# Signals & Systems Questions and Answers – The Complex Exponential

«
»

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

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)

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

Explanation: Fundamental period = 2pi/w, hence fundamental frequency will be w.
Sanfoundry Certification Contest of the Month is Live. 100+ Subjects. Participate Now!

4. Total energy possessed by a signal exp(jwt) is?
a) 2pi/w
b) pi/w
c) pi/2w
d) 2pi/3w

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

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) 1w
d) w

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

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

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

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

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) eσt
b) eΩt
c) est
d) eat

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

Explanation: Let x(t) be the complex exponential signal
⇒ x(t) = est = e(σ+jΩ)t = eσt ejΩt
Now, when Ω = 0 ⇒ x(t) = eσt 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

Explanation: A signal is sinusoidal when σ = 0 and Ω ≠ 0
⇒ x(t) = est = e(σ+jΩ)t = eσt ejΩt = ejΩt = 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

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

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.

To practice all areas of Signals & Systems, here is complete set of 1000+ Multiple Choice Questions and Answers. 