Signals and Systems - Electrical Engineering

300 CHAPTER 5: Frequency Analysis: The Fourier Transform

the frequency content of a signal by processing it with an LTI system with a desired frequency

response.

n Modulation and communications—The idea of changing the frequency content of a signal via modu-

lation is basic in analog communications. Modulation allows us to send signals over the airwaves

using antennas of reasonable sizes. Voice and music are relative low-frequency signals that can-

not be easily radiated without the help of modulation. Continuous-wave modulation changes

the amplitude, the frequency, or the phase of a sinusoidal carrier of frequency much greater than

the frequencies present in the message we wish to transmit.

5.2 From the Fourier Series to the Fourier Transform

In practice there are no periodic signals—such signals would have infinite supports and exact periods,

which are not possible. Since only finite-support signals can be processed numerically, signals in

practice are treated as aperiodic. To obtain the Fourier representation of aperiodic signals, we use the

Fourier series representation in a limiting process.

An aperiodic, or nonperiodic, signalx(t)can be thought of as a periodic signalx ̃(t)with an infinite period.

Using the Fourier series representation of this signal and a limiting process we obtain a pair

x(t) ⇔ X()

where the signalx(t)is transformed into a functionX()in the frequency domain by the

Fourier transform: X()=

∫∞

−∞

x(t)e−jtdt (5.1)

whileX()is transformed into a signalx(t)in the time domain by the

Inverse Fourier transform: x(t)=

1

2 π

∫∞

−∞

X()ejtd (5.2)

Any aperiodic signal can be assumed to be periodic with an infinite period. That is, an aperiodic

signalx(t)can be expressed as

x(t)= lim

T 0 →∞

̃x(t)

wherex ̃(t)is a periodic signal of periodT 0. The Fourier series representation of ̃x(t)is

x ̃(t)=

∑∞

n=−∞

Xnejn^0 t  0 =

2 π

T 0

Xn=

1

T 0

T∫ 0 / 2

−T 0 / 2

x ̃(t)e−jn^0 tdt