Z Transform - Signals and Systems

Q#1 Z Transform GATE EC 2025 (Set 1) MSQ +1 mark -0 marks

Let be a discrete-time signal whose -transform is .

Which of the following statements is/are TRUE?

The discrete-time Fourier transform (DTFT) of always exists

The region of convergence (RoC) of contains neither poles nor zeros

The discrete-time Fourier transform (DTFT) exists if the region of convergence (RoC) contains the unit circle

If , where is the unit impulse and is a scalar, then the region of convergence is the entire -plane

Q#2 Z Transform GATE EC 2024 (Set 1) MSQ +1 mark -0 marks

For a causal discrete-time LTI system with transfer function

Which of the following statements is/are true?

The system is stable.

The system is a minimum phase system.

The initial value of the impulse response is 2.

The final value of the impulse response is 0.

Q#3 Z Transform GATE EC 2021 (Set 1) NAT +2 marks -0 marks

For a unit step input , a discrete-time LTI system produces an output signal

Let be the output of the system for an input . The value of is ________.

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Q#4 Z Transform GATE EC 2020 (Set 1) MCQ +1 mark -0.33 marks

Which one of the following pole-zero corresponds to the transfer function of an LTI system characterized by the input-output difference equation given below?

Q#5 Z Transform GATE EC 2020 (Set 1) NAT +2 marks -0 marks

The transfer function of a stable discrete-time LTI system is . where and are real numbers. The value of (rounded off to one decimal place) with , for which the magnitude response of the system is constant over all frequencies, is __________.

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Q#6 Z Transform GATE EC 2019 (Set 1) MCQ +1 mark -0.33 marks

Let H(z) be the z-transform of a real-valued discrete time signal h[n]. If has a zero at , and P(z) has a total of four zeros, which one of the following poles represents all the zeros correctly?

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Q#7 Z Transform GATE EC 2018 (Set 1) MCQ +1 mark -0.33 marks

A discrete-time all-pass system has two of its poles at and ". Which one of the following statements about the system is TRUE?

It has two more poles at and.

It is stable only when the impulse response is two-sided.

It has constant phase response over all frequencies.

It has constant phase response over the entire z-plane.

Q#8 Z Transform GATE EC 2016 (Set 1) MCQ +1 mark -0.33 marks

Consider the sequence where u[n] denotes the unit-step sequence and 0 < |a| < |b| < 1. The region of convergence (ROC) of the z-transform of x[n] is

|z| > |a|

|z| > |b|

|z| < |a|

|a| < |z| < |b|

Q#9 Z Transform GATE EC 2016 (Set 1) MCQ +1 mark -0.33 marks

A discrete-time signal has z-transform X(z). If Y(z)= X(–z) is the z-transform of another signal y[n], then

y[n] = x[n]

y[n] = x[-n]

y[n] = -x[n]

y[n] = -x[-n]

Q#10 Z Transform GATE EC 2016 (Set 1) MCQ +2 marks -0.66 marks

The ROC (region of convergence) of the z-transform of a discrete-time signal is represented by the shaded region in the z-plane. If the signal then the ROC of its z-transform is represented by

Q#11 Z Transform GATE EC 2015 (Set 1) MCQ +2 marks -0.66 marks

For the discrete-time system shown in the figure, the poles of the system transfer function are located at

2,3

, 3

,

2,

Q#12 Z Transform GATE EC 2015 (Set 1) MCQ +2 marks -0.66 marks

The pole-zero diagram of a causal and stable discrete-time system is shown in the figure. The zero at the origin has multiplicity 4. The impulse response of the system is h[n]. If h[0] = 1, we can conclude

h[n] is real for all n

h[n] is purely imaginary for all n

h[n] is real for only even n

h[n] is purely imaginary for only odd n

Q#13 Z Transform GATE EC 2015 (Set 2) NAT +1 mark -0 marks

Two causal discrete-time signals x[n] and y[n] are related as. If the z-transform of y[n] is , the value of x[2] is _______.

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Q#14 Z Transform GATE EC 2015 (Set 3) NAT +1 mark -0 marks

The value of is ______.

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Q#15 Z Transform GATE EC 2015 (Set 3) MCQ +1 mark -0.33 marks

Consider a four-point moving average filter defined by the equation. The condition on the filter coefficients that results in a null at zero frequency is

Q#16 Z Transform GATE EC 2015 (Set 3) MCQ +2 marks -0.66 marks

Suppose x[n] is an absolutely summable discrete-time signal. Its z-transform is a rational function with two poles and two zeroes. The poles are at . Which one of the following statements is TRUE for the signal x[n]?

It is a finite duration signal.

It is a causal signal.

It is a non-causal signal.

It is a periodic signal.

Q#17 Z Transform GATE EC 2015 (Set 3) MCQ +2 marks -0.66 marks

A realization of a stable discrete time system is shown in the figure. If the system is excited by a unit step sequence input x[n], the response y[n] is

Q#18 Z Transform GATE EC 2014 (Set 1) MCQ +2 marks -0.66 marks

Let . The Region of Convergence (ROC) of the z-transform of x[n]

is .

is

does not exist

Q#19 Z Transform GATE EC 2014 (Set 2) MCQ +1 mark -0.33 marks

Let . Let X(z) be the z-transform of x[n]. If is a zero of X(z), which one of the following must also be a zero of X(z).

Q#20 Z Transform GATE EC 2014 (Set 2) MCQ +2 marks -0.66 marks

The input-output relationship of a causal stable LTI system is given as .

If the impulse response h[n] of this system satisfies the condition, the relationship between α and β is

Q#21 Z Transform GATE EC 2014 (Set 3) MCQ +1 mark -0.33 marks

For an all-pass system , where , for all ω. If Re(a) ≠ 0, , then b equals

a

1/

1/a

Q#22 Z Transform GATE EC 2014 (Set 3) NAT +2 marks -0 marks

Let

The quantities p, q, r are real numbers. Consider

If the zero of H(z) lies on the unit circle, then r = _____.

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Q#23 Z Transform GATE EC 2014 (Set 3) NAT +2 marks -0 marks

The z-transform of the sequence x[n] is given by , with the region of convergence . Then, x[n] is ________.

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Q#24 Z Transform GATE EC 2014 (Set 4) NAT +1 mark -0 marks

The sequence , where u[n] is the unit step sequence, is convolved with itself to obtain y[n].

Then is _______

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Q#25 Z Transform GATE EC 2011 (Set 1) MCQ +2 marks -0.66 marks

Two systems and are connected in cascade as shown below. The overall output y[n] is the same as the input x[n] with a one unit delay. The transfer function of the second system is

Q#26 Z Transform GATE EC 2010 (Set 1) MCQ +1 mark -0.33 marks

Consider the z-transform; .

The inverse z-transform x[n] is

Q#27 Z Transform GATE EC 2010 (Set 1) MCQ +1 mark -0.33 marks

Two discrete time systems with impulse responses and are connected in cascade. The overall impulse response of the cascaded system is

Q#28 Z Transform GATE EC 2010 (Set 1) MCQ +2 marks -0.66 marks

The transfer function of a discrete time LTI system is given by

Consider the following statements:

S1: The system is stable and causal for

S2: The system is stable and not causal for

S3: The system is neither stable nor causal for

Which one of the following statements is valid?

Both S1 and S2 are true

Both S2 and S3 are true

Both S1 and S3 are true

S1, S2, and S3 are all true

Q#29 Z Transform GATE EC 2009 (Set 1) MCQ +1 mark -0.33 marks

The ROC of Z-transform of the discrete time sequence is

Q#30 Z Transform GATE EC 2008 (Set 1) MCQ +2 marks -0.66 marks

In the following network, the switch is closed at t = 0- and the sampling starts from t = 0. The sampling frequency is 10 Hz.

The samples x[n] (n = 0, 1, 2, ………) are given by

Q#31 Z Transform GATE EC 2008 (Set 1) MCQ +2 marks -0.66 marks

In the following network, the switch is closed at t = 0- and the sampling starts from t = 0. The sampling frequency is 10 Hz.

The expression and the region of convergence of the z-transform of the sampled signal are

Q#32 Z Transform GATE EC 2007 (Set 1) MCQ +2 marks -0.66 marks

The z-transform of a sequence is given by . It is given that the region of convergence of includes the unit circle. The value of is

–0.5

0

0.25

0.5

Q#33 Z Transform GATE EC 2006 (Set 1) MCQ +1 mark -0.33 marks

If the region of convergence of is , then the region of convergence of includes

Q#34 Z Transform GATE EC 2005 (Set 1) MCQ +1 mark -0.33 marks

The region of convergence of Z-transform of the sequence must be

Q#35 Z Transform GATE EC 2004 (Set 1) MCQ +1 mark -0.33 marks

The z-transform of a system is . If the ROC is |z|<0.2, then the impulse response of the system is

Q#36 Z Transform GATE EC 2004 (Set 1) MCQ +2 marks -0.66 marks

A causal LTI system is described by the difference equation.

The system is stable only if

|α|= 2, |β|< 2

|α|> 2, |β|> 2

|α|< 2, any value of β

|β|< 2, any value of α

Q#37 Z Transform GATE EC 2003 (Set 1) MCQ +1 mark -0.33 marks

A sequence x[n] with the z-transform is applied as an input to a linear, time-invariant system with the impulse response where .

The output at n=4 is

-6

Zero

2

-4

Q#38 Z Transform GATE EC 2002 (Set 1) MCQ +2 marks -0.66 marks

If the impulse response of a discrete- time system is , then the system function H(z) is equal to

and the system is stable

and the system is unstable

Q#39 Z Transform GATE EC 2001 (Set 1) MCQ +1 mark -0.33 marks

The region of convergence of the z-transform of a unit step function is

|z| > 1

|z| < 1

(Real part of z) > 0

(Real part of z) < 0

Q#40 Z Transform GATE EC 1999 (Set 1) MCQ +1 mark -0.33 marks

The z-transform F(z) of the function is

Q#41 Z Transform GATE EC 1999 (Set 1) MCQ +2 marks -0.66 marks

The z-transform of a signal is given by . Its final value is

Zero

1

Infinity

Q#42 Z Transform GATE EC 1998 (Set 1) MCQ +1 mark -0.33 marks

The Z-transform of the time function is

Q#43 Z Transform GATE EC 1997 (Set 1) MSQ +2 marks -0 marks

In the figure is a linear time invariant discrete system is shown. Blocks labelled D represent unit delay elements. For , you may assume that , , are all zero.

(a) Find the expression for and in terms of x(n)

(b) Find the transfer function in the z – domain

(c) If , find

(a)
$\scriptstyle {y_1[n] + 2y_1[n-1] + y_1[n-2] = x[n-1]}$

$\scriptstyle {y_2[n] = x[n-2] - 2y_2[n-1] - y_2[n-2]}$

(b)
$\scriptstyle {\frac{Y_2(z)}{X(z)} = \frac{1}{[z+1]^2}}$

(c) $\scriptstyle {y_2[n] = (n-1)(-1)^{n-1} u[n-1]}$

(c) $\scriptstyle {y_2[n] = (n+1)(-1)^{n-1} u[n+1]}$

Q#44 Z Transform GATE EC 1992 (Set 1) MCQ +2 marks -0.66 marks

A linear discrete time system has the characteristic equation, . The system

Is stable

Is marginally stable

In unstable

Stability cannot be assessed from the given information

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GATE EC

Signals and Systems
Z Transform

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