Question

For each of the signals below, plot x(t) as a function of time. Then, find their Fourier transform, X(f), and plot ∣X(f)∣ as a function of frequency. For your frequency domain plots, let the frequency axis be on a littear scale and let the vertical axis be in decibels (dB),

a. Square pulse −x(t) = rect(t − 1/2),
b. Half-sine pulse −x(t) = rect(t − 1/2)sin(πt),
c. Full cosine pulse −x(t) = rect(t − 1/2)[1/2​ − 1/2​ cos(2πt)].
The idea in this problem is to observe width of the main lobe and the behavior of the sidelobes of the spectrums as we tweak the pulse shape in the time domain.

Answer 1

The question requires plotting the time-domain signals and their Fourier transforms for a square pulse, half-sine pulse, and full cosine pulse. the Fourier transforms show varying main lobe and sidelobe behaviors, indicating how changes in time-domain pulse shape affect the frequency spectrum.

Step-by-step explanation:

The question involves analyzing three distinct time-domain signals, plotting them, and then determining and plotting their Fourier transforms in terms of magnitude on a linear frequency scale in decibels (dB).

Square Pulse

For x(t) = rect(t - 1/2), the square pulse is centered at t = 1/2 with a width of 1 unit. The Fourier transform X(f) of a square pulse is a sinc function, which constitutes the main lobe at the center and decaying sidelobes as the frequency increases.

Half-Sine Pulse

The half-sine pulse is given as x(t) = rect(t - 1/2)sin(\(\pi\)t). This signal is a product of a square pulse and a sine wave. Its Fourier transform will have a main lobe and sidelobes that are different from the square pulse due to the sine modulation.

Full Cosine Pulse

Lastly, for the full cosine pulse x(t) = rect(t - 1/2)(1/2 \(-\) 1/2 cos(2\(\pi\)t)), which is a square pulse modulated by a cosine wave, the Fourier transform will also depict sidelobes but with different characteristics than the previous two signals. for each Fourier transform, the magnitude |X(f)| will display different lobe widths and sidelobe behaviors, illustrating the impact of time-domain signal shaping on its frequency content.