### 5.25.3 Convolution of two signals : convolution

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 :

zk=

k
∑
i=0

vi wk−i, k=0,1,…,N−1,

such that vj=0 for j≥ n and wj=0 for j≥ m .

For example, input :

convolution([1,2,3],[1,-1,1,-1])

Output :

[1.0,1.0,2.0,-2.0,1.0,-3.0]

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 :

clip:=readwav("/path/to/sounds/guitar.wav"):; ir:=readwav("/path/to/sounds/salon-ir.wav"):;

Input :

plotwav(clip)

Output :

Input :

plotwav(ir)

Output :

Convolving the data from clip with both channels in ir produces a reverberated variant of the recording, in stereo. Input :

data:=channel_data(clip):; L:=convolution(data,channel_data(ir,1)):; R:=convolution(data,channel_data(ir,2)):;

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 :

spatial:=createwav([L,R],normalize=-3):; playsnd(spatial)

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 :

plotwav(spatial)

Output :

Note that the resulting audio is longer than the input (for the length of the impulse response).