Demodulation of DPSK

Differential PSK (DPSK) is the non-coherent form of BPSK. The information is carried by the phase change between consecutive bits, so the receiver uses the previous received bit as its phase reference and needs no locally generated coherent carrier.

DPSK signal (recap)

The data $b_k$ is first differentially encoded:

$$d_k=\overline{b_k\oplus d_{k-1}}$$

i.e. if $b_k=1$ the encoded bit is unchanged, if $b_k=0$ it toggles. Then $d_k$ drives a BPSK modulator (1: phase 0, 0: phase $\pi$).

Demodulation method: delay-and-multiply (correlation) receiver

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

Working:

1. The received signal in bit $k$ is $r_k(t)=A\cos(\omega_ct+\theta_k)$ and in the previous bit, after delay $T_b$, $r_{k-1}(t)=A\cos(\omega_ct+\theta_{k-1})$.
2. The multiplier gives

$$r_k\,r_{k-1}=\frac{A^2}{2}\cos(\theta_k-\theta_{k-1})+\frac{A^2}{2}\cos(2\omega_ct+\theta_k+\theta_{k-1})$$

3. The integrator (low-pass action) removes the $2\omega_c$ term and keeps $y=\tfrac{A^2T_b}{2}\cos(\theta_k-\theta_{k-1})$.
4. If the phase did not change, $y>0$, decide $b_k=1$; if it changed by $\pi$, $y<0$, decide $b_k=0$.

Any unknown fixed carrier phase appears in both $\theta_k$ and $\theta_{k-1}$ and cancels, which is why no carrier recovery is needed.

Example

The output equals the original data.

Remarks

• Error probability: $P_e=\tfrac12 e^{-E_b/N_0}$, about 1 dB worse than coherent BPSK at $P_e=10^{-4}$.
• Errors tend to occur in pairs, because a wrong bit is also the reference for the next bit.
• The other method is coherent detection of $d_k$ (as in BPSK) followed by a differential decoder ($b_k=\overline{d_k\oplus d_{k-1}}$); it gives slightly better $P_e$ but needs carrier recovery.