Properties of Convolution

Introduction

Convolution possesses a set of algebraic properties that parallel those of multiplication — and these properties are not merely theoretical curiosities. They have direct practical applications in system design. Commutativity means you can choose which function to flip in graphical convolution. Associativity means cascaded systems can be analyzed in any order, or combined into a single equivalent system. Distributivity enables parallel system analysis. Understanding these properties transforms convolution from a computational procedure into a flexible analytical tool.

Commutativity

$$x(t) * h(t) = h(t) * x(t)$$

The order of the two signals in convolution does not matter. Mathematically: $$\int_{-\infty}^{\infty} x(\tau)h(t-\tau)d\tau = \int_{-\infty}^{\infty} h(\tau)x(t-\tau)d\tau$$

Practical use: In graphical convolution, you can choose to flip whichever function is simpler. If $h(t)$ is a complicated shape but $x(t)$ is a simple rectangle, flip $x$ instead.

Physical interpretation: Whether you think of the input sliding past the impulse response, or the impulse response sliding past the input, the output is identical.

Associativity

$$[x(t) * h_1(t)] * h_2(t) = x(t) * [h_1(t) * h_2(t)]$$

Three signals convolved together give the same result regardless of the grouping.

Practical use: For a cascade of two LTI systems ($h_1$ followed by $h_2$), you can:

1. Convolve the input with $h_1$, then convolve that result with $h_2$, OR
2. First compute the overall impulse response $h_{total} = h_1 * h_2$, then convolve with the input once.

The second approach is more efficient when processing many different inputs through the same cascade.

In the frequency domain, associativity corresponds to: $H_{total}(j\omega) = H_1(j\omega) \cdot H_2(j\omega)$ — transfer functions of cascaded systems multiply.

Distributivity

$$x(t) * [h_1(t) + h_2(t)] = x(t) * h_1(t) + x(t) * h_2(t)$$

Convolution distributes over addition, just like multiplication distributes over addition.

Practical use: For parallel systems, the total output equals the sum of individual outputs. The equivalent impulse response of parallel systems is $h_{total}(t) = h_1(t) + h_2(t)$.

In the frequency domain: $H_{total}(j\omega) = H_1(j\omega) + H_2(j\omega)$.

Identity Element

$$x(t) * \delta(t) = x(t)$$

The unit impulse is the identity element for convolution — convolving any signal with $\delta(t)$ leaves it unchanged. This is analogous to multiplying by 1.

Shifted impulse: $$x(t) * \delta(t - t_0) = x(t - t_0)$$

Convolution with a shifted impulse simply delays the signal by $t_0$. This is why a system with impulse response $h(t) = \delta(t - T)$ is a pure delay of $T$ seconds.

Shift Property

$$\text{If } y(t) = x(t) * h(t), \text{ then } x(t-t_1) * h(t-t_2) = y(t - t_1 - t_2)$$

Shifts in either input accumulate in the output. This is useful for computing convolutions of shifted signals without re-evaluating the integral.

Differentiation Property

$$\frac{d}{dt}[x(t) * h(t)] = \frac{dx}{dt} * h(t) = x(t) * \frac{dh}{dt}$$

You can differentiate either function before convolving — the result is the derivative of the convolution. This is useful when one function is easier to differentiate than to convolve directly.

Integration Property

$$\int_{-\infty}^{t} [x(\tau) * h(\tau)] d\tau = \left[\int_{-\infty}^{t} x(\tau) d\tau\right] * h(t) = x(t) * \left[\int_{-\infty}^{t} h(\tau) d\tau\right]$$

Integration can be applied to either function before convolution.

Width (Duration) Property

If $x(t)$ has duration $T_x$ and $h(t)$ has duration $T_h$: $$\text{Duration of } y(t) = T_x + T_h$$

In discrete-time: If $x[n]$ has $N$ samples and $h[n]$ has $M$ samples, $y[n]$ has $N + M - 1$ samples.

Area Property

$$\int_{-\infty}^{\infty} y(t) dt = \left(\int_{-\infty}^{\infty} x(t)dt\right) \cdot \left(\int_{-\infty}^{\infty} h(t)dt\right)$$

The area under the convolution equals the product of the individual areas. This provides a quick sanity check for computed convolutions.

Application: Simplifying Cascade Systems

Consider three systems in cascade: $H_1, H_2, H_3$. By associativity: $$h_{total} = h_1 * h_2 * h_3$$ $$H_{total}(j\omega) = H_1(j\omega) \cdot H_2(j\omega) \cdot H_3(j\omega)$$

By commutativity of multiplication, the order of systems in a cascade does not affect the overall transfer function for LTI systems. You can rearrange the order of processing stages freely.

Application: Parallel Decomposition

A bandpass filter can often be implemented as the difference of two low-pass filters: $$h_{BP} = h_{LP2} - h_{LP1}$$

By distributivity: $y_{BP} = x * h_{LP2} - x * h_{LP1}$

Key Takeaways

• Commutative: $x * h = h * x$ — flip whichever function is simpler in graphical convolution
• Associative: cascade systems combine by convolving impulse responses (or multiplying transfer functions)
• Distributive: parallel systems combine by adding impulse responses (or transfer functions)
• Identity: $x * \delta = x$; shifted impulse $x * \delta(t-T)$ produces delay
• Duration of output = sum of input durations
• Area of output = product of input areas
• These properties enable flexible rearrangement and simplification of system architectures