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

Explanation: Continuous time fourier series undergoes periodic convolution.

X(t)*y(t)=z(t) ↔ X

_{n}Y

_{n}= Z

_{n}.

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

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) ↔ X

_{n}Y

_{n}=Z

_{n}.

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

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) ↔ Z

_{n}= X

_{n}Y

_{n-k}.

4. What is the differentiation property of continuous time fourier series?

a) Y_{n} = jnwtX_{n}

b) Y_{n} = jntX_{n}

c) Y_{n} = jnwX_{n}

d) X_{n} = jnwtX_{n}

View Answer

Explanation: x(t) ↔X

_{n}, x(t) is the signal and X

_{n}is the coefficient.

Then, Y

_{n}= jnwX

_{n}.

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

Explanation: The differentiation property of the continuous time fourier series is,

Y(t) = dx(t)/dt ↔ Y

_{n}= jnwX

_{n}.

Hence, the differentiation property of time averaged value of the differentiated signal to be zero, hence, fourier series coefficient for n=0 is zero.

6. What is the integration property of the continuous time fourier series?

a) y(t) ↔ Y_{n} = 1/jnwX_{n}

b) y(t) ↔ Y_{n} = 1/jwX_{n}

c) y(t) ↔ Y_{n} = 1/jnX_{n}

d) y(t) ↔ Y_{n} = 1/jnw

View Answer

Explanation: y(t)↔ Y

_{n}= 1/jnwX

_{n}, 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

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.

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

Explanation: x(t) ↔ X

_{n}

Y(t) = *x(t) ↔Y

_{n}=*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

Explanation: If the signal is real and odd, the fourier series coefficients are conjugate symmetric.

And its fourier series coefficients are real and even.

X

_{n}= X

_{-n}*= X

_{n}.

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

Explanation: If the signal is real and even, the fourier series coefficients are conjugate symmetric.

And its fourier series coefficients are imaginary and even.

X

_{n}= X

_{-n}*= -X

_{n}.

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