GATE 2015 Signals and Systems Questions and Solutions

A moving-average function is given by \(y(t)=\dfrac{1}{T}\displaystyle\int_{t-T}^{t}u(\tau)\,d\tau\). If the input \(u\) is a sinusoid of frequency \(\dfrac{1}{2T}\) Hz, then in steady state the output \(y\) lags \(u\) (in degrees) by (numerical):

Solution

Let \(u(t)=\sin(\omega_0 t)\) with \(\omega_0=2\pi\cdot\tfrac{1}{2T}=\pi/T\). Integrating,

Since \(\omega_0 T=\pi\), \(\cos(\omega_0 t-\pi)=-\cos(\omega_0 t)\), giving \(y(t)=-\dfrac{2}{\pi}\cos(\omega_0 t)=\dfrac{2}{\pi}\sin(\omega_0 t-90^\circ)\). The output lags by \(90^\circ\).

The impulse response \(g(t)\) of a system \(G\) is shown in Figure (a). What is the maximum value attained by the impulse response of two cascaded blocks of \(G\) shown in Figure (b)?

1. \(\tfrac{2}{3}\)
2. \(\tfrac{3}{4}\)
3. \(\tfrac{4}{5}\)
4. \(1\)

Solution

Here \(g(t)=u(t)-u(t-1)\), a unit rectangular pulse of width 1. The cascade impulse response is \(g(t)*g(t)\), a triangular pulse on \((0,2)\) peaking at \(t=1\):

The signum function is \(\operatorname{sgn}(x)=x/|x|\) for \(x\ne0\) and \(0\) at \(x=0\). The Fourier series expansion of \(\operatorname{sgn}(\cos t)\) has:

1. Only sine terms with all harmonics
2. Only cosine terms with all harmonics
3. Only sine terms with even harmonics
4. Only cosine terms with odd harmonics

Solution

\(\operatorname{sgn}(\cos t)\) is an even function, so its series contains only cosine terms. It has half-wave symmetry (changes sign every half period), which eliminates all even harmonics. Hence only odd cosine harmonics remain.

Consider \(x[n]=(-0.25)^n u[n]+(0.5)^n u[-n-1]\). The region of convergence of its Z-transform is:

1. Inside the circle of radius 0.5
2. Outside the circle of radius 0.25
3. Annular region between radii 0.25 and 0.5
4. Entire Z-plane

Solution

The right-sided term \((-0.25)^n u[n]\) has ROC \(|z|>0.25\); the left-sided term \((0.5)^n u[-n-1]\) has ROC \(|z|<0.5\). The combined ROC is the intersection \(0.25<|z|<0.5\).

The Laplace transform of \(f(t)=2\sqrt{\dfrac{t}{\pi}}\) is \(s^{-3/2}\). The Laplace transform of \(g(t)=\dfrac{1}{\sqrt{\pi t}}\) is:

1. \(\tfrac{3}{2}s^{-5/2}\)
2. \(s^{-1/2}\)
3. \(s^{1/2}\)
4. \(s^{3/2}\)

Solution

Note that \(g(t)=\dfrac{1}{\sqrt{\pi t}}=\dfrac{d}{dt}\!\left(2\sqrt{\dfrac{t}{\pi}}\right)=f'(t)\). With \(f(0)=0\), the derivative property gives

Note: the OCR of this problem dropped the square-root signs, printing \(f(t)\) as "\(2t/\pi\)" and \(g(t)\) as "\(1/\pi t\)". Those forms are inconsistent with the stated transform \(s^{-3/2}\); the intended functions are \(2\sqrt{t/\pi}\) and \(1/\sqrt{\pi t}\), restored above. The answer (B) is unaffected.

Consider \(x(t)=e^{j10t}\) for \(|t|\le1\) and \(0\) for \(|t|>1\). Its Fourier transform is:

1. \(\dfrac{2\sin(\omega-10)}{\omega-10}\)
2. \(2e^{j10}\dfrac{\sin(\omega-10)}{\omega-10}\)
3. \(\dfrac{2\sin\omega}{\omega-10}\)
4. \(e^{j10\omega}\dfrac{2\sin\omega}{\omega}\)

Solution

This is the rectangular-pulse sinc spectrum frequency-shifted to centre at \(\omega=10\).

For LTI systems that are Bounded-Input Bounded-Output (BIBO) stable, which statement is TRUE?

1. The impulse response will be integrable, but may not be absolutely integrable.
2. The unit impulse response will have finite support.
3. The unit step response will be absolutely integrable.
4. The unit step response will be bounded.

Solution

BIBO stability requires the impulse response to be absolutely integrable. Since the unit step is a bounded input, the corresponding step response must be bounded. (It need not be absolutely integrable — e.g. it can settle to a nonzero constant.)

The Z-transform of a sequence \(x[n]\) is \(X(z)=2z+4-\dfrac{4}{z}+\dfrac{3}{z^2}\). If \(y[n]\) is the first difference of \(x[n]\), then \(Y(z)\) is:

1. \(2z+2-\tfrac{8}{z}+\tfrac{7}{z^2}-\tfrac{3}{z^3}\)
2. \(-2z+2-\tfrac{6}{z}-\tfrac{1}{z^2}-\tfrac{3}{z^3}\)
3. \(-2z+2-\tfrac{8}{z}-\tfrac{7}{z^2}-\tfrac{3}{z^3}\)
4. \(4z-2-\tfrac{8}{z}+\tfrac{7}{z^2}-\tfrac{3}{z^3}\)

Solution

The first difference \(y[n]=x[n]-x[n-1]\) gives \(Y(z)=(1-z^{-1})X(z)\):

The discrete-time equations from numerical integration of an undamped simple harmonic oscillator are

For this system, which statement is TRUE?

1. Stable for \(h>0\)
2. Stable for \(h>\tfrac{1}{\pi}\)
3. Stable for \(0
4. Stable for \(\tfrac{1}{2\pi}

Solution

The official GATE answer is (A). The state-update matrix is \(\begin{bmatrix}1&h\\-h&1\end{bmatrix}\), whose eigenvalues are \(1\pm jh\); the scheme is treated as a stable (bounded-oscillation) integrator for \(h>0\).