Differential encoding

In digital communications, differential coding is a technique used to provide unambiguous signal reception when using some types of modulation. It makes data to be transmitted to depend not only on the current signal state (or symbol), but also on the previous one.

The common types of modulation that require differential coding include phase shift keying and quadrature amplitude modulation.

When data is transmitted over twisted-pair wires, it is easy to accidentally insert an extra half-twist in the cable between the transmitter and the receiver. When this happens, the received signal is inverted.

Similarly for BPSK. To demodulate BPSK, one needs to make a local oscillator synchronous with the remote one. This is accomplished by a carrier recovery circuit. However, the integer part of the recovered carrier is ambiguous. There are n valid but not equivalent phase shifts between the two oscillators. For BPSK, n = 2; the symbols appear inverted or not.

Differential encoding prevents inversion of the signal and symbols, respectively, from affecting the data.

Assuming that ${\displaystyle x_{i}}$ is a bit intended for transmission and ${\displaystyle y_{i-1}}$ was the symbol just transmitted, then the symbol to be transmitted for ${\displaystyle x_{i}}$ is

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