Conventional Differential Coding
A method illustrated above can deal with a data stream inversion (it is called 180° ambiguity). Sometimes it is enough (e.g. if BPSK is used or if other ambiguities are detected by other circuits, such as a Viterbi decoder or a frame synchronizer) and sometimes it isn't.
Generally speaking, a differential coding applies to symbols (these are not necessary the same symbols as used in the modulator). To resolve 180° ambiguity only, bits are used as these symbols. When dealing with 90° ambiguity, pairs of bits are used, and triplets of bits are used to resolve 45° ambiguity (e.g. in 8PSK).
A differential encoder provides the operation, a differential decoder - the operation.
Both differential encoder and differential decoder are discrete linear time-invariant systems. The former is recursive and IIR, the latter is non-recursive and thus FIR. They can be analyzed as digital filters.
A differential encoder is similar to an analog integrator. It has an impulse response
and a transfer function
A differential decoder is thus similar to an analog differentiator, its impulse response being
and its transfer function
Note that in binary (modulo-2) arithmetic, addition and subtraction (and positive and negative numbers) are equivalent.
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