Signals and Systems - Electrical Engineering

Problems 635

10.18. Power spectral density
Consider an autocorrelation function

c[n]=0.5|n| −∞<n<∞

(a) Find the magnitude square of the DTFTC(ejω)ofc[n], which is called the power spectral density.

(b)Find the Z-transform ofc[n]and determine where its poles and zeros are. Are there any zeros or poles

on the unit circle?

(c) FindC(ej^0 )—that is, the dc value of the power spectral density. Determine the phase ofC(ejω)—is

it linear?

10.19. Convolution sum and product of polynomials
The convolution sum can be seen as a way to compute the coefficients of the product of polynomials.
This is because

[x∗y][n]⇔X(z)Y(z)⇔X(ejω)Y(ejω)

(a) LetX(z)= 1 + 2 z−^1 + 3 z−^2 andY(z)=z−^2 + 4 z−^3 ifx[n]= 1 δ[n]+ 2 δ[n−1]+ 3 δ[n−2]and

y[n]= 1 δ[n−2]+ 4 δ[n−3]are sequences formed by the coefficients of the polynomials. Compute

the convolution sum[x∗y][n]and compare it to the coefficients of the polynomialZ(z)=X(z)Y(z),

orZ(ejω)=X(ejω)Y(ejω).

(b)Suppose that the transfer function of a discrete-time system is

H(z)=

W(z)

V(z)

= 3 z^2 + 2 z+ 2 z−^1 + 3 z−^2

and that it is known that the input isv[n]=u[n]−u[n−3]. Use the connection between the product

of the polynomials and the convolution sum to find the output w[n]of the system.

10.20. Windowing and DTFT—MATLAB
A window w[n]is used to consider the part of a signal we are interested in.
(a) Let w[n]=u[n]−u[n−20]be a rectangular window of length 20. Letx[n]=sin(0.1πn). We are
interested in a period of the infinite lengthx[n], ory[n]=x[n]w[n]. Compute the DTFT ofy[n]and
compare it with the DTFT ofx[n]. Write a MATLAB script to computeY(ejω).
(b)Let w 1 [n]=( 1 +cos( 2 πn/ 11 ))(u[n+5]−un[n−5])be a raised-cosine window that is symmetric
with respect ton= 0 (noncausal). Adapt the script in the previous part to find the DTFT of

z[n]=x[n]w 1 [n]

wherex[n]is the sinusoid given above.

10.21. Z-transform and Fourier series—MATLAB
Let

x 1 [n]=0.5n 0 ≤n≤ 9

be a period of a periodic signalx[n].

(a) Use the Z-transform to compute the Fourier series coefficients.

(b)Use MATLAB to plot the magnitude and the phase line spectrum (i.e.,|Xk|and∠Xkversus frequency

−π≤ω≤π).