could determine the Fourier series representation of g(t) directly from the analysis equation (3.39).
Instead, we will use the relationship of g(t) to the symmetric periodic square wave x(t) in Example 3.5. Referring to that example, we see that, with T=4 and T1=1,
g(t) = x(t-1) - (1)/(2).
The time-shift property in Table 3.1 indicates that, if the Fourier Series coefficients of
x(t) are denoted by ak, the Fourier coefficients of x(t-1) may be expressed as:
b_(k) = a_(k)e^(-jk(pi)/(2)).
Applying the linearity property in Table 3.1, we conclude that the coefficients for g(t)
may be expressed as:
d_(k) = {(a_(k)e^(-jk(pi)/(2)), for k!=0), (a_(0) - (1)/(2), for k=0), ():}
where each ak may now be replaced by the corresponding expression from eqs. (3.45) and (3.46), yielding:
d_(k) = {((sin(pi(k)/(2)))/(kpi)e^(-jk(pi)/(2)), for k!=0), ([0, for k=0].):}
Question to solve by hand:
could de Instead, we will use the relationship of g(t) to the symmetric periodic square wave x(t) in Example 3.5. Referring to that example, we see that, with T = 4 and T1 = 1,
Analytical Calculation of Fourier Series Coefficients Consider the three continuous-time periodic signals defined below. x1(t) = 4 cos(2pi(200t)) + 3 cos(2pi(300t + pi/3))
g(t)
x2(t)
x(t) = (1/6) x(1000t - 0.5)
The time-shift property in Table 3.1 indicates that, if the Fourier Series coefficients of
x(t) are denoted by ak, the Fourier coefficients of x(t-1) may be expressed as:
For each signal:
1.a) Determine the fundamental period T0 and compute the Fourier series coefficients (the s in Equation 3.38 in the textbook) by hand. 1.b) Plot the magnitude spectrum (|S|) and phase spectrum versus k. Specifically, plot the spectra by hand for x(t), and plot the magnitude and phase spectrum for x1(t) and x2(t). Note that all these signals are real-valued. Identify which signals are symmetric with x(t) = x(-t), and which are anti-symmetric with x(t) = -x(-t). What do you observe about the symmetry properties of their magnitude and phase spectra? Hint: you may want to review the examples in the textbook, particularly Example 3.6.
Applying the linearity property in Table 3.1, we conclude that the coefficients for g(t)
may be expressed as:
ae^(-jkm/2) for k != 0, 0 for k = 0.
dr
ak's equation
where each ak may now be replaced by the corresponding expression from eqs. (3.45) and (3.46) yielding:
k != 0,
for k = 0
ar
Example 3.6
Consider the signal g(t) with a fundamental period of 4, shown in Figure 3.10. We