Differential coding; DPSK modulation and detection

Differential coding

Differential coding represents each data bit by a change or no change in the encoded sequence, rather than by an absolute level. The encoder compares the present data bit with the previous encoded bit:

$$d_k=\overline{b_k\oplus d_{k-1}}\quad(\text{1: keep previous symbol, 0: toggle})$$

An initial reference bit is chosen arbitrarily. Decoding uses only two neighbouring received bits, $b_k=\overline{d_k\oplus d_{k-1}}$, so a complete inversion of the received stream (e.g. $180^\circ$ phase ambiguity) does not cause errors.

DPSK modulation

Differential PSK = differential encoding + BPSK. The encoded bit $d_k$ selects carrier phase 0 ($d_k=1$) or $\pi$ ($d_k=0$).

 b_k -->[XNOR]--d_k-->[polar level]-->(X)--> DPSK s(t)
          ^       |                    ^
          |  [delay Tb]           A cos wc t
          +--------+

Example, data 1 0 0 1 0 0 1 1 with reference 1:

Each 0 in the data causes a $\pi$ phase change; each 1 leaves the phase unchanged.

DPSK detection

 r(t)->[BPF]-+------------>(X)->[int 0..Tb]->[decide]->b_k
             |              ^                 y>0 -> 1
             +->[delay Tb]--+                 y<0 -> 0

The received bit is multiplied by the previous bit and integrated:

$$y=\int_0^{T_b}A\cos(\omega_ct+\theta_k)\,A\cos(\omega_ct+\theta_{k-1})\,dt\approx\frac{A^2T_b}{2}\cos(\theta_k-\theta_{k-1})$$

The output equals the data. No carrier recovery is needed because any constant phase offset cancels in the difference.

• $P_e=\tfrac12e^{-E_b/N_0}$ (about 1 dB worse than coherent BPSK).
• Errors tend to come in pairs since each bit is the reference for the next.