Find, using (12.2.2), the Fourier transform \(F(\omega)\) of a single rectangular pulse of amplitude \(a\) extending from \(-\tau\) to \(+\tau\). Substitute this \(F(\omega)\) in (12.2.1) and compare the resulting expression with (12.1.8), obtained from the Fourier series expansion of a sequence of rectangular pulses.

Question: Find the Fourier Transform of a single rectangular pulse of amplitude "a" extending from -τ to +τ and compare the resulting expression with equation (12.1.8). Solution: 1. Understand the given pulse function: The pulse function can be defined as $$ f(t) = \begin{cases} a & \text{if } -\tau \leq t \leq \tau \\ 0 & \text{otherwise} \end{cases} $$ 2. Define the Fourier Transform Formula: The formula for the Fourier Transform is $$ F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-j\omega t} dt $$ 3. Compute the Fourier Transform: Using the pulse function and the Fourier Transform formula, we can calculate the Fourier Transform as $$ F(\omega) = \frac{2a}{\omega}\sin(\omega\tau) $$ 4. Substitute this F(ω) in (12.2.1) and compare with (12.1.8): $$ f(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} \frac{2a}{\omega}\sin(\omega\tau)e^{j\omega t} d\omega $$ We cannot directly compare this with equation (12.1.8) as it is not provided in the problem statement. You must refer to your textbook or source to find equation (12.1.8) and make the necessary comparison.