Signals and Systems - Electrical Engineering

10.2 Discrete-Time Fourier Transform 583

nExample 10.4

Consider the frequency response of an ideal low-pass filter,

H(ejω)=

{

1 −π/ 2 ≤ω≤π/ 2

0 −π≤ω <−π/2 and π/ 2 < ω≤π

which is the DTFT of an impulse responseh[n]. Determineh[n]. Suppose that we downsam-

pleh[n] with a factor ofM=2. Find the downsampled impulse responsehd[n]=h[2n] and its

corresponding frequency responseHd(ejω).

Solution

The impulse responseh[n] corresponding to the ideal low-pass filter is found to be

h[n]=

1

2 π

π/∫ 2

−π/ 2

ejωndω=

{

0.5 n= 0

sin(πn/ 2 )/(πn) n6= 0

The downsampled impulse response is given by

hd[n]=h[2n]=

{

0.5 n= 0

sin(πn)/( 2 πn)= 0 n6= 0

orhd[n]=0.5δ[n], with a DTFT ofHd(ejω)=0.5 for−π < ω≤π(i.e., an all-pass filter). This

agrees with the downsampling theory, which gives that

Hd(ejω)=

1

2

H(ejω/^2 )=

1

2

, −π≤ω < π

That is,H(ejω)multiplied by 1/M= 1 /2 and expanded byM=2. n

nExample 10.5

A discrete pulse is given byx[n]=u[n]−u[n−4]. Suppose we downsamplex[n] by a factor of

M=2, so that the length 4 of the original signal is reduced to 2, giving

xd[n]=x[2n]=u[2n]−u[2n−4]=u[n]−u[n−2]

Find the corresponding DTFTs forx[n] andxd[n], and determine how they are related.

Solution

The Z-transform ofx[n] is

X(z)= 1 +z−^1 +z−^2 +z−^3