Phase-Shift Keying, PSK, is a very prevalent form of wireless communication.
The formation of a PSK signal is straightforward. Choose a carrier frequency,
agreed upon by both the transmitter and receiver. For binary PSK, BPSK, each
individual bit is encoded in a phase shift of an agreed-upon period of the
carrier cosine signal. This phase shift is zero, meaning the sent signal is the
fundamental cosine signal itself, when the sent bit is 0. The phase shift is Pi,
or 180 degrees, a negative of the original signal, when the sent bit is a 1.
For Quaternary PSK, or 4-PSK, two bits at a time are encoded into a phase shift of
the given period of the cosine. If the two bits are 00, no shift is given. If the
two bits are 01, the shift is Pi/2, or 90 degrees. See the picture below.

If the bits are 10, the shift
is Pi, and if the two bits are 11, the shift is 3/2 * Pi. This idea is carried
further, in the same way, for 8-PSK, in which 3 bits at a time are encoded, 16-PSK,
in which 4 bits at a time are encoded, and so on. A general formula for deriving
the signal for M-ary PSK, where M represents the number 2, 4, 8, etc., is given:

Here, Fc is the carrier frequency, M is the number in M-ary, and Em is
the number representing the bits that are encoded, for example 5 for 101 when
using 8-PSK. This research uses only 4- and 8-PSK, but these methods could
easily be extended to other forms of PSK, likely with similar results. In general,
as the phase shifts get more closely spaced, when using a higher number of symbols,
the probability of making a bit error steadily declines for a given SNR, or signal-
to-noise ratio. It would seem, then, that Binary PSK, for example, would be very
preferable to 8-PSK. This is true when bandwidth is no factor. However, in most
cases in the real world, bandwidth is very much a factor, and with higher numbers of
symbols, or closer spacing of the phase shifts, more bits are encoded in every
symbol. This means with essentially the same signal duration and the same bandwidth
as BPSK, three times as many bits are sent using 8-PSK. These tradeoffs are always
present in communications systems.

The optimal receiver for PSK signals is simpler than for many other communications
schemes. The PSK signal encodes the signal in shifts of a cosine. However, recall
that cosine and sine are orthogonal signals, i.e.,

This orthogonality means our receiver can correlate the incoming signal with
the sine and cosine versions of the signal, our sole basis functions, and from
this obtain two orthogonal projections of our signal. From this, we have all the
data we can take from our signal in two parameters, the cosine projection, x, and the
sine projection, y. A diagram of the receiver is shown.

Since neither a real transmitter or receiver could be built in the course of a
couple of months, especially with the flexibility desired, PSK communication
had to be simulated on a computer. Rather than creating sinusoidal waveforms,
a simpler way to simulate was necessary, or the computers never would be able
to handle simulating many millions of bits and symbols, as was necessary to determine
the effectiveness of the changes made during the course of the research. By
investigating the properties of x and y, the sampled cosine and sine projections,
respectively, of the actual signal, a simple mechanism can be seen. For these
simulations, we assume that
the sinusoidal signal, s(t), passes through an additive white Gaussian noise (AWGN)
channel on its way to the receiver. This means the actual received signal that is
processed by the receiver, r(t) is a sum of the sent signal, s(t) and a white
Gaussian noise signal, n(t). The mathematical manipulations below rely on the
trigonometric properties of sinusoids.

These values for expectations and variances simplify simulations greatly.
Rather than creating vectors of signal values as sinusoids, it is necessary only
to create arrays of transmitted x and y (cosine and sine) components of the signal,
then add Gaussian noise to these components of the same variance as the Gaussian
noise desired for the signal itself. Gaussian values, for the simulations, were
derived from large numbers of the uniformly distributed random numbers, already
mentioned, using a Box-Muller method of transformation of random variables. For
a more detailed description of random variables, please see a nearby statistics book
or professor.

A final point necessary for understanding the ideas that follow is representation
of PSK signals on a plane, where the x-coordinates correspond to the cosine
projection of the PSK signal, and the y-coordinates, to sine. A traditional
transmitter, the form currently employed today, spaces the transmission points
evenly around the circle with radius equal to the root of the signal's energy
per symbol, as the white dots illustrate.
The diagram shown below corresponds to 8-PSK, the scheme most studied,
but a picture of the 4-PSK transmitter can be extrapolated from this.

The evenly-spaced wedges which split the plane into eight different regions
correspond to the traditional, currently employed, receiver's decision regions.
Should the received values of the cosine and sine projections of the signal formed
from the correlations fall anywhere in the red region on the right, for example,
the closest transmission point to this is 000. Therefore, the receiver determines
that the most likely bit sequence sent from the transmitter is 000.

With a solid understanding of the above colorful diagram, continue on
to Altering the Reception Scheme.

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