conv

Purpose

Convolution and polynomial multiplication.

Synopsis

c = conv(a,b)

Description

c = conv(a,b) convolves vectors a and b. Convolution can be thought of as multiplying the polynomials whose coefficients are the elements of a and b since, algebraically, this is the same operation.

Let m = length(a) and n = length(b). Then c is the vector of length m+n-1 whose k-th element is

The sum is over all the values of j which lead to legal subscripts for a(j) and b(k+1-j), specifically j = max(1,k+1-n): min(k,m). When m = n, this gives

c(1) = a(1)*b(1)
c(2) = a(1)*b(2)+a(2)*b(1)
c(3) = a(1)*b(3)+a(2)*b(2)+a(3)*b(1)
...
c(n) = a(1)*b(n)+a(2)*b(n-1)+ ... +a(n)*b(1)
...
c(2*n-1) = a(n)*b(n)

Algorithm

The convolution theorem says, roughly, that convolving two sequences is the same as multiplying their Fourier transforms. In order to make this precise, it is necessary to pad the two vectors with zeros and ignore roundoff error. Thus, if 

X = fft([x zeros(1,length(y)-1)])
and

Y = fft([y zeros(1,length(x)-1)])
then

conv(x,y) = ifft(X.*Y)

Examples

Let a = [1 2 3 4] and b = [10 20 30], then c = conv(a,b) is

c =
    10    40    100    160    170    120
The vectors a and b represent the polynomials

Then the product c = conv(a,b) is the polynomial

See Also

deconv, residue, filter

convmtx, xconv2 in the Signal Processing Toolbox

(c) Copyright 1994 by The MathWorks, Inc.