QPSK (Quadrature Phase Shift Keying)

QPSK: Doubling Efficiency Without Extra Power

Quadrature Phase Shift Keying is one of the most widely-used modulation schemes in modern communication, employed in everything from satellite TV broadcasts to GPS navigation and 4G/5G cellular networks. Its popularity stems from a remarkable property: QPSK transmits twice the data rate of BPSK using the same bandwidth and the same power per bit. This doubling of spectral efficiency at no power cost makes QPSK the baseline modulation for virtually all modern wireless standards.

The Key Insight: Orthogonal Carriers

To understand why QPSK achieves this "free lunch," consider two key mathematical facts:

1. cos(2πfct) and sin(2πfct) are orthogonal — their product, integrated over one period, equals zero
2. Two orthogonal signals can coexist on the same frequency without interfering

QPSK exploits this by transmitting two independent BPSK signals simultaneously:

• I-channel (In-phase): BPSK on cos(2πfct) — carries the odd-numbered bits
• Q-channel (Quadrature): BPSK on sin(2πfct) — carries the even-numbered bits

Think of it this way: imagine a highway with two separate lanes going in the same direction but perfectly separated by a barrier. Each lane carries independent traffic (data). QPSK's I and Q channels are those two "lanes" on the same frequency, separated by mathematical orthogonality rather than physical barriers.

Mathematical Representation

The QPSK signal for the kth symbol period:

s(t) = (1/√2) × [dI(k) × cos(2πfct) - dQ(k) × sin(2πfct)]

Where dI(k) and dQ(k) are each ±1 (the I and Q data bits for symbol k).

Equivalently, combining using trigonometric identity:

s(t) = A × cos(2πfct + θk)

Where θk takes one of four values: π/4, 3π/4, 5π/4, 7π/4 (or equivalently 45°, 135°, 225°, 315°).

Mapping (Gray coded):

Gray coding ensures adjacent constellation points differ by only one bit, so the most likely errors (to a neighboring point) cause only one bit error, not two.

QPSK = Two Parallel BPSK Systems

This equivalence is crucial for understanding QPSK performance:

Symbol rate: Rs = Rb/2 (each symbol carries 2 bits) Bandwidth: BW = Rs × (1+α) = Rb(1+α)/2 (half the bandwidth of BPSK at same bit rate) Spectral efficiency: 2/(1+α) bits/s/Hz (twice BPSK)

Power performance: Since each quadrature channel is an independent BPSK system:

• BER per bit = Q(√(2Eb/N₀)) — identical to BPSK!
• Same Eb/N₀ requirement as BPSK

This is the "miracle" of QPSK: double the efficiency, same power per bit. The reason is that while total transmitted power doubles (two BPSK systems), the bit rate also doubles, so energy per bit stays constant.

QPSK Modulator

Implementation steps:

1. Serial-to-parallel: Split incoming bit stream into even bits (I) and odd bits (Q)
2. NRZ mapping: Convert 0→-1, 1→+1 for each branch
3. Pulse shaping: Apply root-raised-cosine filter (controls bandwidth)
4. I modulation: Multiply I data by cos(2πfct)
5. Q modulation: Multiply Q data by -sin(2πfct)
6. Summation: Add I and Q modulated signals

The result is a constant-envelope signal (all four phase states have equal amplitude) with phase transitions every Ts = 2Tb seconds.

QPSK Demodulator

Coherent demodulation:

1. I-branch: Multiply received signal by cos(2πfct), low-pass filter, sample, threshold
2. Q-branch: Multiply received signal by -sin(2πfct), low-pass filter, sample, threshold
3. Parallel-to-serial: Recombine I and Q decisions into original bit stream

Critical requirement: The local oscillator must be phase-locked to the transmitted carrier. A Costas loop or pilot-aided synchronization provides this reference.

The 180° Phase Jump Problem

In standard QPSK, when both bits change simultaneously (e.g., 00→11), the phase jumps by 180°. This causes the signal envelope to pass through zero, creating:

• Large amplitude fluctuations
• Spectral regrowth after non-linear amplification
• Problems with Class C amplifiers and hard-limiting satellite transponders

Offset QPSK (OQPSK)

OQPSK solves the 180° phase jump by offsetting the Q-channel by half a symbol period (Tb):

• I-channel transitions occur at t = 0, 2Tb, 4Tb, ...
• Q-channel transitions occur at t = Tb, 3Tb, 5Tb, ...

Since I and Q never change simultaneously, maximum phase change is limited to 90° (never 180°). This reduces envelope fluctuations significantly.

Performance: Same BER as QPSK (no performance penalty) Bandwidth: Same as QPSK Benefit: Better suited to non-linear channels and satellite communication

π/4-QPSK

π/4-QPSK rotates the constellation by 45° on alternate symbols. The phase change between consecutive symbols is limited to ±45° or ±135° (never 0° or 180°):

Advantages:

• Can be differentially detected (no coherent carrier needed)
• Maximum phase transition is 135° (less than QPSK's 180°)
• Better envelope properties than standard QPSK

Applications: Used in IS-136 (North American TDMA cellular), TETRA (European digital trunked radio), and Japanese PDC systems.

QPSK in Practice: Real Systems

GPS: The GPS C/A code uses BPSK at 1.023 Mbps, but modernized GPS (L2C, L5) uses QPSK to double data capacity on the same bandwidth.

DVB-S (Satellite TV): QPSK at symbol rates up to 45 Msymbols/s delivers standard and high-definition television to millions of satellite dishes worldwide.

4G LTE: QPSK is the baseline modulation — used when channel conditions are poor. As conditions improve, the system upgrades to 16-QAM, 64-QAM, or 256-QAM.

Deep Space: NASA uses QPSK for Mars orbiters and other deep-space missions where every bit of spectral efficiency matters.

CDMA/WCDMA: Spreading codes are modulated onto I and Q channels using QPSK principles.

Error Performance Summary

In AWGN:

• BER = Q(√(2Eb/N₀)) — identical to BPSK
• At BER = 10⁻⁵: requires Eb/N₀ = 9.6 dB
• At BER = 10⁻⁶: requires Eb/N₀ = 10.5 dB

In Rayleigh fading:

• BER ≈ 1/(4Eb/N₀) for high SNR
• Much worse than AWGN — diversity techniques essential

With differential detection (DQPSK):

• BER ≈ 2 × Q(√(2Eb/N₀) × sin(π/4)) — approximately 2.3 dB penalty vs. coherent

Key Takeaways

1. QPSK transmits 2 bits per symbol using two orthogonal carriers (I and Q), doubling spectral efficiency over BPSK at the same Eb/N₀.
2. BER performance is identical to BPSK: Q(√(2Eb/N₀)), because each quadrature channel operates independently.
3. Gray coding limits adjacent-point errors to single bit errors, optimizing overall BER.
4. OQPSK eliminates 180° phase transitions by offsetting Q-channel timing, improving performance through non-linear channels.
5. π/4-QPSK enables differential detection while limiting phase transitions, useful in mobile fading environments.
6. QPSK is the baseline modulation for GPS, satellite TV, 4G/5G cellular, and deep-space communication — the universal starting point for spectral-efficient digital transmission.