Question

.) (f,c) If rx,Y[k] CPSD. You will need a computer to do this. 19.10( ?[k] + 2?[A-11, plot the magnitude and phase of the

Answer 1

If r_X,Y[k] is the cross correlation function of two real jointly WSS random processes X[k] and Y[k], then the cross power spectral density(CPSD) of X[k] and Y[k] is given by the discrete time Fourier transform of r_X,Y[k].

$\small r_{X,Y}[k]\xrightarrow{DTFT} R(j\omega) = \sum_{k = -\infty}^{\infty }\left(\delta[k]+2\delta[k-1] \right )e^{-j\omega k}$

$\small \Rightarrow R(j\omega) = \delta[k]e^{-j\omega k} |_{k=0} + 2 \delta[k-1]e^{-j\omega k} |_{k=1}$

                  $\small = 1 + 2 e^{-j\omega}$

$\small R(j\omega) = 1+ 2cos\omega -j2sin\omega .......(1)\: \; [\because \: e^{-j \theta} = cos\theta + jsin \theta \; \forall \; \theta]$

Magnitude: $\small \left |R(j\omega) \right |= \sqrt{(1+ 2cos\omega)^2 +(2sin\omega)^2 }$

                                  $\small = \sqrt{1+4cos\omega+4cos^2\omega +4sin^2\omega }$

                                  $\small = \sqrt{5+4cos\omega } \: \: [\because \; \; cos^2\theta+sin^2\theta = 1 \; \forall \; \theta]$

Phase :

$\small \angle R(j\omega) = tan^{-1}\left( \frac{-2sin\omega}{1+ 2cos\omega }\right)\: \; \; [from\;\; equation \: (1)]$

To plot this using MATLAB, we use the following piece of code:

%To plot angular frequency vs. magnitude and phase responce

w = [-10000;0.01;10000];

Mag = sqrt(5+4*cos(w));

Phase = atan(-2*sin(w)/(1+2*cos(w)));

%To plot angular frequency in logarithmic axis vs. Magnitude in dB and phase in degree

w1=[0.001;0.01;10000]; %we take initial value of w>0 in order to compute logarithm

logw = log10(w1);

MagdB = 20*log10(sqrt(5+4*cos(w1)));

PhaseDegree = (180/pi)*atan(-2*sin(w1)/(1+2*cos(w1)));

figure;subplot(2,2,1);plot(w,Mag);title('Magnitude plot: w vs |R(jw)|')

subplot(2,2,3);plot(w,Phase);title('Phase plot: w vs <R(jw)')

subplot(2,2,2);plot(logw,MagdB);title('Logarithmic Magnitude plot: log10(w) vs 20log10|R(jw)|')

subplot(2,2,4);plot(logw,PhaseDegree);title('Phase plot: log10(w) vs <R(jw) in degrees')