# Communication Systems/Coherent Receivers

## Receiver Design

It turns out that if we know what kind of signal to expect, we can
better receive those signals. This should be intuitive, because it is
hard to find something if we don\'t know what precisely we are looking
for. How is a receiver supposed to know what is data and what is noise,
if it doesnt know what data looks like?

Coherent transmissions are transmissions where the receiver knows what
type of data is being sent. Coherency implies a strict timing mechanism,
because even a data signal may look like noise if you look at the wrong
part of it. In contrast, noncoherent receivers don\'t know exactly what
they are looking for, and therefore noncoherent communication systems
need to be far more complex (both in terms of hardware and mathematical
models) to operate properly.

This section will talk about coherent receivers, first discussing the
\"Simple Receiver\" case, and then going into theory about what the
optimal case is. Once we know mathematically what an optimal receiver
should be, we then discuss two actual implementations of the optimal
receiver.

It should be noted that the remainder of this book will discuss optimal
receivers. After all, why would a communication\'s engineer use anything
that is less than the best?

## The Simple Receiver

A simple receiver is just that: simple. A general simple receiver will
consist of a low-pass filter (to remove excess high-frequency noise),
and then a sampler, that will select values at certain points in the
wave, and interpolate those values to form a smooth output curve. In
place of a sampler (for purely analog systems), a general envelope
filter can also be used, especially in AM systems. In other systems,
different tricks can be used to demodulate an input signal, and acquire
the data. However simple receivers, while cheap, are not the best choice
for a receiver. Occasionally they are employed because of their price,
but where performance is an issue, a better alternative receiver should
be used.

## The Optimal Receiver

Engineers are able to mathematically predict the structure of the
optimal receiver. Read that sentence again: Engineers are able to
design, analyze, and build the best possible receiver, for any given
signal. This is an important development for several reasons. First, it
means that no more research should go into finding a better receiver.
The best receiver has already been found, after all. Second, it means
any communications system will not be hampered (much) by the receiver.

### Derivation

here we will attempt to show how the coherent receiver is derived.

### Matched Receiver

The matched receiver is the logical conclusion of the optimal receiver
calculation. The matched receiver convolves the signal with itself, and
then tests the output. Here is a diagram:

`s(t)----->(Convolve with r(t))----->`

This looks simple enough, except that convolution modules are often
expensive. An alternative to this approach is to use a correlation
receiver.

### Correlation Receiver

The correlation receiver is similar to the matched receiver, instead
with a simple switch: The multiplication happens first, and the
integration happens second.

Here is a general diagram:

`           r(t)`\
`            |`\
`            v`\
`s(t) ----->(X)----->(Integrator)--->`

In a digital system, the integrator would then be followed by a
threshold detector, while in an analog receiver, it might be followed by
another detector, like an envelope detector.

## Conclusion

To do the best job of receiving a signal we need to know the form of the
signal that we are sending. After all we can\'t design a receiver until
after we\'ve decided how the signal will be sent. This method poses some
problems in that the receiver must be able to line up the received
signal with the given reference signal to work the magic: if the
received signal and the reference signal are out of sync with each
other, either as a function of an error in phase or an error in
frequency, then the optimal receiver will not work.

## Further reading

