Question

Compute the convolution \((f * g)(t)\) (in the Laplace transform sense) and its corresponding Laplace transform \(\mathcal{L}[f * g]\) for the following functions: a. \(f(t)=t^{2}, g(t)=t^{3}\). b. \(f(t)=t^{2}, g(t)=\cos 2 t\). c. \(f(t)=3 t^{2}-2 t+1, g(t)=e^{-3 t}\). d. \(f(t)=\delta\left(t-\frac{\pi}{4}\right), g(t)=\sin 5 t\)

Short answer

The procedure starts by finding the convolution for each pair of functions using the convolution formula. Then, solving the integral expression for each pair. Lastly, computing the Laplace transform for each calculated convolution.

Step-by-step solution

Calculate the convolutions for each function pair

The convolution of two signals \(f\) and \(g\) can be represented as \((f*g)(t)=\int_{-\infty}^{\infty} f(\tau)g(t-\tau)\,d\tau\). Let's apply this formula for each function pair:- For the case \(f(t)=t^{2}, g(t)=t^{3}\), this would become \((f*g)(t)= \int_{0}^{t} \tau^{2}(t-\tau)^{3}d\tau\).- For the case \(f(t)=t^{2}, g(t)=\cos2t\): \((f*g)(t)= \int_{0}^{t} \tau^{2}\cos 2(t-\tau)d\tau\).- For the case \(f(t)=3t^{2}-2t+1, g(t)=e^{-3t}\): \((f*g)(t)= \int_{0}^{t} (3\tau^{2}-2\tau+1)e^{-3(t-\tau)}d\tau\).- For the case \(f(t)=\delta(t-\frac{\pi}{4}), g(t)=\sin 5t\): \((f*g)(t)= \int_{0}^{t} \delta(\tau-\frac{\pi}{4})\sin 5(t-\tau)d\tau\). In this case, convolution with the delta function is unique because it s(t) = ∫f(τ)δ(t - τ)dτ = f(t).

Solve the integral expressions

To find \((f*g)(t)\) for each pair, it is crucial to solve the above integral expressions. These could be solved using standard calculus methods. However, do notice that this can be quite tricky as each integral has a function times a shifted version of the other function as an integrand. Therefore, go ahead and calculate these integrals.

Find the Laplace transforms

Once \((f*g)(t)\) for each pair has been calculated, the next step is to find the corresponding Laplace transforms (denoted as \(\mathcal{L}[f * g]\)). This is done using the formula \(\mathcal{L}[f(t)](s)=\int_{0}^{+\infty}e^{-st}f(t)dt\). Do note that computing the Laplace transform from scratch can be tedious. In many cases, instead of taking the transform directly, make use of the standard Laplace transform table, if the given function is a standard one or it can be reduced or expressed in terms of those standard ones. If not, go ahead and solve the integral.

Key concepts

Integral Calculus

Integral calculus is a fundamental component of calculus concerned with the concept of integration. It involves finding the integral or anti-derivative of a function, which can represent various physical quantities like area under a curve or accumulated value over time. In the context of convolution in Laplace transforms, integration is used to combine two functions into a single one.
The convolution of two functions, expressed as \((f * g)(t) = \int_{0}^{t} f(\tau)g(t-\tau) d\tau\), requires integrating the product of one function and a shifted version of another.

• This application of integration involves shifting and scaling one of the functions, which can make the integral complex to compute manually.
• However, understanding how to set up and solve these integrals is essential for mastering convolution processes in integral calculus.

By learning how to manipulate these integrals, students gain deeper insight into not just mathematical theory but also practical applications in engineering and physics.

Laplace Transform

The Laplace transform is an efficient tool for solving differential equations, transforming functions from the time domain to the complex frequency domain. This transformation simplifies the complexity involved in analyzing linear time-invariant systems.
By converting a function \(f(t)\) to \(F(s)\) in the frequency domain, we replace time-dependent operations with algebraic ones.

• For convolutions, the Laplace transform of a convolution \((f * g)(t)\) is the product of their separate Laplace transforms: \(\mathcal{L}[f * g](s) = F(s)G(s)\).
• This property significantly reduces the complexity of finding the convolution in the time domain by turning it into a simple multiplication problem in the Laplace domain.

Using the Laplace transform tables can save time and effort, especially when dealing with standard transform pairs. This transforms the tedious task of solving differential equations into a straightforward algebraic manipulation.

Delta Function

The delta function, or Dirac delta function, is a mathematical construct that represents an "idealized" impulse. It is not a function in the traditional sense but rather a distribution used to model a very short-duration, high-magnitude event.
Its defining properties are:

• \(\delta(t) = 0\) everywhere except at \(t = 0\), and
• \(\int_{-\infty}^\infty \delta(t) dt = 1\) .

When used in convolution, the delta function acts like a selector that "picks out" the function value at a certain point. For convolution \((f * \delta)(t)\), the result simplifies directly to \(f(t)\).
This property is particularly useful for signal processing, control systems, and solving differential equations, where inserting an impulse yields the system's impulse response. It provides a powerful way to simplify otherwise complex convolution calculations.

Trigonometric Functions

Trigonometric functions, including sine and cosine, are essential tools in mathematics to represent periodic phenomena. In the context of Laplace transforms and convolution, they arise frequently in solving problems involving oscillatory behavior such as vibrations and waves.
These functions can be convolved with other types, like polynomials, leading to integral expressions that model complex systems. Using trigonometric identities, these integrations can be simplified:

• For example, \(\cos(A - B) = \cos A \cos B + \sin A \sin B\).
• Such identities help in breaking down complex integrals into more manageable parts.

Understanding the properties of trigonometric functions and their interactions with exponential or polynomial forms in convolutions enables students to more easily solve for the behavior of mechanical and electrical systems. These concepts are crucial for engineering applications and advanced calculus.