AM with Suppressed Carrier

6.4.1 AM with Suppressed Carrier

Consider a message signal m(t) (e.g., voice or music, or a combination of the two) modulating a cosine carrier cos( c t) to give an amplitude modulated signal

(6.13) The carrier frequency  c >> 2π f 0 where f 0 (Hz) is the maximum frequency in the message (for music

s(t) = m(t) cos( c t)

f 0 is about 22 KHz). The signal s(t) is called an amplitude modulated with suppressed carrier (AM-SC) signal (the last part of this denomination will become clear later). According to the modulation property of the Fourier transform, the transform of s(t) is

S() = [M( −  c ) + M( +  c ) ]

where M() is the spectrum of the message. The frequency content of the message is now shifted to

a much larger frequency  c (rad/sec) than that of the baseband signal m(t). Accordingly, the antenna needed to transmit the amplitude modulated signal is of reasonable length. An AM-SC system is shown in Figure 6.14.

At the receiver, we need to first detect the desired signal among the many signals transmitted by several sources. This is possible with a tunable band-pass filter that selects the desired signal and rejects the others. Suppose that the signal obtained by the receiver, after the band-pass filtering, is exactly s(t)—we then need to demodulate this signal to get the original message signal m(t). This is done by multiplying s(t) by a cosine of exactly the same frequency of the carrier in the transmitter

(i.e.,  c ), which will give r(t) = 2s(t) cos( c t), which again according to the modulation property has

a Fourier transform

R() = S( −  c ) + S( +  c ) = M() + [M( − 2 c ) + M( + 2 c ) ] (6.15)

The spectrum of the message, M(), is obtained by passing the received signal r(t) through a low-pass filter that rejects the other terms M( ± 2 c ) . The obtained signal is the desired message m(t).

The above is a simplification of the actual processing of the received signal. Besides the many other transmitted signals that the receiver encounters, there is channel noise caused by interferences from

cos(Ω c t )

2cos(Ω c t )

FIGURE 6.14 AM-SC transmitter, channel, and receiver.

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 of m(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

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.

A more general characterization than  c >> 2π f 0 where f 0 is the largest frequency in the message is given by  c >> 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:

The half-power or 3-dB bandwidth is 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 to 0.707, the value at the peak. This corresponds to the frequencies for which the power at dc, infinity, or center frequency reduces to half.

The null-to-null bandwidth determines 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.

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 of  c the demodulator uses  c + 1—so that the received signal in that case has the Fourier transform

˜R() = S( −  c − 1) + S( +  c + 1)

1 = [M( + 1) + M( − 1)]

+ [M( − 2( c + 1/2)) + M( + 2( c + 1/2)]

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