Question

Consider the signals \( x[n] = 2^{n-1} u[-n + 2] \) and \( y[n] = 2^{-n+2} u[n + 1] \), where \( u[n] \) is the unit step sequence. Let \( X(e^{j\omega}) \) and \( Y(e^{j\omega}) \) be the discrete-time Fourier transform of \( x[n] \) and \( y[n] \), respectively. The value of the integral \[ \frac{1}{2\pi} \int_0^{2\pi} X(e^{j\omega}) Y(e^{-j\omega}) d\omega \, \text{(rounded off to one decimal place) is} _________. \]

Hint:

To compute the value of an integral involving Fourier transforms, use the convolution theorem and evaluate the result.

Answer 1

Correct Answer: 7.9

The value of the integral is the convolution of the signals \( x[n] \) and \( y[n] \). The continuous convolution formula is given by: \[ \frac{1}{2\pi} \int_0^{2\pi} X(e^{j\omega}) Y(e^{-j\omega}) d\omega = 7.9. \] Thus, the value of the integral is \( \boxed{7.9} \).