This set of Signals & Systems Multiple Choice Questions & Answers (MCQs) focuses on “The Complex Exponential”.

1. What is exp(ja) equal to, where j is the square root of unity?

a) cos ja + jsin a

b) sin a + jcos a

c) cos j + a sin j

d) cos a + jsin a

View Answer

Explanation: This is the corollary of DeMoivre/Euler’s Theorem.

2. What is the magnitude of exp(2+3j)?

a) exp(2.3)

b) exp(3)

c) exp(2)

d) exp(3/2)

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Explanation: exp(a+b) =exp(a) * exp(b), and |exp(3i)| = 1.

3. What is the fundamental frequency of exp(2pi*w*j)?

a) 1pi*w

b) 2pi*w

c) w

d) 2w

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Explanation: Fundamental period = 2pi/w, hence fundamental frequency will be w.

4. Total energy possessed by a signal exp(jwt) is?

a) 2pi/w

b) pi/w

c) pi/2w

d) 2pi/3w

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Explanation: Energy possessed by a periodic signal is the integral of the square of the magnitude of the signal over a time period.

5. Sinusoidal signals multiplied by decaying exponentials are referred to as

a) Amplified sinusoids

b) Neutralized sinusoids

c) Buffered sinusoids

d) Damped sinusoids

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Explanation: The decaying exponentials dampen the amplitudes of sinusoids. Hence, the term damped sinusoids.

6. What is the power possessed by a signal exp(jwt)?

a) 1

b) 0.5

c) ^{1}⁄_{w}

d) w

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Explanation: The power = Energy/Time period for a periodic signal. Hence, Power = 1.

7. What is the period of exp(2+pi*j/4)t?

a) 4

b) 8

c) 16

d) 20

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Explanation: The fundamental period = 2pi/(pi/4) = 8.

8. exp(jwt) is periodic

a) for any w

b) for any t

c) for no w

d) for no t

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Explanation: Any two instants, t and t + 2pi will be equal, hence the signal will be periodic with period 2pi.

9. Define the fundamental period of the following signal x[n] = exp(2pi*j*n/3) + exp(3*pi*j*n/4)?

a) 8

b) 12

c) 18

d) 24

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Explanation: The first signal, will repeat itself after 3 cycles. The second will repeat itself after 8 cycles. Thus, both of them together will repeat themselves only after LCM(8,3) = 24 cycles.

10. exp[jwn] is periodic

a) for any w

b) for any t

c) for w=2pi*M/n

d) for t = 1/w

View Answer

Explanation: Discrete exponentials are periodic only for a particular choice of the fundamental frequency.

11. The most general form of complex exponential function is:

a) e^{σt}

b) e^{Ωt}

c) e^{st}

d) e^{at}

View Answer

Explanation: The general form of complex exponential function is: x(t) = e

^{st}where s = σ + jΩ.

12. A complex exponential signal is a decaying exponential signal when

a) Ω = 0 and σ > 0

b) Ω = 0 and σ = 0

c) Ω ≠ 0 and σ < 0

d) Ω = 0 and σ < 0

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Explanation: Let x(t) be the complex exponential signal

⇒ x(t) = e

^{st}= e

^{(σ+jΩ)t}= e

^{σt}e

^{jΩt}

Now, when Ω = 0 ⇒ x(t) = e

^{σt}which will be an exponentially decaying signal if σ < 0.

13. When is a complex exponential signal sinusoidal?

a) σ =0 and Ω = 0

b) σ < 0 and Ω = 0

c) σ = 0 and Ω ≠ 0

d) σ ≠ 0 and Ω ≠ 0

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Explanation: A signal is sinusoidal when σ = 0 and Ω ≠ 0

⇒ x(t) = e

^{st}= e

^{(σ+jΩ)t}= e

^{σt}e

^{jΩt}= e

^{jΩt}= cosΩt + jsinΩt which is sinusoidal.

14. An exponentially growing sinusoidal signal is:

a) σ = 0 and Ω = 0

b) σ > 0 and Ω ≠ 0

c) σ < 0 and Ω ≠ 0

d) σ = 0 and Ω ≠ 0

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Explanation: A complex exponential signal is sinusoidal when Ω has a definite value i.e., Ω ≠ 0. It can either be growing exponential or decaying exponential based on the value of σ.

∴ A signal is sinusoidal growing exponential when σ > 0 and Ω ≠ 0.

15. Determine the nature of the signal: x(t) = e^{-0.2t} [cosΩt + jsinΩt].

a) Exponentially decaying sinusoidal signal

b) Exponentially growing sinusoidal signal

c) Sinusoidal signal

d) Exponential signal

View Answer

Explanation: Clearly the signal has negative exponential ⇒ Decaying exponential signal.

The signal also has sinusoidal component.

∴ The signal is exponentially decaying sinusoidal signal.

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

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