Signals and Systems - Electrical Engineering

294 C H A P T E R 4: Frequency Analysis: The Fourier Series

4.15. Figuring out Fourier’s idea

Fourier proposed to represent a periodic signal as a sum of sinusoids, perhaps an infinite number of

them. For instance, consider the representation of a periodic signalx(t)as a sum of cosines of different

frequencies

x(t)=

∑∞

k= 0

Akcos(kt+θk)

(a) Ifx(t)is periodic of periodT 0 , what should the frequencieskbe?

(b) Considerx(t)= 2 +cos( 2 πt)−3 cos( 6 πt+π/ 4 ). Is this signal periodic? If so, what is its periodT 0?

Determine its trigonometric Fourier series as given above by specifying the values ofAkandθkfor all

values ofk=0, 1,....

(c) Let the signalx 1 (t)= 2 +cos( 2 πt)−3 cos( 20 t+π/ 4 )(this signal is almost likex(t)given above,

except that the frequency 6 πrad/sec of the second cosine has been approximated by 20 rad/sec).

Is this signal periodic? Can you determine its Fourier series as given above by specifying the values

ofAkandθkfor all values ofk=0, 1,...? Explain.

4.16. DC output from a full-wave rectified signal—MATLAB

Consider a full-wave rectifier that has as output a periodic signalx(t)of periodT 0 = 1 and a period of it is

given as

x 1 (t)=

{

cos(πt) −0.5≤t≤0.5

0 otherwise

(a) Obtain the Fourier coefficientsXk.

(b) Suppose we passx(t)through an ideal filter of transfer functionH(s). Determine the values of this

filter at harmonic frequencies 2 πk,=0,±1,±2,..., so that its output is a constant (i.e., we have a dc

source).

(c) Use MATLAB to plot the signalx(t)and its magnitude line spectrum.

4.17. Fourier series of sum of periodic signals

Suppose you have the Fourier series of two periodic signalsx(t)andy(t)of periodsT 1 andT 2 , respectively.

LetXkandYkbe the Fourier series coefficients corresponding tox(t)andy(t).

(a) IfT 1 =T 2 , what would be the Fourier series coefficients of z(t)=x(t)+y(t)in terms of Xk

andYk?

(b) IfT 1 = 2 T 2 , determine the Fourier series coefficients ofw(t)=x(t)+y(t)in terms ofXkandYk?

4.18. Manipulation of periodic signals

Let the following be the Fourier series of a periodic signalx(t)of periodT 0 (fundamental frequency 0 =

2 π/T 0 ):

x(t)=

∑∞

k=−∞

Xkej^0 kt

Consider the following functions ofx(t), and determine if they are periodic and what are their periods

if so:

n y(t)= 2 x(t)− 3

n z(t)=x(t− 2 )+x(t)

n w(t)=x( 2 t)

Express the Fourier series coefficientsYk,Zk, andWkin terms ofXk.