR, the larger the error after convergence. (SNR value is tuned by changing A without changing s[n].) All the other parameters are the same as those used in Fig.C2.!46

47 (a) (b) (c) (d)!47

48 (e) (f) (g) (h)!48

49 Fig.C6 Comparison between results with different SRN (different A) (all the other parameters are the same as C2). (v) channel impulse response response h Another h sequence with different L is also used to test the designed adaptive equalizer (h=[0.3, 0.2, 0.1, 0.8, 0.5, 0.6, 0.1]). The results (delay=10) are shown in Fig.C7. Compared with the results shown in Fig.C2, they are equally good in terms of the equalization. (a) (b) (c)!49

50 (d) (e) (f) Fig.C7 Results of adaptive equalizer filter with h=[0.3, 0.2, 0.1, 0.8, 0.5, 0.6, 0.1], training sequence of length ~ 5,000, A=100 (SNR~35dB), M=20, channel delay=10, step size µ=10-6.!50

51 Matlab codes in this section are attached below. %% Adaptive Equalizer with Training Sequence % Define sequence s N=5000; A=100; s=randi([0,1],1,n); for m=1:n s(m)=s(m)*a; if s(m)==0 s(m)=-a; end end % Define sequence h and s*h L=4; h=zeros(1,l+1); x=zeros(1,n); h(1)=0.3; h(2)=1; h(3)=0.7; h(4)=0.3; h(5)=0.2; for m=5:n x(m)=h(1)*s(m)+h(2)*s(m-1)+h(3)*s(m-2)+h(4)*s(m-3)+h(5)*s(m-4); end % Define sequence n (SNR~35dB) and x=s*h+n n=randn(1,n); x=x+n; % Initialize d=5; M=20; hh=zeros(1,m+1);!51

52 hh(m/2)=1; y=zeros(1,n); er=zeros(1,n); mu= ; % Main loop for m=100:n for mm=1:m+1 y(m)=y(m)+x(m+1-mm)*hh(mm); end er(m)=s(m-d)-y(m); for mm=1:m+1 hh(mm)=hh(mm)+2*mu*er(m)*x(m+1-mm); end end % Plot results figure(1); plot(s(n-100-d:n-d)); hold plot(y(n-100:n),'r-'); hold off figure(2); plot(er); figure(3); freqz(h,1); figure(4); freqz(hh,1); hf=conv(h,hh); figure(5); freqz(hf,1); figure(6); zplane(h);!52

53 figure(7); zplane(hh); figure(8); zplane(hf); figure(9); stem(h); figure(10); stem(hh); figure(11); stem(hf); C.2 Blind Equalization The scheme is very similar to Fig.C1 with difference in how to update coefficients: no training sequence is available so the error signal now becomes error[n]=(y[n]) 2 -A 2 and the stochastic gradient algorithm is gn+1=g n -error[n]y[n]xn. Also because no intentional delay is made as reference to update coefficients, the final delay is purely due to the FIR filters in this system (M/2 as results indicate). Fig.C8 shows an example of the results of designed blind equalizer for h=[0.3, 1, 0.7, 0.3, 0.2], training sequence of length ~ 100,000, A=100 (SNR~35dB), M=10, step size µ =10-11, where (a) shows output signal after convergence (red) almost identical to the original delayed signal (blue), (b) shows the error signal converges around 50,000 steps, (c-e) are the impulse response, frequency response, and pole-zero plot of channel h, (f-h) are those of converged FIR filter g, (i-k) are those of their cascade, i.e., the equivalent system response. (a)!53

54 (b) (c) (d)!54

55 (e) (f) (g)!55

56 (h) (i) (j)!56

57 (k) Fig.C8 Results of adaptive blind equalizer filter with h=[0.3, 1, 0.7, 0.3, 0.2], training sequence of length ~ 100,000, A=100 (SNR~35dB), M=10, step size µ= Other A values (or SNR) are also chosen for blind equalization. As shown in Fig.C9, as A decreases (SNR decreases), the equalization becomes worse and step size µ increases. All the other parameters are the same as Fig.C8. (a) (b)!57

58 (c) (d) Fig.C9 Results of adaptive blind equalizer filter with different A (SNR). Matlab codes in this section are attached below. %% Adaptive Blind Equalizer % Define sequence s clear; N=100000; A=100; s=randi([0,1],1,n); for m=1:n s(m)=s(m)*a; if s(m)==0 s(m)=-a; end end!58

59 % Define sequence h and s*h L=4; h=zeros(1,l+1); x=zeros(1,n); h(1)=0.3; h(2)=1; h(3)=0.7; h(4)=0.3; h(5)=0.2; for m=5:n x(m)=h(1)*s(m)+h(2)*s(m-1)+h(3)*s(m-2)+h(4)*s(m-3)+h(5)*s(m-4); end % Define sequence n (SNR~35dB) and x=s*h+n n=randn(1,n)/10; x=x+n; % Initialize M=10; g=zeros(1,m+1); g(m/2)=0.1; y=zeros(1,n); er=zeros(1,n); mu= ; % Main loop for m=100:n for mm=1:m+1 y(m)=y(m)+x(m+1-mm)*g(mm); end er(m)=y(m)*y(m)-a*a; for mm=1:m+1 g(mm)=g(mm)-mu*er(m)*y(m)*x(m+1-mm); end!59

60 end % Plot results figure(1); plot(s(n-100-m/2:n-m/2)); hold plot(y(n-100:n),'r-'); hold off figure(2); plot(er); figure(3); freqz(h,1); figure(4); freqz(g,1); hf=conv(h,g); figure(5); freqz(hf,1); figure(6); zplane(h); figure(7); zplane(g); figure(8); zplane(hf); figure(9); stem(h); figure(10); stem(g); figure(11); stem(hf); figure(12); plot(y);!60

61 C.3 4-level Input Amplitude in Blind Equalization 4-level input signal s[n] is selected randomly from {-3A, -A, A, 3A}. Using the same algorithm in C.2, the error signal for 4-level input is error[n]=(y[n]) 2 -ca 2 and the stochastic gradient algorithm is gn+1=g n -error[n]y[n]xn. While identical h=[0.3, 1, 0.7, 0.3, 0.2], training sequence of length ~ 100,000, A=100 (SNR~35dB), and M=10 are used, different c (1, 3, 5, and 10) is applied to test the equalization (µ values also changes correspondingly). Fig. C10 shows the converged output y[n] compared with delayed s[n], the frequency response and impulse response of the equivalent cascade of h and g filters. All of the converged output by different c cluster around 4 equally spaced values around 0 with different scaling factor, which decreases as c decreases. It is also reflected in the hf (=h*g) response: as c decreases, the flat top of its frequency response decreases from 0 db to nearly -10 db and the non-zero impulse response near M/2 decreases from 1.1 to (a) (b)!61

62 (c) (d) (e)!62

63 (f) (g)!63

64 (h) (i) (j)!64

65 (k) (l) Fig.C10 Results of 4-level amplitude {-3, -1, 1, 3} in adaptive blind equalizer filter with different c values while h=[0.3, 1, 0.7, 0.3, 0.2], training sequence of length ~ 100,000, A=100 (SNR~35dB), and M=10 are fixed. Matlab codes in this section are attached below. %% 4-level Amplitude Input in Adaptive Blind Equalizer % Define sequence s clear; N=100000; A=100; s=randi([0,1],1,n); for m=1:n s(m)=s(m)*a; if s(m)==0 s(m)=-a; end!65

66 if s(m)==2*a s(m)=-3*a; end end % Define sequence h and s*h L=4; h=zeros(1,l+1); x=zeros(1,n); h(1)=0.3; h(2)=1; h(3)=0.7; h(4)=0.3; h(5)=0.2; for m=5:n x(m)=h(1)*s(m)+h(2)*s(m-1)+h(3)*s(m-2)+h(4)*s(m-3)+h(5)*s(m-4); end % Define sequence n (SNR~35dB) and x=s*h+n n=randn(1,n)/10; x=x+n; % Initialize M=10; g=zeros(1,m+1); g(m/2)=0.1; y=zeros(1,n); er=zeros(1,n); mu= ; % Main loop for m=100:n for mm=1:m+1 y(m)=y(m)+x(m+1-mm)*g(mm); end!66

67 er(m)=y(m)*y(m)-5*a*a; for mm=1:m+1 g(mm)=g(mm)-mu*er(m)*y(m)*x(m+1-mm); end end % Plot results figure(1); plot(s(n-100-m/2:n-m/2)); hold plot(y(n-100:n),'r-'); hold off figure(2); plot(er); figure(3); freqz(h,1); figure(4); freqz(g,1); hf=conv(h,g); figure(5); freqz(hf,1); figure(6); zplane(h); figure(7); zplane(g); figure(8); zplane(hf); figure(9); stem(h); figure(10); stem(g); figure(11);!67

68 stem(hf); figure(12); plot(y);!68