GATE 2023 Signals and Systems Questions and Solutions

The period \(N\) of the discrete-time signal below is ____ (nearest integer):

Solution

For a discrete sinusoid of frequency \(\omega\), the period is the smallest integer \(N=\tfrac{2\pi}{\omega}k\). For \(\omega_1=\tfrac{15\pi}{8}\): \(N_1=\tfrac{16k}{15}\Rightarrow N_1=16\) (with \(k=15\)). For \(\omega_2=\tfrac{\pi}{3}\): \(N_2=6\). The overall period is

For the signals \(x(t)\) and \(y(t)\) shown, \(z(t)=x(t)*y(t)\) is maximum at \(t=T_1\). Then \(T_1\) in seconds is ____ (nearest integer).

Solution

The convolution \(z(t)=\int y(\tau)x(t-\tau)\,d\tau\) is largest when the flipped-and-shifted copy of \(x\) maximally overlaps \(y\). Aligning the leading edges, the overlap is greatest when \(-1+t=3\), i.e. \(t=4\) s.

The DTFT of \(x[n]\) is \(X(\Omega)=(1+\cos\Omega)e^{-j\Omega}\). A periodic signal of period \(N=5\) is built with \(x_p[n]=x[n]\) for \(n=0,1,2\) and \(0\) for \(n=3,4\), with \(x_p[n]=\sum_{k=0}^{N-1}a_k e^{j(2\pi/N)kn}\). The magnitude \(|a_3|\) is ____ (3 decimal places).

Solution

Sampling the DTFT of one period, \(a_k=\tfrac{1}{N}X\!\left(\tfrac{2\pi k}{N}\right)\). Hence

The Fourier transform of \(x(t)\) is \(X(\omega)=1\) for \(|\omega|W_0\). Which statement is true?

1. \(x(t)\) tends to an impulse as \(W_0\to\infty\)
2. \(x(0)\) decreases as \(W_0\) increases
3. At \(t=\tfrac{\pi}{2W_0}\), \(x(t)=-\tfrac{1}{\pi}\)
4. At \(t=\tfrac{\pi}{2W_0}\), \(x(t)=\tfrac{1}{\pi}\)

Solution

Inverting the rectangular spectrum gives \(x(t)=\dfrac{\sin W_0 t}{\pi t}=\dfrac{W_0}{\pi}\operatorname{Sa}(W_0t)\). Then \(x(0)=W_0/\pi\) grows with \(W_0\) (so B is false). As \(W_0\to\infty\), \(X(\omega)\to1\) for all \(\omega\), whose inverse transform is \(\delta(t)\) — statement (A) is true.

Which of the following statement(s) is/are true? (Multiple-select)

1. If an LTI system is causal, it is stable.
2. A discrete-time LTI system is causal if and only if its step response is \(0\) for \(n<0\).
3. If a discrete-time LTI system has a finite-duration impulse response, the system is stable.
4. If \(0<|h[n]|<1\) for all \(n\), then the LTI system is stable.

Solution

(A) is false — causality does not imply stability. (B) is true: \(h[n]=0\) for \(n<0\) (causality) is equivalent to step response \(s[n]=\sum_{k\le n}h[k]=0\) for \(n<0\), since \(h[n]=s[n]-s[n-1]\). (C) is true: a finite-duration \(h[n]\) gives a finite sum \(\sum|h[n]|<\infty\), so it is BIBO stable. (D) is false: \(|h[n]|>0\) for all \(n\) allows non-summable cases (e.g. \(h[n]=\tfrac12\) for all \(n\)).

Note: this is a multiple-select question; the supplied solution discussed only statement (B). Both (B) and (C) are correct.

The signal \(x(t)=2\cos(180\pi t)\cos(60\pi t)\) is sampled at \(200\) Hz and passed through an ideal low-pass filter with cut-off \(100\) Hz. The maximum frequency present in the filtered signal (Hz) is ____ (nearest integer).

Solution

Using \(2\cos A\cos B=\cos(A+B)+\cos(A-B)\), \(x(t)=\cos240\pi t+\cos120\pi t\), i.e. tones at \(120\) Hz and \(60\) Hz. Sampling at \(f_s=200\) Hz produces images at \(|nf_s\pm f|\):

Below the \(100\) Hz cut-off only \(60\) Hz and \(80\) Hz survive, so the maximum frequency is \(80\) Hz.

The \(z\)-transform of \(x[n]\) is \(X(z)=\dfrac{4z}{\left(z-\tfrac13\right)\left(z-\tfrac23\right)(z-3)}\) with ROC \(R\). Which statement is true?

1. The DTFT of \(x[n]\) converges if \(R\) is \(|z|>3\)
2. The DTFT of \(x[n]\) converges if \(R\) is \(\tfrac23<|z|<3\)
3. The DTFT converges if \(R\) is such that \(x[n]\) is left-sided
4. The DTFT converges if \(R\) is such that \(x[n]\) is right-sided

Solution

The DTFT exists only when the ROC contains the unit circle \(|z|=1\). Among the pole-bounded annuli, only \(\tfrac23<|z|<3\) includes \(|z|=1\).