Signals & Systems Questions and Answers – Fourier Series Properties – 1

This set of Signals & Systems Multiple Choice Questions & Answers (MCQs) focuses on “Fourier Series Properties – 1”.

1. How do we represent a pairing of a periodic signal with its fourier series coefficients in case of continuous time fourier series?
a) x(t) ↔ Xn
b) x(t) ↔ Xn+1
c) x(t) ↔ X
d) x(n) ↔ Xn

Explanation: In case of continuous time fourier series, for simplicity, we represent a pairing of a periodic signal with its fourier series coefficients as,
x(t) ↔ Xn
here, x(t) is the signal and Xn is the fourier series coefficient.

2. What are the properties of continuous time fourier series?
a) Linearity, time shifting
b) Linearity, time shifting, frequency shifting
c) Linearity, time shifting, frequency shifting, time reversal, time scaling, periodic convolution
d) Linearity, time shifting, frequency shifting, time reversal, time scaling, periodic convolution, multiplication, differentiation

Explanation: Linearity, time shifting, frequency shifting, time reversal, time scaling, periodic convolution, multiplication, differentiation are some of the properties followed by continuous time fourier series. Integration and conjugation are also followed by continuous time fourier series.

3. Integration and conjugation are also followed by continuous time fourier series?
a) True
b) False

Explanation: Linearity, time shifting, frequency shifting, time reversal, time scaling, periodic convolution, multiplication, differentiation are some of the properties followed by continuous time fourier series. Integration and conjugation are also followed by continuous time fourier series.

4. If x(t) and y(t) are two periodic signals with coefficients Xn and Yn then the linearity is represented as?
a) ax(t) + by(t) = aXn + bYn
b) ax (t) + by(t) = Xn + bYn
c) ax(t) + by(t) = aXn + Yn
d) ax(t) + by(t) = Xn + Yn

Explanation: ax(t) + by(t) = aXn + bYn, x(t) and y(t) are two periodic signals with coefficients Xn and Yn.

5. How is time shifting represented in case of periodic signal?
a) If x(t) is shifted to t0, Xn is shifted to t0
b) x(t-t0), Yn = Xn e-njwt0
c) Xn = x(t-t0), Yn = Xn e-njwt0
d) Xn = x(-t0), Yn = Xn e-njwt0

Explanation: If x(t) and y(t) are two periodic signals with coefficients Xn and Yn, then if a signal is shifted to t0, then the property says,
Xn = x(t-t0), Yn = Xne-njwt0
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6. What is the frequency shifting property of continuous time fourier series?
a) Multiplication in the time domain by a real sinusoid
b) Multiplication in the time domain by a complex sinusoid
c) Multiplication in the time domain by a sinusoid
d) Addition in the time domain by a complex sinusoid

Explanation: If x(t) and y(t) are two periodic signals with coefficients Xn and Yn,
Then y(t)= ejmwtx(t)↔Yn=Xn-m.
Hence, we can see that a frequency shift corresponds to multiplication in the time domain by complex sinusoid whose frequency is equal to the time shift.

7. What is the time reversal property of fourier series coefficients?
a) Time reversal of the corresponding sequence of fourier series
b) Time reversal of the last term of fourier series
c) Time reversal of the corresponding term of fourier series
d) Time reversal of the corresponding sequence

Explanation: x(t)↔ Xn
Y(t) = x(-t)↔Yn=X-n.
That is the time reversal property of fourier series coefficients is time reversal of the corresponding sequence of fourier series.

8. It does not depend whether the signal is odd or even, it is always reversal of the corresponding sequence of fourier series.
a) True
b) False

Explanation: It does depend whether the signal is odd or even.
If the signal is even, the reversal is positive and if the signal is odd, the reversal is negative.

9. Why does the signal change while time scaling?
a) Because the frequency changes
b) Time changes
c) Length changes
d) Both frequency and time changes

Explanation: x(t)↔Xn
Y(t) = x(at)↔Yn = Xn
Hence, the fourier coefficients have not changed but the representation has changed because of changes in fundamental frequency.

10. What is the period of the signal when it is time shifted?
a) Changes according to the situation
b) Different in different situation
c) Remains the same
d) Takes the shifted value

Explanation: The period of the periodic signal does not change even if it is time shifted.
If x(t) and y(t) are two periodic signals with coefficients Xn and Yn, then if a signal is shifted to t0, then the property says,
Xn = x(t-t0), Yn = Xne-njwt0.

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