ulation of a carrier given by Equation generation of a local carrier at the demodulator. In DPSK, the 19.38, the output voltage is given by Equation 19.39. information is conveyed by changes in phase between digits, instead of by the phase deviation of each from a reference carrier. (19.38) The principle of differential encoding is shown in Figure 19.8. The coder is fed with a sequence of binary symbols s,(1),s,(2), ... s,(n), where (n) = +1. Its output signal, s,(n), is obtained by multi- plying ,(n) by the previous output from the coder, i.e. S2(n - 1), cowc() = for Vm= (19.39) resulting in Equation 19.40. 19/10 Digital modulation S1(n) S2(n) Transmission system S2(n) S3(n) S2(n-1) S2(n-1) One digit delay One digit delay S5(n) S1(n) S2(n) S4(n) = S2(n) cos wc t S2(n-1) S4(n-1) One digit delay One digit delay Figure 19.8 Principle of differential phase shift keying (DPSK): (a) differential coding and decoding; (b) DPSK system S2(m) (19.40) the values of two output digits. Consequently, a noise peak that would produce a single error with PSK can produce two errors with Thus,s,only changes sign ifs,(n)=-1 DPSK. Since s,(n) ands,(n-1) are single digit binary numbers, their Calculation of the bit error probability for DPSK is complicated. multiplication can be carried out simply by means of an exclusive- It has been shown (Schwartz et al., 1966) that the error probability OR logic gate. is given by Equation 19.43. Assuming error free transmission, the output signal from the decoder is given by Equation 19.41, regardless of whether s2(n-1) is +1 or -1 and whether the initial condition of was s2(0) or (19.43) s2(0)=-1. 19.3.4 Comparative performance of ASK, FSK and PSK (19.41) Figure 19.9 shows, for signals perturbed by added Gaussian noise, In differentially encoded pulse phase modulation (DEPSK), the how the bit error probability varies with the signal to noise ratio for coder shown in Figure 19.8(a) is followed by a PSK modulator and binary systems using coherent and non-coherent ASK, coherent and the demodulator i