# Phase-Shift Keying.

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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.