Signals and Systems - Electrical Engineering

380 C H A P T E R 6: Application to Control and Communications

equipment in the transmission path and interference from other signals being transmitted around

the carrier frequency. This noise will also be picked up by the band-pass filter and a perfect recovery

ofm(t)will not be possible. Furthermore, the sent signal has no indication of the carrier frequency

c, which is suppressed in the sent signal, and so the receiver needs to guess it and any deviation

would give errors.

Remarks

n The transmitter is linear but time varying. AM-SC is thus called a linear modulation. The fact that the

modulated signal displays frequencies much higher than those in the message indicates the transmitter is

not LTI—otherwise it would satisfy the eigenfunction property.

n A more general characterization thanc>> 2 πf 0 where f 0 is the largest frequency in the message is given

byc>>BW where BW (rad/sec) is the bandwidth of the message. You probably recall the definition

of bandwidth of filters used in circuit theory. In communications there are several possible definitions for

bandwidth. The bandwidth of a signal is the width of the range of positive frequencies for which some

measure of the spectral content is satisfied. For instance, two possible definitions are:

nThehalf-poweror3-dB bandwidthis the width of the range of positive frequencies where a peak

value at zero or infinite frequency (low-pass and high-pass signals) or at a center frequency (band-pass

signals) is attenuated to0.707, the value at the peak. This corresponds to the frequencies for which the

power at dc, infinity, or center frequency reduces to half.

nThenull-to-null bandwidthdetermines the width of the range of positive frequencies of the spectrum

of a signal that has a main lobe containing a significant part of the energy of the signal. If a low-pass

signal has a clearly defined maximum frequency, then the bandwidth are frequencies from zero to the

maximum frequency, and if the signal is a band-pass signal and has a minimum and a maximum

frequency, its bandwidth is the maximum minus the minimum frequency.

n In AM-SC demodulation it is important to know exactly the carrier frequency. Any small deviation would

cause errors when recovering the message. Suppose, for instance, that there is a small error in the carrier

frequency—that is, instead ofcthe demodulator usesc+ 1 —so that the received signal in that case

has the Fourier transform

R ̃()=S(−c−1)+S(+c+1)

=

1

2

[M(+1)+M(−1)]

+

1

2

[M(− 2 (c+1/ 2 ))+M(+ 2 (c+1/ 2 )]

The low-pass filtered signal will not be the message.

6.4.2 Commercial AM

In commercial broadcasting, the carrier is added to the AM signal so that information of the carrier

is available at the receiver helping in the identification of the radio station. For demodulation, such

information is not important, as commercial AM usesenvelope detectorsto obtain the message. By

making the envelope of the modulated signal look like the message, detecting this envelope is all