Question

Prove the following properties of convolution: (a) \(f * g=g * f\). (b) \(f \cdot(g * h)=(f * g) * h .\) (c) \(f *(g+h)=f * g+f * h .\) (d) \(\quad f * 0=0 * f\).

Short answer

Based on the step-by-step solution provided, answer the following short question: Question: Prove that the commutative property of convolution holds, i.e., \(f * g = g * f\). Answer: To prove the commutative property of convolution, we started with the definition of convolution: \((f * g)(t) = \int_{-\infty}^{\infty} f(\tau)g(t-\tau) d\tau\). We then made a change of variables: \(\xi = t - \tau\), \(d \xi = -d \tau\), and when \(\tau \to -\infty , \xi \to \infty\). Substituting these changes into the integral and flipping the integration limits, we obtained \((g * f)(t) = \int_{-\infty}^{\infty} g(t-\xi)f(\xi) d\xi\). Since \((f * g)(t) = (g * f)(t)\), the commutative property of convolution holds.

Step-by-step solution

Property (a): Commutative Property of Convolution (\(f * g = g * f\))

To prove this, we'll start with the definition of convolution: $$ (f * g)(t) = \int_{-\infty}^{\infty} f(\tau)g(t-\tau) d\tau $$ We will make a change of variables to prove commutativity: Let \(\xi = t - \tau\). Then, \(d \xi = -d \tau\) and when \(\tau \to -\infty , \xi \to \infty\). Now, substitute these changes into the integral: $$ (g * f)(t) = \int_{-\infty}^{\infty} g(\tau)f(t-\tau) d\tau = \int_{\infty}^{-\infty} g(t-\xi)f(\xi) (-d\xi) $$ The negative sign can be used to flip the integration limits: $$ (g * f)(t) = \int_{-\infty}^{\infty} g(t-\xi)f(\xi) d\xi $$ Thus, we've shown that \((f * g)(t) = (g * f)(t)\).

Property (b): Associative Property of Convolution (\(f \cdot (g * h) = (f * g) * h\))

To show this, we start with the left-hand side: $$ (f \cdot (g * h))(t) = \int_{-\infty}^{\infty} f(\tau)(g * h)(t-\tau) d\tau $$ Now insert the convolution definition on \((g * h)\): $$ (f \cdot (g * h))(t) = \int_{-\infty}^{\infty} f(\tau) \left(\int_{-\infty}^{\infty} g(\xi)h(t-\tau-\xi) d\xi\right) d\tau $$ We can change the order of integration and combine the integrals: $$ (f \cdot (g * h))(t) = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} f(\tau) g(\xi)h(t-\tau-\xi) d\xi d\tau $$ Using the change of variables \(u = t - \tau - \xi\), we get: $$ (f \cdot (g * h))(t) = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} f(\tau) g(u+\tau)h(u) d(u+\tau) d\tau $$ Using the convolution definition, we can rewrite this as: $$ (f \cdot (g * h))(t) = \int_{-\infty}^{\infty} ((f * g)(u))h(u) d(u) $$ Thus, we've shown that \(f \cdot (g * h) = (f * g) * h\).

Property (c): Distributive Property of Convolution (\(f * (g + h) = f * g + f * h\))

Let's start by working on the left-hand side: $$ (f * (g + h))(t) = \int_{-\infty}^{\infty} f(\tau)(g(t-\tau) + h(t-\tau)) d\tau $$ Using the properties of integrals, we can split the integral: $$ (f * (g + h))(t) = \int_{-\infty}^{\infty} f(\tau)g(t-\tau) d\tau + \int_{-\infty}^{\infty} f(\tau)h(t-\tau) d\tau $$ By applying the convolution definition, we get: $$ (f * (g + h))(t) = (f * g)(t) + (f * h)(t) $$ Thus, we've shown that \(f * (g + h) = f * g + f * h\).

Property (d): Convolution with Zero (\(f * 0 = 0 * f\))

We'll start with the left-hand side: $$ (f * 0)(t) = \int_{-\infty}^{\infty} f(\tau)0(t-\tau) d\tau $$ Since multiplication by zero results in zero, the integral will equal zero: $$ (f * 0)(t) = 0(t) $$ Similarly, we have: $$ (0 * f)(t) = \int_{-\infty}^{\infty} 0(\tau)f(t-\tau) d\tau = 0(t) $$ Thus, we've shown that \(f * 0 = 0 * f\).

Key concepts

Commutative Property of Convolution

Understanding the commutative property of convolution is essential as it assures us that the order in which we convolve two functions does not affect the result. Simply put, if you have two functions, let's call them function A and function B, convolving A with B will yield the same output as convolving B with A.

In our exercise, we proved this property by starting with the integral definition of the convolution and making a clever change of variables. The critical part here is recognizing that the integral is still valid even after we swap the functions and variables because the operation over the entire real line doesn't depend on the order of application. This fundamental concept has far-reaching implications in fields such as signal processing and systems engineering, where the response of a system should remain consistent irrespective of the order of signal inputs.

Associative Property of Convolution

The associative property of convolution tells us that when we have a series of convolutions to perform, we can group them in any order and the final result will remain unaffected. It's like saying when you have a set of numbers to add, it does not matter if you add the first two and then add the third, or if you add the last two and then add it to the first; the sum is the same.

To establish the associative property in our exercise, we inserted the definition of convolution within another convolution integral. Pay attention to the change of integration order—it's a critical step! Through this property, we can solve complex convolution problems by breaking them down into simpler, more manageable parts, a valuable strategy for tackling real-world signal processing challenges.

Distributive Property of Convolution

Like distribution in basic arithmetic, where you can 'distribute' a multiplication over an addition operation, the distributive property of convolution allows you to distribute a convolution over an addition operation involving functions.

In our example, we took an integral involving a sum inside and 'distributed' the convolution across each term, ending up with the sum of two separate convolutions. This is powerful because it enables us to break down more complex convolutions into simpler ones, making it much easier to analyze or compute the effect of a system with multiple inputs. Remember, the key takeaway here is that convolution can be 'opened up' over addition, which can greatly simplify the analysis of systems with multiple signals interacting.

Convolution with Zero

In convolution, whenever one of the functions is zero everywhere, the result of the convolution will also be zero, regardless of the other function. This property is analogous to multiplying any number by zero in basic arithmetic—the result is always zero.

The 'convolution with zero' property we demonstrated in the exercise has significant implications for practical applications. For example, in signal processing, if a system's impulse response is zero, it means the system will not respond to any input signal, translating this property directly into a real-world phenomenon.