# Topic: Signals and Systems (Test 4)

Topic: Signals and Systems
Q.1
The following discrete-time equations result from the numerical integration of the differential equations of an un-damped simple harmonic oscillator with state variables x and y. The integration time step is h. For this discrete-time system, which one of the following statements is TRUE?
A. The system is not stable for h>0
B. The system is stable for h>1/π
C. The system is stable for 0<h<1/2π
D. The system is stable for 1/2π<h<1/π
Explaination / Solution:
No Explaination.

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Q.2
The z −transform X(z) of a sequence x n is given by X It is given that the region of convergence of X(z) includes the unit circle. The value of x is
A. −0.5
B. 0
C. 0.25
D. 05
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Q.3
Consider a continuous-time system with input x(t) and output y(t) given by y(t) = x(t) cos(t) This system is
A. linear and time-invariant
B. non-linear and time-invariant
C. linear and time-varying
D. non-linear and time-varying
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Q.4
The output of a continuous-time, linear time-invariant system is denoted by T{x(t)} where x(t) is the input signal. A signal z(t) is called eigen-signal of the system T , when T{z(t)} = γz(t), where γ is a complex number, in general, and is called an eigenvalue of T. Suppose the impulse response of the system T is real and even. Which of the following statements is TRUE?
A. cos(t) is an eigen-signal but sin(t) is not
B. cos(t) and sin(t) are both eigen-signals but with different eigenvalues
C. sin(t) is an eigen-signal but cos(t) is not
D. cos(t) and sin(t) are both eigen-signals with identical eigenvalues
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Q.5
Step responses of a set of three second-order underdamped systems all have the same percentage overshoot. Which of the following diagrams represents the poles of the three systems ?
A. B. C. D. Explaination / Solution:

Transfer function for the given pole zero plot is: From the plot Re (P1 and P2)>(Z1 and Z2
So, these are two lead compensator.
Hence both high pass filters and the system is high pass filter

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Q.6
A periodic voltage waveform observed on an oscilloscope across a load is shown. A permanent magnet moving coil (PMMC) meter connected across the same load reads A. 4 V
B. 5 V
C. 8 V
D. 10 V
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Q.7
Let x(t) be the input and y(t) be the output of a continuous time system.
Match the system properties P1, P2 and P3 with system relations R1, R2, R3, R4
Properties                                               Relations
P1 : Linear but NOT time - invariant R1 : y(t) = t2x(t)
P2 : Time - invariant but NOT linear R2 : y(t) = t|x(t)|
P3 : Linear and time - invariant        R3 : y(t) = |x(t)|
R4 : y(t) = x(t-5)
A. (P1, R1), (P2, R3), (P3, R4)
B. (P1, R2), (P2, R3), (P3, R4)
C. (P1, R3), (P2, R1), (P3, R2)
D. (P1, R1), (P2, R2), (P3, R3)
Explaination / Solution:

Mode function are not linear. Thus y(t) = |x(t)|  is not linear but this functions is time invariant. Option (A) and (B) may be correct.
The y(t) = t|x(t)| is not linear, thus option (B) is wrong and (a) is correct. We can see that
R1 : y(t) = t2x(t) Linear and time variant.
R2 : y(t) = t|x(t)| Non linear and time variant.
R3 : y(t) = |x(t)| Non linear and time invariant.
R4 : y(t) = x(t-5) Linear and time invariant.

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Q.8
Consider the system with following input-output relation where, x[n] is the input and y[n] is the output. The system is
A. invertible and time invariant
B. invertible and time varying
C. non-invertible and time invariant
D. non-invertible and time varying
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Q.9
Let the signal be passed through an LTI system with frequency response H(𝜔), as given in the figure below The Fourier series representation of the output is given as
A. 4000 + 4000cos(2000πt) + 4000cos(4000πt)
B. 2000 + 2000cos(2000πt) + 2000cos(4000πt)
C. 4000cos(2000πt)
D. 2000cos(2000πt)
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Q.10
A system is defined by its impulse response The system is
A. stable and causal
B. causal but not stable
C. unstable and non-causal
D. stable but not causal