This set of Signals & Systems Multiple Choice Questions & Answers (MCQs) focuses on “Fourier Series”.

1. What is Fourier series?

a) The representation of periodic signals in a mathematical manner is called a Fourier series

b) The representation of non periodic signals in a mathematical manner is called a Fourier series

c) The representation of non periodic signals in terms of complex exponentials or sinusoids is called a Fourier series

d) The representation of periodic signals in terms of complex exponentials or sinusoids is called a Fourier series

View Answer

Explanation: The Fourier series is the representation of non periodic signals in terms of complex exponentials, or equivalently in terms of sine and cosine waveform leads to Fourier series. In other words, Fourier series is a mathematical tool that allows representation of any periodic wave as a sum of harmonically related sinusoids.

2. Who discovered Fourier series?

a) Jean Baptiste de Fourier

b) Jean Baptiste Joseph Fourier

c) Fourier Joseph

d) Jean Fourier

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Explanation: The Fourier series is the representation of non periodic signals in terms of complex exponentials or sine or cosine waveform. This was discovered by Jean Baptiste Joseph Fourier in 18th century.

3. Fourier series representation can be used in case of Non-periodic signals too. True or false?

a) True

b) False

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Explanation: False. The Fourier series is the representation of periodic signals in terms of complex exponentials, or equivalently in terms of sine and cosine waveform leads to Fourier series. In other words, Fourier series is a mathematical tool that allows representation of any periodic wave as a sum of harmonically related sinusoids. They are for periodic signals only.

4. What are the conditions called which are required for a signal to fulfil to be represented as Fourier series?

a) Dirichlet’s conditions

b) Gibbs phenomenon

c) Fourier conditions

d) Fourier phenomenon

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Explanation: When the Dirichlet’s conditions are satisfied, then only for a signal, the fourier series exist. Fourier series is of two types- trigonometric series and exponential series.

5. Choose the condition from below that is not a part of Dirichlet’s conditions?

a) If it is continuous then there are a finite number of discontinuities in the period T1

b) It has a finite average value over the period T

c) It has a finite number of positive and negative maxima in the period T

d) It is a periodic signal

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Explanation: Even if the Fourier series demands periodicity as the major necessity for its formation still it is not a part of Dirichlet’s condition. It is the basic necessity for Fourier series.

6. What are the two types of Fourier series?

a) Trigonometric and exponential

b) Trigonometric and logarithmic

c) Exponential and logarithmic

d) Trigonometric only

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Explanation: The two types of Fourier series are- Trigonometric and exponential. The exponential is more convenient for Fourier series calculations.

7. How is a trigonometric Fourier series represented?

a) A_{0} +∑[ancos(w_{0}t)+ ansin(w_{0}t)]

b) ∑[ancos(w_{0}t)+ ansin(w_{0}t)]

c) A_{0} *∑[ancos(w_{0}t)+ ansin(w_{0}t)]

d) A_{0} +∑[ancos(w_{0}t)+ ansin(w_{0}t)] + sinwt

View Answer

Explanation: A

_{0}+ ∑[ancos(w

_{0}t)+ ansin(w

_{0}t)] is the correct representation of a trigonometric Fourier series. Here A

_{0}= 1/T∫x(t)dt and an =2/T∫x(t)cos(w

_{0}t)dt and bn= 2/T∫x(t)sin(w

_{0}t)dt.

8. How is the exponential Fourier series represented?

a) X(t) = ∑X_{n}e^{jnwt} + wt

b) X(t) = 1/T∑Xne^{jnwt}

c) X(t) = ∑X_{n}e^{jnwt}

d) X(t) = T*∑X_{n}e^{jnwt}

View Answer

Explanation: The exponential Fourier series is represented as – X(t)=∑X

_{n}e

^{jnwt}. Here, the X(t) is the signal and X

_{n}=1/T∫x(t)e

^{-jnwt}.

9. What is the equation – X(t)=∑X_{n}e^{jnwt} called?

a) Synthesis equation

b) Analysis equation

c) Frequency domain equation

d) Discrete equation

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Explanation: The equation – X(t) = ∑X

_{n}e

^{jnwt}called the synthesis equation of an exponential Fourier series. It is because it is used to synthesize the Fourier series.

10. What is the equation – X_{n}=1/T∫x(t) e^{jwtn} called?

a) Synthesis equation

b) Analysis equation

c) Frequency domain equation

d) Discrete equation

View Answer

Explanation: The equation – X

_{n}=1/T∫x(t)e

^{-jwtn}called the analysis equation of an exponential Fourier series. It is because it is used to synthesize the Fourier series.

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