Signals and Systems - Electrical Engineering

322 CHAPTER 5: Frequency Analysis: The Fourier Transform

Solution

The energy of the pulse isE=2 (the area under the pulse). But according to Parseval’s result the

energy computed in the frequency domain is given by

1

2 π

∫∞

−∞

(

2 sin()



) 2

d=Ex

since 2 sin()/=P()=F(p(t)). ReplacingEx, we obtain the interesting and not obvious result

∫∞

−∞

(

sin()



) 2

d=π

This is one more way to computeπ! n

5.6.4 Symmetry of Spectral Representations........................................

Now that the Fourier representation of aperiodic and periodic signals is unified, we can think of

just one spectrum that accommodates both finite-energy as well as infinite-energy signals. The word

spectrumis loosely used to mean different aspects of the frequency representation. In the following

we provide definitions and the symmetry characteristic of the spectrum of real-valued signals.

IfX()is the Fourier transform of a real-valued signalx(t), periodic or aperiodic, the magnitude|X()|is an

even function of:

|X()|=|X(−)| (5.16)

and the phase∠X()is an odd function of:

∠X()=−∠X(−) (5.17)

We then have:

Magnitude spectrum: |X()|versus

Phase spectrum: ∠X()versus

Energy/power spectrum: |X()|^2 versus

To show this, consider the inverse Fourier transform of a real-valued signalx(t),

x(t)=

∫∞

−∞

X()ejtd

which, because of being real, is identical to

x∗(t)=

∫∞

−∞

X∗()e−jtd=

∫∞

−∞

X∗(−)ejtd