# Communication Systems/Pulse Amplitude Modulation
## Definition

What is PAM? Pulse-Amplitude Modulation is "pulse shaping". Essentially, communications engineers realize that the shape of the pulse in the time domain can positively or negatively affect the characteristics of that pulse in the frequency domain. There is no one way to shape a pulse, there are all sorts of different pulse shapes that can be used, but in practice, there are only a few pulse shapes that are worth the effort. These chapters will discuss some of the common pulses, and will develop equations for working with any generic pulse.

## Square Wave

The most logical way to transmit a digital signal is through a stream of pulses. One distinct pulse for a digital "1", and another distinct pulse for a digital "0". Intuitively, a square pulse will transmit this data, and there are a number of different ways to transmit the data using
The square wave is a basic choice for transmitting digital data because it is easy to transmit, and is generally easy to receive. If we take the fourier transform of a square wave, we get a sinc function. A sinc function is a never-ending function, which means that a square wave in the time domain has a very wide bandwidth. When using a square wave, there will always be a trade-off, because high-frequency components of the square wave will be attenuated by the channel, and the resultant waveform will be more prone to error on the other end.

### Unipolar Square Wave

A unipolar square wave is a wave where a logical 1 is transmitted using a square pulse of height A. Then a logical 0 is transmitted with a 0 voltage.

### Bipolar Square Wave

A bipolar square wave is a square wave where a 1 is transmitted with a pulse of height A/2, and a 0 is transmitted with a pulse of -A/2.

## Other pulses

It turns out that simply by changing the shape of the pulse (changing it away from a square wave), we can create a signal with better properties: lower bandwidth, lower error rate, etc... This section will talk about a few different pulses.

## Sinc

By the property of duality, however, we can see that if we have a sinc wave in the time domain, we will have a square-shape in the frequency domain. This is an interesting result, because it means that we can transmit a sinc shape with definite bandwidth bounds, and it can travel through a channel in its entirety without being attenuated, or losing any harmonics!

## Comparison

Here we will show a basic comparison between square pulses and sinc pulses:
| Metric | Square Pulse | Sinc Pulse |
| Bandwidth | Large bandwidth | Small bandwidth |
| Jitter | Not susceptible to Jitter | Very susceptible to Jitter |
| Noise | Very susceptible to Noise | Susceptible to Noise |
| ISS | Not affected by ISS | Affected heavily by ISS |

## slew-rate-limited pulses

Most systems that use RS232, RS485, or high speed clocks use slew-rate-limited drivers. Some use driver chips with integrated slew-rate limiting; other systems use digital driver chips that put sharp square pulses on their output pins, then an external slew-rate-limiting resistor between that output pin and the rest of the system.

## Raised-Cosine Rolloff

Sinc pulses use less bandwidth, but they are very susceptible to jitter. Conversely, Square pulses have a large bandwidth, but are very resistant to jitter. The decision seems hopeless, but the intrepid communications engineer will try to find a 3rd option. The 3rd option is called the "Raised-Cosine Rolloff" pulse, and has the best characteristics of each wave. Raised Cosine Rolloff (RCR) pulses are resistant to jitter, and at the same time have reasonably good bandwidth. Granted the bandwidth of an RCR pulse is wider than a sinc pulse, and the jitter resistance isn't as good as with a square wave, but this is a compromise, after all.
| Pulse | Bandwidth | Jitter Resistance |
| Rectangular | rb/2 | Excellent |
| Sinc | | Bad |
| RCR | | Good |

## Binary symmetric pulses

A system uses binary symmetric pulses if it transmits only 2 kinds of pulse, s$_1$ and s$_0$, and s$_1$(t) = -s$_0$(t). In a symmetric case, our comparator circuits are very easy: If the correlator outputs a positive value, it's a binary 1. If it outputs a negative value, it's a binary 0.

## Asymmetric Pulses

Asymmetric pulses are more difficult for many reasons:
* The threshold where the comparator should test the value may not be zero
* The correlation receiver needs to correlate two different signals.
* The different pulses may have different power, and they may be different susceptible to noise.

## Asymmetric Correlation Receiver

When asymmetric pulses are used, the receiver system must actually employ 2 coherent optimal receivers, each one tuned to receive one of the pulses. When two coherent optimal receivers are used, the receiver that is outputting the highest voltage value at the end of time T has received it's pulse.
... why not employ only 1 coherent optimal receiver, tuned to receiver neither s0 nor s1, but instead tuned to receive the difference (s1 - s0) ? ...
Some PAM systems, such as Ethernet 100BASE-T2, use PAM-5 ...

## References

* ↑ slew-rate-limiting resistor