# Signals & Systems Questions and Answers – Fourier Series

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

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

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

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

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.
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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

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

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) A0 +∑[ancos(w0t)+ ansin(w0t)]
b) ∑[ancos(w0t)+ ansin(w0t)]
c) A0 *∑[ancos(w0t)+ ansin(w0t)]
d) A0 +∑[ancos(w0t)+ ansin(w0t)] + sinwt

Explanation: A0 + ∑[ancos(w0t)+ ansin(w0t)] is the correct representation of a trigonometric Fourier series. Here A0 = 1/T∫x(t)dt and an =2/T∫x(t)cos(w0t)dt and bn= 2/T∫x(t)sin(w0t)dt.

8. How is the exponential Fourier series represented?
a) X(t) = ∑Xnejnwt + wt
b) X(t) = 1/T∑Xnejnwt
c) X(t) = ∑Xnejnwt
d) X(t) = T*∑Xnejnwt

Explanation: The exponential Fourier series is represented as – X(t)=∑Xnejnwt. Here, the X(t) is the signal and Xn=1/T∫x(t)e-jnwt.

9. What is the equation – X(t)=∑Xnejnwt called?
a) Synthesis equation
b) Analysis equation
c) Frequency domain equation
d) Discrete equation

Explanation: The equation – X(t) = ∑Xnejnwt 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 – Xn=1/T∫x(t) ejwtn called?
a) Synthesis equation
b) Analysis equation
c) Frequency domain equation
d) Discrete equation

Explanation: The equation – Xn=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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