Test: Continuous Time Signal Analysis - Electrical Engineering (EE) MCQ

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

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

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.

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.

Explanation: 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
c_k=|c_k|e^jθkand c_k=|c_k|e^-jθk
Consequently, we obtain the Fourier series as

Explanation: cos(2πkF₀ t+θ_k)= cos2πkF₀ t.cosθ_k-sin2πkF₀ 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

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

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.

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

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,

Explanation: Let us consider a signal x(t) whose Fourier transform X(F) is given as