15.2.8  Convolution of two signals or functions: convolution

The convolution of two real vectors v=[v₁,…,v_n] and w=[w₁,…,w_m] is the complex vector z=v∗ w of length n+m−1 given by

such that v_j=0 for j≥ n and w_j=0 for j≥ m.

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

variable x as an optional third argument, in which case the 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:

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.

Example.
Input:

Output:

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.

  The functions 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.

Examples.

• Compute the convolution of f(x)=25 e^2 x u(x) and g(x)=x e^−3 x θ(x), where θ is the Heaviside function:
  Input:
  convolution(25*exp(2x),x*exp(-3x))
  Output:

  ⎛ ⎝

  −5 x e−3 x−e−3 x+e2 x

  ⎞ ⎠

  θ

  ⎛ ⎝

  x

  ⎞ ⎠

• Compute the convolution of f(t)=ln(1+t) u(t) and g(t)=1/√t.
  Input:
  convolution(ln(1+t),1/sqrt(t),t)
  Output:

  −	⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 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 this 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:
  Input:

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

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

  Then: 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)
  Output: A sound that sounds as if 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).