Differential PSK

Differential Phase Shift Keying (DPSK) is a phase modulation scheme where information is encoded in the phase difference between consecutive symbols rather than the absolute carrier phase. This approach eliminates the need for a coherent phase reference at the receiver, making DPSK systems simpler and more robust in practice. It is an important topic in GATE digital communications.

Core Concept of DPSK

In conventional BPSK, the receiver must know the exact carrier phase to correctly decode symbols. Generating this coherent reference is complex and costly, especially in time-varying channels. Differential encoding solves this by making information reside in the transition between symbols rather than in the absolute phase of each symbol. If the phase changes by 180 degrees from one symbol to the next, a binary 1 is transmitted. If the phase stays the same, a binary 0 is transmitted.

The encoding rule for DBPSK is: the differentially encoded bit d(k) = d(k-1) XOR m(k) where m(k) is the current input bit and d(k-1) is the previously transmitted encoded bit. An initial reference bit (typically 0) is required to start the encoding chain. The modulator then maps each d(k) to a BPSK symbol in the usual way.

At the receiver, detection is performed by comparing the current received symbol with the previous one. This is called non-coherent detection because no explicit phase reference is needed. The receiver simply multiplies the current received signal with a delayed version of itself (delayed by one symbol period T), integrates over T, and passes the output through a threshold. This self-referencing approach makes DPSK simpler than coherent BPSK at the cost of slightly higher bit error rate.

Mathematical Expression

For DBPSK, the transmitted signal in the k-th symbol interval is s(t) = A cos(2*pi*fc*t + phi_k) where phi_k is either 0 or pi depending on d(k). The received signal after passing through an AWGN channel is r(t) = s(t) + n(t). The non-coherent correlator multiplies r(t) with r(t - T) and integrates. The decision variable is positive for phase agreement (decoded 0) and negative for phase reversal (decoded 1).

The Bit Error Probability (BEP) for DPSK in AWGN is:

Pe_DPSK = (1/2) * exp(-Eb/N0)

For coherent BPSK the BEP is Pe_BPSK = (1/2) * erfc(sqrt(Eb/N0)). The DPSK performance is approximately 3 dB worse than coherent BPSK at high SNR. This 3 dB penalty is the price paid for avoiding coherent detection. For DQPSK, the BEP expression is more complex and SNR penalty is similarly around 2 to 3 dB.

Practical Understanding

DPSK is useful in fast-fading channels where maintaining a coherent phase reference is difficult or impractical. It was used in older Bluetooth versions (pi/4-DQPSK) and some satellite links. The key practical advantage is receiver simplicity: no phase locked loop is required, reducing cost and complexity.

A practical consideration is error propagation. Since each symbol decision depends on the previous symbol, a single error in the differential decoding chain can cause two bit errors in succession. However, this burst error behavior is limited to at most two consecutive errors per single channel error event.

Given:
System: DBPSK in AWGN channel
Eb/N0 = 10 dB

Why this formula applies:
DBPSK uses differential encoding with non-coherent detection
BEP formula: Pe = 0.5 * exp(-Eb/N0)

Formula:
Pe_DPSK = (1/2) * exp(-Eb/N0)
Eb/N0 (linear) = 10^(10/10) = 10

Substitution:
Pe = 0.5 * exp(-10)

Calculation:
exp(-10) = 4.54 x 10^-5
Pe = 0.5 * 4.54 x 10^-5

Final Answer:
Pe_DPSK = 2.27 x 10^-5
Compare: Coherent BPSK at same Eb/N0 gives Pe approximately 7.83 x 10^-6
DPSK is worse by approximately factor of 3 (close to 3 dB penalty)

Exam Tip: GATE frequently asks to compare BEP of BPSK and DPSK. Remember: DPSK BEP = 0.5 * exp(-Eb/N0) and BPSK BEP = 0.5 * erfc(sqrt(Eb/N0)). DPSK is always worse by approximately 3 dB. Also, DPSK does not need a coherent reference, which is its main advantage.

Mechanism of Differential Encoding and Decoding

• Differential encoder computes d(k) = d(k-1) XOR m(k), converting absolute bits to relative phase transitions.
• BPSK modulator maps d(k) = 0 to phase 0 degrees and d(k) = 1 to phase 180 degrees.
• Non-coherent receiver multiplies received signal with its delayed version (delay = T) and integrates. No local oscillator phase reference is required.
• Decision rule: positive correlator output means no phase change (decoded 0), negative output means phase reversal (decoded 1).
• A single channel error causes at most two decoded bit errors due to the differential dependency on adjacent symbols.

Quick Revision

• DPSK encodes information in phase difference between adjacent symbols, not absolute phase.
• DBPSK encoding rule: d(k) = d(k-1) XOR m(k). Initial reference bit is needed.
• BEP of DPSK in AWGN: Pe = 0.5 * exp(-Eb/N0).
• DPSK is approximately 3 dB worse than coherent BPSK but does not require a phase reference.
• Receiver uses 1-symbol delay and multiply (autocorrelation) for non-coherent detection.
• Trap: Do not confuse differential encoding (transmitter) with differential decoding (receiver). Both operate on adjacent symbols.
• Error propagation in DPSK is limited to 2 consecutive bit errors per single channel error event.