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208 Chapter 5 THE DISCRETE FOURIER TRANSFORM
P5.32 The overlap-add method of block convolution is an alternative to the overlap-save method.
Let x(n) bealong sequence of length ML where M,L ≫ 1. Divide x(n) intoM segments
{x m(n), m=1,...,M} each of length L
{
x(n), mL≤ n ≤ (m +1)L − 1
x m(n) =
0, elsewhere
Let h(n) beanL-point impulse response. Then
so that x(n) =
M−1
∑
m=0
x m(n)
y(n) =x(n) ∗ h(n) =
M−1
∑
m=0
x m(n) ∗ h(n) =
M−1
∑
m=0
y m(n);
y m(n) △ = x m(n) ∗ h(n)
Clearly, y m(n) isa(2L − 1)-point sequence. In this method we have to save the
intermediate convolution results and then properly overlap these before adding to form the
final result y(n). To use DFT for this operation we have to choose N ≥ (2L − 1).
1. Develop a MATLAB function to implement the overlap-add method using the circular
convolution operation. The format should be
function [y] = ovrlpadd(x,h,N)
% Overlap-Add method of block convolution
% [y] = ovrlpadd(x,h,N)
%
% y = output sequence
% x = input sequence
% h = impulse response
% N = block length >= 2*length(h)-1
2. Incorporate the radix-2 FFT implementation in this function to obtain a high-speed
overlap-add block convolution routine. Remember to choose N =2 ν .
3. Verify your functions on the following two sequences
P5.33 Given the following sequences x 1(n) and x 2(n):
P5.34 Let
x(n) =cos (πn/500) R 4000(n), h(n) ={1, −1, 1, −1}
x 1(n) ={2, 1, 1, 2} , x 2(n) ={1, −1, −1, 1}
1. Compute the circular convolution x 1(n) N○ x 2(n) for N =4,7,and 8.
2. Compute the linear convolution x 1(n) ∗ x 2(n).
3. Using results of calculations, determine the minimum value of N necessary so that linear
and circular convolutions are same on the N-point interval.
4. Without performing the actual convolutions, explain how you could have obtained the
result of P5.33.3.
x(n) =
{
A cos (2πln/N), 0 ≤ n ≤ N − 1
0, elsewhere
= A cos (2πln/N) R N (n)
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