PYQs Based Test - 04 : Signals & Systems

Consider a continuous-time signal defined as:

$x\left(t\right)=\left(\frac{\sin \left(\frac{\pi t}{2}\right)}{\left(\frac{\pi t}{2}\right)}\right)\ast \sum _{n=-\infty }^{\infty }\delta \left(t-10n\right)$

where ‘∗’ denotes the convolution operation and t is in seconds. The Nyquist sampling rate (in samples/sec) for x(t) is ______.

The outputs of four systems (S₁, S₂, S₃, and S₄) corresponding to the input signal sin(t), for all time t, are shown in the figure.

Based on the given information, which of the four systems is/are definitely NOT LTI (linear and time-invariant)?

A continuous time signal x(t) = 4 cos(200πt) + 8 cos(400πt), where t is in seconds, is the input to a linear time invariant (LTI) filter with the impulse response

h≤ft( t ) = ≤ft\{ {*20c {{{2≤ft( 300π t )}}π t}&t ≠ 0\\ 600,&t = 0 } .

Let y(t) be the output of this filter. The maximum value of |y(t)| is ________.

A finite duration discrete-time signal x[n] is obtained by sampling the continuous-time signal x(t) = cos (200πt) at sampling instants t = n/400, n = 0, 1, …, 7. The 8-point discrete Fourier transform (DFT) of x[n] is defined as:

$X\left[k\right]=\sum _{n=0}^{7}x\left[n\right]e^{-j\frac{\pi kn}{4}},k=0,1,\dots ,7.$

Which one of the following statements is TRUE?

Let h[n] be the impulse response of a discrete-time linear time invariant (LTI) filter. The impulse response is given by

$h\left[0\right]=\frac{1}{3};h\left[1\right]=\frac{1}{3};h\left[2\right]=\frac{1}{3};$ h[n] = 0 for n < 0 and n > 2

Let H(ω) be the discrete-time Fourier transform (DTFT) of h[n]. where ω is the normalized angular frequency in radians. Given that H(ω₀) = 0 and 0 < ω₀ < π, the value of ω₀ (in radians) is equal to ________.