Phase Shift Keying (PSK)
Theoretical Framework:
Phase Shift Keying (PSK) is a digital modulation paradigm where information is encoded by modulating the instantaneous phase of a continuous-wave carrier. In Binary PSK (BPSK), the most fundamental implementation, the bitstream is mapped onto two discrete phases separated by $\pi$ radians—typically representing binary '1' and '0'—resulting in an antipodal signaling scheme where each state change induces a phase reversal in the carrier.
BPSK Transmitter Synthesis:
The analytic representation of a BPSK signal $s(t)$ is defined as:
\[ s(t) = \begin{cases} A_c \cos(2\pi f_c t), & \text{for symbol } m_1 \\ A_c \cos(2\pi f_c t + \pi) = -A_c \cos(2\pi f_c t), & \text{for symbol } m_0 \end{cases} \]
Interactive BPSK Simulator
Coherent Demodulation:
Recovery of the baseband information from a BPSK signal necessitates a coherent receiver. The observed signal is cross-correlated with a locally synthesized carrier that maintains phase-synchronicity with the transmitter.
Signal Space Representation (Constellation)
Fig 3: Signal Constellation for BPSK
BPSK signaling resides in a one-dimensional vector space spanned by the orthonormal basis function:
\[ \phi_1(t) = \sqrt{\frac{2}{T_b}} \cos(2\pi f_c t), \quad 0 \le t \le T_b \]
Mean symbol energy: \( E_b \)
Euclidean distance: \( d_{12} = 2\sqrt{E_b} \)
The constellation is confined to the in-phase (I) axis, reflecting the strictly phase-dependent nature of the modulation.