Visualize the big picture of the AM radio transmitter, receiver, and interfering signals with a system block diagram. Each block in the diagram has an underlying mathematical model.

## Start with the AM signal model

The signal model for an AM signal is

where *A** _{c}* is the carrier amplitude,

*f*

*is the carrier frequency,*

_{c}*m*(

*t*) is the

*message*signal, and the

*modulation index*is

The* *modulation index* *controls how much of the signal is made up of the message and how much is pure carrier.

The AM scheme also requires that min[*m*(*t*)] ≥ –1. A simple decomposition of *x** _{c}*(

*t*) shows that a pure carrier term and a double sideband modulation term are involved. The term

*double sideband*comes from viewing

*x*

*(*

_{c}*t*) in the frequency domain.

The second term follows from the modulation theorem for Fourier transforms (FT). For *m*(*t*) real, the magnitude spectrum |*M*(*f*)| is symmetrical about *f* = 0. Multiplying by cos(2π*f*_{c}*t*) shifts *M*(*f*) up and down in frequency by the carrier frequency *f** _{c}*.

The spectrum of *M*(*f* – *f** _{c}*) just above

*f*

*is known as the*

_{c}*upper sideband*

*(USB)*

*,*and the one just below

*f*

*is known as the*

_{c}*lower sideband*

*(LSB)*

*.*The same relation holds for the

*M*(

*f*+

*f*

*) term. The USB and LSB contain the same information — hence the term*

_{c}*double sideband.*

The figure shows the frequency domain details of the AM signal.

## Look at the waveform

The AM waveform, shownhere for a particular *m*(*t*), motivates a simple receiver design.

The envelope of *x** _{c}*(

*t*) is the dashed line, given by

*R*(

*t*) =

*A*

*(1 +*

_{c}*am*(

*t*)). As long as min[

*am*(

*t*)] > –1, retaining just the positive half cycles of

*x*

*(*

_{c}*t*) allows

*R*(

*t*) to accurately contain

*m*(

*t*). In electronic circuit form, the recovery of

*m*(

*t*) from

*x*

*(*

_{c}*t*) is accomplished with an

*envelope detector*as shown.

You need a direct current (DC) block or level shift to finally get a term proportional to *m*(*t*). Circuit theory tells you that a DC block can be as simple as placing a coupling capacitor in a series with the detector output and the next stage of signal processing, which may be an amplifier used to drive headphones or a speaker, for example.

## Complete the diagram

The system block diagram for the transmitter and receiver *(transceiver)* is shown here.

There's an additional component, namely a band-pass filter (BPF), in front of the envelope detector. The filter is expected to combat the adjacent channel interference to the receiver model. A tunable BPF filter centered at *f** _{c}* can become a BPF with center frequency fixed at

*f*

*if you place a mixer (multiplier) in front of it.*

_{IF}Now you need a local oscillator (LO) to shift the frequency of the received signal from *f** _{c}* to

*f*

*. The local oscillator signal is actually a cosine signal at frequency*

_{IF}*f*

*+/–*

_{c}*f*

*. The math behind the frequency shifting is based on the FT modulation theorem. The inclusion of the frequency shifting front-end is known as a*

_{IF}*superheterodyne*

*receiver.*