Phase Shift Keying (PSK)

Theory:

Phase Shift Keying (PSK) is a digital modulation technique in which information is transmitted by changing the phase of a carrier signal. The simplest form, Binary PSK (BPSK), uses two phases: one phase for binary '1' and the opposite phase (180° shift) for binary '0'. Each change in input data directly shifts the phase of the carrier.

PSK Transmitter:

In BPSK:

\[ s(t) = \begin{cases} A_c \cos(2\pi f_c t), & \text{for binary 1} \\ A_c \cos(2\pi f_c t + \pi) = -A_c \cos(2\pi f_c t), & \text{for binary 0} \end{cases} \]

Fig 1: BPSK Transmitter

PSK Receiver:

BPSK can be demodulated using a coherent receiver. The received signal is multiplied by a locally generated carrier and integrated over the bit duration \(T_b\).

Fig 2: Coherent BPSK Receiver

In a coherent Binary Phase Shift Keying (BPSK) receiver, the incoming signal is first multiplied by a locally generated carrier signal that is synchronized in frequency and phase with the transmitter. This process is known as coherent detection. The multiplication shifts the signal to baseband, producing a component proportional to the transmitted data along with high-frequency terms.

The output of the multiplier is then passed through a Low-Pass Filter (LPF), which removes the high-frequency components and retains the baseband signal. The resulting signal is a positive or negative voltage level depending on whether a '1' or '0' was transmitted.

Finally, the filtered signal is fed into a threshold comparator. If the input voltage is greater than zero, the receiver decides that a binary '1' was transmitted. If the voltage is less than zero, it decides a binary '0'. This decision process converts the analog signal back into digital data.

Constellation Diagram of BPSK

Fig 3: Constellation Diagram of BPSK

BPSK uses a single orthonormal basis function:

\[ \phi_1(t) = \sqrt{\frac{2}{T_b}} \cos(2\pi f_c t), \quad 0 \le t \le T_b \]

• Binary '1': \( s_1(t) = \sqrt{E_b} \, \phi_1(t) \)
• Binary '0': \( s_2(t) = -\sqrt{E_b} \, \phi_1(t) \)

Energy of both symbols: \( E_b \)
Distance between points: \( d_{12} = 2\sqrt{E_b} \)
The constellation lies on a single axis (in-phase), since BPSK only varies phase.

Effect of Noise

BPSK with AWGN:
The received signal:
\( y(t) = s(t) + n(t) \)
where \( n(t) \) is additive white Gaussian noise. Noise can cause phase shifts, leading to bit errors. Detection reliability depends on SNR.

BPSK with Rayleigh Fading:
The received signal:
\( y(t) = h \cdot s(t) + n(t) \)
where \( h \) is a complex fading coefficient. Fading introduces random amplitude and phase changes, affecting symbol detection. Robust demodulation requires considering channel variations and SNR.

Bit Error Rate (BER) and the Q-function

For BPSK under AWGN with coherent detection, the theoretical BER is expressed using the Gaussian Q-function:
\( Q(x) = \frac{1}{\sqrt{2\pi}} \int_{x}^{\infty} e^{-t^2/2} \, dt \)
This is simply the probability that a standard normal random variable exceeds \( x \) -- it is what determines how likely noise is to push a received symbol across the decision boundary and cause a bit error.

The Q-function is closely related to the complementary error function, \( \text{erfc} \), which appears in some textbooks instead:
\( Q(x) = \frac{1}{2} \, \text{erfc}\left(\frac{x}{\sqrt{2}}\right) \)
Both forms describe the same underlying probability; which one is used is mostly a matter of textbook convention.

For BPSK specifically, the bit error probability works out to:
\( P_b = Q\left(\sqrt{\frac{2E_b}{N_0}}\right) \)
where \( E_b \) is the energy per bit and \( N_0 \) is the noise power spectral density. As \( E_b/N_0 \) (the SNR per bit) increases, \( P_b \) decreases -- higher signal power relative to noise means fewer bit errors.