A QPSK modulator produces a phase-imbalanced signal of the form

(a) Show that the integrator outputs of Figure 10.2, instead of (10.5) and (10.7), are now given by

where the ± signs depend on whether the data bits d₁ (t) and d₂ (t) are + 1 or -1.

Figure 10.2

(b) Show that the probability of error per quadrature channel is 

For no phase imbalance, the correlator outputs were V₁, V₂ = ± 1/2 AT_s = ± AT_b giving E_b = V²₁ / T_b = V²₂ / T_b and 

With phase imbalance, the best- and worst-case values for E^'_b are 

and

These occur with equal probability.

(c) Plot P_E given by (10.16) and the above result for P^'_{E,quad chan} on the same plot for β = 0, 2.5, 5, 7.5, and 10 degrees. Estimate and plot the degradation in E_b / N₀, expressed in dB, due to phase imbalance at a symbol error probability of 10^-4 and 10^-6 from these curves.

Transcribed Image Text:

x.(t) = Ad, (t) cos ( 27f t- -Ad, (1) sin ( 2af.1 - 5) 2 = Ar, (zcos sin - AT, V; = AT, ± sin ± cos 2