Convolution properties

The convolution operation has a handful of algebraic properties that make it usable. Throughout, $x,y,z,h$ are continuous-time signals, $A$ is a constant, $a$ is a nonzero constant, $t_0$ is real, and $\delta$ is the unit impulse.

Commutativity

$x(t)\ast y(t)=y(t)\ast x(t).$

Order of the two signals doesn’t matter. Falls out of the change of variable $\tau \to t-\tau$ in the convolution integral. Practical use: if one of the two signals is easier to flip than the other, flip that one.

Associativity

$(x(t)\ast y(t))\ast z(t)=x(t)\ast (y(t)\ast z(t)).$

Order of convolving three signals doesn’t matter. This is what makes cascaded systems clean: if input $x$ feeds a system with impulse response $h_1$, whose output feeds a system with response $h_2$, the overall I/O relation is $(x\ast h_1)\ast h_2=x\ast (h_1\ast h_2)$. So a cascade is equivalent to a single system with impulse response $h_1\ast h_2$.

Distributivity

$(x(t)+y(t))\ast z(t)=x(t)\ast z(t)+y(t)\ast z(t).$

Convolution distributes over addition. Practical use: parallel-connected systems with shared input have overall impulse response $h_1+h_2$.

Scalar multiplication

$x(t)\ast (Ay(t))=A(x(t)\ast y(t)).$

Constants pass through convolution.

Identity (the impulse)

$x(t)\ast \delta (t)=x(t).$

The unit impulse is the convolution identity. Quick proof: $\int x(\tau )\delta (t-\tau )d\tau =x(t)$ by the sifting property.

More generally, $x(t)\ast \delta (t-t_0)=x(t-t_0)$. Convolving with a shifted impulse shifts the signal — a “system” with impulse response $\delta (t-t_0)$ is just a pure delay.

A useful corollary: if $g(t)=g_0(t)\ast \delta (t)$, then $g(t-t_0)=g_0(t-t_0)\ast \delta (t)=g_0(t)\ast \delta (t-t_0)$. The shift can be applied to either side of the convolution.

Differentiation

If $y=x\ast h$, then

$y^{\prime }(t)=x^{\prime }(t)\ast h(t)=x(t)\ast h^{\prime }(t).$

The derivative of a convolution is the convolution of either factor’s derivative with the other. Useful when one factor has a simpler derivative than itself.

Area

$(\text{Area of }y)=(\text{Area of }x)\times (\text{Area of }h).$

Total area under a convolution equals the product of the areas of the factors. Fast sanity check: if your computed $y$ has the wrong total area, there’s an error.

Proof: integrating $y(t)=\int x(\tau )h(t-\tau )d\tau$ over all $t$ and swapping the order, ${\int }_{\tau }x(\tau )[{\int }_{t}h(t-\tau )dt]d\tau ={\int }_{\tau }x(\tau )\cdot (\text{Area of }h)d\tau =(\text{Area of }h)(\text{Area of }x)$.

Time-scaling

If $y(t)=x(t)\ast h(t)$, then

$y(at)=|a|(x(a\cdot )\ast h(a\cdot ))(t),$

where $x(a\cdot )$ and $h(a\cdot )$ are the time-scaled signals $\tau \mapsto x(a\tau )$ and $\tau \mapsto h(a\tau )$, convolved and then evaluated at $t$ (not at $at$). The $|a|$ factor out front comes from the change of integration variable.

Smoothing and continuity

Two more properties that don’t have one-line formulas but are worth knowing:

• Convolution is smoothing. The output is at least as smooth as the smoother of the two factors. Rough signals become rough only if both are rough; convolving with a smooth signal smooths the result.
• Convolution is continuous. Small changes in the inputs produce small changes in the output.

For the frequency-domain consequence — convolution in time becoming multiplication in frequency — see the Convolution theorem. This is what makes the Fourier transform and Laplace transform so useful for LTI analysis.