# Signals & Systems Questions and Answers – Fourier Series and LTI Systems

This set of Signals & Systems Multiple Choice Questions & Answers (MCQs) focuses on “Fourier Series and LTI Systems”.

1. Which system among the following is a time invariant system?
a) y(n) = n x(n)
b) y(n) = x(n) – x(n-1)
c) y(n) = x(-n)
d) y(n) = x(n) cos 2nf

Explanation: We know that, for any system y (n) = k x (n), to be a time invariant system, it must satisfy the relation, y (n-n1) = k x (n-n1) [where k is a constant or a function of n].
For y (n) = n x (n), y (n-n1) = (n-n1) x (n-n1)
This does not satisfy the criteria as stated above. Hence not time invariant.
For y (n) = x (-n), y (n-n1) = x (-n+n1)
This also does not satisfy the criteria as stated above. Hence not time invariant.
For y (n) = x (n) cos 2nf, y (n-n1) = x (n-n1) cos 2(n-n1) f
This also does not satisfy the criteria as stated above. Hence not time invariant.
For y (n) = x (n) – x (n-1), y (n-n1) = x (n-n1) – x (n-n1-1)
This satisfies the above criteria. Hence given system is time invariant.

2. Which of the following is a causal system?
a) y(n) = 3x(n) – 2x(n-1)
b) y(n) = 3x(n) + 2x(n+1)
c) y(n) = 3x(n+1) + 2x(n-1)
d) y(n) = 3x(n+1) + 2x(n-1) + x(n)

Explanation: We know that, for a causal system, output must depend on present and past but not on future.
For y (n) = 3x (n) + 2x (n+1), we can observe that output depends on future because of the term x (n+1). Hence, not a causal system.
For y (n) = 3x (n+1) + 2x (n-1), we can observe that output depends on future because of the term x (n+1). Hence not a causal system.
For y (n) = 3x (n+1) + 2x (n-1) + x (n), we can observe that output depends on future because of the term x (n+1). Hence not a causal system.
For y (n) = 3x (n) – 2x (n-1), we can observe that output depends on present and past but not on the future. Hence, it is a causal system.

3. Which of the following is a dynamic system?
a) y(n) = y(n-1) + y(n+1)
b) y(n) = y(n-1)
c) y(n) = x(n)
d) y(n) + y(n+1) + y(n+3) = 0

Explanation: We know that for a dynamic system, the present output of the system should depend only on the past output and the future output.
For y (n) = y (n-1), we can observe that output depends only on the past but not on the future. Hence it is not a dynamic system.
For y (n) = x (n), we can observe that output depends on the present. Hence it is not a dynamic system.
For y (n) + y (n+1) + y (n+3) = 0, we can observe that output is a constant. Hence it is not a dynamic system.
For y (n) = y (n-1) + y (n+1), we can observe that output depends only on past and future outputs. Hence it is a dynamic system.

4. A series RC circuit excited by voltage V is __________
a) A memory less system
b) A causal system
c) A dynamic system
d) Static system

Explanation: Dynamic systems are those systems which consist of memory. In the series RC circuit excited by voltage V, the capacitor C is an energy storing element which acts as a memory for the circuit. Therefore since the system has memory it is not a memoryless system. Also, a causal system depends only on the past and present value. But since the future value of the charge is also under consideration in this type of circuit, so the system is not causal. Since charge moves about in the circuit due to the applied voltage V, hence the system is not a static system. Therefore the system is a dynamic system.

5. A LTI system is characterized by ___________
a) Unit impulse response
b) Time shifted impulses
c) Unit step response
d) Response to any signal(bounded)

Explanation: The response of an LTI system to an arbitrary input is given by,
$$Y[n] = ∑_{k=-∞}^∞ x[k]h[n]$$, where $$x[n] = ∑_{k=-∞}^∞ x[k]δ[n-k]$$ represents sequences as a linear combination of shifted unit impulses δ[n-k]. Thus, any LTI system is completely characterized by its unit impulse response.
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6. Given a stable and causal system with impulse response h (t) and system function H(s). Let us suppose H(s) is rational, contains a pole at s=-2 and does not have a zero at origin and location of all other zeroes is unknown, poles are present at some unknown location other than the origin. Then,
i) H(s) = H(-s)
ii) $$\frac{dh(t)}{dt}$$ Contains at least one pole in its Laplace transform.
Which of the following options is correct?
a) i – true, ii – true
b) i – false, ii – true
c) i – true, ii – false
d) i – false, ii – false

Explanation: H (s) = H (-s)
If it were true, then H(s) has a pole at s=-2, it must also have a pole at s=2. This is contradicting the fact that all the poles of a causal and stable system must be in the left half of the s-plane.
For $$\frac{dh(t)}{dt}$$, if it has Laplace transform s H(s), but multiplication with s does not eliminate pole at s=-2.

7. From the given conditions, what are the Dirichlet conditions?
i. X(t) should be absolutely integrable
ii. X(t) should have finite discontinuities
iii. X(t) should have a finite number of maxima as well as minima in its domain
a) i, ii and iii
b) i and ii
c) i and iii
d) ii and iii

Explanation: For both periodic and non-periodic signals to have their Laplace transforms, they must satisfy the Dirichlet conditions. Dirichlet conditions state that a system should be absolutely integrable, have finite discontinuities, and have a finite number of maxima and minima in its domain.

8. If y(t) = ex(t), then the relation is _________
a) Dynamic
b) Static
c) Memory
d) Memoryless but not static

Explanation: Given relation, y (t) = ex(t).
The system represented by the above relation is static (memoryless) since the output at time t is dependent on t only. Further, the input-output relation is not an integrodifferential relation. Hence it is a static system.

9. A signal e-at sin (ωt) is the input to a real linear time-invariant system. Given K and ∅ are constants, the output of the system will be of the form Ke-bt sin (vt + ∅). The correct statement among the following is __________
a) b need not be equal to a but v must be equal to ω
b) v need not be equal to ω but b must be equal to a
c) b must be equal to a and v must be equal to ω
d) b need not be equal to a and v need not be equal to ω

Explanation: For a system with input e-at sin (ωt) and output Ke-bt sin (v t + ∅), frequency (v) to output must be equal to input frequency (ω) while b will depend on system parameters and need not be equal to a.

10. Which of the following is true about the bounded signal?
a) A finite signal is always bounded
b) A bounded signal always possess finite energy
c) A bounded signal is always zero outside a given interval
d) A bounded signal is always finite

Explanation: A bounded signal always possesses finite energy as a matter of fact. Mathematically on integrating it can be also shown that the energy of the bounded signal is finite while power is zero.

11. A system is said to be dynamic if the output of the system depends on ___________
a) The past Input
b) The Future Input
c) The Present Input
d) Both the Present and future Inputs

Explanation: A dynamic system is a system whose present output depends only on past inputs. Mathematically we can say that for a dynamic system y (t), the condition y (t) = y (t-1) should be always satisfied.

12. The system $$x(k) = 7(\frac{1}{3})^k \,u(-k-1) – 6(\frac{1}{2})^k \,u(k)$$ is ___________
a) Causal
b) Anti-causal
c) Non-causal
d) Cannot be determined

Explanation: Taking the z-transform, we get,
$$X (z) = 7\left(\frac{1}{z-\frac{1}{3}}\right) – 6\left(\frac{1}{z-\frac{1}{2}}\right)$$
∴ the ROC for given condition is as derived above.
∴ the bounded signal as a whole is non-causal.

13. The element which can weaken a signal system is ___________
a) Attenuation
b) Distortion
c) Noise
d) Attenuation, Distortion & Noise

Explanation: We know that distortion is the alteration of the original shape, attenuation is the reduction of signal strength and noise is the unwanted output of a signal or system. So, attenuation, distortion and noise all can impair a signal.

14. The difference between the highest and lowest frequencies of a signal is _________
a) Frequency
b) Period
c) Bandwidth
d) Amplitude

Explanation: Bandwidth is the name for that frequency range that a signal requires for transmission and is also a name for the frequency capacity of a particular transmission medium.

15. A system is said to be causal if the output of the system depends on ___________
a) The Past and Present Inputs
b) The Present Inputs
c) The Future Inputs
d) The Past and Future Inputs

Explanation: A system is causal if and only if the current output is only a function of present and past inputs. The current output does not depend on the future inputs y (k) = f(u(k)); u(k-1); u(k-2);).

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