GATE 2019 Signals and Systems Questions and Solutions

The symbols \(a\) and \(T\) represent positive quantities, and \(u(t)\) is the unit step. Which one of the following is NOT a valid impulse response of a causal LTI system?

1. \(e^{+at}u(t)\)
2. \(e^{-a(t+T)}u(t)\)
3. \(1+e^{-at}u(t)\)
4. \(e^{-a(t-T)}u(t)\)

Solution

Causality requires \(h(t)=0\) for \(t<0\). Options A, B and D are all multiplied by \(u(t)\), so they vanish for \(t<0\) (growth or decay does not affect causality). In option C, the constant term \(1\) is present for all \(t\), including \(t<0\), so \(h(t)\ne0\) for \(t<0\) — this is non-causal.

A periodic function \(f(t)\) of period \(2\pi\) is written as \(f(t)=a_0+\sum a_n\cos nt+\sum b_n\sin nt\), with \(f(t)=A\sin t\) for \(0\le t\le\pi\) and \(0\) for \(\pi

1. \(a_1=\tfrac{4}{\pi},\ b_1=0\)
2. \(a_1=\tfrac{A}{2},\ b_1=0\)
3. \(a_1=0,\ b_1=\tfrac{A}{\pi}\)
4. \(a_1=0,\ b_1=\tfrac{A}{2}\)

Solution

With \(\omega_0=1\):

Note: the OCR corrupted several option values (writing "A" as "4" or "Δ"). The computed result \(a_1=0,\ b_1=A/2\) corresponds to option (D).

The output response \(y(t)\) has Laplace transform \(Y(s)=\dfrac{10}{s\left(s^{2}+s+100\sqrt2\right)}\). The steady-state value of \(y(t)\) is:

1. \(\dfrac{1}{10\sqrt2}\)
2. \(10\sqrt2\)
3. \(\dfrac{1}{100\sqrt2}\)
4. \(100\sqrt2\)

Solution

By the final value theorem,

A transfer function is \(H(s)=\dfrac{a_1 s^2+b_1 s+c_1}{a_2 s^2+b_2 s+c_2}\). If \(a_1=b_1=0\) and all other coefficients are positive, it represents a:

1. low-pass filter
2. high-pass filter
3. band-pass filter
4. notch filter

Solution

With \(a_1=b_1=0\), \(H(s)=\dfrac{c_1}{a_2 s^2+b_2 s+c_2}\). At \(s=0\), \(H(0)=c_1/c_2\) (finite, nonzero); as \(s\to\infty\), \(H\to0\). A response that is constant at DC and rolls off at high frequency is a low-pass filter.

The inverse Laplace transform of \(H(s)=\dfrac{s+3}{s^{2}+2s+1}\) for \(t\ge0\) is:

1. \(3te^{-t}+e^{-t}\)
2. \(3e^{-t}\)
3. \(2te^{-t}+e^{-t}\)
4. \(4te^{-t}+e^{-t}\)

Solution