The above explanation assumes that the input signal consists of just a single frequency component. In general, we can represent any input as some
appropriate combination or spectrum of frequency components. The
details of this spectrum can provide us with information about the signal
source. It can also be a pattern deliberately used to convey messages
— e.g. to send radio messages or TV pictures. The action of the
heterodyne system when presented with such an input is illustrated in figure
1.4.
We can represent the input field as the sum of a series of components
The heterodyne system illustrated in figure 1.3 would produce a ‘down converted’ output component for each input component. The resulting signal voltage across the IF amplifier's input resistance, 
, will therefore be
where 
is its frequency of the local oscillator and
There are two important feature of this result. Consider initially a signal which only contains frequencies in the range 
. This region is called the heterodyne system's Upper Sideband. The output IF spectrum produced by such an input is ‘shifted down in frequency’ by an amount 
and its amplitude scaled by an amount proportional the the LO field amplitude, 
. As a result, we can determine the details of the input spectrum by examining that of the IF output.
For components in the Lower Sideband frequency range — i.e. where 
— we find that 
. An input signal component at, say, 100·1 GHz will — when mixed with a 100 GHz LO — produce an output at 100 MHz. . Hence an input at 99·9 GHz mixed with a 100 GHz LO will also produce an output at 100 MHz. You can prove this result mathematically by noting it is a standard trig identity that 
. As a result, an input which consists only of lower sideband components produces a down converted output spectrum which is a sort of ‘mirror image’ of the input signal's spectrum. Like the upper sideband result, the frequencies and amplitudes of the IF components depend on the signal, hence we can use the IF spectrum to determine the details of the input signal.
A problem can arise when the input signal simultaneously presents the heterodyne mixer with frequency components both above and below the LO frequency. When this happens the output IF spectrum is a combination of the down converted upper sideband and the down converted mirror imaged lower sideband. This can lead to confusion. How can we tell, when using a 100 GHz LO, whether an 100 MHz IF output component was produced by an input signal at 100·1 GHz or one at 99·9 GHz? Worse, how can we tell if it was produced by a combination of input components at both these frequencies?
This problem arises because the fact that 
leads to a sideband ambiguity. This effect is inherent in the heterodyne mixing process. Fortunately, there are ways around it. The most common solution is to employ a single sideband filter . This is a filter placed between the mixer and the signal source. The filter is built to only allow through frequency components from one, chosen, sideband. When such a filter is in place we can know that a given IF output component can only have come from a signal component at one of the two possible signal frequencies because the other would not have been able to reach the mixer.
The addition of such a filter turns the heterodyne system into a Superheterodyne Receiver. The single sideband filter is conventionally called an rf filter. (Here ‘rf’ stands for ‘radio frequency’ filter, even if we aren't talking about normal radio frequencies!) Using a superheterodyne receiver we can collect information about a laser, or a distant star, or we can process the modulated signals produced by a distant TV station. In each case the information we require is obtained by measuring the details of the signal spectrum as conveyed by the IF output.
Summary
It should now be clear that we can use either a conventional diode or a photodetector as part of a heterodyne arrangement to measure the amplitude, frequency, and phase of an input signal. In general, any nonlinear device can be used as the basis of a heterodyne system when combined with a suitable local oscillator. You should also now understand that the main advantage of a heterodyne system is that is allows us to ‘shift down’ a signal to a lower, more convenient, frequency range without losing information about the amplitude, frequency, and phase of the original signal. This allows us to take in very high frequency signals and transfer their spectral pattern to lower frequencies which are easier to amplify and examine.
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