38)
where u has been inserted to represent an arbitrary phase offset between the two FSK signals. If we let
these signals be represented as vectors in Cartesian coordinate system, then the orthogonality constraint
is viewed as the two vectors being perpendicular to one another. Alternatively, it can be stated as
(2.39)
After substituting the previous expressions and carrying out the integration, we arrive with an expression
for the correlation of two sinusoids separated in frequency by .
In Fig. 2.59, we plot the cross-correlation function for BFSK modulation as a function of the modulation
index [23]. Similarly, we have shown the cross-correlation for BPSK modulation or antipodal
signaling, which is a constant value of 1.
Below we see the orthogonality condition is met for modulation indices that are multiples of 0.5. We
would like to take this opportunity to relate this finding to what was presented earlier for BPSK. The
BPSK modulation scheme has waveforms that are antipodal to one another, that is, the correlation
between the 2 possible sinusoids is 1. This is drawn in the Fig. 2.59 as a dashed line. So, it was earlier
stated the antipodal signaling provided the best performance in an AWGN channel. Hence modulation
indices that produce a correlation factor close to 1 should produce better BER results.
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