Signals & Systems Questions and Answers – Fourier Series Properties – 2

This set of Signals & Systems Problems focuses on “Fourier Series Properties – 2”.

1. Can continuous time fourier series undergo periodic convolution?
a) They cannot undergo periodic convoluion
b) They can undergo in certain situations
c) They undergo periodic convolution
d) Only even signals undergo periodic convolution
View Answer

Answer: c
Explanation: Continuous time fourier series undergoes periodic convolution.
X(t)*y(t)=z(t) ↔ XnYn = Zn.

2. What is the outcome of a periodic convolution of signals in case of continuous time fourier series?
a) Division in frequency domain
b) Multiplication in frequency domain
c) Convolution is easier
d) Addition of signals in frequency domain
View Answer

Answer: b
Explanation: This is a very important property of continuous time fourier series, it leads to the conclusion that the outcome of a periodic convolution is the multiplication of the signals in frequency domain representation.
X(t)*y(t)=z(t) ↔ XnYn=Zn.

3. What is the multiplication property of continuous time fourier series?
a) Convolution of the signals
b) Multiplication of the elements of the signal
c) Division of the frequency domain
d) Addition of the signals in frequency domain
View Answer

Answer: a
Explanation: In the case of continuous time fourier series, the multiplication property leads to discrete time convolution of the signals.
z(t)=x(t)y(t) ↔ Zn = XnYn-k.
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4. What is the differentiation property of continuous time fourier series?
a) Yn = jnwtXn
b) Yn = jntXn
c) Yn = jnwXn
d) Xn = jnwtXn
View Answer

Answer: c
Explanation: x(t) ↔Xn, x(t) is the signal and Xn is the coefficient.
Then, Yn = jnwXn.

5. What is the fourier series coefficient for n=0?
a) Zero
b) Unity
c) Depends on the situation
d) Non zero positive
View Answer

Answer: a
Explanation: The differentiation property of the continuous time fourier series is,
Y(t) = dx(t)/dt ↔ Yn = jnwXn.
Hence, the differentiation property of time averaged value of the differentiated signal to be zero, hence, fourier series coefficient for n=0 is zero.
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6. What is the integration property of the continuous time fourier series?
a) y(t) ↔ Yn = 1/jnwXn
b) y(t) ↔ Yn = 1/jwXn
c) y(t) ↔ Yn = 1/jnXn
d) y(t) ↔ Yn = 1/jnw
View Answer

Answer: a
Explanation: y(t)↔ Yn = 1/jnwXn, here x(t) is the signal and y(t) is the output.
This is the integration property of the signal.

7. What is the smoothing operation?
a) Differentiation property
b) Multiplication property
c) Integration property
d) Conjugation property
View Answer

Answer: c
Explanation: The integration attenuates the magnitude of the high frequency components of the signal. High frequency contributors cause sharp details such as occurring at the points of discontinuity. Hence, integration smoothens the signal, hence it is called a smoothening operation.
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8. What is the complex conjugate property of a fourier series?
a) It leads to convolution
b) It leads to time reversal
c) It leads to multiplication
d) It leads to addition of signals
View Answer

Answer: b
Explanation: x(t) ↔ Xn
Y(t) = *x(t) ↔Yn=*X-n
It leads to time reversal.

9. If the signal x(t) is odd, what will be the fourier series soeffiients?
a) Real and even
b) Odd
c) Real only
d) Real and odd
View Answer

Answer: a
Explanation: If the signal is real and odd, the fourier series coefficients are conjugate symmetric.
And its fourier series coefficients are real and even.
Xn = X-n*= Xn .
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10. If the signal x(t) is even, what will be the fourier series coefficients?
a) Real and even
b) Odd
c) Real only
d) Imaginary and odd
View Answer

Answer: d
Explanation: If the signal is real and even, the fourier series coefficients are conjugate symmetric.
And its fourier series coefficients are imaginary and even.
Xn = X-n*= -Xn.

Sanfoundry Global Education & Learning Series – Signals & Systems.

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Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He lives in Bangalore, and focuses on development of Linux Kernel, SAN Technologies, Advanced C, Data Structures & Alogrithms. Stay connected with him at LinkedIn.

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