The Fourier series representation of any signal x(t) is defined as:
Explanation: 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
Which of the following is the equation for the Fourier series coefficient?
Explanation: When we apply integration to the definition of Fourier series representation, we get
Which of the following is a Dirichlet condition with respect to the signal x(t)?
Explanation: 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.
The equation is known as analysis equation.
Explanation: 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.
Which of the following is the Fourier series representation of the signal x(t)?
d) None of the mentioned
Explanation: In general, Fourier coefficients ck are complex valued. Moreover, it is easily shown that if the periodic signal is real, ck and ck are complex conjugates. As a result
c_{k}=c_{k}e^{jθk}and c_{k}=c_{k}e^{jθk}
Consequently, we obtain the Fourier series as
The equationis the representation of Fourier series.
Explanation: cos(2πkF_{0} t+θ_{k})= cos2πkF_{0} t.cosθ_{k}sin2πkF_{0} t.sinθ_{k}
θ_{k} is a constant for a given signal.
So, the other form of Fourier series representation of the signal x(t) is
What is the spectrum that is obtained when we plot c_{k} ^{2} as a function of frequencies kF_{0}, k=0,±1,±2..?
Explanation: When we plot a graph of c_{k} ^{2} as a function of frequencies kF_{0}, k=0,±1,±2… the following spectrum is obtained which is known as Power density spectrum.
What is the spectrum that is obtained when we plot ck as a function of frequency?
Explanation: We know that, Fourier series coefficients are complex valued, so we can represent ck in the following way.
c_{k}=c_{k}e^{jθk}
When we plot c_{k} as a function of frequency, the spectrum thus obtained is known as Magnitude voltage spectrum.
What is the equation of the Fourier series coefficient ck of an nonperiodic signal?
Explanation: We know that, for an periodic signal, the Fourier series coefficient is
If we consider a signal x(t) as nonperiodic, 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,
Which of the following relation is correct between Fourier transform X(F) and Fourier series coefficient c_{k}?
Explanation: Let us consider a signal x(t) whose Fourier transform X(F) is given as
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