Signals and Systems - Electrical Engineering

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

since cos(πk)=(− 1 )k. The DC value of the full-wave rectified signal isX 0 = 2 /π. Notice that the

Fourier coefficients are real given that the signal is even.

The MATLAB script used in the previous example can be used again with the following

modification for the generation of a period ofx(t). The results are shown in Figure 4.7.

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% Example 4.6---Fourier series of full-wave rectified signal

%%%%%%%%%%%%%%%%

% period generation

T0 = 1;

m = heaviside(t)−heaviside(t−T0);x = abs(cos(pi∗t))∗m n

nExample 4.7

Computing the derivative of a signal enhances higher harmonics. To illustrate this consider the

train of triangular pulsesy(t)(Figure 4.8) with fundamental periodT 0 =2. Letx(t)=dy(t)/dt. Find

its Fourier series and compare|Xk|with|Yk|to determine which of these signals is smoother—that

is, which one has lower frequency components.

Solution

A period ofy(t),− 1 ≤t≤1, is given by

y 1 (t)=r(t+ 1 )− 2 r(t)+r(t− 1 )

with a Laplace transform

Y 1 (s)=

1

s^2

[

es− 2 +e−s

]

FIGURE 4.8

(a) Train of triangular

pulsesy(t)and (b) its

derivativex(t). Notice

thaty(t)is a continuous

function whilex(t)is

discontinuous. (a) (b)

···

y(t)

t

− 2 − (^1012)
1
···
− 2 − (^1) 01 2
t
1
−1
··· ···
x(t)=dydt(t)