Digital Signal Processing Multiple Choice Questions on “Frequency Analysis of Continuous Time Signals”.
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
Clarification: 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
Clarification: When we apply integration to the definition of Fourier series representation, we get
ckTp=(int_{t_0}^{t_0+T_p} x(t)e^{-j2πkF_0 t} dt)
=>ck=(frac{1}{T_p} int_{t_0}^{t_0+T_p} x(t)e^{-j2πkF_0 t} dt)
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
Clarification: 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 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
Clarification: Since we are synthesizing the Fourier series of the signal x(t), we call it as synthesis equation, where as the equation giving the definition of Fourier series coefficients is known as analysis equation.
5. Which of the following is the Fourier series representation of the signal x(t)?
A. (c_0+2sum_{k=1}^{infty}|c_k|sin(2πkF_0 t+θ_k))
B. (c_0+2sum_{k=1}^{infty}|c_k|cos(2πkF_0 t+θ_k))
C. (c_0+2sum_{k=1}^{infty}|c_k|tan(2πkF_0 t+θ_k))
D. None of the mentioned
Answer: B
Clarification: 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
ck=|ck|ejθkand ck=|ck|e-jθk
Consequently, we obtain the Fourier series as x(t)=(c_0+2sum_{k=1}^{infty}|c_k|cos(2πkF_0 t+θ_k))
6. The equation x(t)=(a_0+sum_{k=1}^∞(a_k cos2πkF_0 t – b_k sin2πkF_0 t)) is the representation of Fourier series.
A. True
B. False
Answer: A
Clarification: cos(2πkF0 t+θk) = cos2πkF0 t.cosθk-sin2πkF0 t.sinθk
θk is a constant for a given signal.
So, the other form of Fourier series representation of the signal x(t) is
(a_0+sum_{k=1}^∞(a_k cos2πkF_0 t – b_k sin2πkF_0 t)).
7. The equation of average power of a periodic signal x(t) is given as ___________
A. (sum_{k=0}^{infty}|c_k|^2)
B. (sum_{k=-infty}^{infty}|c_k|)
C. (sum_{k=-infty}^0|c_k|^2)
D. (sum_{k=-infty}^{infty}|c_k|^2)
Answer: D
Clarification: The average power of a periodic signal x(t) is given as
The average power of a periodic signal x(t) is given as (frac{1}{T_p} int_{t_0}^{t_0+T_p}|x(t)|^2 dt)
=(frac{1}{T_p} int_{t_0}^{t_0+T_p}x(t).x^* (t) dt)
=(frac{1}{T_p} int_{t_0}^{t_0+T_p}x(t).sum_{k=-infty}^{infty} c_k * e^{-j2πkF_0 t} dt)
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