Presented by: Idris AjiaAssessor: Dr. Tareq Al-Naffouri

Date: Saturday, 12 December 2009

Increasing dependence on Digital Communication.

Need for accurate data TXCalculating BER. Improving the

System Aim of Paper:

Use of grey coding to map the symbols on the Constellation plotThen use the geometric property of the constellation plot.

The generic equation for an M-PSK message is:

Where

ETP is the symbol energy, T is the symbol period.

Use Mathematical induction. Start with 4-PSK Each bit = a0a1 Pi is the probability of

an error occurring based on the decision made with respect to the i-axis

Constellation of 4-PSK

Given symbols are equiprobable. Then: And Hence, probability of error for 4-psk is:

For an 8-PSK, there are 3 bits per symbol as shown below:

It follows that:

Which approximates to:

•Let the bits in each symbol be a0a1a2. •Notice two new Axes C1 and C2

Then:

It can now be seen that by means of induction,

Now the BER of M-PSK can be determined by calculating Pa. To do this, we know that: For any MPSK signal space representation, there are four quadrants and each quadrant

contains the same number of constellation point. This means there are M/4 possible values for the vertical distance of each constellation to

the a-axis. The value for these distances, by observing the constellation plot is given by:

Since all symbols are equiprobable, the probability for any given symbol over an AWGN channel is given by:

So the probability of error due to the decision made with respect to the a-axis, given any symbol is given by:

Then

Where di = distance between each symbol and the a-axis.

Now,

By means of substituting (7) into (6), we get the BER of MPSK as:

But

The Ber for M-PSK is:

My Approximation Real Value

Don’t blame me, Blame the likes of Fourier, Mr

Gauss and Dr. Tareq. I will now entertain your

questions...