Signals and Systems - Electrical Engineering

Problems 631

(a) From the eigenfunction property find the frequency response of the two filters atω= 0 ,π/ 2 , andπ

radians. Use the MATLAB functionsfreqzandabsto compute the magnitude responses of the two

filters. Plot them to verify that the filters are low pass and high pass.

(b)CallH 1 (ejω)the frequency response of the first filter andH 2 (ejω)the frequency response of the

second filter. Show that

H 2 (ejω)=H 1 (ej(π−ω))

and relate the impulse responseh 2 [n]toh 1 [n].

(c) Use the MATLAB functionzplaneto find and plot the poles and the zeros of the filters and determine

the relation between the poles and the zeros of the two filters.

10.3. Computations from definition of DTFT and IDTFT
Consider the discrete-time signalx[n]=0.5|n|, and find its DTFTX(ejω). From the direct and the inverse
DTFT ofx[n]:
(a) Determine the infinite sum
∑∞

k=−∞

0.5|n|

(b)Find the integral

∫π

−π

X(ejω)dω

(c) Find the phase ofX(ejω).

(d)Determine the sum

∑∞

k=−∞

(− 1 )n0.5|n|

10.4. Frequency shift of FIR filters—MATLAB
Consider a moving-average FIR filter with an impulse response

h[n]=

1

3

(δ[n]+δ[n−1]+δ[n−2])

LetH(z)be the Z-transform ofh[n].

(a) Find the frequency responseH(ejω)of the FIR filter.

(b)Let the impulse response of a new filter be given by

h 1 [n]=h[n]ejπn

Use the eigenfunction property to find the frequency responseH 1 (ejω)of the new FIR filter.

(c) Use the MATLAB functionsfreqzandabsto compute the magnitude response of the two filters. Plot

them and determine the location of the poles and the zeros of the two filters. What type of filters are

these?

10.5. Duality of DTFT
The DTFT of a discrete-time signalx[n]is given as

X(ejω)=ejπ/^4 δ(ω−0.5π)+e−jπ/^4 δ(ω+0.5π)− 2 πe−jπ/^8 δ(ω−0.71)− 2 πejπ/^8 δ(ω+0.71)

(a) Is the signalx[n]periodic? If so, indicate its period.

(b)Determine the signalx[n], and verify your answer above.