Question

You are given a single "pulse" (height =10 and width W ) as defined below.

x(t)=10 for −0.5W≤t≤0.5W
x(t)=0 for all other t
​a. Sketch this pulse.
b. Is this a periodic signal? Do you need Fourier series expansion or Fourier transform for this signal?
c. Using applicable technique you decided in problem 4-b above and suitable formula from "Table of Fourier Transform Pairs" or textbook, determine the spectrum of this signal in f or ω. Simplify the results.

Answer 1

The given pulse is not periodic and thus requires a Fourier transform for analysis. The sketch would show a rectangle with height 10 and width W. The spectrum of the signal is a sinc function that represents the frequency content of the pulse.

Step-by-step explanation:

The question requires us to analyze a signal represented by a rectangular pulse. Firstly, to sketch this pulse, we would draw a rectangle centered at the origin t = 0 on the time axis; its amplitude is 10 units high, and it extends from −0.5W to 0.5W on the time axis. Beyond these points, the function is 0.

Upon analyzing the signal, we can determine that this is not a periodic signal because it does not repeat at regular intervals. Thus, we would use the Fourier transform instead of the Fourier series to analyze this signal, as the Fourier series is applicable for periodic functions.

To find the spectrum of this signal, we would apply the Fourier transform, typically using the formula for a rectangular pulse, which is:

X(f) = 10W × sinc(πWf)

where ‘sinc’ represents the cardinal sine function, which is defined as sinc(x) = sin(x)/x. The result is a sinc function that shows the frequency content or spectrum of the pulse in terms of frequency (f) or angular frequency (ω).