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Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE) PDF Download

Q1: The input x(t) and the output y(t) of a system are related as       (2024)
Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)The system is       (2024)
(a) nonlinear
(b) linear and time-invariant
(c) linear but not time-invariant
(d) noncasual
Ans: 
(b)
Sol: System will follow law of superposition. Therefore, system will be linear.
Time-invariance check:
Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)Since,
y(t) = y(t − t0)
So, the system is time-invariant.

Q2: Which of the following statement(s) is/are true?      (2023)
(a) If an LTI system is causal, it is stable
(b) A discrete time LTI system is causal if and only if its response to a step input u[n] is 0 for n < 0
(c) If a discrete time LTI system has an impulse response h[n] of finite duration the system is stable  
(d) If the impulse response 0 < ∣h[n]∣ < 1 for all n, then the LTI system is stable.
Ans: 
(b)
Sol: For causal system, impuse response
h(n) = 0; n < 0
Therefore, for step input also
h(n) = 0; n < 0

Q3: Consider the system as shown below
Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)where y(t) = x(et). The system is       (2022)
(a) linear and causal.
(b) linear and non-causal.
(c) non-linear and causal
(d) non-linear and non-causal
Ans:
(b)
Sol: We know, a linear system follows the law of superposition.
It is a combination of two laws:
(i) Law of additivity:
Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)Both results are same, hence, it follows law of additivity.
(ii) Law of Homogeneity:
Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)Here also both results are same, hence it follows law of Homogeneity.
Therefore, System is linear.
We know, a causal system is independent of future values of input at each & every instant of time them system will be causal.
Given : y(t) = x(et)
Put t = 0
 y(0) = x(e0) = x(1)
Because its depends on future value.
Therefore, system is non-causal.

Q4: Let a causal LTI system be governed by the following differential equation  Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE) where x(t) and x(t) are the input and output respectively. Its impulse response is       (2022)
(a) Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)

(b) Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)
(c) Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)
(d) Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)
Ans: (d)
Sol: Given:
Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)Taking Laplace transform,
Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)Taking inverse Laplace, transform,
Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)
Q5: If the input x(t) and output y(t) of a system are related as y(t) = max(0, x (t)), then the system is       (2021)
(a) linear and time-variant
(b) linear and time-invariant
(c) non-linear and time-variant
(d) non-linear and time-invariant
Ans:
(d)
Sol: Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)Linearity check:
at input x1(t) = −2, output y1(t) = 0
at input x2(t) = 1, output y2(t) = 1
Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)∴ system is non-linear because it violates law of additivity.
Check for time-invariance :
Delayed O/P:
Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)Therefore, system is time-invariant.

Q6: Which of the following options is true for a linear time-invariant discrete time system that obeys the difference equation:
y[n] − ay[n − 1] = b0x[n] − b1x[n − 1]     (2020)
(a)  y[n] is unaffected by the values of x[n − k]; k > 2
(b) The system is necessarily causal.
(c) The system impulse response is non-zero at infinitely many instants.
(d) When x[n] = 0, n < 0, the function y[n]; n > 0 is solely determined by the function x[n].
Ans: 
(c)
Sol: Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)By taking right-sided inverse ZT,
h(n) = b0anu(n)−b1an−1u(n − 1)
By taking left-sided inverse ZT,
 h(n) = −b0anu(−n − 1) + b1an−1u(−n)
Thus system is not necessarily causal.
The impulse response is non-zero at infinitely many instants.  

Q7: A continuous-time input signal x(t) is an eigenfunction of an LTI system, if the output is       (2018)
(a) k x (t) , where k is an eigenvalue
(b) ke jωt x(t), where k is an eigenvalue and ejωt is a complex exponential signal
(c) x(t) ejωt, where ejωt is a complex exponential signal
(d) k H(ω) ,where k is an eigenvalue and H(ω) is a frequency response of the system
Ans: 
(a)
Sol: Eigen function is a type of input for which output is constant times of input.
i.e.
Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)Where,
x(t) = System input = eigen function
H(s) = transfer function of system
y(t) =  system output
Here,
Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)where,
k =  eigen-value = H(s)∣s = a
x(t) = eigen-function input  

Q8: Let z(t) = x(t) * y(t) , where "*" denotes convolution. Let c be a positive real-valued constant. Choose the correct expression for z(ct).       (SET-1 (2017))
(a) c x(ct)*y(ct)
(b) x(ct)*y(ct)
(c) c x(t)*y(ct)
(d) c x(t)*y(ct)
Ans:
(a)
Sol: Time scaling property of convolution.
Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)
Q9: Consider a causal LTI system characterized by differential equation Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE) The response of the system to the input Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE) where u(t) denotes the unit step function, is         (SET-2  (2016))
(a) Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)

(b) Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)
(c) Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)
(d) Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)
Ans: (d)
Sol: The differential equation
Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)
Q10: 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?        (SET-1 (2016))
(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
Ans:
(d)
Sol: Given that impulse response is real and even, Thus H(jω) will also be real and even.
Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)Since, H(jω) is real and even thus,
Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)So, sin(t) and cos(t) are eigen signal with same eigen values.  

Q11: Consider the following state-space representation of a linear time-invariant system.
Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)The value of y(t) for t = loge2 is______.       (SET-1 (2016))
(a) 4
(b) 5
(c) 6
(d) 7
Ans:
(c)

Q12: Consider a continuous-time system with input x(t) and output y(t) given by
y(t) = x(t) cos(t)
This system is      (SET-1  (2016))
(a) linear and time-invariant
(b) non-linear and time-invariant
(c) linear and time-varying
(d) non-linear and time-varying
Ans: 
(c)
Sol: Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)So, the system is linear, to check time invariance.
The delayed output,
Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)System is time varying.

Q13: 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.
Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)For this discrete-time system, which one of the following statements is TRUE?      (SET-2 (2015))
(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) 
Ans: (a)

Q14: For linear time invariant systems, that are Bounded Input Bounded Output stable, which one of the following statements is TRUE?      (SET-2 (2015))
(a) The impulse response will be integrable, but may not be absolutely integrable.
(b) The unit impulse response will have finite support.
(c) The unit step response will be absolutely integrable
(d) The unit step response will be bounded.
Ans: 
(d)

Q15: The impulse response g(t) of a system, G , is as shown in Figure (a). What is the maximum value attained by the impulse response of two cascaded blocks of G as shown in Figure (b)?      (SET-1 (2015))
Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)

Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)
(a) 2/3
(b) 3/4
(c) 4/5
(d) 1
Ans:
(d)
Sol: Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)Maximum value = 1

Q16: A moving average function is given by Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE) If the input u is a sinusoidal signal of frequency (1/2τ) Hz, then in steady state, the output y will lag u (in degree) by ______ .      (SET-1 (2015))
(a) 30
(b) 60
(c) 90
(d) 120
Ans:
(c)
Sol: System input: sin ω0t
Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)Therefore,
Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)
Q17: Consider an LTI system with impulse response h(t) = e−5tu(t). If the output of the system is y(t) = e−3tu(t) − e−5tu(t) then the input, x(t), is given by       (SET-2 (2014))
(a) e−3t u(t)
(b) 2e−3t u(t)
(c) e−5t u(t)
(d) 2𝑒5𝑡𝑢(𝑡)2e−5t u(t)
Ans:
(b)
Sol: Impulse response of an LTI system = transfer function Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)
Q18: x(t) is nonzero only for T< t < Tx′, and similarly, y(t) is nonzero only for  T< t < Ty′. Let z(t) be convolution of x(t) and y(t). Which one of the following statements is TRUE ?         (SET-1 (2014))
(a) z(t) can be nonzero over an unbounded interval.
(b) z(t) is nonzero for t < T+ Ty
(c) z(t) is zero outside of Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)

(d) z(t) is nonzero for Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)
Ans: (c)
Sol: x(t) is non-zero for Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)and  y(t) is non-zero for Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)then the limits of the resultant signal is
Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)
Q19: The impulse response of a continuous time system is given by h(t) = δ(t − 1)δ(t − 3). The value of the step response at t = 2 is        (2013)
(a) 0
(b) 1
(c) 2
(d) 3
Ans:
(b)
Sol: For step response, impulse response can be integrated.
Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)
Q20: Two systems with impulse responses h1(t) and h2(t) are connected in cascade. Then the overall impulse response of the cascaded system is given by     (2013)
(a) product of h1(t) and h2(t)
(b) sum of h1(t) and h2(t)
(c) convolution of h1(t) and h2(t)
(d) subtraction of h1(t) and h2(t)
Ans:
(c)
Sol: In cascade connection,
Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)Previous Year Questions- Linear Time Invariant Systems - 1 | Signals and Systems - Electrical Engineering (EE)

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FAQs on Previous Year Questions- Linear Time Invariant Systems - 1 - Signals and Systems - Electrical Engineering (EE)

1. What is a linear time-invariant system?
Ans. A linear time-invariant system is a system that follows the principles of linearity and time invariance. Linearity means the system obeys the superposition principle, and time invariance means the system's behavior does not change over time.
2. How can we determine if a system is linear or time-invariant?
Ans. To determine if a system is linear, we can check if it satisfies the properties of additivity and homogeneity. To check for time-invariance, we can verify if the system's response remains the same regardless of when the input is applied.
3. What are the key characteristics of a linear time-invariant system?
Ans. The key characteristics of a linear time-invariant system include superposition, homogeneity, time-invariance, and causality. These properties help in analyzing and understanding the system's behavior.
4. How can we mathematically represent a linear time-invariant system?
Ans. A linear time-invariant system can be represented using a differential equation, transfer function, impulse response, or frequency response. These mathematical representations help in analyzing the system's input-output relationship.
5. Why is the study of linear time-invariant systems important in electrical engineering?
Ans. The study of linear time-invariant systems is essential in electrical engineering as it helps in analyzing and designing various systems such as filters, amplifiers, and control systems. Understanding these systems' behavior is crucial for ensuring their stability and performance.
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