Editing Phase-shift keying (section)

===Example: Differentially-encoded BPSK===
[[File:Differential Codec.png|center|500px|thumb|Differential encoding/decoding system diagram]]

At the <math>k^{\textrm{th}}</math> time-slot call the bit to be modulated <math>b_k</math>, the differentially encoded bit <math>e_k</math> and the resulting modulated signal <math>m_k(t)</math>. Assume that the constellation diagram positions the symbols at ±1 (which is BPSK). The differential encoder produces:

:<math>\,e_k = e_{k-1} \oplus b_k</math>

where <math>\oplus{}</math> indicates [[binary addition|binary]] or [[modular arithmetic|modulo-2]] addition.

[[File:Diff enc BPSK BER curves.svg|thumb|right|280px|BER comparison between BPSK and differentially encoded BPSK operating in white noise]]

So <math>e_k</math> only changes state (from binary "0" to binary "1" or from binary "1" to binary "0") if <math>b_k</math> is a binary "1". Otherwise it remains in its previous state. This is the description of differentially encoded BPSK given above.

The received signal is demodulated to yield <math>e_k = \pm 1</math> and then the differential decoder reverses the encoding procedure and produces

:<math>b_k = e_k \oplus e_{k-1},</math>

since binary subtraction is the same as binary addition.

Therefore, <math>b_k=1</math> if <math>e_k</math> and <math>e_{k-1}</math> differ and <math>b_k=0</math> if they are the same. Hence, if both <math>e_k</math> and <math>e_{k-1}</math> are ''inverted'', <math>b_k</math> will still be decoded correctly. Thus, the 180° phase ambiguity does not matter.

Differential schemes for other PSK modulations may be devised along similar lines. The waveforms for DPSK are the same as for differentially encoded PSK given above since the only change between the two schemes is at the receiver.

The BER curve for this example is compared to ordinary BPSK on the right. As mentioned above, whilst the error rate is approximately doubled, the increase needed in <math>E_b/N_0</math> to overcome this is small.  The increase in <math>E_b/N_0</math> required to overcome differential modulation in coded systems, however, is larger{{snd}} typically about 3&nbsp;dB. The performance degradation is a result of [[noncoherent transmission]]{{snd}} in this case it refers to the fact that tracking of the phase is completely ignored.