1. The Fourier series representation of any signal x(t) is defined as ___________

A. \(\sum_{k = -\infty}^{\infty}c_k e^{j2πkF_0 t}\)

B. \(\sum_{k = 0}^{\infty}c_k e^{j2πkF_0 t}\)

C. \(\sum_{k = -\infty}^{\infty}c_k e^{-j2πkF_0 t}\)

D. \(\sum_{k = -\infty}^{\infty}c_{-k} e^{j2πkF_0 t}\)

Answer: A

If the given signal is x(t) and F0 is the reciprocal of the time period of the signal and ck is the Fourier coefficient then the Fourier series representation of x(t) is given as

\(\sum_{k = -\infty}^{\infty}c_k e^{j2πkF_0 t}\).

2. Which of the following is the equation for the Fourier series coefficient?

A. \(\frac{1}{T_p} \int_0^{t_0+T_p} x(t)e^{-j2πkF_0 t} dt\)

B. \(\frac{1}{T_p} \int_{t_0}^∞ x(t)e^{-j2πkF_0 t} dt\)

C. \(\frac{1}{T_p} \int_{t_0}^{t_0+T_p} x(t)e^{-j2πkF_0 t} dt\)

D. \(\frac{1}{T_p} \int_{t_0}^{t_0+T_p} x(t)e^{j2πkF_0 t} dt\)

Answer: C

When we apply integration to the definition of Fourier series representation, we get

3. Which of the following is a Dirichlet condition with respect to the signal x(t)?

A. x(t) has a finite number of discontinuities in any period
B. x(t) has finite number of maxima and minima during any period
C. x(t) is absolutely integrable in any period
D. all of the mentioned

Answer: D

For any signal x(t) to be represented as Fourier series, it should satisfy the Dirichlet conditions which are x(t) has a finite number of discontinuities in any period, x(t) has a finite number of maxima, and minima during any period and x(t) is absolutely integrable in any period.

4. The equation x(t) = \(\sum_{k = -\infty}^{\infty}c_k e^{j2πkF_0 t}\) is known as analysis equation.

A. True
B. False

Answer: B

Since we are synthesizing the Fourier series of the signal x(t), we call it a synthesis equation, whereas the equation giving the definition of Fourier series coefficients is known as the analysis equation.

5. Which of the following is the Fourier series representation of the signal x(t)?

A. \(c_0+2\sum_{k = 1}^{\infty}|c_k|sin(2πkF_0 t+θ_k)\)

B. \(c_0+2\sum_{k = 1}^{\infty}|c_k|cos(2πkF_0 t+θ_k)\)

C. \(c_0+2\sum_{k = 1}^{\infty}|c_k|tan(2πkF_0 t+θ_k)\)

D. None of the mentioned

Answer: B

In general, Fourier coefficients ck are complex valued. Moreover, it is easily shown that if the periodic signal is real, ck and c-k are complex conjugates. As a result

By interchanging the positions of integral and summation and by applying the integration, we get

= \(\sum_{k = -\infty}^{\infty}|c_k |^2\)

8. What is the spectrum that is obtained when we plot |ck |2 as a function of frequencies kF0, k = 0,±1,±2..?

A. Average power spectrum
B. Energy spectrum
C. Power density spectrum
D. None of the mentioned

Answer: C

When we plot a graph of |ck|2 as a function of frequencies kF0, k = 0,±1,±2… the following spectrum is obtained which is known as the Power density spectrum.

9. What is the spectrum that is obtained when we plot |ck| as a function of frequency?

A. Magnitude voltage spectrum
B. Phase spectrum
C. Power spectrum
D. None of the mentioned

Answer: A

We know that Fourier series coefficients are complex-valued, so we can represent ck in the following way.
ck = |ck|ejθk
When we plot |ck| as a function of frequency, the spectrum thus obtained is known as the Magnitude voltage spectrum.

10. What is the equation of the Fourier series coefficient ck of an non-periodic signal?

A. \(\frac{1}{T_p} \int_0^{t_0+T_p} x(t)e^{-j2πkF_0 t} dt\)

B. \(\frac{1}{T_p} \int_{-\infty}^∞ x(t)e^{-j2πkF_0 t} dt\)

C. \(\frac{1}{T_p} \int_{t_0}^{t_0+T_p} x(t)e^{-j2πkF_0 t} dt\)

D. \(\frac{1}{T_p} \int_{t_0}^{t_0+T_p} x(t)e^{j2πkF_0 t} dt\)

Answer: B

We know that, for an periodic signal, the Fourier series coefficient is

ck = \(\frac{1}{T_p} \int_{-T_p/2}^{T_p/2} x(t)e^{-j2πkF_0 t} dt\)

If we consider a signal x(t) as non-periodic, it is true that x(t) = 0 for |t|>Tp/2. Consequently, the limits on the integral in the above equation can be replaced by -∞ to ∞. Hence,

ck = \(\frac{1}{T_p} \int_{-\infty}^∞ x(t)e^{-j2πkF_0 t} dt\)