Question

Establish the commutative, distributive, and associative properties of the convolution integral. (a) \(f * g=g * f\) (b) \(f *\left(g_{1}+g_{2}\right)=f * g_{1}+f * g_{2}\) (c) \(f *(g * h)=(f * g) * h\)

Short answer

Based on the step-by-step solution, provide a short answer to the problem: To prove the commutative, distributive, and associative properties of the convolution integral, we manipulate the integral formula for convolution, which is given by \((f*g)(\tau) = \int_{-\infty}^{\infty} f(t)g(\tau-t)\, dt\). (a) For the commutative property, we substitute a variable change and rearrange the integral to demonstrate that \(f * g = g * f\). (b) For the distributive property, we distribute and separate the terms inside the integral, showing that \(f *(g_1 + g_2) = f * g_1 + f * g_2\). (c) For the associative property, we substitute the definition of convolution and apply a variable change, resulting in the proof that \(f *(g * h) = (f * g) * h\). In conclusion, we have successfully proven the commutative, distributive, and associative properties of the convolution integral through integral manipulation and the use of substitution.

Step-by-step solution

(a) Prove the commutative property: \(f * g = g * f\)

Start by writing down the formula for \(f * g\): \((f*g)(\tau) = \int_{-\infty}^{\infty} f(t)g(\tau-t)\, dt\) Now, introduce a variable change: substitute \(u = \tau-t\). This gives, \(dt = -du\) and \(t = \tau-u\). Substituting these expressions in the convolution definition, we get: \((f*g)(\tau) = \int_{\infty}^{-\infty} f(\tau-u)g(u)(-du)\) Change the limits and remove the negative sign: \((f*g)(\tau) = \int_{-\infty}^{\infty} f(\tau-u)g(u)\, du\) Observe that the integral is now in the form of a convolution, but with the order of \(f\) and \(g\) switched: \((f*g)(\tau) = (g*f)(\tau)\) Hence, the commutative property is proven.

(b) Prove the distributive property: \(f *(g_1 + g_2) = f * g_1 + f * g_2\)

Start by writing down the formula for \(f*(g_1+g_2)\): \((f*(g_1+g_2))(\tau) = \int_{-\infty}^{\infty} f(t)(g_1+g_2)(\tau-t)\, dt\) Now, distribute the terms inside the integral: \((f*(g_1+g_2))(\tau) = \int_{-\infty}^{\infty} [f(t)g_1(\tau-t) + f(t)g_2(\tau-t)]\, dt\) Now, separate the integral into two parts: \((f*(g_1+g_2))(\tau) = \int_{-\infty}^{\infty} f(t)g_1(\tau-t)\, dt + \int_{-\infty}^{\infty} f(t)g_2(\tau-t)\,dt\) Recognizing the convolution definition, we obtain: \((f*(g_1+g_2))(\tau) = (f*g_1)(\tau) + (f*g_2)(\tau)\) Hence, the distributive property is proven.

(c) Prove the associative property: \(f *(g * h) = (f * g) * h\)

Start by writing down the formula for the convolution of \(f*(g*h)\): \((f*(g*h))(\tau) = \int_{-\infty}^{\infty} f(t)(g*h)(\tau-t)\, dt\) Now, substitute the definition of convolution for \((g*h)(\tau-t)\): \((f*(g*h))(\tau) = \int_{-\infty}^{\infty} f(t)\left[\int_{-\infty}^{\infty} g(u)h(\tau-t-u)\, du\right] dt\) Now, swap the order of integration: \((f*(g*h))(\tau) = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} f(t)g(u)h(\tau-t-u)\, dtdu\) Introduce a variable change: \(v = \tau - u\) \(v-u = t\) The Jacobian for this transformation is \(1\). Apply the inverse mapping of the Jacobian and substitute the variables: \((f*(g*h))(\tau) = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} f(v-u)g(u)h(v)\, dudv\) Now, group the terms related to each variable: \((f*(g*h))(\tau) = \int_{-\infty}^{\infty} \left[\int_{-\infty}^{\infty} f(v-u)g(u)\, du\right]h(v)\, dv\) Applying the definition of convolution, we get: \((f*(g*h))(\tau) = ((f*g)*h)(\tau)\) Hence, the associative property is proven.

Key concepts

Commutative Property

The commutative property for the convolution integral states that the order of functions can be switched without affecting the result. We express this property as \( f * g = g * f \).

To understand this, let's think of the convolution as an integral over the entire domain. Originally, we start with the convolution of two functions, \( f \) and \( g \), given by the integral:

\[(f*g)(\tau) = \int_{-\infty}^{\infty} f(t)g(\tau-t)\, dt\]

By changing the variable of integration using \( u = \tau - t \), we essentially reverse the roles of \( f \) and \( g \). The reformed integral becomes equivalent in form, but with \( g \) and \( f \) swapped:

\[(g*f)(\tau) = \int_{-\infty}^{\infty} g(t)f(\tau-t)\, dt\]

This symmetry confirms that convolution is commutative. No matter which function we start with, the result of the convolution operation remains the same. This fundamental characteristic simplifies the analysis of linear systems, making them more predictable and easier to manage.

Distributive Property

The distributive property in convolution describes how it spreads across addition. The key takeaway is that convolving a function with a sum of functions is the same as convolving the function with each individual function and then summing the results. Mathematically, we write this as \( f * (g_1 + g_2) = f * g_1 + f * g_2 \).

This is more intuitive upon simplification. Start with:

\[(f*(g_1+g_2))(\tau) = \int_{-\infty}^{\infty} f(t)(g_1+g_2)(\tau-t)\, dt\]

By distributing the terms inside the integral, we see:

\[= \int_{-\infty}^{\infty} (f(t)g_1(\tau-t) + f(t)g_2(\tau-t))\, dt\]

Breaking this down into separate, simple integrals reveals:

\[= \int_{-\infty}^{\infty} f(t)g_1(\tau-t)\, dt + \int_{-\infty}^{\infty} f(t)g_2(\tau-t)\, dt\]

And from here, we recognize the original convolutions with \( g_1 \) and \( g_2 \). The sum of these two results provide the same outcome as the entire process performed initially with the sums.

Understanding this property aids in managing complex systems where functions can be decomposed into simpler components. This makes convolutions especially useful in signal processing and other engineering applications.

Associative Property

The associative property implies that when three functions are being convolved, the order in which the convolution is performed does not affect the outcome, described as \( f * (g * h) = (f * g) * h \).

To illustrate, consider that to convolve \( f \) with the convolution of \( g \) and \( h \), is expressed as:

\[(f*(g*h))(\tau) = \int_{-\infty}^{\infty} f(t)(g*h)(\tau-t)\, dt\]

Replace \((g*h)(\tau-t)\) with its integral definition and change the order of integration:

\[= \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(t)g(u)h(\tau-t-u)\, du\, dt\]

Now we switch the order of integration to reveal:

\[= \int_{-\infty}^{\infty} \left( \int_{-\infty}^{\infty} f(v-u)g(u)\, du \right) h(v)\, dv\]

The inner integral forms a new convolution \((f*g)(v)\), that, when convolved with \( h \), succinctly completes our associative property illustration.

This property is pivotal in the analysis of convolution, enabling flexibility in computation and allowing for rearrangement of operations without impacting the result, a crucial feature in digital systems and filtering applications.