A delay-and-multiply receiver for DPSK as shown in Figure 9.17 should have an input filter bandwidth of B = 2/T = 2R Hz to achieve the asymptotic probability of error given by (9.113). If the received signal has a frequency error of Δf (due to Doppler shift, for example), the input filter must have a bandwidth of 2R + | Δf | to accommodate this frequency error.

(a) Show that the error probability as a result of this frequency error is given by 

(b) Plot P_{E, |Δf|} versus z in dB for |Δf| / 2R = 0, 0.1, 0.2, 0.3, and 0.4. Estimate the degradation in dB at a probability of error of 10^-6.
(c) Give the mathematical expression for the degradation and compare it with the results estimated in part (b).

Figure 9.17

E, PE, JASI , J 3 0 No 1+ |Af\/2R = ( 1+|Afl/2R t = to + T Received to + T ()dt signal Decision Threshold to One-bit delay