Solutions for examination in TSRT78 Digital Signal Processing,

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1 Solutions for examination in TSRT78 Digital Signal Processing, s(t) is generated by s(t) = 1 w(t), q 1 Var(w(t)) = σ 2 w = 2. It is measured as y(t) = s(t) + n(t) where n(t) is white noise with Var(n(t)) = σ 2 n = 1. The noise processes are independent. a) 1 2 ( Φ ss (z) = σ 2 1 )( 1 ) z 1 w = z 1 = z (z + z 1 ) Since w(t) and e(t) are independent and white, it holds that Φ yy (z) = Φ ss (z) + Φ ee (z) = Φ ss (z) + σ 2 n = (z + z 1 ) (z + z 1 ). Furthermore, Φ sy (z) = Φ ss (z) follows from the independence of w(t) and e(t). The noncausal Wiener filter is given by H(z) = Φ sy(z) Φ yy (z) = (z + z 1 ) = 6.667z z z + 1 = 6.667z (z 10.20)(z 0.098) A separation into a causal and a anti-causal part is given below. b) Var(s(t) ŝ(t)) = R ss (0) i h(i)r sy(i) where R sy (i) = R ss (i). We begin with the covariance function R ss (k) = E [s(t)s(t k)]. For k = 0 and k = 1 we obtain R ss (0) = E [s(t)( 0.3s(t 1) + w(t))] = 0.3R ss (1) + 2, R ss (1) = E [( 0.3s(t 1) + w(t))s(t 1)] = 0.3R ss (0). Elimination of R ss (1) in the above equations yields R ss (0) = For general k > 0 we have R ss (k) = E [s(t + k)s(t)] = E [( 0.3s(t + k 1) + w(t + k))s(t)] = 0.3R ss (k 1) =... = 0.3 k R ss (0) and R ss ( k) = R ss (k) by symmetry. From this we can conclude that R ss (k) = ( 0.3) k. After decomposing H(z) into partial fractions we obtain H(z) = z z z z ( ) 0.098z = z z ( 1 ) = ( 0.098z) k + ( 0.098z 1 ) k = k=1 1 k= where we have used the geometric series results k=0 ( 0.098) k z k ( 0.098) k z k k=0 ( r) k = r k=0 and ( r) k = r 1 + r k=1 for r < 1. From the definition of z-transform, H(z) = k= h(k)z k, we get h(k) = ( 0.098) k. The error variance is then 1 Ver: 2014/05/06

2 Var(s(t) ŝ(t)) = R ss (0) k h(k)r sy (k) = k= = = ( ( ( 0.098) k )( ( 0.3) k ) k= ( ) k ( ) ( ) = = a) The zero-padded DTFT can be plotted by the following MATLAB code: % Generate the signal N = 128; x = 0:1:(N-1); A = 0.03; y = sin( 2 * pi * 0.29 * x ) + A * sin( 2 * pi * 0.25 * x); y1 = sin( 2 * pi * 0.29 * x ); y2 = A * sin( 2 * pi * 0.25 * x); %% Task 2a % Compute the zero-padded DFT and create the frequency grid yzeropadded = [y zeros(1,7*n)]; Yzeropadded = fft(yzeropadded); yzeropadded1 = [y1 zeros(1,7*n)]; Yzeropadded1 = fft(yzeropadded1); yzeropadded2 = [y2 zeros(1,7*n)]; Yzeropadded2 = fft(yzeropadded2); grid = 2 * pi / ( 8 * N * 1) * (0:1:(8*N-1) ); % Plot the DFT figure(1) subplot(3,2,[1 2]) semilogy(grid,abs(yzeropadded), k, LineWidth,2) set(gca, xlim,[2*pi*0.2 2*pi*0.35]) set(gca, ylim,[ ]) xlabel( Frequency (rad/s) ); ylabel( DTFT of signal ); hold on; plot(2*pi*0.29*[1 1],[ ], g, LineWidth,2) plot(2*pi*0.25*[1 1],[ ], g, LineWidth,2) subplot(3,2,3) semilogy(grid,abs(yzeropadded1), k, LineWidth,2) set(gca, xlim,[2*pi*0.2 2*pi*0.35]) set(gca, ylim,[ ]) xlabel( Frequency (rad/s) ); ylabel( DTFT of signal 1 ); hold on; plot(2*pi*0.29*[1 1],[ ], g, LineWidth,2) plot(2*pi*0.25*[1 1],[ ], g, LineWidth,2) subplot(3,2,4) semilogy(grid,abs(yzeropadded2), k, LineWidth,2) set(gca, xlim,[2*pi*0.2 2*pi*0.35]) set(gca, ylim,[ ]) xlabel( Frequency (rad/s) ); ylabel( DTFT of signal 2 ); ) ( ) k k=1 2 Ver: 2014/05/06

3 hold on; plot(2*pi*0.29*[1 1],[ ], g, LineWidth,2) plot(2*pi*0.25*[1 1],[ ], g, LineWidth,2) Running this code with A = 0.05 results in Figure 1, where we see that the peak of the weaker signal barely sticks out from the side lobes of the main peak. The limit of A to give a lobe that is detectable in the DTFT is around If the value is smaller than that it is difficult to distinguish from the side lobes of the main frequency. b) The difficulty in this case is mainly due to the leakage from the main lobe in the DTFT approximation. This can be solved to some extent by adding a window to the signal before computing the DTFT. c) We apply a Hanning window to the signal, zero-pad it and compute the DTFT using the following Matlab code: %% Task 2c % Windowing w = hanning(n); yw = w(:).*y(:) ; % Zero-pad and Approximate DTFT yzeropaddedw = [yw zeros(1,7*n)]; YzeropaddedW = fft(yzeropaddedw); subplot(3,2,[5 6]) semilogy(grid,abs(yzeropaddedw), k, LineWidth,2) set(gca, xlim,[2*pi*0.2 2*pi*0.35]) set(gca, ylim,[ ]) xlabel( Frequency (rad/s) ); ylabel( DTFT of signal ); hold on; plot(2*pi*0.29*[1 1],[ ], g, LineWidth,2) plot(2*pi*0.25*[1 1],[ ], g, LineWidth,2) In the lower part of Figure 1, we see that the windowing helps and both frequencies can now be seen in the DTFT. Here, we have also used A = 0.05 but it is possible to detect the small peak in the DTFT even if A = Therefore, we conclude that windowing helps and improves the frequency separation somewhat. 3. We start with the residual perspective and compute the loss function of the AR model at different model orders p on validation data. That is, we fit the model using the first 2/3 of the data and compute the prediction error on the last 1/3 of the data. This can be done in Matlab by: %% Generate the data N = 1000; w = randn(1,n); e = randn(1,n); y = filter([1 0.5],[1,-1,0.09],w) + filter([1 0.9],[ ],e); figure(1) subplot(3,2,[1 2]) plot(y); ylabel( signal ); xlabel( time ); %% Model order selection by residual analysis % Divide the data into estimation and validation sets yest = y(1:667); yval = y(668:1000); % Fit the AR(p) for different values of p. Make predictions of the % validation data and compute the prediction error variance resvar = zeros(20,1); for pp = 1:20 m = ar(yest,pp); yhat = predict(m,yval,1); resvar(pp) = mean( (yhat-yval ).^2 ); end subplot(3,2,[3 4]) 3 Ver: 2014/05/06

4 Figur 1: The DTFT of the signal with two sinusoids (upper), the two components (middle) and the windowed signal (lower). The green lines indicate the frequencies of the sinusoids. 4 Ver: 2014/05/06

5 % Plot the prediction error variance. Model order 5 seems nice! plot(1:20,resvar); axis([ ]); ylabel( Prediction error variance ); xlabel( model order (p) ); % Check whiteness of residuals m = ar(yest,5); yhat = predict(m,yval,1); acfresid = xcorr(yhat-yval,25); acfresid = acfresid(26:end); acfresid = acfresid./acfresid(1); subplot(3,2,5); plot(acfresid); % seems uncorrelated hold on; plot([0 30],2/sqrt(N)*[1 1], k: ); plot([0 30],-2/sqrt(N)*[1 1], k: ); xlabel( lag ); ylabel( autocorrelation of residuals ); subplot(3,2,6); qqplot(yhat); % look likes Gaussian (no deviation in the tail behaviour) In Figure 2, we present the data and the corresponding loss function for different values of p. We see that it has a knee point at around p = 5. We check the prediction residuals at this point and conclude that they are uncorrelated and Gaussian. Note that, in a QQ-plot the data (+) should not deviate to much from the theoretical quantiles (red dashed line) especially in the tails. Consequently, we have validated the model assumptions. The model order selection by frequency analysis can be done by the following Matlab code: figure(2) gam = 50; G = etfe(y,gam); [Per,wp]=spectrum(G); Per = squeeze(per); loglog(wp,per, k, LineWidth,2); [mag4,phase,w4] = bode(ar(y,4)); [mag6,phase,w6] = bode(ar(y,6)); [mag8,phase,w8] = bode(ar(y,8)); [mag10,phase,w10] = bode(ar(y,10)); hold on; plot(w4,squeeze(mag4(:)), r, LineWidth,2); plot(w6,squeeze(mag6(:)), b, LineWidth,2); plot(w8,squeeze(mag8(:)), c, LineWidth,2); plot(w10,squeeze(mag10(:)), g, LineWidth,2); legend({ B.-T., AR(4), AR(6), AR(8), AR(10) }, fontsize,16, location, best ) where we try some different model orders. The resulting output is presented in Figure 3, where it is quite difficult to see if any of the models are a better match than the other. As there are a number of peaks in the spectrum of the data, we require at least an AR(8) to cover the first four. Therefore, we conclude that an AR(8) or AR(10) is probably a good choice for this data. 4. a) We select a constant velocity model for this application (see p. 203 in the course book). In two dimensions the state vector is given by x = (p x, p y, v x, v y ) where p and v denote the position and the velocity in each dimension, respectively. The corresponding state space form is given by 1 0 T 0 T 2 / T 0 T 2 /2 x t+1 = x t ( ) y t = x t + e t, T 0 0 T v t, where T denotes the sampling time. Here, we assume that the noise sources are Gaussian, i.e. v t N (0, σ 2 v) and e t N (0, σ 2 e). For this application, we have T = 1 according to the data. b) We make use of a Kalman filter to estimate the position and velocity of the rescue leader given the constant velocity model. This can be done using the following Matlab code: 5 Ver: 2014/05/06

6 Figur 2: The generated data (upper), the loss function at different model orders (middle), the ACF (lower left) and the normal QQ-plot (lower right) of the prediction errors. 6 Ver: 2014/05/06

7 %% Exercise 4 load Apr14; % Noise variances and uncertainty in initial position Q0=0.1; R0=1; P0=[1 1 1e6 1e6]; T=1; % Setup the motion model and load the data N=length(Apr14Meas(:,2)); A=[1 0 T 0; T; ; ]; B=[T^2/2 0; 0 T^2/2; T 0; 0 T]; C=[ ; ]; Q=B*eye(2)*B *Q0; R=eye(2)*R0; P=diag(P0); % Set the initial conditions (zero velocity) xhat=zeros(n,4); yhat=zeros(n,2); xhat(1,:)=[apr14meas(1,1) Apr14Meas(1,2) 0 0]; % Kalman filter for k=2:n % Update Kalman Gain K=P*C /(C*P*C +R); % Eq % Estimate filtered state xhat(k,:)=(a-k*c*a)*xhat(k-1,:) +K*[Apr14Meas(k,1) Apr14Meas(k,2)] ; % Eq yhat(k,:)=c*xhat(k,:) ; % Update covariance estimate P=P-P*C /(C*P*C +R)*C*P; % Eq. 8.17e P=A*P*A +Q; % Eq. 8.17b end % Plot the results figure(4) subplot(2,2,[1 2]); plot(apr14meas(:,1),apr14meas(:,2), k ); hold on; plot(yhat(2:end,1),yhat(2:end,2), k-*, LineWidth,2); title( measured and filtered rescue leader position ); xlabel( x ); ylabel( y ); legend( measured, filtered ); subplot(2,2,3); plot(xhat(1:end,3), k, LineWidth,2); xlabel( time ); ylabel( est. velocity in x-direction ); subplot(2,2,4); plot(xhat(1:end,3), k, LineWidth,2); xlabel( time ); ylabel( est. velocity in y-direction ); The results obtained by running this code is presented in Figure 4. The Kalman filter manages to track the rescue leader s position quite well and gives reasonable estimates of the velocities in each direction. 7 Ver: 2014/05/06

8 Figur 3: The estimated spectrum from the data (black line) and for some different AR models (coloured lines). 8 Ver: 2014/05/06

9 Figur 4: The estimated and filtered position of the rescue leader (upper) and the estimated velocity in each direction (lower). 9 Ver: 2014/05/06