GATE EC Continuous-time Signals questions

Practise Continuous-time Signals questions for GATE EC. Explore official PYQs and original topic-wise quizzes, check answers and view available solutions. Free, no sign-in.

Identify whether the signal and system are continuous-time or discrete-time. Check linearity, time invariance and stability before selecting a transform or response method.

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SubjectTopicYearType

Consider a continuous-time signal defined as

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

’\ast’

denotes the convolution operation and

t

is in seconds. The Nyquist sampling rate

\text{(in samples/sec)}

x(t)

M(f)

is the Fourier transform of the message signal

m(t)

A = 100

B = 40

v(t)= \cos (2\pi f_{c}t)

w(t)= \cos (2\pi (f_{c}+A)t)

f_{c}>A

. The cutoff frequencies of both the filters are

f_{c}

. The bandwidth of the signal at the output of the modulater (in Hz) is ______.

For a periodic signal

v(t) = 30\sin100\:t + 10\cos300\:t + 6\sin(500\:t+\pi/4),

the fundamental frequency in

rad/s

A band-limited signal with a maximum frequency of

5\: kHz

is to be sampled. According to the sampling theorem, the sampling frequency which is not valid is

g(t) = e^{-\pi t^{2}},

h(t)

is a filter matched to

g(t).

g(t)

is applied as input to

h(t),

then the Fourier transformation of the output is

The Fourier transform of a signal

h(t)

H(j\omega)=(2\cos\omega)(\sin2\omega)/\omega

h(0)

Four messages band limited to

\text{W, W, 2W}

\text{3W}

respectively are to be multiplexed using

\text{Time Division Multiplexing (TDM)}.

The minimum bandwidth required for transmission of this

\text{TDM}

In delta modulation, the slope overload distortion can be reduced by

In the following scheme, if the spectrum

M(f)

m(t)

is as shown, then the spectrum

Y(f)

y(t)

x(t) \longleftrightarrow \mathrm{X}(j \omega)

be Fourier Transform pair. The Fourier Transform of the signal

x(5 t-3)

X(j \omega)

The Dirac delta function

\delta(t)

m(t)

500 \mathrm{~Hz}

is first multiplied by a signal

g(t)

g(t)=\sum_{\mathrm{R}=-\infty}^{\infty}(-1)^{k} \delta\left(t-0.5 \times 10^{-4} k\right)

The resulting signal is then passed through an ideal lowpass filter with bandwidth

1 \; \mathrm{kHz}

. The output of the lowpass filter would be

The minimum sampling frequency (in samples/sec) required to reconstruct the following signal from its samples without distortion

x(t)=5\left(\frac{\sin 2 \pi 1000 t}{\pi t}\right)^{3}+7\left(\frac{\sin 2 \pi 1000 t}{\pi t}\right)^{2}

The minimum step-size required for a Delta-Modulator operating at

32 \mathrm{~K}

\mathrm{sec}

to track the signal (here

u(t)

is the unit-step function)

\begin{array}{l}x(t)=125 t(u(t)-u(t-1))+(250-125 t)(u(t-1) -u(t-2)) \end{array}

so that slope-overload is avoided, would be

In the following figure the minimum value of the constant '

C

', which is to be added to

y_{1}(t)

y_{1}(t)

y_{2}(t)

are different, is

Choose the function

f(t); – \infty < 1 < \infty,

for which a Fourier series cannot be defined.

x(t)

is shown in the figure. Even and odd parts of a unit-step function

u(t)

are respectively,

The power in the signal

s(t)=8 \cos \left(20 \pi t-\frac{\pi}{2}\right)+

4 \sin (15 \pi t)

y(t)

of a linear time invariant system is related to its input

x(t)

by the following equation:

y(t)=0.5 x\left(t-t_{d}+\mathrm{T}\right)+x\left(t-t_{d}\right)+0.5 x\left(t-t_{d}-\mathrm{T}\right)

. The filter transfer function

\mathrm{H}(\omega)

of such a system is given by

Match the following and choose the correct combination. …