convolution takes two arguments, a real vector v of length n and a real vector w of length m , and returns their convolution z=v∗w which is the vector of length N=n+m−1 defined as :
|vi wk−i, k=0,1,…,N−1,|
such that vj=0 for j≥ n and wj=0 for j≥ m . The two arguments may also be real functions f(x) and g(x) , with the variable x as an optional third argument, in which case the integral
|f(t) g(x−t) dt|
is computed. It is assumed that f and g are causal functions, i.e. f(x)=g(x)=0 for x<0 . Therefore both f and g are multiplied by Heaviside function prior to integration.
For example, input :
To compute the convolution of f(x)=25 e2 x u(x) and g(x)=x e−3 x u(x) , where u is the Heaviside function, input :
To compute the convolution of f(t)=ln(1+t) u(t) and g(t)=1/√t , input :
In the following example convolution is used for reverberation. Assume that the directory sounds contains two files, a dry, mono recording of a guitar stored in guitar.wav and a two-channel impulse response recorded in a French 18th century salon and stored in salon-ir.wav. Files are loaded with the following command lines :
Convolving the data from clip with both channels in ir produces a reverberated variant of the recording, in stereo. Input :
The convolved signals L and R now become the left and right channel of a new audio clip, respectively. The normalize option is used because convolution usually results in a huge increase of sample values (which is clear from the definition). Input :
The result sounds as it was recorded in the same salon as the impulse response. Furthermore, it is a true stereo sound. To visualize it, input :
Note that the resulting audio is longer than the input (for the length of the impulse response).