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19.2.4  Convolution of two signals or functions

The convolution of two real vectors v=[v1,…,vn] and w=[w1,…,wm] is the complex vector z=vw (note the difference between ∗ and the cross-correlation operation ⋆, see Section 19.2.2) of length n+m−1 given by

  zk=
k
i=0
vi wki,   k=0,1,…,N−1,

such that vj=0 for jn and wj=0 for jm.

The convolution of two real functions f(x) and g(x) is the integral

  
+∞


−∞
f(tg(xtdt.

The convolution command finds the convolution of two vectors or two functions.

Examples

To find the convolution of two lists [1,2,3] and [1,−1,1,−1], enter:

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

1.0,1.0,2.0,−2.0,1.0,−3.0
          

Compute the convolution of f(x)=25 ex θ(x) and g(x)=x e−3 x θ(x), where θ is the Heaviside function:

convolution(25*exp(2x),x*exp(-3x))
     

−5 x e−3 xe−3 x+ex
θ
x
          

Compute the convolution of f(t)=ln(1+t) θ(t) and g(t)=1/√t.

convolution(ln(1+t),1/sqrt(t),t)
     





t ln






t
−2 
t+1


t
+2 
t+1





+4 
t
 
t+1
+2 ln






t
−2 
t+1


t
+2 
t+1










θ
t
t+1
          

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.

Load the files:

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

Then:

plotwav(clip)
plotwav(ir)

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

data:=channel_data(clip):; L:=convolution(data,channel_data(ir,left)):; R:=convolution(data,channel_data(ir,right)):;

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).

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

Output: a music that sounds as if it was recorded in the same salon as the impulse response. Moreover, it is a true stereo sound. To visualize it:

plotwav(spatial)

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


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