# Signals & Systems Questions and Answers – Properties of Systems – 3

This set of Signals & Systems Interview Questions and Answers for Experienced people focuses on “Properties of Systems – 3”.

1. Which one of the following is an example of a system with memory?
a) Identity System
b) Resistor
c) y(n)=x(n)-2x(n)
d) Accumulator

Explanation: An identity system gives the output same as input hence it totally depends on the present state of the input. Therefore, it is memory less. Similarly, a resistor and the expression in option c are memory less systems as they depend upon the present state of the input. An accumulator sums up the values of all past and present states of input. Therefore, it is a system with memory.

2. Which among the following is a memory less system?
a) Delay
b) Summer
c) Resistor
d) Capacitor

Explanation: Options Delay, Summer and Capacitor are all systems with memory as they depend upon past, past and present, past and present values of input respectively. Whereas, a resistor is a memory less system as its relationship with output always depends upon the current or present state of the input.

3. In a continuous-time physical system, memory is directly associated with _____________
a) Storage registers
b) Time
c) Storage of energy
d) Number of components in the system

Explanation: Memory is directly associated with storage of energy such as electric charge in the capacitor or kinetic energy in an automobile. Storage registers are for discrete time systems such as microprocessor etc. Time and number of components of a system have got nothing to do with memory.

4. A system with memory which anticipates future values of input is called _________
a) Non-causal System
b) Non-anticipative System
c) Causal System
d) Static System

Explanation: A system which anticipates the future values of input is called a non-causal system. A causal depends only on the past and present values of input. Non-anticipative is another name for the causal system. A static system is memory less system.

5. Determine the nature of the system: y(n)=x(-n).
a) Causal
b) Non-causal
c) Causal for all positive values of n
d) Non-causal for negative values of n

Explanation: The given system gives negative values of input i.e., past values of input when we feed positive integers to LHS. However, it gives positive values for negative values of n i.e., future values. Therefore, the system depends upon past values for some integers and future values for some other. A system cannot be called partially causal or non-causal, therefore, the system is non-causal.
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6. Which among the following is an application of non-causal system?
a) Image processing
b) RC circuit
c) Stock market Analysis
d) Automobile

Explanation: Image processing, RC circuit, and an automobile are all causal systems as they do not anticipate the future values of an image, RC circuit and future actions of a driver respectively. Instead, they function upon either the stored information or on the current values of the input. Whereas, in the stock market, analysts try to figure out a trend in the future based upon the stored information. Therefore, it is non-causal.

7. Determine the nature of the given system: y(t)=x(sint)
a) Causal, Non-linear
b) Causal, Linear
c) Non-Causal, Non-linear
d) Non-causal, Linear

Explanation: The system is non-causal as it gives future values for some inputs.
E.g. y (- π) = x (sin (-π)) = x (0)
For linearity, it needs to satisfy superposition principle,
⇒ y1 (t) = x1 (sint)
⇒ y2 (t) = x2 (sint)
⇒ ay1 (t) + by2 (t) = ax1 (sint) + bx1 (sint) Equation 1
Now, y3 (t) = x3 (sint) = (ax1 + bx2)(sint) = ax1 (sint) + bx1 (sint) Equation 2
Clearly, Equation 1 and 2 are equal, hence the system is linear.

8. Is the system y[n]=2x[n]+2 linear?
a) YES
b) NO

Explanation: The system needs to satisfy superposition principle for linearity:
For input x1[n], y1 [n] = 2x1 [n] + 2
For input x2[n], y2 [n] = 2x2 [n] + 2
⇒ ay1 [n]+ by2 [n] = 2(ax1 [n]+ bx2 [n]) + 2(a+b) Equation 1
For, x3[n], y3 [n]=2x3 [n]+2 = 2(ax1 [n]+ bx2 [n]) + 2 Equation 2
Clearly, Equation 1 is not equal to equation
∴ The system does not satisfy superposition principle ⇒ The system is not linear.

9. An inverse system with the original system gives an output equal to the input. How is the inverse system connected to the original system?
a) Series
c) parallel
d) No connection

Explanation: An inverse system when cascaded with the original system gives an output equal to the input.

10. Which among the following is an invertible system?
a) y[n] = 0
b) y[n] = 2x[n]
c) y(t) = x2(t)
d) y(t) = dx(t)/dt

Explanation: A system is said to be invertible if it’s input can be found out from its output. Implying, if a system has same outputs for several inputs then it is impossible to find the correct input as output is same for many. Therefore, a system is invertible if it gives distinct outputs to distinct inputs. It is non-invertible if it gives same outputs for many inputs.
Option a produces 0 output for any input → Non-invertible
Option b produces different outputs for different inputs and also it’s inverse system is (1/2)y[n] → Invertible
Option c, we get same output for both positive and negative values → Non-invertible
Option d, we get 0 for all constant input values → Non-invertible.

11. Is the system time invariant: y(t) = x(4t)?
a) YES
b) NO

Explanation: A system is said to be time invariant if a change input causes the same change in output.
For change in input by T
⇒ y(t, T) = x(4(t – T)) = x(4t – 4T) Equation 1
For the same change in output
⇒ y(t – T) = x(4t – T) Equation 2
Equation 1 is not equal to equation 2.
∴ The system is not time invariant or is time variant.

12. Determine the nature of the system: y[n] = x[n]x[n – 1] with unit impulse function as an input.
a) Dynamic, output always zero, non-invertible
b) Static, output always zero, non-invertible
c) Dynamic, output always 1, invertible
d) Dynamic, output always 1, invertible

Explanation: Since the system depends on present and past values, therefore, it is not memory less(dynamic).
Now, input is a unit impulse function. Unit impulse function = 1 at n = 0, otherwise it is equal to 0.
For, y[0] = x[0]x[-1] = 1 × 0 = 0
For, y[1] = x[1]x[0] = 0 × 1 = 0
For, y[2] = x[2]x[1] = 0 × 0 = 0
∴ For any time, output is always zero.
Since, the output is always same, the system is non-invertible.

13. Determine the nature of the system: y(t)= t2 x(t-1)
a) Linear, time invariant
b) Linear, time variant
c) Non-linear, time invariant
d) Non-linear, time variant

Explanation: For linearity:
For input x1(t): y1 (t)= t2 x1 (t-1)
For input x2(t): y2 (t)= t2 x2 (t-1)
⇒ ay1 (t)+by2 (t)= t2 [x1 (t-1)+ bx2 (t-1)] Equation 1
For input x3(t): y3 (t)= t2 x3 (t-1) = t2 [ax1 (t-1)+ bx2 (t-1)] Equation 2
∴ The system is linear.
For time invariancy: Shift in input:
⇒y(t,T)= t2 x(t-1-T)
Shift in output: y(t- T)= (t-T)2 x(t-1-T)
∵ The shift in output is not equal to the shift in input, therefore, the system is time variant.

14. y[n]=rn x[n] is ________ system.
a) LTI
b) Time varying
c) Linear and time invariant
d) Causal and time invariant

Explanation: The input-output relationship of the given system shows it does not satisfy the condition of time-invariant system. Hence it is time varying system.

15. A system is said to be linear if _______
a) It satisfies only the principle of superposition theorem
b) It satisfies only amplitude scaling
c) It satisfies both amplitude scaling and principle of superposition theorem
d) It satisfies amplitude scaling but not the principle of superposition theorem

Explanation: By the definition of linearity a system is said to be linear if it satisfies the condition y1(t) + y2(t) = ax1(t) + bx2(t).

16. If the input-output relationship is given by y(t) = 2x(t) ddx x(t). What kind of system it represents?
a) Linear system
b) Non linear system
c) LTI system
d) Linear but time-invariant system

Explanation: The given input-output relationship of the system does not satisfy the principle of superposition theorem hence it is an example for non linear system.

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