Signals and Systems - Electrical Engineering

632 C H A P T E R 10: Fourier Analysis of Discrete-Time Signals and Systems

10.6. Chirps for jamming—MATLAB

A chirp signal is a sinusoid of continuously changing frequency. Chirps are frequently used to jam

communication transmissions. Consider the chirp

x[n]=cos(θn^2 )u[n] θ=

π

2 L

0 ≤n≤L− 1

(a) A measure of the frequency of the chirp is the so-called instantaneous frequency, which is defined

as the derivative of the phase in the cosine—that is,

IF(n)=

dθn^2

dn

Find the instantaneous frequency of the given chirp. Use MATLAB to plotx[n]forL= 256.

(b) LetL= 256 and use MATLAB to compute the DTFT ofx[n]and to plot its magnitude. Indicate the

range of discrete frequencies that would be jammed by the given chirp.

10.7. Time specifications for FIR filters—MATLAB

When designing discrete filters the specifications can be given in the time domain. One can think of

converting the frequency-domain specifications into the time domain. Assume you wish to obtain a filter

that approximates an ideal low-pass filter with a cut-off frequencyωc=π/ 2 and that has a linear phase

−Nω. Thus, the frequency response is

H(ejω)=

{

1 e−jNω −π/ 2 ≤ω≤π/ 2

0 −π≤ω < π/ 2 andπ/ 2 < ω≤π

(a) Find the corresponding impulse response using the inverse DTFT ofH(ejω).

(b) IfN= 50 , ploth[n]using the MATLAB functionstemfor 0 ≤n≤ 100. Comment on the shape of the

plot.

(c) Suppose we want a band-pass filter of center frequencyω 0 =π/ 2. Use the above impulse response

h[n]to obtain the impulse response of the desired band-pass filter.

10.8. Z-transform and DTFT—MATLAB

Consider a discrete pulsep[n]=u[n]−u[n−N].

(a) Use the definition of the DTFT to determineP(ejω)and then use the Z-transformP(z)ofp[n]to verify

your result.

(b) ForN= 5 , 10 , and 20 , use the MATLAB functionzplaneto find the zeros ofP(z)and indicate at what

frequenciesP(ejω)is zero. Verify your response usingfreqz.

(c) Suppose the impulse response of a filter ish[n]=u[n]−u[n−4], and its input is

v[n]=

∑^2

k= 1

cos(kω 0 n)

For what value ofω 0 is the steady-state response of the filter zero?

10.9. Downsampling and DTFT—MATLAB

Consider pulses x 1 [n]=u[n]−u[n−20] and x 2 [n]=u[n]−u[n−10], and their product x[n]=

x 1 [n]x 2 [n].

(a) Plot the three pulses. Could you say thatx[n]is a downsampled version ofx 1 [n]? What would be the

downsampling rate? FindX 1 (ejω).

(b) Find directly the DTFT ofx[n]and compare it toX 1 (ejω/M)whereMis the downsampling rate found

above. If we downsamplex 1 [n]to getx[n], would the result be affected by aliasing? Use MATLAB

to plot the magnitude DTFT ofx 1 [n]andx[n]to provide an answer.