PSK Modulation

Phase Shift Keying (PSK): Maximum Power Efficiency

Phase Shift Keying is one of the most important digital modulation techniques in modern communications. It encodes information by changing the phase of a carrier signal — keeping amplitude and frequency constant. This constant-envelope property, combined with excellent power efficiency, makes PSK the modulation of choice for satellite communication, deep-space probes, and many wireless standards.

The Core Concept

Think of it this way: imagine a spinning wheel viewed from the side — you see it oscillating back and forth. If someone nudges the wheel to shift its timing (phase) by a specific amount, the oscillation looks identical in speed and height but is offset in time. PSK uses exactly this principle: the carrier wave continues at the same frequency and amplitude, but its starting point (phase) is shifted to carry information.

Binary PSK (BPSK):

s₁(t) = A × cos(2πfct) for bit "1" (phase = 0°) s₀(t) = A × cos(2πfct + π) = -A × cos(2πfct) for bit "0" (phase = 180°)

Notice that BPSK is simply flipping the sign of the carrier. When you transmit a "1," the carrier goes positive at time zero. When you transmit a "0," it goes negative. The two signals are exact mirror images of each other.

Why PSK Beats ASK and FSK in Power Efficiency

The key to PSK's power advantage lies in geometry. In signal space:

• BPSK places constellation points at +√Eb and -√Eb on the real axis
• The distance between them is 2√Eb
• Binary ASK (OOK) places points at 0 and √(2Eb), distance = √(2Eb)

Since error probability depends on the minimum distance between constellation points, and BPSK has distance 2√Eb compared to ASK's √(2Eb), BPSK achieves the same BER with 3 dB less power. This 3 dB advantage is fundamental and cannot be overcome by any receiver design.

BPSK Error Performance

For BPSK in an AWGN channel with optimal coherent detection:

BER = Q(√(2Eb/N₀)) = (1/2) × erfc(√(Eb/N₀))

This is the best achievable performance for any binary modulation scheme — no binary system can do better than BPSK in AWGN.

Practical numbers:

The Constellation Diagram

A constellation diagram plots the signal points in two-dimensional signal space (I-Q plane — In-phase and Quadrature axes):

• BPSK: Two points on the real axis (I-axis), separated by 180°
• QPSK: Four points at ±45°, ±135° (or equivalently at 0°, 90°, 180°, 270°)
• 8-PSK: Eight points equally spaced at 45° intervals around a circle

The constellation diagram reveals everything about a modulation scheme: power (distance from origin), noise margin (distance between nearest points), and spectral efficiency (log₂ of number of points).

BPSK Modulator and Demodulator

Modulator:

1. Binary data stream {0, 1} mapped to {-1, +1}
2. Multiply by carrier: output = (±1) × A×cos(2πfct)
3. This is simply a balanced modulator (multiplier)

Coherent Demodulator:

1. Multiply received signal by local carrier copy: r(t) × cos(2πfct)
2. Low-pass filter: extracts ±A/2 (plus noise)
3. Sample at bit intervals
4. Threshold detector: positive → "1," negative → "0"

The critical challenge is generating the local carrier with correct frequency AND phase. A Costas loop or squaring loop is typically used for carrier recovery.

Differential PSK (DPSK): Avoiding Carrier Recovery

DPSK solves the carrier recovery problem by encoding information in phase *changes* rather than absolute phase:

• "1" → No phase change from previous symbol
• "0" → 180° phase change from previous symbol

The demodulator compares each received symbol with the previous one — no absolute phase reference needed.

DPSK BER: BER = (1/2) × exp(-Eb/N₀)

DPSK requires approximately 1 dB more Eb/N₀ than coherent BPSK for the same BER — a small penalty for greatly simplified receiver design.

M-ary PSK: More Bits Per Symbol

Higher-order PSK places M points equally spaced around a circle:

sᵢ(t) = A × cos(2πfct + 2π(i-1)/M) for i = 1, 2, ..., M

Each symbol carries log₂(M) bits. Spectral efficiency increases, but points get closer together (smaller minimum distance), requiring more power.

Performance comparison (BER = 10⁻⁵):

Notice the remarkable result: QPSK requires the same Eb/N₀ as BPSK while doubling spectral efficiency! This is because QPSK is mathematically equivalent to two independent BPSK signals on orthogonal carriers (I and Q channels).

Beyond 8-PSK, the required power penalty becomes severe. For 16 and higher, QAM (which varies both amplitude and phase) is preferred since it provides better distance properties.

The Phase Ambiguity Problem

Coherent PSK demodulators can lock onto the carrier with a phase offset that is a multiple of the constellation spacing (360°/M). For BPSK, a 180° lock error inverts ALL bits. For QPSK, a 90° error swaps I and Q channels.

Solution: Differential encoding — Encode data as phase differences rather than absolute phases. Gray-coded QPSK with differential encoding limits the damage of phase ambiguity to at most one symbol error.

Applications of PSK

PSK modulation dominates applications where power efficiency is paramount:

1. Deep space communication — NASA's Deep Space Network uses BPSK for Voyager, New Horizons (every fraction of dB matters at billions of km)
2. Satellite TV — DVB-S uses QPSK; DVB-S2 uses 8PSK and higher
3. GPS — BPSK at 1.023 Mbps (C/A code) and 10.23 Mbps (P code)
4. WiFi — 802.11 uses BPSK/QPSK for lower data rates
5. 4G LTE/5G — QPSK as the baseline modulation
6. Military — Spread spectrum systems with BPSK/QPSK

Key Takeaways

1. PSK encodes data in carrier phase while keeping amplitude constant, achieving optimal power efficiency among binary schemes.
2. BPSK achieves BER = Q(√(2Eb/N₀)) — the theoretical best for any binary modulation in AWGN.
3. QPSK doubles BPSK's spectral efficiency with zero power penalty by using two orthogonal carriers.
4. DPSK avoids carrier recovery complexity at a cost of approximately 1 dB in performance.
5. Higher-order M-PSK (beyond 8-PSK) becomes increasingly power-inefficient; QAM is preferred for M > 8.
6. PSK dominates power-limited applications: satellites, deep space, GPS, and as the baseline in all modern wireless standards.