Understanding the Fourier Transform of a Rectangular Pulse

The question asks us to identify the Fourier transform of a rectangular pulse. The Fourier transform is a mathematical tool that decomposes a function of time (or space) into its constituent frequencies.

Let's first define a rectangular pulse. A common way to represent a rectangular pulse centered at time \(t=0\) with amplitude \(A\) and duration \(\tau\) is:

\( f(t) = \begin{cases} A & \text{if } |t| \le \frac{\tau}{2} \\ 0 & \text{if } |t| > \frac{\tau}{2} \end{cases} \)

Alternatively, using the rectangular function notation:

\(f(t) = A \cdot \text{rect}\left(\frac{t}{\tau}\right)\)

The Fourier transform \(F(\omega)\) of a function \(f(t)\) is given by the integral:

\(F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-j\omega t} dt\)

Let's compute the Fourier transform of the rectangular pulse \(f(t)\) defined above:

\( F(\omega) = \int_{-\infty}^{\infty} A \cdot \text{rect}\left(\frac{t}{\tau}\right) e^{-j\omega t} dt \)

Since the rectangular function is non-zero only for \(-\frac{\tau}{2} \le t \le \frac{\tau}{2}\), the integral becomes:

\( F(\omega) = \int_{-\tau/2}^{\tau/2} A e^{-j\omega t} dt \)

We can pull the constant \(A\) out of the integral:

\( F(\omega) = A \int_{-\tau/2}^{\tau/2} e^{-j\omega t} dt \)

Now, integrate with respect to \(t\):

\( F(\omega) = A \left[ \frac{e^{-j\omega t}}{-j\omega} \right]_{-\tau/2}^{\tau/2} \)

\( F(\omega) = \frac{A}{-j\omega} \left( e^{-j\omega \tau/2} - e^{j\omega \tau/2} \right) \)

We can rewrite \((e^{-j\theta} - e^{j\theta})\) using Euler's formula, \(e^{j\theta} = \cos\theta + j\sin\theta\), which gives \(e^{-j\theta} - e^{j\theta} = (\cos\theta - j\sin\theta) - (\cos\theta + j\sin\theta) = -2j\sin\theta\). Letting \(\theta = \omega\tau/2\), we get \((e^{-j\omega \tau/2} - e^{j\omega \tau/2}) = -2j\sin(\omega\tau/2)\).

Substitute this back into the expression for \(F(\omega)\):

\( F(\omega) = \frac{A}{-j\omega} (-2j\sin(\omega\tau/2)) \)

\( F(\omega) = A \frac{2j\sin(\omega\tau/2)}{j\omega} \)

\( F(\omega) = A \frac{2\sin(\omega\tau/2)}{\omega} \)

To relate this to the sinc function, recall the definition of the normalized sinc function:

\( \text{sinc}(x) = \frac{\sin(\pi x)}{\pi x} \)

And the unnormalized sinc function (often used in signal processing):

\( \text{sinc}(x) = \frac{\sin(x)}{x} \)

Our result is \(F(\omega) = A \frac{2\sin(\omega\tau/2)}{\omega}\). We can rewrite the denominator \(\omega\) as \(\frac{2(\omega\tau/2)}{\tau}\).

\( F(\omega) = A \frac{\sin(\omega\tau/2)}{(\omega\tau/2)/\tau} = A\tau \frac{\sin(\omega\tau/2)}{\omega\tau/2} \)

This matches the form of the unnormalized sinc function, \(\frac{\sin(x)}{x}\), where \(x = \omega\tau/2\). Therefore, the Fourier transform is \(F(\omega) = A\tau \text{sinc}(\omega\tau/2)\).

This result shows that the Fourier transform of a rectangular pulse is a sinc function.

Analyzing the Options for the Fourier Transform

Let's consider why the other options are incorrect:

• Another rectangular pulse: The Fourier transform of a rectangular pulse in the time domain is not another rectangular pulse in the frequency domain. This is a key Fourier transform pair, and they are different shapes.
• Triangular pulse: The Fourier transform of a triangular pulse is a squared sinc function. Conversely, the Fourier transform of a sinc function is a rectangular pulse.
• Impulse function: The Fourier transform of a constant (DC signal) is an impulse function at \(\omega=0\). The Fourier transform of an impulse function in the time domain is a constant in the frequency domain. A rectangular pulse is not an impulse or a constant.
• Sinc function: As derived above, the Fourier transform of a rectangular pulse is indeed a sinc function.

Based on the derivation and the properties of Fourier transforms, the Fourier transform of a rectangular pulse is a sinc function.

Revision Table: Fourier Transform Fundamentals

Additional Information: Properties and Applications

The relationship between the rectangular pulse and the sinc function in the Fourier domain is very important in signal processing and communications. Here are a few related points:

• Duality: There is a duality property in Fourier transforms. If \(f(t) \leftrightarrow F(\omega)\), then \(F(t) \leftrightarrow 2\pi f(-\omega)\). The Fourier transform of a sinc function in the time domain is a rectangular pulse in the frequency domain, which is consistent with the result found.
• Spectral Content: The sinc function in the frequency domain shows the distribution of frequencies present in the rectangular pulse. The main lobe of the sinc function contains most of the energy, and its width is inversely proportional to the duration \(\tau\) of the rectangular pulse. A wider rectangular pulse (\(\tau\) is large) has a narrower sinc function main lobe (smaller bandwidth), and a narrower rectangular pulse (\(\tau\) is small) has a wider sinc function main lobe (larger bandwidth). This illustrates the time-bandwidth product concept.
• Filtering: The sinc function is related to the ideal low-pass filter's impulse response. An ideal low-pass filter has a rectangular frequency response, and its inverse Fourier transform (the impulse response) is a sinc function.

Understanding these fundamental Fourier transform pairs is crucial for analyzing and designing systems in various engineering fields.