UploadFile_6417

Linear Convolution Using the DFT 183
Solution
Clearly, the linear convolution x 3(n) isstill the same.
x 3(n) ={1, 1, −1, −2, −1, 1, 1}
When N =6,weobtain a 6-point sequence.
x 4(n) =x 1(n) 6○ x 2(n) ={2, 1, −1, −2, −1, 1}
Therefore
e(n) ={2, 1, −1, −2, −1, 1}−{1, 1, −1, −2, −1, 1} , 0 ≤ n ≤ 5
= {1, 0, 0, 0, 0, 0}
= x 3(n +6)
as expected. When N =5,weobtain a 5-point sequence,
x 4(n) =x 1(n) 5○ x 2(n) ={2, 2, −1, −2, −1}
and
e(n) ={2, 2, −1, −2, −1}−{1, 1, −1, −2, −1} , 0 ≤ n ≤ 4
= {1, 1, 0, 0, 0}
= x 3(n +5)
Finally, when N =4,weobtain a 4-point sequence,
x 4(n) =x 1(n) 4○ x 2(n) ={0, 2, 0, −2}
and
e(n) ={0, 2, 0, −2}−{1, 1, −1, −2} , 0 ≤ n ≤ 3
= {−1, 1, 1, 0}
= x 3(n +4)
The last case of N =4also provides the following useful observation.
Observation: When N = max(N 1,N 2)ischosen for circular convolution, then
the first (M − 1) samples are in error (i.e., different from the linear convolution),
where M = min(N 1,N 2). This result is useful in implementing long convolutions
in the form of block processing.
□
5.5.2 BLOCK CONVOLUTIONS
When we want to filter an input sequence that is being received continuously,
such as a speech signal from a microphone, then for practical
purposes we can think of this sequence as an infinite-length sequence. If
we want to implement this filtering operation as an FIR filter in which
the linear convolution is computed using the DFT, then we experience
some practical problems. We will have to compute a large DFT, which is
generally impractical. Furthermore, output samples are not available until
all input samples are processed. This introduces an unacceptably large
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.