PYQs Based Test - 03 : Signals & Systems

Let u(t) denote the unit step function. The bilateral Laplace transform of the function f(t) = e${}^t$ u(−t) is ____.

Consider a causal and stable LTI system with rational transfer function H(z), whose corresponding impulse response begins at n = 0. Furthermore, $H\left(1\right)=\frac{5}{4}$. The poles of H(z) are $p_k=\frac{1}{\sqrt{2}}\exp \left(j\frac{\left(2k-1\right)}{4}\pi \right)$ for k = 1, 2, 3, 4. The zeros of H(z) are all at z = 0. Let g[n] = jⁿh[n]. The value of g[8] equals __________. (Give the answer up to three decimal places.)

For the sequence x[n] = {1, -1, 1, -1}, with n = 0, 1, 2, 3, the DFT is computed as $X\left(k\right)=\sum _{n=0}^{3}x\left[n\right]e^{-j\frac{2\pi }{4}nk}$, for k = 0, 1, 2, 3. The value of k for which X(k) is not zero is

The output of a continuous-time system y(t) is related to its input x(t) as y(t) = x(t) + $\frac{1}{2}$ x(t − 1). If the Fourier transforms of x(t) and y(t) are X(ω) and Y(ω) respectively, and |X(0)|² = 4, the value of |Y(0)|² is ______

Let f(t) be an even function i.e. f(-t) = f(t) for all t. Let the Fourier transform of f(t) be defined as $F(\omega )={\int }_{-\infty }^{\infty }f(t)e^{-j\omega t}dt$. Suppose $\frac{dF(\omega )}{d\omega }=-\omega F(\omega )$ for all ω, and F(0) = 1. Then