--- title: "Differential coding; DPSK modulation and detection" subject: "Communication Systems" chapter: 7 key: "semester_6/communication_systems:differential-coding-dpsk" source: https://www.bctengineer.com/q/semester_6/communication_systems/differential-coding-dpsk site: BCT Engineering --- # Differential coding; DPSK modulation and detection Communication Systems · Chapter 7: Digital Modulation Techniques - Asked in: 2071 Shrawan (CS II), 2+5 marks - Video: [Differential Phase Shift Keying DPSK (Definition, Waveform, DPSK transmitter & DPSK receiver)](https://www.youtube.com/watch?v=ZTKDZ6QPMlE) — Engineering Funda - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/communication_systems/differential-coding-dpsk - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-7 ## Question What do you understand by differential coding? Explain differential phase shift keying modulation and detection with example and diagrams. ## Short answer Differential coding represents each data bit by a change or no change in the encoded sequence. DPSK modulation uses an XNOR gate to encode bits (1=keep, 0=toggle) then drives a BPSK modulator. Detection multiplies the signal by a one-bit delayed copy and integrates; positive output means no phase change (bit 1), negative means a change (bit 0). ## Answer #### 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 $\pi$ ($d_k=0$). ```text 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: | $k$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |---|---|---|---|---|---|---|---|---|---| | $b_k$ | | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | | $d_k$ | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | | Phase | 0 | 0 | $\pi$ | 0 | 0 | $\pi$ | 0 | 0 | 0 | Each 0 in the data causes a $\pi$ phase change; each 1 leaves the phase unchanged. #### DPSK detection ```text 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: $$ y=\int_0^{T_b}A\cos(\omega_ct+\theta_k)\,A\cos(\omega_ct+\theta_{k-1})\,dt\approx\frac{A^2T_b}{2}\cos(\theta_k-\theta_{k-1}) $$ | $k$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |---|---|---|---|---|---|---|---|---| | $\theta_k-\theta_{k-1}$ | 0 | $\pi$ | $\pi$ | 0 | $\pi$ | $\pi$ | 0 | 0 | | Sign of $y$ | + | − | − | + | − | − | + | + | | Output | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 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. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. In differential coding with d_k = XNOR(b_k, d_(k-1)), what do data bits 1 and 0 do? A. 1 toggles the previous symbol, 0 keeps it B. 1 keeps the previous symbol, 0 toggles it C. Both keep the previous symbol D. 1 gives level +1 and 0 gives level -1 absolutely Answer: B. Differential coding sends change/no change: bit 1 keeps the previous encoded bit, bit 0 toggles it. (Key point: Meaning of differential coding) 2. With reference d_0 = 1 and data 0 1 0, what is the encoded sequence d_1 d_2 d_3? A. 0 0 1 B. 1 0 1 C. 0 1 1 D. 1 1 0 Answer: A. 0 toggles 1 to 0, 1 keeps 0, 0 toggles 0 to 1, giving 0 0 1. (Key point: Encoding example) 3. Why does a 180° phase ambiguity (complete inversion) not cause errors in differential decoding? A. Decoding uses only the difference of two neighbouring bits, which is unchanged by inverting both B. The decoder uses a larger threshold C. Inverted bits are detected by a parity check D. The receiver resets the reference every bit Answer: A. b_k = XNOR(d_k, d_(k-1)); inverting both d_k and d_(k-1) gives the same result. (Key point: Immunity to phase inversion) 4. In the DPSK detector, the integrated product is approximately (A^2 Tb/2)cos(theta_k - theta_(k-1)). A phase difference of pi therefore gives which output? A. y > 0, output 1 B. y = 0, no decision C. y < 0, output 0 D. y > 0, output 0 Answer: C. cos(pi) = -1 makes y negative, which means a phase change, i.e. data bit 0. (Key point: DPSK detection) ## Related in Communication Systems - [Need for shift keying; ASK vs FSK vs PSK](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ashift-keying-ask-fsk-psk) — Communication Systems, chapter 7 - [DSB-FC vs DSB-SC; phase shift method for SSB](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Adsbfc-dsbsc-ssb-phase-shift) — Communication Systems, chapter 2 - [Phase shift method of SSB generation: pros and cons](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Assb-phase-shift-pros-cons) — Communication Systems, chapter 2, asked 2× - [PSK generation and detection; GMSK](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Apsk-generation-detection-gmsk) — Communication Systems, chapter 7 - [Phase shift method of SSB generation](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Assb-phase-shift) — Communication Systems, chapter 2 - [Phase error in synchronous DSB-SC detection; synchronized receiver](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aphase-error-dsbsc-costas) — Communication Systems, chapter 2 ## Related in other subjects - [RC phase shift oscillator](https://www.bctengineer.com/question.md?key=semester_3%2Felectronics_devices_and_circuit%3Arc-phase-shift-oscillator) — Electronics Devices and Circuit, chapter 5, asked 6× - [Barkhausen criteria and RC phase shift oscillator](https://www.bctengineer.com/question.md?key=semester_3%2Felectronics_devices_and_circuit%3Abarkhausen-rc-phase-shift) — Electronics Devices and Circuit, chapter 5, asked 3× - [Cavity magnetron with 45° phase shift and applications](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Acavity-magnetron-45-applications) — RF and Microwave Engineering, chapter 5 - [Cross-field cavity amplifier with 90° phase shift](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Acrossfield-device-90-degree) — RF and Microwave Engineering, chapter 5