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Extra info for Third Wave Project Management: A Handbook for Managing the Complex Information System for the 1990's
This requires averaging over all the pairs weighting each by 1/M; the integral in Eq. 5-5) is simply T, so Each QAM symbol interval generates an independent two-dimensional constellation, provided that the basic pulse is orthogonal. Both constellations in the examples of Fig. 18 have 16 points, but the in Fig. 18(a) are such that the I and Q signals are not independent of each other. When and are independent and each takes values in the PAM set a rectangular QAM constellation such as Fig. 18(b) results.
The BPSK signal attributed to the nth interval is and the vector representation of a signal in that interval alone is and This is the antipodal signaling constellation in Fig. 6(a), whatever the details of may be. Consequently, signal space analysis tells us that the error probability of the nth symbol is directly Eq. 4, the simple linear receiver is ML for each orthogonal carrier pulse, during whose time it ignores all the others. QPSK is a generalization of BPSK to two dimensions per symbol interval.
For these reasons, Nyquist Criterion pulses are generally not used over noisy channels. Nyquist  in 1924 proposed the pulse criterion that bears his name2 and gave a condition for the pulses in terms of their Fourier transform. He showed that a necessary condition for the zero-crossings was that V(f) had to be symmetrical about the points (1/2T, 1/2) and ( – 1 / 2 T , 1/2), assuming that V(f) has peak value 1. This symmetry is illustrated by the two transforms in Fig. 1, with square blocks marking the symmetry points.
Third Wave Project Management: A Handbook for Managing the Complex Information System for the 1990's by Rob Thomsett