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Evaluating the Bit Error Rate for M-PSK

Evaluating the Bit Error Rate for M-PSK. Presented by: Idris Ajia Assessor: Dr. Tareq Al- Naffouri Date: Saturday, 12 December 2009. Introduction: Importance of Digital Communications. Increasing dependence on Digital Communication. Need for accurate data TX Calculating BER.

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Evaluating the Bit Error Rate for M-PSK

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  1. Evaluating the Bit Error Rate for M-PSK Presented by: IdrisAjia Assessor: Dr. Tareq Al-Naffouri Date: Saturday, 12 December 2009

  2. Introduction:Importance of Digital Communications. • Increasing dependence on Digital Communication. • Need for accurate data TX • Calculating BER. • Improving the System • Aim of Paper:

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

  4. The generic equation for an M-PSK message is: Where Pre-Analysis ETP is the symbol energy, T is the symbol period.

  5. Computing BER • 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

  6. 4-PSK • Given symbols are equiprobable. • Then: • And • Hence, probability of error for 4-psk is:

  7. Then 8-psk • 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

  8. 8-PSK cont. • Then: • It can now be seen that by means of induction,

  9. BER for M-psk • 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:

  10. BER for m-psk • 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

  11. Finally... • The Ber for M-PSK is:

  12. Comparison My Approximation Real Value

  13. Confused yet??? • Don’t blame me, • Blame the likes of Fourier, Mr Gauss and Dr. Tareq. • I will now entertain your questions...

  14. Applauds

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