11.8. Example Problems

Example 1: Find the Fourier Transform of f(t)=e−2tu(t)

Z ∞ Z ∞ 1 F(ω)= e−2te−iωtdt= e−(2+iω)tdt= 2+iω 0 0

In this example, we are tasked with finding the Fourier transform of the function f(t) = e^(-2t)u(t). The u(t) indicates that this function is defined for t >= 0 and is zero for t < 0. To find the Fourier transform, we set up the integral of the function multiplied by e^(-iωt) over the limits from 0 to infinity. This leads to the expression e^(-(2+iω)t) evaluated from 0 to ∞. When we solve this integral, we manage to evaluate it by recognizing that as t approaches infinity, the exponential term converges to zero, simplifying our calculations. The answer is then represented as 1/(2+iω).

Think of this as trying to capture the 'sound' or 'frequency' signature of an electronic signal that decays over time, like a speaker gradually silencing after being switched off. The Fourier transform tells us how much of each frequency is present in that signal, and in this case, the function's decay (modulated by e^(-2t)) translates to how quickly sounds fade away.

Example 2: Find the Fourier Sine Transform of f(t)=e−at, a>0

r2 Z ∞ F (ω)= e−atsin(ωt)dt s π 0 This is a standard integral: r2 ω F (ω)= · s π a2+ω2

In this example, we are calculating the Fourier Sine Transform of the function f(t) = e^(-at), where a > 0. To find the transform, we set up the integral of e^(-at) multiplied by sin(ωt) over the limits from 0 to infinity. This results in a standard form integral that, when solved, yields a formula for the Fourier Sine Transform reflecting how frequencies relate to the decaying exponential function.

Imagine a dampened pendulum, where the speed of oscillation decreases over time due to friction. As the pendulum slows, it never truly stops; similarly, this transform captures the frequency components as the pendulum's oscillation fades away, represented mathematically by the exponential decay factor.

Example 3: Use the time-scaling property to find the Fourier Transform of f(t)=rect(2t)

We know:

(cid:16)ω(cid:17) F{rect(t)}=sinc 2

Using time scaling: f(at)⇒ 1 F (cid:0)ω(cid:1) |a| a

1 (cid:16)ω(cid:17) F{rect(2t)}= ·sinc 2 4

In this example, we're applying the time-scaling property of the Fourier Transform to the rectangular function defined as rect(2t). The time-scaling property states that if we scale a function in time (by a factor of 'a'), it affects the frequency domain representation by a certain factor. This leads us to determine that the Fourier Transform of the scaled rectangle function results in a sinc function, revealing how the rectangle's width and height transform within the frequency domain.

Think of scaling a graphic that represents sound waves. If you stretch or compress the time axis of the sound waves (like speeding up or slowing down music), the resulting sound (or frequency representation) changes as well. This example allows us to understand how actions affecting time directly influence what we hear, represented mathematically through the sinc function.

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