BPSK Modulation

Binary Phase Shift Keying is the most power-efficient binary modulation scheme and forms the foundation of all higher-order PSK and QAM constellations. BPSK encodes data in the phase of the carrier, and its BER formula Q(sqrt(2Eb/N0)) is the benchmark that every other scheme is compared against. Mastery of BPSK signal space geometry is essential for solving GATE problems involving constellation diagrams and BER analysis.

Core Concept: Phase as Information Carrier

In Binary Phase Shift Keying (BPSK), bit 1 is mapped to s1(t) = Ac cos(2πfct) and bit 0 to s0(t) = Ac cos(2πfct + π) = -Ac cos(2πfct). The two waveforms are antipodal: they are exact negatives of each other. This antipodal nature is what makes BPSK the most power-efficient binary scheme. The amplitude is constant and the frequency is constant; only the phase toggles between 0 and 180 degrees.

In the signal space (or constellation diagram) representation, each signal is a point on the real axis. s1 maps to +sqrt(Eb) and s0 maps to -sqrt(Eb), where Eb = Ac²Tb/2 is the energy per bit. The Euclidean distance between the two symbols is d = 2 sqrt(Eb). This is the largest possible distance for a two-symbol constellation of fixed energy, which directly produces the minimum BER among all binary schemes.

Mathematical Expression

The bit error rate for coherent BPSK is derived from the signal space geometry. The decision threshold is at 0. The noise component along the signal axis is Gaussian with zero mean and variance N0/2. The probability that noise pushes the received symbol past the threshold is:

Pb = Q(d / sqrt(2 × N0/2)) = Q(2 sqrt(Eb) / sqrt(N0)) = Q(sqrt(2Eb/N0))

where Q(x) = (1/sqrt(2π)) integral from x to infinity of exp(-t²/2) dt is the Q-function. At Eb/N0 = 9.6 dB, the BER of BPSK is approximately 10⁻⁵. The bandwidth is BT = 2Rb for rectangular pulses and BT = Rb(1+α) for raised cosine filtering.

Practical Understanding

BPSK is used wherever the link budget is tight and power efficiency is the top priority. GPS signals (L1 and L2 bands), deep space probes (Voyager, Cassini), and the downlink of DVB-S satellite systems all use BPSK or its differential variant DBPSK. DBPSK does not require absolute phase reference but has about 1 dB worse BER performance compared to coherent BPSK.

BPSK is also the building block of QPSK. QPSK transmits two independent BPSK streams on cosine and sine carriers simultaneously, doubling the spectral efficiency with no BER penalty compared to BPSK when measured against Eb/N0. Understanding BPSK signal space is therefore the necessary prerequisite to understanding QPSK and higher-order constellations.

Given:
Carrier frequency fc = 1 MHz, Bit rate Rb = 100 kbps, Ac = 1 V, Eb/N0 = 8.5 dB

Why this formula applies:
Coherent BPSK uses antipodal signaling with maximum symbol distance, giving the optimal binary BER formula.

Formula:
Pb = Q(sqrt(2 Eb/N0))
Eb = Ac^2 * Tb / 2 = Ac^2 / (2 Rb)
Bandwidth BT = 2 Rb

Substitution:
Eb/N0 (linear) = 10^(8.5/10) = 7.079
2 Eb/N0 = 14.16
sqrt(14.16) = 3.763
BT = 2 × 100 kbps = 200 kHz

Calculation:
Pb = Q(3.763)
Using Q-function table: Q(3.763) ≈ 8.4 × 10^-5

Final Answer with units:
BER ≈ 8.4 × 10^-5, Transmission bandwidth = 200 kHz

Exam Tip: Remember the exact BER formula for BPSK is Q(sqrt(2Eb/N0)), not Q(sqrt(Eb/N0)). The factor of 2 comes from antipodal signaling where d = 2sqrt(Eb). If the question gives SNR (not Eb/N0), use Pb = Q(sqrt(2 SNR × BW/Rb)) — be careful about what ratio is given.

• The received signal r(t) = ±Ac cos(2πfct) + n(t) is multiplied by the local carrier and integrated over Tb. The sign of the output determines the decoded bit.
• The integrator output when s1(t) is sent equals +Eb with additive Gaussian noise. When s0(t) is sent, the output equals -Eb plus noise. The threshold is zero.
• An error occurs when noise magnitude exceeds sqrt(Eb), giving Pb = Q(sqrt(2Eb/N0)) since the SNR at the integrator output is 2Eb/N0.
• BPSK consistently outperforms coherent OOK and coherent BFSK by 3 dB for the same BER, as seen from the BER curves. This advantage comes purely from the antipodal (maximum distance) symbol arrangement.

Quick Revision

• BPSK: s1(t) = Ac cos(2πfct) for bit 1, s0(t) = -Ac cos(2πfct) for bit 0.
• BER: Pb = Q(sqrt(2Eb/N0)). This is the best BER achievable for any binary scheme at given Eb/N0.
• Signal space: s1 = +sqrt(Eb), s0 = -sqrt(Eb) on 1D axis. Decision threshold = 0.
• Symbol distance d = 2 sqrt(Eb). Larger distance means lower BER. No other binary scheme can do better.
• Bandwidth: BT = 2Rb (null-to-null rectangular), BT = Rb(1+α) (raised cosine).
• BPSK is 3 dB better than OOK and BFSK. Coherent detection required — no non-coherent BPSK equivalent.
• Common trap: If given SNR = S/N (not Eb/N0), must convert using Eb/N0 = SNR × (B/Rb) before applying BER formula.