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Q1: If the Z-transform of a finite-duration discrete-time signal x[n] is X(z), then the Z-transform of the signal y[n] = x[2n] is       (2024)
(a) ๐‘Œ(๐‘ง)=๐‘‹(๐‘ง2)Y(z) = X(z2)
(b) ๐‘Œ(๐‘ง)=12[๐‘‹(๐‘งโˆ’1/2)+๐‘‹(โˆ’๐‘งโˆ’1/2)]Y(z) = (1/2) [X(zโˆ’1/2) + X(โˆ’zโˆ’1/2)]
(c) Y(z) = (1/2) [X(z1/2)+X(โˆ’z1/2)]
(d) Y(z) = (1/2) [X(z2) + X(โˆ’z2)]
Ans:
(c)
Sol: Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)Let
2n = m
then
Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)
Q2: The Z-transform of a discrete signal x[n] is
Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)Which one of the following statements is true?       (2023)
(a) Discrete-time Fourier transform of x[n] converges if R is โˆฃzโˆฃ > 3
(b) Discrete-time Fourier transform of x[n] converges if R is (2/3) < โˆฃ๐‘งโˆฃ < 3
(c) Discrete-time Fourier transform of x[n] converges if R is such that x[n] is a leftsided sequence
(d) Discrete-time Fourier transform of x[n] converges if R is such that x[n] is a rightsided sequence
Ans:
(b)
Sol:  Z = 1, unit circle include by ROC : (2/3) < โˆฃzโˆฃ < 3.
Therefore DIFT of x[n] conveys for this ROC.

Q3: The casual signal with z-transformer Z2(Z โˆ’ a)โˆ’2 is
(u[n] is the unit step signal)        (2021)
(a) ๐‘Ž2๐‘›๐‘ข[๐‘›]a2nu[n]
(b) (n + 1) anu [n]
(c) nโˆ’1 anu [n]  
(d) n2anu[n]
Ans:
(b)
Sol: As we know,
Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)Time-shifting property,
Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)Thus,
Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)
Q4: The causal realization of a system transfer function H(s) having poles at (2, -1), (-2, 1) and zeroes at (2, 1), (-2, -1) will be      (2020)
(a) stable, real, allpass
(b) unstable, complex, allpass
(c) unstable, real, highpass
(d) stable, complex, lowpass
Ans:
(b)
Sol: Since pole zero plot of given transfer function
Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)Since, given pole zero is symmetrical about origin hence it is a all pass system.
Since, one pole on the RHS thus unstable.
Since, pole doesnot have complex conjugate poles and zeros present thus system is not real means system is complex.

Q5: Consider a signal Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE) where 1[n] = 0 if n < 0, and 1[n] = 1 if n โ‰ฅ 0. The z-transform of x[n โˆ’ k], k > 0 is Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE) with region of convergence being       (2020)
(a) โˆฃ๐‘งโˆฃ<2โˆฃzโˆฃ < 2
(b) โˆฃzโˆฃ > 2
(c) โˆฃzโˆฃ < 1/2
(d)  โˆฃzโˆฃ > 1/2
Ans:
(d)
Sol: Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)
Q6: A cascade system having the impulse responses
Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)is shown in the figure below, where symbol โ†‘ denotes the time origin.
Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)The input sequence x(n) for which the cascade system produces an output sequence Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE) is      (SET-2  (2017))
(a) Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)

(b) Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)
(c) Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)
(d) Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)
Ans: (d)
Sol: Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)Now,
Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)
Q7: The pole-zero plots of three discrete-time systems P, Q and R on the z-plane are shown below. Which one of the following is TRUE about the frequency selectivity of these systems?      (SET-2 (2017))
Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)(a) All three are high-pass filters.
(b) All three are band-pass filters.
(c) All three are low-pass filters.
(d) P is a low-pass filter, Q is a band-pass filter and R is a high-pass filter.
Ans:
(b)
Sol: Since all 3 pole-zero plots have zero at z = 1 and z = -1.
So, filter will not pass low frequency and high frequency components.
Therefore all are bandpass filter.
Note: in digital filter
For low frequency z = 1
For high frequency z = -1

Q8: Consider a causal and stable LTI system with rational transfer function H(z). Whose corresponding impulse response begins at n = 0. Furthermore, H(1) = 5/4. The poles of H(z) are Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)for k = 1, 2, 3, 4. The zeros of H(z) are all at z = 0. Let g[n] = jnh[n]. The value of g[8] equals ___________.          (SET-1 (2017))
(a) 0.01
(b) 0.03
(c) 0.06
(d) 0.09
Ans:
(d)
Sol: Pole locationof H(z) are given as,
Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)Now,
Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)[As h(n) is causal and it starts from n=0, so numerator will have same order as denominator is having]. by solving equation (i)
Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)
Q9: Let Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)where โˆฃฮฑโˆฃ < 1. The value of ฮฑ in the range 0 < ฮฑ < 1, such that S = 2ฮฑ is _______.          (SET-1 (2016))
(a) 0.1
(b) 0.9
(c) 0.6
(d) 0.3
Ans: 
(d)
Sol: The Z-transform of
Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)If we put Z = 1 in above equation, we get,
Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)
Q10: The z-Transform of a sequence x[n] is given as X(z) = 2z + 4 โˆ’ (4/z) + (3/z2). If y[n] is the first difference of x[n], then Y(z) is given by       (SET-2  (2015))
(a) 2z + 2 โˆ’ 8/z + 7/z2 โˆ’ 3/z3
(b) โˆ’2z + 2 โˆ’ 6/z โˆ’ 1/z2 โˆ’ 3/z3 
(c) โˆ’2z + 2 โˆ’ 8/z โˆ’ 7/z2 โˆ’ 3/z3 
(d) 4z โˆ’ 2 โˆ’ 8/z + 7/z2 โˆ’ 3/z3 
Ans:
(a)
Sol: y(n) is first difference of x(n)
Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)
Q11: Consider a discrete time signal given by
 x[n] = (โˆ’0.25)nu[n] + (0.5)nu[โˆ’n โˆ’ 1]
The region of convergence of its Z-transform would be      (SET-1 (2015))
(a) the region inside the circle of radius 0.5 and centered at origin
(b) the region outside the circle of radius 0.25 and centered at origin
(c) the annular region between the two circles, both centered at origin and having radii 0.25 and 0.5
(d) the entire Z plane.
Ans: 
(c)
Sol: x[n] = (0.25)nu(n) + (0.5)nu(โˆ’n โˆ’ 1)
Signal x[n] is sum of two signals, one is right sided [(โˆ’0.25)nu(n)] and other is left sided [(0.5)nu(โˆ’n โˆ’ 1)]. The right sided signal will have pole at location with magnitude 0.25. So, ROC is โˆฃzโˆฃ > 0.25. The left sided signal will have pole at location with magnitude 0.5. So, ROC is  โˆฃzโˆฃ < 0.5. So, ROC of X(z) (Z-transform of x(n) will be )  0.25 < โˆฃzโˆฃ < 0.5.

Q12: An input signal x(t) = 2 + 5sin(100ฯ€t) is sampled with a sampling frequency of 400 Hz and applied to the system whose transfer function is represented by
Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)where, N represents the number of samples per cycle. The output y(n) of the system under steady state is       (SET-2 (2014))
(a) 0
(b) 1
(c) 2
(d) 5
Ans
: (c)
Sol: Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)Therefore, N = time-period of Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)Therefore, for sinusoidal part of input, system output is zero. For dc part of input,
Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)Thus, steady state output = 2.

Q13: A discrete system is represented by the difference equation
Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)It has initial conditions X1(0) = 1; X2(0) = 0. The pole locations of the system for a = 1, are       (SET-2 (2014))
(a) 1 ยฑ j0
(b) โˆ’1 ยฑ  j0
(c) ยฑ 1 + j0
(d) 0  ยฑ  j1
Ans:
(a)
Sol: Given that,
Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)With initial conditions,
X1(0) = 1, X2(0) = 0
For a = 1, we can write,
Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)Applying z-transform on equation (i),
Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)from equation (ii),
Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)Thus, transfer function, Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)
Therefore, pole location is z = 1.

Q14: Let Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE) be the Z -transform of a causal signal x [n]. Then, the values of x[2] and x[3] are       (SET-1 (2014))
(a) 0 and 0
(b) 0 and 1
(c) 1 and 0
(d) 1 and 1
Ans:
(b)
Sol: Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)From z-transform defination,
Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)By comparision x(2) = 0 and x(3) = 1.

Q15: If x[n] = (1/3)โˆฃnโˆฃ โˆ’ (1/2)nu[n], then the region of convergence (ROC) of its Z-transform in the Z-plane will be       (2012)
(a) (1/3) < โˆฃzโˆฃ < 3
(b) (1/3) < โˆฃzโˆฃ < (1/2)
(c) 12<โˆฃ๐‘งโˆฃ<3(1/2) < โˆฃzโˆฃ < 3
(d) (1/3) < โˆฃ๐‘งโˆฃ
Ans:
(c)
Sol: Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)
Q16: The z-transform of a signal x[n] is given by 4zโˆ’3 + 3zโˆ’1 + 2 โˆ’ 6z2 + 2z3. It is applied to a system, with a transfer function H(z) = 3zโˆ’1 โˆ’ 2. Let the output be y(n). Which of the following is true ?      (2009)
(a) y(n) is non causal with finite support
(b) y(n) is causal with infinite support
(c) y(n) = 0; โˆฃnโˆฃ > 3
(d) Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)

Ans: (a)
Sol: Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)Therefore, it is noncausal with finite support.

Q17: Given Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE) with โˆฃzโˆฃ > a, the residue of X(z) zn โˆ’ 1 at z = a for n โ‰ฅ 0 will be        (2008)
(a) ๐‘Ž๐‘›โˆ’1anโˆ’1 
(b) an
(c) nan
(d) nanโˆ’1 
Ans:
(d)
Sol: Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)Since, z - a is a pole of second order, therefore residue at z = a
Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)
Q18: H(z) is a transfer function of a real system. When a signal x[n] = (1 + j)n is the input to such a system, the output is zero. Further, the Region of convergence (ROC) of (1 โˆ’ (1/2)z โˆ’1)H(z) is the entire Z-plane (except z = 0). It can then be inferred that H(z) can have a minimum of      (2008)
(a) one pole and one zero
(b) one pole and two zeros
(c) two poles and one zero
(d) two poles and two zeros
Ans: 
(d)

Q19: A signal is processed by a causal filter with transfer function G(s).
G(z) = ฮฑzโˆ’1 + ฮฒzโˆ’3 is a low pass digital filter with a phase characteristics same as that of the above question if      (2007)
(a) ฮฑ = ฮฒ
(b) ฮฑ = โˆ’ฮฒ
(c) ฮฑ = ฮฒ(1/3)
(d) ฮฑ = ฮฒ(โˆ’1/3) 
Ans:
(a)
Sol: G(z) = ฮฑzโˆ’1 + ฮฒzโˆ’3
For, low frequency (z = 1) โ†’ G(1) = ฮฑ+ฮฒ...(i)
For, high frequency (z = โˆ’1) โ†’ G(โˆ’1) = โˆ’(ฮฑ + ฮฒ)...(ii)
For now, g(n)  = {0, ฮฑ, 0, ฮฒ}
For linear phase FIR filter, g(n) should be either even symmetric or odd symmetric about virtual y-axis.
To, satisfy the above condition.
CASE: (i) ฮฑ = โˆ’ฮฒ
( odd symmetric about virtual y-axis)
But in this case,
G(1) = G(-1) = 0
i.e. at low frequency โ‡’ G(z) = 0 and at high frequency โ‡’ G(z) = 0
Thus in this case filter will be band-pass.
CASE: (ii) ฮฑ = ฮฒ
(even symmetric about virtual y-axis)
In this case,
At low frequency โ‡’ G(z) = 2ฮฑ and at high frequency โ‡’ G(โˆ’1) = โˆ’2ฮฑ
Filter will be either all pass or band-stop.
Now,  
Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)
Q20: A signal is processed by a causal filter with transfer function G(s).
For a distortion free output signal wave form, G(s) must       (2007)
(a) provides zero phase shift for all frequency
(b) provides constant phase shift for all frequency
(c) provides linear phase shift that is proportional to frequency
(d) provides a phase shift that is inversely proportional to frequency
Ans: 
(c)

Q21: X(z) = 1 โˆ’ 3zโˆ’1,  Y(z) = 1 + 2zโˆ’2 are Z transforms of two signals x[n], y[n] respectively. A linear time invariant system has the impulse response h[n] defined by these two signals as
h[n] = x[n - 1] * y[n]
where * denotes discrete time convolution. Then the output of the system for the input ฮด[n-1]      (2007)
(a) has Z-transform zโˆ’1 X(z)Y(z)
(b) equals ฮด[n โˆ’ 2] โˆ’ 3ฮด[n โˆ’ 3] + 2ฮด[n โˆ’ 4] โˆ’ 6ฮด[n โˆ’ 5]
(c) has Z-transform 1 โˆ’ 3zโˆ’1 + 2zโˆ’2 โˆ’ 6zโˆ’3
(d) does not satisfy any of the above three
Ans:
(b)
Sol: Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)
Q22: Consider the discrete-time system shown in the figure where the impulse response of G(z) is g(0) = 0, g(1) = g(2) = 1, g(3) = g(4) =...= 0
Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)This system is stable for range of values of K         (2007)
(a) [-1, 1/2]
(b) [-1, 1]
(c) [-1/2, 1]
(d) [-1/2, 2]
Ans
: (a)
Sol: Given: g(1) = g(2) = 1
i.e.  g[n] = ฮด[nโˆ’1] + ฮด[nโˆ’2]
Therefore,
 G(z) = zโˆ’1 + zโˆ’2
Therefore overall transfer function of closed loop system,
Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)So the system will be stable if it's outer most pole will lie inside the unit circle.
Location of poles,
Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)
Q23: The discrete-time signal
Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)where โ†” denotes a transform-pair relationship, is orthogonal to the signal      (2006)
(a) Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)

(b) Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)
(c) Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)
(d) Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)
Ans: (b)
Sol: Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)In X(z), power of z are even. Therefore, samples in x(n) are available at even instant of time.
By observing all the options.
Option (B):
Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)In y2(z), powers of z are odd. Therefore, samples in y2(n) are avialable only at odd instant of time.
Hence, Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)Thus, x(n) is orthogonal to y2(n).  

Q24: A discrete real all pass system has a pole at z = 2 โˆ 30ยฐ: it, therefore       (2006)
(a) also has a pole at (1/2) โˆ  30ยฐ
(b) has a constant phase response over the z -plane: arg โˆฃH(z)โˆฃ = constant
(c) has a constant phase response over the unit circle: arg |H(๐‘’๐‘–ฮฉ)| = constant
(d) is stable only if it is anti-causal
Ans:
(c)
Sol: For causal system, all the poles are inside the unit circle then system is stable, and converse in true for anti-causal system.
Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)
Q25: If u(t) is the unit step and ฮด(t) is the unit impulse function, the inverse z-transform of F(z) = Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE) for k > 0 is       (2005)
(a) (โˆ’1)ฮด (k)
(b) ๐›ฟ(๐‘˜)โˆ’(โˆ’1)๐‘˜ฮด(k) โˆ’ (โˆ’1)k
(c)  (โˆ’1)ku(k)
(d) ๐‘ข(๐‘˜)โˆ’(โˆ’1)๐‘˜u(k) โˆ’ (โˆ’1)k 
Ans:
(b)
Sol: Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Z-Transform | Signals and Systems - Electrical Engineering (EE)

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FAQs on Previous Year Questions- Z-Transform - Signals and Systems - Electrical Engineering (EE)

1. What is the Z-transform in electrical engineering?
Ans. The Z-transform is a mathematical technique used in digital signal processing and control system analysis to convert a discrete-time signal into a complex function of a complex variable.
2. How is the Z-transform related to the Laplace transform?
Ans. The Z-transform is the discrete-time counterpart of the Laplace transform, which is used for continuous-time signals. Both transforms help analyze the behavior of signals and systems in the time and frequency domains.
3. What are the key properties of the Z-transform?
Ans. Some key properties of the Z-transform include linearity, time shifting, scaling, time reversal, convolution, and initial value theorem, which are essential for analyzing discrete-time signals and systems.
4. How is the Z-transform used in digital filter design?
Ans. The Z-transform is used to represent the transfer function of digital filters, allowing for the analysis of filter characteristics such as frequency response, stability, and poles and zeros placement.
5. How does the region of convergence (ROC) affect the Z-transform?
Ans. The region of convergence (ROC) determines the range of values for which the Z-transform converges, affecting the stability and causality of the system represented by the Z-transform.
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