Fourier Transform of a Rectangular Pulse

The Fourier Transform is a mathematical tool used to transform a function from the time domain to the frequency domain. It provides a way to analyze the frequency components present in a given signal. The Fourier Transform of a rectangular pulse is a sinc function.

Rectangular Pulse

A rectangular pulse, also known as a rectangular function, is a function that has a constant value of 1 over a certain interval and 0 elsewhere. Mathematically, it can be represented as:

rect(t) = 1, -T/2 <= t=""><=>
rect(t) = 0, otherwise

where T is the width of the pulse.

Fourier Transform

The Fourier Transform of a function f(t) is defined as:

F(w) = ∫[f(t) * e^(-jwt)] dt

where F(w) represents the Fourier Transform of f(t), and e^(-jwt) is the complex exponential function.

Fourier Transform of a Rectangular Pulse

To find the Fourier Transform of a rectangular pulse, we substitute the rectangular pulse function rect(t) into the Fourier Transform equation:

F(w) = ∫[rect(t) * e^(-jwt)] dt

We can simplify this by splitting the integral into two parts, over the intervals where the rectangular pulse is non-zero:

F(w) = ∫[1 * e^(-jwt)] dt, -T/2 <= t=""><=>
= ∫[e^(-jwt)] dt, -T/2 <= t=""><=>

The integral of the complex exponential function e^(-jwt) can be evaluated using standard mathematical techniques. The result is a sinc function:

F(w) = (1/jw) * [e^(-jwt)] , -T/2 <= t=""><=>
= (1/jw) * [e^(-jwt)] , -T/2 <= t=""><=>
= sinc(wT/2)

where sinc(x) = sin(x)/x.

Answer: Option C - Sinc Function

Therefore, the Fourier Transform of a rectangular pulse is a sinc function. The sinc function is a fundamental function in signal processing and is used to characterize the frequency response of systems and analyze the frequency content of signals.