11.7.2. Fourier Transform of Exponential Decay

Let f(t)=e−atu(t), where a>0, u(t) is the unit step function.

This chunk introduces the exponential decay function, denoted as f(t). The expression f(t) = e^{-at}u(t) indicates that the function decays exponentially over time. Here, 'a' is a positive constant that determines the rate of decay, and 'u(t)' is the unit step function, which ensures that the function is defined and starts from zero at t < 0. This means f(t) is 0 for negative time and follows the exponential decay for t ≥ 0.

Imagine a tank of water where the water level decreases over time due to a small hole at the bottom. If we let 'a' represent the speed at which the water leaks out, the water level (analogous to f(t)) will decay exponentially after the hole is created. Before the hole is made, the water level remains unchanged (u(t) = 0), but once it’s opened, the level starts to decrease (u(t) = 1).

Z ∞ Z ∞ F(ω)= e−ate−iωtdt= e−(a+iω)tdt 0 0

In this part, we calculate the Fourier Transform of the function f(t). The initial expression combines the exponential decay term with a complex exponential, e^{-iωt}, which represents a rotating vector in the complex plane. As we integrate from 0 to infinity (the limits of the integral), we simplify our expression into e^{-(a+iω)t}. This transformation allows us to consider both the decay and oscillation in the frequency domain.

Think of tuning a musical instrument. The sound emitted is complex and changes over time, similar to the decay we observe in the function. When we analyze the sound waves using Fourier Transform, we can break down the complex sounds (like e^{-iωt}) into base frequencies, just like analyzing how the water level changes from its original state.

F(ω)= (e−(a+iω)t | ∞ 1 −(a+iω) 0

This chunk outlines the result of integrating the function we derived. When substituting the limits, the term e^{-(a+iω)t} approaches zero as t approaches infinity because of the exponential decay. Thus, we are left with the expression F(ω) = 1/(a+iω). This result shows how the frequencies (ω) interact with our decay rate (a) and gives us the final Fourier Transform of the signal.

Returning to our tank analogy, think of how once the hole is open, the water quickly drains away. After a while, there’s almost no water left—this concept mirrors the behavior of our function as time goes to infinity. The remaining factors show how quickly things change in relation to how they decay (a) and how they oscillate (ω), akin to persistent sound fainter over time as it dissipates.

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