ty by (2.15) In the case of QPSK, symbol errors occur when the noise pushes the received phasor into the wrong quadrant as illustrated in Figure 2.4. In this figure it is assumed that the transmitted symbol has a phase of /4 rad, corresponding to the demodulator I and Q values of VI = V and VQ = V volts (i.e. noise-free case). Thus, if we consider that the noise phasors (n1 and n2) are pointing in directions that are most likely to cause errors, then a symbol error will occur if either n1 or N2 exceeds V. Transmitted noise signal Received signal noise I Axis Figure 2.4 Transmitted and received signal vectors [3]. Now, if for simplicity we also assume that a QPSK signal is transmitted without Nyquist filtering and demodulated with hard-decisions, the probability of a correctly demodulate symbol value is equal to the product of the probabilities that each demodu- lator low-pass filter output lies in the correct quadrant. Then the probability that the demodulated symbol value is correct is given by P.=(1-Pa)(1-Po2), (2.16) where Pel and Pe2 are the probabilities that the two filter output sample values are in the wrong quadrant. When showing that the low-pass filters are equivalent to integrators, which is the optimum choice if Nyquist filtering is not used, Pel and Pe2 can be ex- pressed as (2.17) System Analysis Fundamentals where Es = A2T/2 is the energy per symbol, No/2 is the two-sided noise power noise spectral density (in V2/Hz) at the demodulator input, and the function Q(x) is the com- plementary integral Gaussian function. The error function erf(x) given by (2.18) and complementary error function erfc(x) expressed as erfc(x)=1-erf(x) (2.19) are not fully identical to the integral Gaussian function G(x), and the complementary integral G(x) or Q(x) in our case. Now if we assume G(x) = Q(x), we can use the fol- lowing (2.20) function to evaluate our error probabilities. Then since Pe1=Pe2, the symbol error prob- ability could be written as P,=1-P,=2P (2.21) which at Pel becomes Ps 2Pe1. (2.22) Substituting Pe