Question

Let \(f(t)\) be a function defined and continuous on \(0 \leq t<\infty\). Determine $$ f * \delta=\int_{0}^{t} f(t-\lambda) \delta(\lambda) d \lambda . $$

Short answer

Answer: The convolution of the continuous function \(f(t)\) with the Dirac delta function \(\delta(\lambda)\) is \(f(t)\).

Step-by-step solution

Recall the properties of the Dirac delta function

The Dirac delta function is not a function in the traditional sense, but rather a distribution. It has some unique properties, such as: $$ \delta(\lambda) = \begin{cases} +\infty, & \lambda = 0 \\ 0, & \lambda \neq 0 \end{cases} $$ and satisfies the identity: $$ \int_{-\infty}^{\infty} \delta(\lambda) d\lambda = 1. $$ The Dirac delta function is used to model a system where there is an impulse at a specific point, in this case, at \(\lambda = 0\).

Setting up the convolution integral

According to the problem, we need to find the convolution of \(f(t)\) and \(\delta(\lambda)\). The convolution can be represented as: $$ (f * \delta)(t) = \int_{0}^{t} f(t - \lambda) \delta(\lambda) d \lambda. $$

Use the "sifting" property of the Dirac delta function

One of the most important properties of the Dirac delta function is its "sifting" property, which states that: $$ \int_{-\infty}^{\infty} f(\lambda) \delta(\lambda - a) d\lambda = f(a). $$ This property can be used to compute the convolution integral. When the Dirac delta function is integrated with any other function, the result is simply the value of that function evaluated at the point where the Dirac delta function has an impulse (in this case \(\lambda = 0\)).

Compute the convolution integral

By applying the sifting property, we can simplify the convolution integral as follows: $$ (f * \delta)(t) = \int_{0}^{t} f(t - \lambda) \delta(\lambda) d \lambda = f(t - 0) = f(t). $$ So, the convolution of \(f(t)\) and the Dirac delta function \(\delta(\lambda)\) is simply \(f(t)\).

Key concepts

Convolution

Convolution is a mathematical operation used to blend two functions. It is widely used in signal processing, probability, and differential equations to analyze how one function affects another. The operation involves integrating the product of the two functions after one is reversed and shifted. This might sound a bit complex, but it's much like sliding one shape over another and computing the overlap.
For functions \(f(t)\) and \(g(t)\), the convolution is denoted by \((f * g)(t)\) and is defined as:

• \((f * g)(t) = \int_{-\infty}^{\infty} f(\tau) g(t - \tau) \, d\tau\)

In more practical terms, convolution can be used to determine how a system, represented by \(g(t)\), affects an input signal \(f(t)\).
The convolution operation has several useful properties, such as commutativity—meaning \(f * g = g * f\), associativity, and distributivity over addition. Understanding convolution is key to grasping advanced topics in engineering and physics, such as filtering and system response.

Sifting Property

The sifting property is a unique and crucial characteristic of the Dirac delta function. This property effectively "sifts out" a single value from a function. Mathematically, it is expressed as:

• \(\int_{-\infty}^{\infty} f(\lambda) \delta(\lambda - a) \, d\lambda = f(a)\).

The way it works is that the Dirac delta function spikes at a single point, \(\lambda = a\), with an infinite amplitude.
Everywhere else it is zero. Thus, when it multiplies another function, it "picks" the value of that function at the point \(\lambda = a\). This simplifies many complex integrals involving the Dirac delta function.

• It turns an integral into a simple function evaluation.
• This makes it a powerful tool for modeling systems with sudden impulses or point changes.

The sifting property is often employed in signal processing and systems theory to simplify the analysis of systems interacting with instantaneous impulses.

Distribution Theory

Distribution theory is a framework in mathematics that extends the concept of functions for more complex and generalized models. Traditional functions can't handle certain behaviors like impulses, jumps, or oscillations properly. Instead, distribution theory allows mathematicians and engineers to handle such phenomena seamlessly.
The Dirac delta "function" is one of the most famous examples of a distribution. Although it is commonly called a function, it is technically not one because it doesn't have a finite value at every point. It acts more like an operator that defines action in an integrated context.

• Distributions can be added, multiplied by numbers, and transformed just like functions.
• They elegantly handle mathematical operations involving discontinuities and impulses.

This theory is essential in fields like quantum mechanics, where wave functions often require impulse interpretations via delta functions. Distribution theory brings the necessary precision and flexibility to maintain the integrity of such complex systems.