UploadFile_6417

478 Chapter 9 SAMPLING RATE CONVERSION
FIGURE 9.3
x(n)
↓ D
Rate F x
A downsampling element
y(m)
Rate F y = F x
D
previously. In fact, we can show that a system containing a downsampling
element is shift varying. However, this fact does not prohibit the
frequency-domain analysis of y(m) interms of x(n) asweshall see later.
□ EXAMPLE 9.1 Using D =2and x(n) ={1, 2, 3, 4, 3, 2, 1} verify that the downsampler is time
↑
varying.
Solution The downsampled signal is y(m) ={1, 3, 3, 1}. Ifwenow delay x(n) byone
↑
, 1, 2, 3, 4, 3, 2, 1}. The corresponding downsampled
sample, we get x(n − 1) = {0
↑
signal is y 1(m) ={0
↑
, 2, 4, 2}, which is different from y(m − 1). □
MATLAB Implementation MATLAB provides the function [y] =
downsample(x,D) that downsamples input array x into output array y
by keeping every D-th sample starting with the first sample. An optional
third parameter “phase” specifies the sample offset which must be an
integer between 0 and (D-1). For example,
>> x = [1,2,3,4,3,2,1]; y = downsample(x,2)
y =
1 3 3 1
downsamples by a factor of 2 starting with the first sample. However,
>> x = [1,2,3,4,3,2,1]; y = downsample(x,2,1)
y =
2 4 2
produces an entirely different sequence by downsampling, starting with
the second sample (i.e., offset by 1).
The frequency-domain representation of the downsampled signal
y(m) We now express Y (ω) interms of X(ω) using z-transform
relations. Toward this we introduce a high-rate sequence ¯x(n), which is
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.