19
Notice that the greater the sequence length the quicker the SESS system converges to
BPSK in an Additive White Gaussian Noise (AWGN) channel. Conversely, for lower N values
the BER stays closer to 0.5 longer.
This makes sense when you consider that if there is an error for a small N of 1 or 2, the
next bit will be demodulated with the wrong spreading code leaving with either 100 or 50
percent of the wrong bits in the demodulating spreading sequence. This leads to the 50 percent
error rate that can been seen in these two values on Figure 5. The chip errors in the receiver
registers attenuate the de-spreaded signal strength. This can be regarded as a form of self
interference introduced by self encoding. Let X be the random variable denoting the number of
chip errors within the receiver register of length N. For l chip errors, the amplitude attenuation
can be expressed as:
(7)
|_{= }
= 1−
2
Then the conditional probability of error becomes:
(8)
_{ }
_{| }
=
1−
2
2
The probability of error will be greater than 0.5 if the argument of the Q-function is
negative; this corresponds to the situation that X > N/2 [24]. This suggests that there exists an
undesirable situation caused by error propagation: a sufficient number of chip errors may