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Properties of Convolution Integrals

Module by: Carlos E. Davila. E-mail the author

We list several important properties and their proofs.

  1. Commutative Property:
    x(t)*h(t)=h(t)*x(t)x(t)*h(t)=h(t)*x(t)
    (1)
    Lets start with
    x(t)*h(t)=∫-∞∞x(τ)h(t-τ)dτx(t)*h(t)=∫-∞∞x(τ)h(t-τ)dτ
    (2)
    and make the substitution γ=t-τγ=t-τ. It follows that
    x(t)*h(t)=∫-∞∞x(t-γ)h(γ)dγdτ=h(t)*x(t)x(t)*h(t)=∫-∞∞x(t-γ)h(γ)dγdτ=h(t)*x(t)
    (3)
  2. Associative Property:
    x(t)*h1(t)*h2(t)=x(t)*h1(t)*h2(t)x(t)*h1(t)*h2(t)=x(t)*h1(t)*h2(t)
    (4)
    To prove this property we begin with an expression for the left-hand side of Equation 4
    ∫-∞∞x(τ)h1(t-τ)dτ*h2(t)∫-∞∞x(τ)h1(t-τ)dτ*h2(t)
    (5)
    where we have expressed x(t)*h1(t)x(t)*h1(t) as a convolution integral. Expanding the second convolution gives
    ∫-∞∞∫-∞∞x(τ)h1(γ-τ)dτh2(t-γ)dγ∫-∞∞∫-∞∞x(τ)h1(γ-τ)dτh2(t-γ)dγ
    (6)
    Reversing the order of integration gives
    ∫-∞∞x(τ)∫-∞∞h1(γ-τ)h2(t-γ)dγdτ∫-∞∞x(τ)∫-∞∞h1(γ-τ)h2(t-γ)dγdτ
    (7)
    Using the variable substitution φ=γ-τφ=γ-τ and integrating over φφ in the inner integral gives the final result:
    ∫-∞∞x(τ)∫-∞∞h1(φ)h2(t-τ-φ)dγdτ∫-∞∞x(τ)∫-∞∞h1(φ)h2(t-τ-φ)dγdτ
    (8)
    where the inner integral is recognized as h1(t)*h2(t)h1(t)*h2(t) evaluated at t=t-τt=t-τ, which is required for the convolution with x(t)x(t).
  3. Distributive Property:
    x(t)*h1(t)+h2(t)=x(t)*h1(t)+x(t)*h2(t)x(t)*h1(t)+h2(t)=x(t)*h1(t)+x(t)*h2(t)
    (9)
    This property is easily proven from the definition of the convolution integral.
  4. Time-Shift Property: If y(t)=x(t)*h(t)y(t)=x(t)*h(t) then x(t-t0)*h(t)=y(t-t0)x(t-t0)*h(t)=y(t-t0) Again, the proof is trivial.

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