Signals and Systems - Electrical Engineering

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

with the wholez-plane (except for the origin) as its region of convergence. Thus, the DTFT of

x[n] is

X(ejω)=e−j(

(^32) ω)[
ej(
(^32) ω)
+ej(
(^12) ω)
+e−j(
(^12) ω)
+e−j(
(^32) ω)]
= 2 e−j(
(^32) ω)

[

cos

(ω

2

)

+cos

(

3 ω

2

)]

The Z-transform of the downsampled signal (M=2) is

Xd(z)= 1 +z−^1

and the DTFT ofxd[n] is

Xd(ejω)=e−j(

(^12) ω)[
ej(
(^12) ω)
+e−j(
(^12) ω)]
= 2 e−j(
1
2 ω)cos
(ω
2

)

Clearly, this is not equal to 0.5X(ejω/^2 ). This is caused by aliasing: The maximum frequency ofx[n]

is notπ/M=π/2 and soXd(ejω)is the sum of superposed and shiftedX(ejω).

Suppose we passx[n] through an ideal low-pass filterH(ejω)with cut-off frequencyπ/2. Its output

would be a signalx 1 [n] with a maximum frequency ofπ/2, and downsampling it withM= 2

would give a signal with a DTFT of 0.5X 1 (ejω/^2 ). n

nExample 10.6

Discuss the effects of downsampling a discrete signal that is not band limited versus the case of

one that is. Consider a unit rectangular pulse of lengthN=10. Downsample it by a factor of

M=2, and compute and compare the DTFTs of the pulse and its downsampled version. Do a

similar procedure to a sinusoid of discrete frequencyπ/4 and comment on the results. Explain the

difference between the above two cases. Use the MATLAB functiondecimate(low-pass filtering is

used to avoid aliasing followed by downsampling) to perform similar operations and comment on

the differences with downsampling. Use the MATLAB functioninterpto interpolate (upsampling

with smoothing by a low-pass filter) the downsampled signals. See Figure 10.3 for illustrations of

downsampler and upsampler and decimator and interpolator.

Solution

As indicated, when we downsample a discrete-time signalx[n] by a factor ofM, in order not to

have aliasing in frequency the signal must be band limited toπ/M. If the signal satisfies this

condition, the spectrum of the downsampled signal is an expanded version of the spectrum of

x[n]. To illustrate this in the following script we downsample by a factor ofM=2 first a signal that

is not band limited toπ/2, and then another that is.