BPSK

Binary phase-shift keying (BPSK) is a binary digital modulation scheme in which the phase of a carrier signal is shifted between two discrete values, typically 0 and pi, to represent binary symbols. In coherent BPSK, a bit 0 is mapped to a carrier phase of 0 and bit 1 to a carrier phase of pi. The transmitted signal over a symbol interval T can be written as s(t) = A a_k cos(2π f_c t) for t in kT to (k+1)T, where a_k ∈ {+1, -1} and A is the amplitude. Thus each symbol conveys one bit.

Detection in coherent BPSK uses a local reference carrier and correlator to recover the sign of a_k. The received signal is mixed with a locally generated carrier and integrated over the symbol interval; the decision is +1 if the result is positive and -1 otherwise. BPSK can also be implemented non-coherently via differential encoding (DBPSK) to avoid the need for carrier phase recovery, at the cost of performance.

Performance and characteristics: BPSK is robust to additive white Gaussian noise and has a simple optimal detector under coherent conditions. The bit-error rate in AWGN is BER = Q(√(2 Eb/N0)), where Eb is the energy per bit and N0/2 is the single-sided noise spectral density. The scheme is bandwidth-efficient for a binary modulation and offers good power efficiency compared with higher-order modulations, though its spectral efficiency is lower than that of schemes with more than two constellation points.

Applications: BPSK is widely used in communications where reliability under noise is important, including satellite and space communications, RF links, and optical coherent systems. It also serves as a baseline for comparing other modulation formats and as the basis for differential schemes such as DBPSK.