Generalization to Higher Order Pulse Amplitude Modulation

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It is easy to generalize to a level Pulse Amplitude Modulation scheme, known as M-ary Pulse Amplitude Modulation (where ).  This merely involves allowing the coefficients in the waveform to take on equally spaced values. Each value then represents a sequence of bits. Here is a function that generates the set of amplitudes needed for M-ary Pulse Amplitude Modulation.

As an example we consider . The amplitudes needed are,

In this case each pulse will transmit a sequence of two bits. We need to change a data sequence into the appropriate amplitudes. This can be done easily with substitution rules. First we show how to do this with an example, then we build a function from what we have learned. First a sample data sequence:

Now partition the data into pairs of bits.

We assign symbols according to Gray Encoding whereby the bit sequence of adjacent symbols differs by at most one bit.

For a baseband QPAM (Quaternary Pulse Amplitude Modulated) waveform the amplitudes are assigned according to the Gray Encoding used above. Here is a function that creates the QPAM waveform corresponding to a user supplied bit sequence.

Here is a version where a random bit sequence is generated that is nBits long.

This is the waveform for a random bit sequence as a function of time (we only have to plot it out to since there are two bits transmitted for each symbol).

As in the BPAM case above we can build the eye diagram. Here is the function that makes an eye diagram from a random bit sequence that is nBits long.

And here is the version that takes a user supplied bit sequence.

Here is an example of the eye diagram.

As in the BPAM case, let's look at the eye diagram for different values of the parameter in the raised cosine pulse. First generate a random bit sequence.

Now show the eye diagrams for this particular bit sequence for .

The helpful effects of the increase in excess bandwidth with increasing is evident from these graphs.

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