Signals and Systems - Electrical Engineering

5.6 Spectral Representation 325

The Fourier transform ofx 2 (t)=e−tu(t)is

X 2 ()=

1

1 +j

The magnitude and phase are given by

|X 2 ()|=

1

√

1 +^2

θ()=−tan−^1 

When we compute these in terms of, we have

 |X 2 ()| θ()

0 1 0

1

1

√

2

−π/ 4

∞ 0 −π/ 2

That is, the magnitude spectrum decays asincreases. The signalx 2 (t)is calledlow-passgiven that

the magnitude of its Fourier transform is concentrated in the low frequencies. This also implies

that the signalx 2 (t)is rather smooth. See Figure 5.7 for results. n

nExample 5.12

It is not always the case that the Fourier transform is a complex-valued function. Consider the

signals

(a)x(t)=0.5e−|t|

(b)y(t)=e−|t|cos( 0 t)

Find their Fourier transforms. Discuss the smoothness of these signals.

Solution

(a) The Fourier transform ofx(t)is

X()=

1

^2 + 1

which is a real-valued function of. Indeed,

 |X()|=X() θ()

0 1 0

1 0.5 0

∞ 0 0