10.2. Laplace Transform of the Dirac Delta Function

Let us compute the Laplace Transform of δ(t - a), where 𝑎 ≥ 0:

∞ ℒ{𝛿(𝑡−𝑎)} = ∫ 𝛿(𝑡−𝑎)𝑒^{−𝑠𝑡} 𝑑𝑡 0

Using the sifting property, we get: ℒ{𝛿(𝑡−𝑎)} = 𝑒^{−𝑎𝑠}, 𝑎 ≥ 0

The Laplace Transform is a powerful mathematical tool used to analyze systems, especially in engineering fields. We start by computing the Laplace Transform of the Dirac Delta Function, denoted as δ(t - a). In this context, 'a' represents the point in time where the impulse occurs. The formula for the Laplace Transform is defined as the integral of the function multiplied by an exponential decay factor e^{-st} from 0 to infinity. By applying the sifting property of the Dirac Delta Function, which allows us to focus on the value of the function at the specific point where the impulse occurs, we can simplify our calculations. This leads us to the result that the Laplace Transform of δ(t - a) is e^{-as} for non-negative values of 'a'.

Imagine a microphone capturing the sound of a single clap (the impulse). The Dirac Delta Function models this clap, which happens instantaneously. Using the Laplace Transform is like turning this instantaneous sound into a smooth waveform that we can analyze, such as understanding how the echo behaves over time.

When 𝑎 = 0, the Laplace Transform becomes:

∞ ℒ{𝛿(𝑡)} = ∫ 𝛿(𝑡)𝑒^{−𝑠𝑡} 𝑑𝑡 = 𝑒^{−𝑠⋅0} = 1 0

In the case where 'a' equals zero, we consider the Dirac Delta Function at the origin. The formula simplifies considerably because we are looking at the impulse occurring at the starting point in time. The Laplace Transform of δ(t) can be computed using the same integral formula, leading to the calculation of e^{0}, which equals 1. This result highlights that the impulse at the beginning of time has a constant transform of 1 in the Laplace domain.

Think of throwing a stone into a calm pond. The immediate point where the stone hits the water can be seen as δ(t) at t=0, causing ripples (impulses). The initial impact creates a single, definite moment—captured as 1 in our analysis. This is like saying the action of throwing the stone is fundamental and always has an effect.

• The function δ(t − a) represents an impulse of infinite height and zero width at time t = a. • Its area under the curve is always 1, regardless of the location of the impulse. • In Laplace domain, it is transformed into an exponential decay term 𝑒^{−𝑎𝑠}.

Graphically, the Dirac Delta Function is represented as a spike at time t = a, with an infinite height and no width. This is a bit abstract, but it allows us to capture the concept of an instantaneous event effectively. Despite appearing as an infinitely tall spike, the area under this spike is precisely 1, which conveys that the total 'impulse' effect remains consistent across different time stamps. When we perform the Laplace Transform, this spike turns into an exponential decay factor, which is more manageable for analysis in engineering applications.

Consider a firecracker that explodes instantaneously at a specific moment, producing a loud sound. The firecracker's explosion can be thought of as the Dirac Delta Function—a burst of noise occurring at one point in time with immediate aftermath. The total noise produced (the area under the curve) is unusually captured as '1' in our modeling, and when analyzed through Laplace, we can find its effects over time, showing how the sound dissipates.

Example 1: Laplace Transform of δ(t - 3) ℒ{𝛿(𝑡−3)} = 𝑒^{−3𝑠}

Example 2: Laplace Transform of 5δ(t - 2) ℒ{5𝛿(𝑡−2)} = 5𝑒^{−2𝑠}

These examples illustrate the application of the Laplace Transform to specific instances of the Dirac Delta Function. In Example 1, we compute the Laplace Transform of the function δ(t - 3). Here, the impulse occurs at t = 3, leading us to the result e^{-3s}, which reflects the timing of that impulse in the transformed domain. Example 2 scales the impulse by a factor of 5, which adds a multiplicative factor to the output: 5e^{-2s} when the impulse occurs at t = 2. The computed results showcase how different timing and scaling of impulses affect their representation in the Laplace domain.

If you think of a train arriving at a station, each train car represents an impulse. If three cars arrive simultaneously at 3 seconds, that’s a '3' in our calculations. If a train with multiple cars (5) arrives at 2 seconds, it similarly affects our model—adding weight to what we observe over time. We thus use these calculations to grasp the timing and magnitude of impacts and how they propagate.

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