BER CALCULATIONS

For 8-psk constellation M=8. Order of modulation.k=0,1,2.m=0,1,2M.

If an ideal PSK signal is optimally demodulated. Then using complex phasor notation. Each of the complex decision variables takes one of the following M value.

OPTIMALLY DEMODULATED THE USING COMPLEX PHASOR NOTATION

..RECEIVED SIGNAL AT RECEIVER:

NOISE VARIANCE:

Where is the one sided power spectral density.

BINARY BIT MAPPING OF WEIGHTS: GRAY-CODED BIT MAPPING OF WEIGHTS:

ROTATING THE RECEIVED SYMBOL:

WHERE

== == == == == == == ==

TRANSFORM THE WIEGHT OF BIT WITH RESPECTIVELY:

=1 2 1 2 3 2 1=1 2 3 2 1 2 1=1 2 3 2 1 2 1=1 2 1 2 3 2 1=1 2 1 2 3 2 1=1 2 3 2 1 2 1=1 2 3 2 1 2 1=1 2 1 2 3 2 1

ROTATED TO THE DECISION REGION:

EXPRESSED IN VECTOR NOTATION:

INDIVIDUAL ERROR EVENT PROBABILITIES GIVEN BY A SET OF HALF-PLANE, QUARTER-PLANE, AND CORRECTION PLANE PROBABILITES IN TEMS OF DESIRED

HALF-PLANE CALCULATION:

QUADRATURE - PLANE CALCULATIONS:

CORRECTION-PLANE CALCULATION:

. .x = (0 to 20)*.

EXPRESSED IN LINEAR EQUATION FORM:

THE SYSTEM OF LINEAR EQUATIONS SPECIFIED CAN BE EXPRESSED COMPACTTLY IN MATRIX-VECTOR NOTATION AS

WHERE A= Z

THE SYSTEM OF EQUATIONS SPECIFIED BY CAN BE SOLVED FOR THE UNKNOWN PRABABILITES:

1 0 0 0 0 -1 -1 -1 0 0 0 1 1 1 1 0 0 0 -1 0 -1 = 0 0 0 0 0 0 1 0 1 0 0 0 -1 -1 -1 -1 1 0 1 1 1 1 0 -1 1 -1 0 -1

Z=

THE BIT ERROR PROBABILTY :