1. Continuous time signal x(t) is periodic with period To and fundamental frequency ωo and has complex Fourier Series coefficients ak for the kth harmonic. Find the Fourier Series coefficients bk for the signals y(t) in terms of ak in each of the following scenarios:
d2x(t)
(a) y(t) = x(2t − 3) + 4 dt2
(b) y(t) = t+2α ejωoτ x(τ + 1)dτ
−∞
(c) y(t) = dx3(t)
dt
2. Consider the continuous time signal x(t) shown below
(a)
(a) Determine the complex Fourier series representation or each of the signal x(t).
(b) Plot the magnitude and phase spectrum of the Fourier Series coefficients for x(t).
(c) Derive the Fourier Series coefficients of x(t) from its Laplace transform.
(d) Express the signal x(t) as a trigonometric fourier series. That is, find ak and bk which satisfy
∞ ∞
x(t) = X0 + 2 ak cos (kΩ0t) − 2 bk sin (kΩ0t)
k=1 k=1
3. Consider a continuous-time LTI system with impulse response h(t) = e−4tu(t) Find the Fourier series representation of the output y(t) for each of the following inputs:
(a) x(t) =
(b) x(t) =
+∞
n=−∞
+∞
n=−∞
δ(t − n)
(−1)nδ(t − n)
4. (a) Consider the periodic real signal x(t) with the following properties:
i. The signal period is T = 6
ii. The DC component of the signal is zero
iii. x(t) = −x(t − 3)
1
iv. The Fourier series coefficients Xk = 0 for k > 2
v. X1 is a positive real number
vi. −33 |x(t)|2 = 3
Find the exact expression of the time-domain signal x(t).
j
2
|
|
,
else
(b) Let x(t) be a periodic signal whose Fourier series coefficients are ak =
2,
1
k
k = 0
(i) Is x(t) real?
(ii) Is x(t) even?
(iii) Is dx(t)/dt even?
5. The smoothness of signal in time domain determines how its spectrum will look in the frequency domain. Consider two signals with period T0 = 2sec. The signals are represented as below in the first period: 0 ≤ t ≤ T0 :
x1(t) = u(t) − u(t − 1)
x2(t) = r(t) − 2r(t − 1) + r(t − 2)
Find Fourier series coefficients of x1(t) and x2(t) analytically using Fourier series formula. Then, in MATLAB plot magnitude spectra for both signals for k = −20, −19, . . . 0, . . . , 19, 20 in MAT-LAB using stem function. Determine which spectra decays faster as k increases and explain how it relates to smoothness of the signal in time domain.
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