This set of Signals & Systems Multiple Choice Questions & Answers (MCQs) focuses on “Properties of ROC”.

1. Given a system function H(s) = \(\frac{1}{s+3}\). Let us consider a signal sin 2t. Then the steady state response is ___________

a) \(\frac{1}{8}\)

b) Infinite

c) 0

d) 8

View Answer

Explanation: H(s) = \(\frac{V(s)}{J(s)}\)

= \(\frac{1}{s+3}\)

V(s) = \(\frac{1}{s+3}\). J(s)

J(s) = L (sin 2t) = \(\frac{2}{s^2+4}\)

V (s) = \(\frac{1}{s^2+4} . \frac{2}{s+3}\)

VSS = lim

_{s→0}sV(s)

= 0.

2. If H(f) = \(\frac{y(t)}{x(t)}\), then for this to be true x(t) is ___________

a) exp\((\frac{j2nf}{t})\)

b) exp\((-\frac{j2nf}{t})\)

c) exp(j2nft)

d) exp(-j2nft)

View Answer

Explanation: Let us consider x (t) = e

^{2jnft}

So, y (t) = \(\int_{-∞}^∞ h(τ)x(t-τ)dτ\)

= \(\int_{-∞}^∞ h(τ)e^{j2nπf(t-τ)} \,dτ\)

= e

^{j2nft}H (f)

Or, H (f) = \(\frac{y(t)}{e^{j2nft}}\)

So, x (t) = e

^{j2nft}.

3. The z-transform of –u(-n-1) is ___________

a) \(\frac{1}{1-z}\)

b) \(\frac{z}{1-z}\)

c) \(\frac{1}{1-z^{-1}}\)

d) \(\frac{z}{1-z^{-1}}\)

View Answer

Explanation: z [-u (-n-1)] = \(∑_{n=-1}^∞ [u(-n-1)] z^{-n}\)

= – [z + z

^{2}+ z

^{3}+ z

^{4}…]

= \(\frac{z}{z-1}\)

= \(\frac{1}{1-z^{-1}}\).

4. The value of z(a^{k} u[-k]) is _______________

a) \(\frac{z}{z-a}\)

b) \(\frac{z}{a-z}\)

c) \(\frac{z^2}{z-a}\)

d) \(\frac{a}{a-z}\)

View Answer

Explanation: z (a

^{k}u [-k]) = \(∑_{k=-∞}^{-1} a^k z^{-k}\)

= \(∑_{m=1}^∞ a^{-m} z^m\)

= \(∑_{m=1}^∞ (a^{-1} z)^m \)

= \(\frac{a^{-1} z}{1-a^{-1} z} = \frac{z}{a-z}\).

5. If a system has N different poles, then the system can have ______________

a) N ROC’s

b) (N-1) ROC’s

c) (N+1) ROC’s

d) 2N ROC’s

View Answer

Explanation: Let us consider 2 poles. For 2 poles, we will have 3 ROC conditions. Hence, if a system has N poles then the system will have (N+1) ROC’s.

6. Given 2 signals (-3)^{k} u(k) and u (k-1). These two signals are superimposed. This superimposed signal is _______________

a) \(\frac{z}{z+3} + \frac{1}{z-1}\)

b) \(\frac{z}{z+3} – \frac{1}{z-1}\)

c) \(\frac{z}{z-3} + \frac{1}{z-1}\)

d) \(\frac{z}{z+3} + \frac{1}{z+1}\)

View Answer

Explanation: We know that superposition means addition of these 2 signals.

So, superimposed f[k] = (-3)

^{k}u(k) + u(k-1)

Hence, z[k] = \(\frac{z}{z+3} + \frac{1}{z-1}\).

7. X_{1}(z) = 2z + 1 + z^{-1} and X_{2}(z) = z + 1 + 2z^{-1} is ________________

a) Even signal

b) Odd signal

c) In time power signal

d) In time energy signal

View Answer

Explanation: Given X

_{1}(z) = 2z + 1 + z

^{-1}and X

_{2}(z) = z+1+2z

^{-1}.

So, x

_{1}(k) = {2, 1, 1}; x

_{2}(k) = {1, 1, 2}.

8. The value of z{[k-1] u(k)} is _______________

a) \(\frac{z(z+2)}{(z-1)^2}\)

b) \(\frac{2z-z^2}{(z-1)^2}\)

c) \(\frac{z^2}{(z-1)^2}\)

d) \(\frac{z(z-2)}{(z+1)^2}\)

View Answer

Explanation: z {[k-1] u (k)} = z {k u (k) – u (k)}

= \(\frac{z}{(z-1)^2} – \frac{z}{z-1}\)

= \(\frac{z-z(z-1)}{(z-1)^2}\)

= \(\frac{z-z^2+z}{(z-1)^2}\)

= \(\frac{2z-z^2}{(z-1)^2}\).

9. The area under Gaussian pulse \(\int_{-∞}^∞ e^{{-π}^{{t}^2}} \,dt \) is ___________

a) Unity

b) Infinity

c) Pulse

d) Zero

View Answer

Explanation: Putting \(π^{t^2}\) = x, we get,

\(\int_{-∞}^∞ e^{{-π}^{{t}^2}} \,dt = \int_{-∞}^∞ e^{-x} 2π \sqrt{\frac{x}{π}} \,dx\)

\(= 2\sqrt{π} \int_{-∞}^∞ \sqrt{x} e^{-x} \,dx\)

= 1.

10. 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

View Answer

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.

11. The spectral density of white noise is ____________

a) Exponential

b) Uniform

c) Poisson

d) Gaussian

View Answer

Explanation: The distribution of White noise is homogeneous over all frequencies. Power spectrum is the Fourier transform of the autocorrelation function. Therefore, the power spectral density of white noise is uniform.

12. In the circuit given below, the value of Z_{L} for maximum power to be transferred is _____________

a) R

b) R + jωL

c) R – jωL

d) jωL

View Answer

Explanation: The value of load for maximum power transfer is given by the complex conjugate of Z

_{AB}

Z

_{AB}= R + jX

_{L}

= R + jωL

∴ Z

_{L}for maximum power transfer is given by Z

_{L}= R – jωL.

13. A current I given by I = – 8 + 6\(\sqrt{2}\) (sin (ωt + 30°)) A is passed through three meters. The respective readings (in ampere) will be?

a) 8, 6 and 10

b) 8, 6 and 8

c) – 8, 10 and 10

d) -8, 2 and 2

View Answer

Explanation: PMMC instrument reads only DC value and since it is a center zero type, so it will give – 8 values.

So, rms = \(\sqrt{8^2 + (\frac{6\sqrt{2}}{\sqrt{2}})^2}\) = 10 A

Moving iron also reads rms value, so its reading will also be 10 A.

14. The CDF for a certain random variable is given as F(x) = {0, -∞<x≤0; kx2, 0<x≤10; 100k, 10<x<∞;} The value of k is __________

a) 100

b) 50

c) 1/50

d) 1/100

View Answer

Explanation: From the given F(x),

We get, \(\frac{dF(x)}{dx}\) = 0 + 2kx + 0

= 2kx

∴ \(\int_0^{10}\) 2 kx dx =1

Or, 100k = 1

Or, k = 1/100.

15. For a stable system which of the following is correct?

a) |z| < 1

b) |z| = 1

c) |z| > 1

d) |z| ≠ 1

View Answer

Explanation: We know that, for the system to be stable, the ROC should include the unit circle.

|z| = 1, represents the unit circle but does not include it.

|z| > 1, represents the region outside the unit circle. In other words, it excludes the region of the unit circle.

|z| ≠ 1 does not represent a unit circle.

**Sanfoundry Global Education & Learning Series – Signals & Systems.**

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