This set of Advanced Signals & Systems Questions and Answers focuses 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

View Answer

Explanation: 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

View Answer

Explanation: 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

View Answer

Explanation: 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

View Answer

Explanation: 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

View Answer

Explanation: 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

View Answer

Explanation: \(X_C = \frac{1}{jCω} = \frac{-j}{2 × 3.14 × 100 × 10^3 × 10 × 10^{-12}}\)

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) \(\frac{X[k]-X^* [-k]}{2}\)

View Answer

Explanation: Re{x (t)} = \(\frac{x(t)+x^* (-t)}{2}\)

The Fourier coefficient of x* (t) is

X

_{1}[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, X

_{1}[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 f_{0} is given by g_{p} (t) = \(∑_{n=-∞}^∞ C_n e^{j 2πnf_0 t}\). Given that C_{3} = 3 + j5. The value of C_{-3} is ______________

a) 5 + 3j

b) -5 + 3j

c) -3 – j5

d) 3 – j5

View Answer

Explanation: Given that C

_{3}= 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

View Answer

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. 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

View Answer

Explanation: 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) \(\frac{1}{T_0} ∑_{n=-∞}^∞ exp(\frac{j2πnt}{T_0}) \)

View Answer

Explanation: s (t) = \(∑_{n=-∞}^∞ C_n e^{jnω_0 t}\), where ω

_{0}=(2T/T

_{0})

And C

_{n}= \(\frac{1}{T_0} \displaystyle\int_{-\frac{T_0}{2}}^{\frac{T_0}{2}} δ(t) e^{-jnω_0 t} \,dt\)

= \(\frac{1.e^{-jnω_0 t}}{T_0}\)

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

View Answer

Explanation: 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 T_{1} (< T) and −V for the duration T − T_{1} = T_{2}, equals _____________

a) V

b) \(\frac{V}{\sqrt{2}}\)

c) \(\frac{T_1-T_2}{T}\) V

d) \(\frac{T_1}{T_2}\) V

View Answer

Explanation: Period = T = T

_{1}+ T

_{2}

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

View Answer

Explanation: 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}

View Answer

Explanation: Time-scaling property of 6(t)

We know that by this property,

6(at) = \(\frac{1}{a}\) 6(t), a>0

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