250+ TOP MCQs on Miscellaneous Examples on Fourier Series and Answers

Advanced Signals & Systems Questions on “Miscellaneous Examples on Fourier Series”.

1. A signal g (t) = 10 sin (12πt) is ___________
A. A periodic signal with period 6 s
B. A periodic signal with period (frac{10}{6}) s
C. A periodic signal with period (frac{1}{6}) s
D. An aperiodic signal

Answer: C
Clarification: 10 sin (12πt) = A sin ωt
∴ω = 12π
∴ 2πf = 1
So, fundamental frequency = 6
Hence, fundamental period = (frac{1}{6}) s.

2. A voltage having the waveform of a sine curve is applied across a capacitor. When the frequency of the voltage is increased, what happens to the current through the capacitor?
A. Increases
B. Decreases
C. Remains same
D. Is zero

Answer: A
Clarification: The current through the capacitor is given by,
I_C = ωCV cos (ωt + 90°).
As the frequency is increased, I_C also increases.

3. What is the steady state value of F (t), if it is known that (F(s) = frac{2}{s(S+1)(s+2)(s+3)})?
A. (frac{1}{2})
B. (frac{1}{3})
C. (frac{1}{4})
D. Cannot be determined

Answer: B
Clarification: From the equation of F(s), we can infer that, a simple pole is at origin and all other poles are having negative real part.
∴ F(∞) = lim s→0 sF(s)
= lim s→0 (frac{2s}{s(S+1)(s+2)(s+3)})
= (frac{2}{s(S+1)(s+2)(s+3)})
= (frac{2}{6} = frac{1}{3}).

4. What is the steady state value of F (t), if it is known that F(s) = (frac{10}{(s+1)(s^2+1)})?
A. -5
B. 5
C. 10
D. Cannot be determined

Answer: D
Clarification: The steady state value of this Laplace transform is cannot be determined since; F(s) is having two poles on the imaginary axis (j and –j). Hence the answer is that it cannot be determined.

5. A periodic rectangular signal X (t) has the waveform as shown below. The frequency of the fifth harmonic of its spectrum is ______________

A. 40 Hz
B. 200 Hz
C. 250 Hz
D. 1250 Hz

Answer: D
Clarification: Periodic time = 4 ms = 4 × 10^-3
Fundamental frequency = (frac{10^3}{4}) = 250 Hz
∴ Frequency of the fifth harmonic = 250 × 5 = 1250 Hz.

6. A CRO probe has an impedance of 500 kΩ in parallel with a capacitance of 10 pF. The probe is used to measure the voltage between P and Q as shown in the figure. The measured voltage will be?

A. 3.53 V
B. 3.47 V
C. 5.54 V
D. 7.00 V

Answer: B
Clarification: (X_C = frac{1}{jCω} = frac{-j}{2 × 3.14 × 100 × 10^3 × 10 × 10^{-12}})
Applying KCL at the node,
(frac{V_a-10}{100} + frac{V_a}{100} + frac{V_a}{500} + frac{V_a}{-j159})
∴ V_a = 4.37∠-15.95°.

7. The Fourier series coefficient of the signal z(t) = Re{x(t)} is ____________
A. (frac{X[k]+X[-k]}{2})
B. (frac{X[k]-X[-k]}{2})
C. (frac{X[k]+X^* [-k]}{2})
D. (frac{X[k]-X^* [-k]}{2})

Answer: C
Clarification: Re{x (t)} = (frac{x(t)+x^* (-t)}{2})
The Fourier coefficient of x* (t) is
X₁[k] = (frac{1}{T}) ∫ x* (t)e^-jkωt dt = (X_1^*) [-k]
Or, (X_1^*) [k] = (frac{1}{T}) ∫ x(t)e^jkωt dt = X [-k]
So, X₁[k] = (X_1^*) [-k]
∴ Z[k] = (frac{X[k]+X^* [-k]}{2}).

8. The Fourier series expansion of a real periodic signal with fundamental frequency f₀ is given by g_p (t) = (∑_{n=-∞}^∞ C_n e^{j 2πnf_0 t}). Given that C₃ = 3 + j5. The value of C_-3 is ______________
A. 5 + 3j
B. -5 + 3j
C. -3 – j5
D. 3 – j5

Answer: D
Clarification: Given that C₃ = 3 + j5.
We know that for real periodic signal C_-k = (C_k^*)
So, C_-3 = (C_3^*) = 3 – j5.

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

Answer: A
Clarification: 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. Let us suppose that the impulse response of a causal LTI system is given as h (t). Now, consider the following two statements:
Statement 1- Principle of superposition holds
Statement 2- h (t) = 0(for t<0)
Which of the following is correct?
A. Statement 1 is correct and Statement 2 is wrong
B. Statement 2 is correct and Statement 1 is wrong
C. Both Statement 1 and Statement 2 are wrong
D. Both statement 1 and Statement 2 are correct

Answer: D
Clarification: As we know that linear system possesses superposition theorem.
Hence, Statement 1 satisfies the given equation. We also know that time invariant condition depend on time.
Hence, Statement 2 satisfies the given equation.

11. The Fourier series representation of an impulse train denoted by s(t) = (∑_{n=-∞}^∞ δ(t-nT_0)) is given by _____________
A. (frac{1}{T_0} ∑_{n=-∞}^∞ exp⁡(-frac{j2πnt}{T_0}) )
B. (frac{1}{T_0} ∑_{n=-∞}^∞ exp⁡(-frac{jπnt}{T_0}))
C. (frac{1}{T_0} ∑_{n=-∞}^∞ exp⁡(frac{jπnt}{T_0}) )
D. (frac{1}{T_0} ∑_{n=-∞}^∞ exp⁡(frac{j2πnt}{T_0}) )

Answer: B
Clarification: s (t) = (∑_{n=-∞}^∞ C_n e^{jnω_0 t}), where ω₀=(2T/T₀)
And C_n = (frac{1}{T_0} displaystyleint_{-frac{T_0}{2}}^{frac{T_0}{2}} δ(t) e^{-jnω_0 t} ,dt)
= (frac{1.e^{-jnω_0 t}}{T_0})
So, Fourier series representation = (frac{1}{T_0} ∑_{n=-∞}^∞ exp⁡(-frac{jπnt}{T_0})).

12. For which of the following a Fourier series cannot be defined?
A. 3 sin (25t)
B. 4 cos (20t + 3) + 2 sin (710t)
C. exp(-|t|) sin (25t)
D. 1

Answer: C
Clarification: 3 sin (25) = 25
4 cos (20 + 3) + 2sin (710) sum of two periodic function is also periodic function
For 1 which is a constant, Fourier series exists.
For exp (-|t|) sin (25t), due to decaying exponential decaying function, it is not periodic. So Fourier series cannot be defined for it.

13. The RMS value of a rectangular wave of period T, having a value of +V for a duration T₁ (< T) and −V for the duration T − T₁ = T₂, equals _____________
A. V
B. (frac{V}{sqrt{2}})
C. (frac{T_1-T_2}{T}) V
D. (frac{T_1}{T_2}) V

Answer: A
Clarification: Period = T = T₁ + T₂
RMS value = (sqrt{frac{1}{T} ∫_0^T x^2 (t)dt})
= (sqrt{frac{1}{T}[V^2.(T_1 – 0) + V^2 (T – T_1)]})
= (sqrt{V^2})
= V.

14. A periodic square wave is formed by rectangular pulses ranging from -1 to +1 and period = 2 units. The ratio of the power in the 7th harmonic to the power in the 5th harmonic for this waveform is equal to ____________
A. 1
B. 0.5
C. 2
D. 2.5

Answer: B
Clarification: For a periodic square wave nth harmonic component ∝ (frac{1}{n})
Thus the power in the nth harmonic component is ∝ (frac{1}{n^2})
∴ Ratio of power in 7th harmonic to 5th harmonic for the given wage form = (frac{frac{1}{7^2}}{frac{1}{5^2}})
= (frac{25}{4}) ≅ 0.5.

15. A useful property of the unit impulse 6 (t) is ________________
A. 6 (at) = a 6 (t)
B. 6 (at) = 6 (t)
C. 6 (at) = (frac{1}{a}) 6(t)
D. 6(at) = [6(t)]^a

Answer: C
Clarification: Time-scaling property of 6(t)
We know that by this property,
6(at) = (frac{1}{a}) 6(t), a>0