Convolution of two signals or functions

### 21.2.5 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 21.2.3) 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 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 restricted to [0,+∞⟩.

##### Remark.

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 with θ(x) 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)

− |
θ | ⎛ ⎝ | t | ⎞ ⎠ | ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ | 2 t ln | ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ |
− | √ |
t
| + | √ |
t+1
√ |
t
| + | √ |
t+1
| ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ | +4 | √ |
t
|  | √ |
t+1
| +2 ln | ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ |
− | √ |
t
| + | √ |
t+1
√ |
t
| + | √ |
t+1
| ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ | ⎞ ⎟ ⎟ ⎟ ⎟ ⎠
√ |
t+1

#### Application

A “dry” signal can be reverberated by convolving it with the impulse response of a particular acoustic space. The latter is typically obtained by recording a short, popping sound at the location, such as firing a starter gun.

The following demonstration requires two files: a music mono recording msmn4.wav downloaded from here and a two-channel impulse response French 18th Century Salon.wav downloaded from here. To load the files in Xcas, enter:

clip:=readwav("/home/luka/Downloads/msmn4.wav")

a sound clip with 110250 samples at 22050 Hz (16 bit, mono) |  |  |  |  |  |  |  |  |  |

ir:=readwav("/home/luka/Downloads/French 18th Century Salon.wav")

a sound clip with 88300 samples at 44100 Hz (16 bit, stereo) |  |  |  |  |  |  |  |  |  |

plotwav(clip)

plotwav(ir)

Convolving data from clip with both channels in ir produces a reverberated variant of the recording, in stereo. Since the two clips have different sample rates, clip should be up-sampled to 44100 Hz before convolving (see Section 28.2.11).

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

Evaluation time: 35.87

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.

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

The music being played back appears as it has been recorded in the same salon as the impulse response. Moreover, it is a true stereo sound at 44100 Hz as opposed to the original mono recording at 22050 Hz. To visualize it, enter:

plotwav(spatial)

The length of the resulting audio equals the length of the original clip plus the length of the impulse response minus 1. Indeed:

length(spatial)==length(data)+length(ir)-1

true |  |  |  |  |  |  |  |  |  |