Question

Consider the following transformation from x(t) to y(t) where

p(t) = ∑ k=−[infinity] -> [infinity].δ(t − k)
is an impulse train (also called ""Dirac comb"") and
x(t) = sin( πt/2 )/πt .

Compute and make a sketch of the FT X(jω) of x(t).

Answer 1

To compute the Fourier Transform of the given signal x(t) = sin(πt/2)/πt, we can use the well-known result for the sinc function, which shows that its Fourier Transform is a rectangular function centered at the origin with a width determined by the frequency component in the sinc function.

Step-by-step explanation:

The student has provided an expression for a signal x(t) and has asked to compute and sketch its Fourier Transform (FT), denoted as X(jω). The signal is x(t) = sin(πt/2)/πt. To find the Fourier Transform of this signal, one would use the definition of the Fourier Transform for continuous-time signals. However, this specific signal corresponds to the sinc function, whose Fourier Transform is well-known. The FT of a sinc function is a rectangular function. Therefore, without delving into the detailed integral computation, we know that the Fourier Transform of x(t) = sin(πt/2)/πt will be a rectangle centered at the origin with a width that corresponds to twice the frequency in the sinc function, which in this case corresponds to π/2 rad/s.

Answer 2

To find the Fourier Transform \( X(j\omega) \) of \( x(t) = \frac{\sin(\frac{\pi t}{2})}{\pi t} \), we can use the following formula:

\[ X(j\omega) = \int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt \]

Substitute \( x(t) \) into this expression and solve the integral:

\[ X(j\omega) = \int_{-\infty}^{\infty} \frac{\sin(\frac{\pi t}{2})}{\pi t} e^{-j\omega t} dt \]

This integral may require some complex analysis techniques or tables for trigonometric integrals to solve. Once you obtain \( X(j\omega) \), you can sketch its magnitude and phase.

Note that the presence of the impulse train \( p(t) \) (Dirac comb) in your question seems to be mentioned, but it's not directly involved in the computation of the Fourier Transform of \( x(t) \). If the Dirac comb is meant to modify the signal in some way, please provide additional details or clarify its role in the transformation.