Signals and Systems - Electrical Engineering

310 CHAPTER 5: Frequency Analysis: The Fourier Transform

FIGURE 5.3

Magnitude and phase

spectrum of two-sided

signalx(t)=e−|t|. The

magnitude spectrum

indicatesx(t)is low

pass. Notice the phase

is zero.

− 10 − 5 0 5 10

0

0.2

0.4

0.6

0.8

1

t(sec)

x(

t)

− 10 − 5 0 5 10

0

0.5

1

1.5

2

Ω

|X

(Ω

)|

− 20 0 20

− 1

0

1

Ω

<X

(Ω

)

and according to the given resultx 1 (−t) (α=− 1 ), we have that

F[x 1 (−t)]=

1

−j+a

so that

X()=

1

j+a

+

1

−j+a

=

2 a

a^2 +^2

Ifa=1, using MATLAB the signalx(t)=e−|t|and its magnitude and phase spectra are com-

puted and plotted as shown in Figure 5.3. SinceX()is real and positive, the corresponding

phase spectrum is zero. This signal is calledlow-passsince its energy is concentrated in the low

frequencies. n

5.5.3 Duality

Besides the inverse relationship of frequency and time, by interchanging the frequency and the time

variables in the definitions of the direct and the inverse Fourier transform (see Eqs. 5.1 and 5.2)

similar equations are obtained. Thus, the direct and the inverse Fourier transforms are dual.

To the Fourier transform pair

x(t) ⇔ X() (5.7)

corresponds the following dual–Fourier transform pair

X(t) ⇔ 2 πx(−) (5.8)