This set of Digital Signal Processing Multiple Choice Questions & Answers (MCQs) focuses on “Frequency Analysis of Continuous Time Signals”.

1. The Fourier series representation of any signal x(t) is defined as:

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Explanation: If the given signal is x(t) and F0 is the reciprocal of the time period of the signal and ck is the Fourier coefficient then the Fourier series representation of x(t) is given as .

2. Which of the following is the equation for the Fourier series coefficient?

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Explanation: When we apply integration to the definition of Fourier series representation, we get

3. Which of the following is a Dirichlet condition with respect to the signal x(t)?

a) x(t) has a finite number of discontinuities in any period

b) x(t) has finite number of maxima and minima during any period

c) x(t) is absolutely integrable in any period

d) All of the mentioned

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Explanation: For any signal x(t) to be represented as Fourier series, it should satisfy the Dirichlet conditions which are x(t) has a finite number of discontinuities in any period, x(t) has finite number of maxima and minima during any period and x(t) is absolutely integrable in any period.

4. The equation is known as analysis equation.

a) True

b) False

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Explanation: Since we are synthesizing the Fourier series of the signal x(t), we call it as synthesis equation, where as the equation giving the definition of Fourier series coefficients is known as analysis equation.

5. Which of the following is the Fourier series representation of the signal x(t)?

d) None of the mentioned

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Explanation: In general, Fourier coefficients ck are complex valued. Moreover, it is easily shown that if the periodic signal is real, ck and c-k are complex conjugates. As a result

c

_{k}=|c

_{k}|e

^{jθk}and c

_{k}=|c

_{k}|e

^{-jθk}

Consequently, we obtain the Fourier series as

6. The equation is the representation of Fourier series.

a) True

b) False

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Explanation: cos(2πkF

_{0}t+θ

_{k})= cos2πkF

_{0}t.cosθ

_{k}-sin2πkF

_{0}t.sinθ

_{k}

θ

_{k}is a constant for a given signal.

So, the other form of Fourier series representation of the signal x(t) is

7. The equation of average power of a periodic signal x(t) is given as:

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Explanation: The average power of a periodic signal x(t) is given as

By interchanging the positions of integral and summation and by applying the integration, we get

8. What is the spectrum that is obtained when we plot |c_{k} |^{2} as a function of frequencies kF_{0}, k=0,±1,±2..?

a) Average power spectrum

b) Energy spectrum

c) Power density spectrum

d) None of the mentioned

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Explanation: When we plot a graph of |c

_{k}|

^{2}as a function of frequencies kF

_{0}, k=0,±1,±2… the following spectrum is obtained which is known as Power density spectrum.

9. What is the spectrum that is obtained when we plot |ck| as a function of frequency?

a) Magnitude voltage spectrum

b) Phase spectrum

c) Power spectrum

d) None of the mentioned

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Explanation: We know that, Fourier series coefficients are complex valued, so we can represent ck in the following way.

c

_{k}=|c

_{k}|e

^{jθk}

When we plot |c

_{k}| as a function of frequency, the spectrum thus obtained is known as Magnitude voltage spectrum.

10. What is the equation of the Fourier series coefficient ck of an non-periodic signal?

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Explanation: We know that, for an periodic signal, the Fourier series coefficient is

If we consider a signal x(t) as non-periodic, it is true that x(t)=0 for |t|>Tp/2. Consequently, the limits on the integral in the above equation can be replaced by -∞ to ∞. Hence,

11. Which of the following relation is correct between Fourier transform X(F) and Fourier series coefficient c_{k}?

a) c_{k}=X(F_{0}/k)

b) c_{k}= 1/T_{P} (X(F_{0}/k))

c) c_{k}= 1/T_{P}(X(kF_{0}))

d) None of the mentioned

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12. According to Parseval’s Theorem for non-periodic signal,

d) All of the mentioned

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Explanation: Let x(t) be any finite energy signal with Fourier transform X(F). Its energy is

**Sanfoundry Global Education & Learning Series – Digital Signal Processing.**

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