GATE EC Discrete-time Signals questions

Practise Discrete-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.

20 on this page

SubjectTopicYearType

x[n]

be a discrete-time signal whose

z

X(z)

. Which of the following statements is/are TRUE?

The radian frequency value(s) for which the discrete time sinusoidal signal

x[n]=A \cos (\Omega n+\pi / 3)

has a period of

40

\_\_\_\_\_\_

The relationship between any N-length sequence

x[n]

and its corresponding

\mathrm{N}

-point discrete Fourier transform

X[k]

X[k]=\mathcal{F}\{x[n]\} .

Another sequence

y[n]

is formed as below

y[n]=\mathcal{F}\{\mathcal{F}\{\mathcal{F}\{\mathcal{F}\{x[n]\}\}\}\} .

For the sequence

x[n]=\{1,2,1,3\}

, the value of …

Consider a discrete-time periodic signal with period

N=5

. Let the discrete-time Fourier series

\text{(DTFS}

) representation be

x[n]=\sum_{k=0}^4 a_k e^{\frac{j k 2 \pi n}{5}}

a_0=1, a_1=

3 j, a_2=2 j, a_3=-2 j

a_4=-3 j

. The value of the sum

\sum_{n=0}^4 x[n] \sin \frac{4 \pi n}{5}

\overline{x} = \left[ x [0], x[1], \dots, x[7] \right],

\text{8-point}

discrete Fourier transform

\text{(DFT)}

\overline{X} = DFT (\overline{x}) = \left[ X [0], X[1], \dots, X[7] \right],

X[k] = \displaystyle \sum_{n=0}^{7} x[n] \; \text{exp} \left(-j \frac{2 \pi}{8} nk \right).

16

-point sequences

x\left [ n \right ]

h\left [ n \right ]

. Let the linear convolution of

x\left [ n \right ]

h\left [ n \right ]

y\left [ n \right ]

z\left [ n \right ]

16

-point inverse discrete Fourier transform

\text{(IDFT)}

of the product of the

16

Consider the signals

x\left [ n \right ]=2^{n-1}u\left [ -n+2 \right ]

y\left [ n \right ]=2^{-n+2}u\left [ n+1 \right ]

u[n]

is the unit step sequence. Let

X(e^{jw})

Y(e^{jw})

be the discrete-time Fourier transform of

x[n]

y[n]

, respectively. The value of the integral …

A finite duration discrete-time signal

x[n]

is obtained by sampling the continuous-time signal

x\left ( t \right )=\cos\left ( 200\pi t \right )

at sampling instants

t=n/400, n=0, 1, \dots ,7.

8

-point discrete Fourier transform

\text{(DFT)}

x[n]

is defined as …

H(z)

z-

transform of a real-valued discrete-time signal

h[n].

P(z) = H(z) H(\frac{1}{z})

z= \frac{1}{2}+\frac{1}{2}j,

P(z)

has a total of four zeros, which one of the following plots represents all the zeros correctly?

Consider a six-point decimation-in-time Fast Fourier Transform

(FFT)

algorithm, for which the signal-flow graph corresponding to

X[1]

is shown in the figure. Let

W_{6}=exp\left(-\:\dfrac{j2\pi}{6}\right).

In the figure, what should be the values of the coefficients

a_{1},a_{2},a_{3}

W_{6}

X[1]

is obtained …

h[n]

be a length –

7

discrete-time finite impulse response filter, given by

h[0]=4, \quad h[1]=3,\quad h[2]=2,\quad h[3]=1,

\quad h[-1]=-3, \quad h[-2]=-2, \quad h[-3]=-1,

h[n]

|n|\geq4.

3

finite impulse response approximation

g[n]

h[n]

has to be obtained such that …

X\left[ k \right ] = k + 1,0\leq k\leq 7

8

\:\text{DFT}\:

x[n]

X\left [ k \right ]=\sum_{n=0}^{N-1}x \left [ n \right ]e^{-j2\pi nk/N}

. The value (correct to two decimal places) of

\sum_{n=0}^{3}x \left [ 2n \right ]

h[n]

be the impulse response of a discrete-time linear time invariant(LTI) filter. The impulse response is given by

h[0]=\frac{1}{3}; \, h[1]=\frac{1}{3}; \, h[2]=\frac{1}{3} \text{; and } h[n]=0 \text{ for } n<0 \text{ and } n>2.

H(\omega)

be the discrete-time Fourier transform (DTFT) of

h[n]

\omega

is the normalized …

Consider the signal

x[n] = 6 \delta[n + 2] + 3 \delta[n + 1] + 8 \delta[n] + 7 \delta[n - 1] + 4 \delta[n - 2]

X(e^{jw})

is the discrete-time Fourier transform of

x[n]

\frac{1}{\pi} \int\limits_{-\pi}^{\pi} X(e^{jw}) \sin^2(2\omega) d\omega

is equal to _______

A continuous-time filter with transfer function

H\left ( s \right )= \frac{2s+6}{s^{2}+6s+8}

is converted to a discrete-time filter with transfer function

G\left ( z\right )= \frac{2z^{2}-0.5032 \: z}{z^{2}-0.5032 \: z+k}

so that the impulse response of the continuous-time filter, sampled at

2

Hz

, is identical at the sampling instants to the …

The Discrete Fourier Transform (DFT) of the

4

-point sequence

x\left [ n \right ]=\left \{ x\left [ 0 \right ],x\left [ 1 \right ], x\left [ 2 \right ], x\left [ 3 \right ] \right \}= \left \{ 3,2,3,4 \right \}

A discrete-time signal

x[n] = \delta[n – 3] + 2 \delta[n – 5]

z

X(z)

Y(z) = X(-z)

z

-transform of another signal

y[n]

A continuous-time speech signal

x_a(t)

is sampled at a rate of

8\:kHz

and the samples are subsequently grouped in blocks, each of size

N

. The DFT of each block is to be computed in real time using the radix-

2

decimation-in-frequency FFT algorithm. If the processor performs all operations sequentially, and takes

20\:\mu s

for computing …

Two causal discrete-time signals

x[n]

y[n]

y[n] = \displaystyle{}\sum _{m=0}^{n} x[m]

z

y[n]

\dfrac{2}{z(z-1)^{2}},

x[2]

Consider two real sequences with time-origin marked by the bold value,

x_{1}[n] = \{\textbf{1},2,3,0\},\:\:x_{2}[n] = \{\textbf{1},3,2,1\}

X_{1}(k)

X_{2}(k)

4

x_{1}[n]

x_{2}[n]

, respectively. Another sequence

x_{3}[n]

is derived by taking

4

-point inverse DFT of

X_{3}(k) = X_{1}(k)X_{2}(k).