Signals and Systems - Electrical Engineering

5.6 Spectral Representation 319

so that the Fourier coefficients ofx(t)are (T 0 =1):

Xk=

1

T 0

X 1 (s)|s=j 2 πk=

1

(j 2 πk)^2

2 (cos(πk)− 1 )e−jπk

=(− 1 )(k+^1 )

cos(πk)− 1

2 π^2 k^2

=(− 1 )k

sin^2 (πk/ 2 )

π^2 k^2

after using the identity cos( 2 θ)− 1 =−2 sin^2 (θ). The DC term isX 0 =0.5. The Fourier transform

ofx(t)is then

X()= 2 πX 0 δ()+

∑∞

k=−∞,6= 0

2 πXkδ(− 2 kπ)

To compute the Fourier transform using symbolic MATLAB, we approximatex(t)by its Fourier

series by means of its average andN=10 harmonics (the Fourier coefficients are computed using

thefourierseriesfunction from Chapter 4). We then create a sequence{ 2 πXk}and the correspond-

ing harmonic frequencies{k=k 0 }and plot them as the spectrumX()(see Figure 5.6). The

following script gives some of the necessary steps to generate the periodic signal and to find its

Fourier transform. The MATLAB functionfliplris used to reflect the Fourier coefficients.

FIGURE 5.6
(a) Triangular periodic signalx(t),
and (b) its Fourier transformX(),
which is zero except at harmonic
frequencies where it is an impulse
of magnitude 2 πXkwhereXkis a
Fourier coefficient ofx(t).

0 1 2 3 4 5

0

0.5

1

t

x(

t)

(a)

− 50 0 50

− 1

0

1

2

3

4

Ω(rad/sec)

X

(Ω

)

(b)