# Signals & Systems Questions and Answers – Fourier Series & Coefficients

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

1. What are fourier coefficients?
a) The terms that are present in a fourier series
b) The terms that are obtained through fourier series
c) The terms which consist of the fourier series along with their sine or cosine values
d) The terms which are of resemblance to fourier transform in a fourier series are called fourier series coefficients

Explanation: The terms which consist of the fourier series along with their sine or cosine values are called fourier coefficients. Fourier coefficients are present in both exponential and trigonometric fourier series.

2. Which are the fourier coefficients in the following?
a) a0, an and bn
b) an
c) bn
d) an and bn

Explanation: These are the fourier coefficients in a trigonometric fourier series.
a0 = 1/T∫x(t)dt
an = 2/T∫x(t)cos(nwt)dt
bn = 2/T∫x(t)sin(nwt)dt

3. Do exponential fourier series also have fourier coefficients to be evaluated.
a) True
b) False

Explanation: The fourier coefficient is : Xn = 1/T∫x(t)e-njwtdt.

4. The fourier series coefficients of the signal are carried from –T/2 to T/2.
a) True
b) False

Explanation: Yes, the coefficients evaluation can be done from –T/2 to T/2. It is done for the simplification of the signal.

5. What is the polar form of the fourier series?
a) x(t) = c0 + ∑cncos(nwt+ϕn)
b) x(t) = c0 + ∑cncos(ϕn)
c) x(t) = ∑cncos(nwt+ϕn)
d) x(t) = c0+ ∑cos(nwt+ϕn)

Explanation: x(t) = c0 + ∑cncos(nwt+ϕn), is the polar form of the fourier series.
C0=a0 and cn = √a2n+ b2n for n≥1
And ϕn = tan-1 bn/an .

6. What is a line spectrum?
a) Plot showing magnitudes of waveforms are called line spectrum
b) Plot showing each of harmonic amplitudes in the wave is called line spectrum
c) Plot showing each of harmonic amplitudes in the wave is called line spectrum
d) Plot showing each of harmonic amplitudes called line spectrum

Explanation: The plot showing each of harmonic amplitudes in the wave is called line spectrum. The line rapidly decreases for waves with rapidly convergent series.

7. Fourier series is not true in case of discrete time signals.
a) True
b) False

Explanation: Fourier series is also true in case of discrete time signals. They just need to follow the dirichlet’s conditions.

8. What is the disadvantage of exponential Fourier series?
a) It is tough to calculate
b) It is not easily visualized
c) It cannot be easily visualized as sinusoids
d) It is hard for manipulation

Explanation: The major disadvantage of exponential Fourier series is that it cannot be easily visualized as sinusoids. Moreover, it is easier to calculate and easy for manipulation leave aside the disadvantage.

9. Fourier series uses which domain representation of signals?
a) Time domain representation
b) Frequency domain representation
c) Both combined
d) Neither depends on the situation

Explanation: Fourier series uses frequency domain representation of signals. X(t)=1/T∑Xnejnwt. Here, the X(t) is the signal and Xn = 1/T∫x(t)e-jwtn.

10. How does Fourier series make it easier to represent periodic signals?
a) Harmonically related
b) Periodically related
c) Sinusoidally related
d) Exponentially related

Explanation: Fourier series makes it easier to represent periodic signals as it is a mathematical tool that allows the representation of any periodic signals as the sum of harmonically related sinusoids.

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