1
1.1
Convolution theorem
Convolution
Let us introduce concept of convolution by an intuitive physical consideration.
Consider some physical system. Denote an input (input signal) to the system by x(x) and system’s
response to the input by y(t).
x(t) → SYSTEM → y(t)
Let us assume the following properties of the system :
Linearity
Let y(t) be the system’s response to the input x(t). For brevity we will use the symbolic notation
x(t) → y(t)
⇔
a x(t) → a y(t)
Let x1 (t) → y1 (t) and x2 (t) → y2 (t). Then
x1 (t) + x2 (t) → y1 (t) + y2 (t)
Consequently
a1 x1 (t) + a2 x2 (t) → a1 y1 (t) + a2 y2 (t)
Invariability with respect to time
Let x(t) → y(t), then x(t − τ ) → y(t − τ )
Causality
Let x(t) = 0 for t < t0 , then y(t) = 0 for t < t0 .
Consider now Dirac delta function (or Dirac delta impulse) δ(t) as input to the system. Denote the
system’s response h(t), i.e.,
δ(t) → h(t)
Since F[δ(t)] = 1, i.e., all frequencies are equally present in the Dirac delta impulse, the system’s response
h(t) is called the impulse response of the system and characterizes transfer properties of the system in
the time domain.
Due to invariability of the system with respect to time
δ(t − τ ) → h(t − τ )
Consider again some general input signal x(t). It can be represented using the Dirac Delta function :
Zt
x(t) =
x(τ )δ(t − τ )dτ
(1)
0
Relation (1) means, that the general signal x(t) is an ’integral linear combination’ of the Dirac delta
impulses δ(t − τ ). Relation (1) and linearity of the system imply, that the response to the input signal to
the input signal x(t) is the same integral combination of the impulse responses h(t − τ ) :
Zt
y(t) =
x(τ )h(t − τ )dτ
(2)
0
Relation (2) is symbolically denoted as
y(t) = x(t) ∗ h(t)
1
(3)
The integral in relation (2) is called the convolutory integral, or simply, the convolution. Relation (2)
means that once we know the impulse response of a system we can compute the output of the system for
an arbitrary input using the convolution.
Now we define the convolution of two functions x1 (t) and x2 (t) as
Z ∞
x1 ,x2
ρ
(t) =
x1 (τ )x2 (t − τ )dτ
(4)
−∞
and symbolically denote
ρx1 ,x2 (t) = x1 (t) ∗ x2 (t)
(5)
Relation (4) is valid for general functions x1 (t) and x2 (t) defined in the interval t ∈ (−∞, ∞). If one or
both of the functions are causal, relation (4) can be modified.
R∞
A) x1 (t) is causal : ρx1 ,x2 (t) = 0 x1 (τ )x2 (t − τ )dτ
Rt
B) x2 (t) is causal : ρx1 ,x2 (t) = −∞ x1 (τ )x2 (t − τ )dτ
C) both x1 (t) and x2 (t) are causal :
ρx1 ,x2 (t) =
Rt
0
x1 (τ )x2 (t − τ )dτ
Basic properties of the convolution
1. x1 (t) ∗ x2 (t) = x2 (t) ∗ x1 (t)
2. x1 (t) ∗ [x2 (t) + x3 (t)] = x1 (t) ∗ x2 (t) + x1 (t) ∗ x3 (t)
3. x1 (t) ∗ [x2 (t) ∗ x3 (t)] = [x1 (t) ∗ x2 (t)] ∗ x3 (t)
4. x(t) ∗ δ(t) = x(t)
x(t) ∗ δ(t − τ ) = x(t − τ )
x(t − t0 ) ∗ δ(t − τ ) = x(t − t0 − τ )
δ(t − t0 ) ∗ δ(t − τ ) = δ(t − t0 − τ )
5. ρx1 ,x2 (t − τ ) = x1 (t − τ ) ∗ x2 (t) = x1 (t) ∗ x2 (t − τ )
6.
7.
dx1 (t)
d
2 (t)
∗ x2 (t) = x1 (t) ∗ dxdt
dt [x1 (t) ∗ x2 (t)] =
dt
Rt
Rt
x (η) ∗ x2 (η)dη = [ −∞ x1 (η)dη] ∗ x2 (t) = x1 (t)
−∞ 1
∗[
Rt
−∞
x2 (η)dη]
Convolution theorem
Find the Fourier transform of the convolution (4).
¸
Z ∞ ·Z ∞
F[x1 (t) ∗ x2 (t)] =
x1 (η)x2 (τ − η)dη e−i2πf τ dτ
−∞
−∞
·Z ∞
¸
Z ∞
x1 (η)
x2 (τ − η)e−i2πf τ dτ dη
−∞
−∞
Z ∞
x1 (η)e−i2πf η X2 (f )dη
=
=
=
X1 (f )X2 (f )
−∞
This allows us to formulate the convolution theorem Let X1 (f ) = F{x1 (t)} and X2 (f ) = F{x2 (t)}. Then
F[x1 (t) ∗ x2 (t)] = X1 (f )X2 (f )
(6)
Relation (6) is one of the most important relations in the spectral analysis and is frequently used in
mathematics, physics, geophysics and other scientific and engineering disciplines.
2
It is easy to show (analogously with the proof of relation (6) that
F[x1 (t)x2 (t)] = X1 (f ) ∗ X2 (f )
(7)
The r.h.s. of relation (7) is called the spectral or frequency convolution while that in relation (4) is called
the time convolution.
Graphical interpretation of convolution
1.
2.
3.
4.
new variable τ
f2 (τ ) → f2 (−τ ) - horizontal mirror
f2 (−τ ) → f2 (t − τ ) - horizontal translation by t
multiply f2 (τ )f2 (t − τ ), area under the graph is the value of convolution in the time t
3