Convolution of two signals or functions

### 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=v∗ w (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 wk−i, k=0,1,…,N−1,

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

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

| ∫ |
+∞
−∞
| f(t) g(x−t) dt.

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

- For the convolution of two vectors, convolution takes two arguments: v,w, two vectors (not necessarily the same length).
- convolution(v,w) returns the convolution correlation v∗ w.
- For the convolution of two functions, convolution takes two mandatory arguments and one optional argument:
  - f,g, two expressions of a variable.
  - Optionally, x, the variable name.
- convolution(f,g ⟨,x⟩) returns the convolution of f and g.

  Note: f and g are assumed to be 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.

#### 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 e2 x 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 x−e−3 x+e2 x | ⎞ ⎠ | 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)

− |
⎛ ⎜ ⎜ ⎜ ⎜ ⎝ | 2 t ln | ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ |
⎪ ⎪ ⎪ | 2 | √ |
t
| −2 | √ |
t+1
| ⎪ ⎪ ⎪
2 | √ |
t
| +2 | √ |
t+1
| ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ | +4 | √ |
t
|  | √ |
t+1
| +2 ln | ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ |
⎪ ⎪ ⎪ | 2 | √ |
t
| −2 | √ |
t+1
| ⎪ ⎪ ⎪
2 | √ |
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).