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# 1 Consider a binary digital communication system with received signal levels 1311 and mu for a 1 hit and [l hit1 respectively. Let drift and .53...

This is the textbook solution to it: HOWEVER I do not know how the solution got those steps as its not very clear the steps. Ive highlighted in red. Could you please write the step by step solution of how the red parts were achieved. PLEASE ONLY ANSWER STEP BY STEP. Thank you

1 Consider a binary digital communication system with received signal levels 1311
and mu for a 1 hit and [l hit1 respectively. Let drift and .53 denote the noise variances for a 1 and [I hit, respectively. Assume that the noise is Gaussian and that a 1 and [I
hit are equally likely. In this lease1 the bit error rate is given by —T 1 T -
“11 s) +—Q( d Win)l
0'1 Jr: 1 where T5 is the reoeiv'er‘s decision threshold. Show that the value of Ta that minimizes
the bit error rate is given by T mm: + ms+1/a§a§[&lt;m1 — mar + 2 (a? — a3) Intense]
. = —. 2_ 2
51 an

BER=lQ ”PT“ +19 T‘f‘m” ,
2 m 2 an From this expression, for large lel, BER —3~ 1,32, so that an optimum Tar that minimizes the BER
exists. Setting BEERfﬂTd = I], we get, which can be written as {0'12 — ﬁngj + 2(mm'5 — mwaTd +mﬁo'l2 — nailing” — 253512 lncrlfa'g = I]. Solving this quadratic equation for Ta, we get u u —m1rr§ + motif + ago? [(711.1 — ﬁle? + 2 (of — Hg) 111(01fogj] 2_ a
51 ﬁre Td=

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