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Vm / for coherent orthogonal signals 24 1 Channel Codes and Modulation p and log ŒI0 . 8Es Rm =N0 / for noncoherent orthogonal signals, where the index m ranges p over the symbol alphabet. Since the latter function increases monotonically and 8Es =N0 is a constant, optimal symbol metrics or decision variables for 2 noncoherent orthogonal signals are Rm or Rm for m D 1; 2; : : : ; q. j 2 fl t/= p Ts , 0 Ä t Ä Ts , l D 1; 2; : : : ; q. 81) where an irrelevant constant have been omitted. 62) is satisfied if the adjacent frequencies are separated by k=Ts , where k is a nonzero integer.

This truncation gives a tight upper bound in Pb for Pb Ä 10 2 . However, the truncation may exclude significant contributions to the upper bound when Pb > 10 2 , and the bound itself becomes looser as Pb increases. The figures indicate that the code performance improves with increases in the constraint length and as the code rate decreases if K 4. The decoder complexity is almost exclusively dependent on K because there are 2K 1 encoder states. However, as the code rate decreases, more bandwidth is required and bit synchronization becomes more challenging due to a reduced energy per symbol.

A frequency translation or downconversion to baseband is followed by matched filtering. 64) p where the factor 2 has been inserted for mathematical convenience. 66) Since the real and imaginary components of nki are jointly Gaussian, this random process is a complex-valued Gaussian random variable. t/g indicate that nki is zero-mean and that EŒnki nlm  D 0. Thus, the random variables nki , k D 1; 2; : : : ; q, i D 1; 2; : : : ; n, are circularly symmetric. 62) indicate that EŒjnki j2  D N0i , k D 1; 2; : : : ; q, and that EŒnki nlm  D 0, l ¤ k or i ¤ m.

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