This set of Digital Signal Processing Multiple Choice Questions & Answers (MCQs) focuses on “Frequency Analysis of Discrete Time Signals-1”.

1. What is the Fourier series representation of a signal x(n) whose period is N?

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Explanation: Here, the frequency F0 of a continuous time signal is divided into 2π/N intervals.

So, the Fourier series representation of a discrete time signal with period N is given as

where c

_{k}is the Fourier series coefficient

2. What is the expression for Fourier series coefficient ck in terms of the discrete signal x(n)?

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Explanation: We know that, the Fourier series representation of a discrete signal x(n) is given as

Now multiply both sides by the exponential e^(-j2πln/N) and summing the product from n=0 to n=N-1. Thus,

If we perform summation over n first in the right hand side of above equation, we get

Therefore, the right hand side reduces to Nck

So, we obtain

3. Which of the following represents the phase associated with the frequency component of discrete-time Fourier series(DTFS)?

a) e^{j2πkn/N}

b) e^{-j2πkn/N}

c) e^{j2πknN}

d) None of the mentioned

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Explanation: We know that,

In the above equation, ck represents the amplitude and ej2πkn/N represents the phase associated with the frequency component of DTFS.

4. The Fourier series for the signal x(n)=cos√2πn exists.

a) True

b) False

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Explanation: For ω

_{0}=√2π, we have f

_{0}=1/√2. Since f

_{0}is not a rational number, the signal is not periodic. Consequently, this signal cannot be expanded in a Fourier series.

5. What are the Fourier series coefficients for the signal x(n)=cosπn/3?

a) c1=c2=c3=c4=0,c1=c5=1/2

b) c0=c1=c2=c3=c4=c5=0

c) c0=c1=c2=c3=c4=c5=1/2

d) None of the mentioned

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Explanation: In this case, f0=1/6 and hence x(n) is periodic with fundamental period N=6.

Given signal is x(n)= cosπn/3=cos2πn/6=1/2 e^(j2πn/6)+1/2 e^(-j2πn/6)

We know that -2π/6=2π-2π/6=10π/6=5(2π/6)

So, we get c1=c2=c3=c4=0 and c1=c5=1/2.

View Answer

Explanation: Here, the frequency F0 of a continuous time signal is divided into 2π/N intervals.

So, the Fourier series representation of a discrete time signal with period N is given as

where c

_{k}is the Fourier series coefficient

7. What is the average power of the discrete time periodic signal x(n) with period N ?

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Explanation: Let us consider a discrete time periodic signal x(n) with period N.

The average power of that signal is given as

8. What is the equation for average power of discrete time periodic signal x(n) with period N in terms of Fourier series coefficient ck?

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9. What is the Fourier transform X(ω) of a finite energy discrete time signal x(n)?

d) None of the mentioned

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Explanation: If we consider a signal x(n) which is discrete in nature and has finite energy, then the Fourier transform of that signal is given as

10. What is the period of the Fourier transform X(ω) of the signal x(n)?

a) π

b) 1

c) Non-periodic

d) 2π

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Explanation: Let X(ω) be the Fourier transform of a discrete time signal x(n) which is given as

Now

So, the Fourier transform of a discrete time finite energy signal is periodic with period 2π.

11. What is the synthesis equation of the discrete time signal x(n), whose Fourier transform is X(ω)?

d) None of the mentioned

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Explanation: We know that the Fourier transform of the discrete time signal x(n) is

By calculating the inverse Fourier transform of the above equation, we get

The above equation is known as synthesis equation or inverse transform equation.

a) ω

_{c}.π

b) -ω

_{c}/π

c) ω

_{c}/π

d) None of the mentioned

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Explanation: We know that,

At n=0,

Therefore, the value of the signal x(n) at n=0 is ω_c/π.

13. What is the value of discrete time signal x(n) at n≠0 whose Fourier transform is represented as below?

d) None of the mentioned

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14. The oscillatory behavior of the approximation of XN(ω) to the function X(ω) at a point of discontinuity of X(ω) is known as Gibbs phenomenon.

a) True

b) False

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Explanation: We note that there is a significant oscillatory overshoot at ω=ωc, independent of the value of N. As N increases, the oscillations become more rapid, but the size of the ripple remains the same. One can show that as N→∞, the oscillations converge to the point of the discontinuity at ω=ωc. The oscillatory behavior of the approximation of XN(ω) to the function X(ω) at a point of discontinuity of X(ω) is known as Gibbs phenomenon.

15. What is the energy of a discrete time signal in terms of X(ω)?

d) None of the mentioned

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

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