SOLVED PROBLEMS #5
• Find Fourier series of the following periodic
signal with half-wave symmetry as shown
in Figure
• Find Fourier series of the following periodic
signal with half-wave symmetry
• Consider the periodic signal 𝑥𝑥(𝑡𝑡) given by
𝑥𝑥 𝑡𝑡 = 2 + 𝑗𝑗𝑗 𝑒𝑒 −3𝑗𝑗𝑗𝑗 − 𝑗𝑗𝑗𝑒𝑒 −2𝑗𝑗𝑗𝑗 +6 + 𝑗𝑗𝑗𝑒𝑒 2𝑗𝑗𝑗𝑗 + 2 + 𝑗𝑗𝑗 𝑒𝑒 3𝑗𝑗𝑗𝑗
a) Determine
the fundamental
frequency of 𝑥𝑥(𝑡𝑡).
b) Show that 𝑥𝑥(𝑡𝑡) is a real signal
c) Find energy of the signal
period
and
• Find the Fourier series coefficients for
each of the following signals:
𝑥𝑥 𝑡𝑡 = cos Ω0 𝑡𝑡 + sin(2Ω0 𝑡𝑡)
𝑥𝑥 𝑡𝑡 = 2 cos Ω0 𝑡𝑡 + sin2 (2Ω0 𝑡𝑡)
• Find Fourier series coefficients of the following
continuous-time periodic signal and plot the
magnitude and phase spectrum of it:
𝑥𝑥 𝑡𝑡 = 2 sin 2𝜋𝜋𝜋𝜋 − 3 + sin(6𝜋𝜋𝑡𝑡)
• Obtain 𝑥𝑥(𝑡𝑡) for the following non-zero Fourier
series coefficients of a continuous time real
valued periodic signal 𝑥𝑥(𝑡𝑡) with fundamental
period of 8.
∗
= 𝑗𝑗, 𝑎𝑎5 = 𝑎𝑎−5 = 1
𝑎𝑎1 = 𝑎𝑎−1
• Consider a continuous periodic signal with the
following magnitude spectra shown in Figure.
• Find the DC component and average power of
the signal.
• Which
of the following signals
represented by the Fourier series?
𝑥𝑥(𝑡𝑡) = 4 cos(𝑡𝑡) + 6 cos(𝑡𝑡)
𝑥𝑥(𝑡𝑡) = 3 cos(𝜋𝜋𝑡𝑡) + 6 cos(𝑡𝑡)
𝑥𝑥(𝑡𝑡) = cos(𝑡𝑡) + 0.75
𝑥𝑥(𝑡𝑡) = 2 cos(3𝜋𝜋𝑡𝑡) + 3 cos(7𝜋𝜋𝑡𝑡)
𝑥𝑥(𝑡𝑡) = 𝑒𝑒 𝑡𝑡 sin(5𝜋𝜋𝑡𝑡)
cannot
be
• Find the Fourier series of a periodic square wave
with period 𝑇𝑇0 defined over one period by
1 𝑡𝑡 < 𝑇𝑇0 /4
𝑥𝑥 𝑡𝑡 = �
0 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜
• Find the Fourier series of the periodic signal
shown in Figure
• Find the trigonometric Fourier series representation of
the periodic signal shown in Figure with 𝐴𝐴 = 3 and
period 𝑇𝑇0 = 2𝜋𝜋.
• Find the trigonometric Fourier series representation of
the periodic triangle wave shown in Figure 3 with 𝐴𝐴 = 3
period 𝑇𝑇0 = 2.
• Find the trigonometric Fourier series representation of
the following periodic signal with period 𝑇𝑇0 = 2.
0
𝑥𝑥 𝑡𝑡 = cos(3𝜋𝜋𝜋𝜋)
0
1
−1 ≤ 𝑡𝑡 ≤ −
2
1
1
− ≤ 𝑡𝑡 <
2
2
1
≤ 𝑡𝑡 < 1
2
• If the input to the half-wave rectifier is an AC signal
𝑥𝑥 𝑡𝑡 = cos(2𝜋𝜋𝜋𝜋), find trigonometric Fourier series
representation of output signal of the half-wave
rectifier.