Phase Modulation and PSK Techniques

In the vast ecosystem of digital modulation schemes, two techniques stand out for their elegant simplicity, robust performance, and near-universal deployment: Binary Phase Shift Keying (BPSK) and Quadrature Phase Shift Keying (QPSK). These schemes form the backbone of virtually every major communication standard—from deep-space missions transmitting precious scientific data across billions of kilometers to the Wi-Fi networks connecting billions of devices worldwide.

BPSK and QPSK share a remarkable property: they achieve optimal or near-optimal bit error performance among all digital modulation schemes at their respective spectral efficiencies. Understanding these two schemes deeply provides the foundation for understanding all other modulation techniques.

Binary Phase Shift Keying (BPSK) is the simplest form of PSK, using only two phase states separated by 180° to represent binary data (0 and 1).

BPSK Signal Definition:

$$s(t) = \sqrt{\frac{2E_b}{T_b}} \cos(2\pi f_c t + \phi_i), \quad 0 \leq t \leq T_b$$

Where:

• $\phi_i = 0$ for bit '1' (or binary 0, depending on convention)
• $\phi_i = \pi$ for bit '0' (or binary 1)

Alternatively, using polar representation:

$$s(t) = \pm\sqrt{\frac{2E_b}{T_b}} \cos(2\pi f_c t) = d_k \sqrt{\frac{2E_b}{T_b}} \cos(2\pi f_c t)$$

Where $d_k \in {+1, -1}$ represents the k-th bit.

BPSK Constellation Diagram:

The BPSK constellation consists of two points on the real (I) axis of the complex plane:

The signal space is one-dimensional despite the two basis functions; the Q component is always zero.

Minimum Distance: $$d_{min} = 2\sqrt{E_b}$$

This is the maximum possible distance for two signals with total energy E_b each, making BPSK optimally robust against Gaussian noise.

Decision Boundary:

The optimal detector uses a simple threshold at zero: if the matched filter output > 0, decide '0'; otherwise, decide '1'. The decision regions are:

• Bit '0': I > 0 (right half-plane)
• Bit '1': I < 0 (left half-plane)

BPSK's error performance in AWGN channels can be derived exactly, providing a foundational result against which all other modulation schemes are compared.

The Decision Problem:

At the matched filter output (sampled at symbol time), we receive:

$$r = \pm\sqrt{E_b} + n$$

Where n is a Gaussian random variable with mean 0 and variance N_0/2 (single-sided noise spectral density).

Probability of Error Derivation:

Assume bit '0' was sent (received signal = +√E_b + n). An error occurs if r < 0:

$$P(error|'0' sent) = P(n < -\sqrt{E_b}) = P\left(\frac{n}{\sqrt{N_0/2}} < -\sqrt{\frac{2E_b}{N_0}}\right)$$

Since n/√(N_0/2) is a standard normal random variable:

$$P_b = Q\left(\sqrt{\frac{2E_b}{N_0}}\right)$$

Where Q(x) is the Q-function: $Q(x) = \frac{1}{\sqrt{2\pi}}\int_x^{\infty}e^{-t^2/2}dt$

Numerical BER Values:

Key Observations:

• At E_b/N_0 = 9.6 dB, BER ≈ 10⁻⁵ (a common reference point)
• Each 3 dB increase in E_b/N_0 roughly decreases BER by 10×
• Very low BER (10⁻⁹ for fiber-class links) requires E_b/N_0 ≈ 12 dB

Quadrature Phase Shift Keying (QPSK) uses four phase states to encode two bits per symbol, doubling spectral efficiency compared to BPSK while maintaining the same bit error performance.

QPSK Signal Definition:

$$s(t) = \sqrt{\frac{2E_s}{T_s}} \cos\left(2\pi f_c t + \frac{(2i-1)\pi}{4}\right), \quad i = 1, 2, 3, 4$$

Alternatively, using I/Q representation:

$$s(t) = \sqrt{\frac{E_s}{2}} \left[I_k \cdot \sqrt{\frac{2}{T_s}} \cos(2\pi f_c t) - Q_k \cdot \sqrt{\frac{2}{T_s}} \sin(2\pi f_c t)\right]$$

Where $(I_k, Q_k) \in {(+1,+1), (-1,+1), (-1,-1), (+1,-1)}$

QPSK Constellation Diagram:

(Normalization: I, Q ∈ {±1} for simplicity; actual amplitude is √(Es/2) per component)

Gray Coding Benefit:

With Gray coding, adjacent symbols differ by only one bit. Since most errors cause detection of an adjacent symbol, Gray coding nearly halves the bit error rate compared to natural binary mapping.

Signal Space Representation:

The four QPSK symbols form a square constellation at distance √Es from the origin:

• Symbol 1 (00): (√(Es/2), √(Es/2))
• Symbol 2 (01): (-√(Es/2), √(Es/2))
• Symbol 3 (11): (-√(Es/2), -√(Es/2))
• Symbol 4 (10): (√(Es/2), -√(Es/2))

One of the most remarkable results in digital communications is that QPSK achieves the same bit error rate as BPSK despite transmitting twice the data in the same bandwidth. This seeming impossibility is resolved by understanding the orthogonality of the I and Q channels.

Derivation of QPSK BER:

Step 1: View QPSK as two orthogonal BPSK streams

The I-channel transmits one bit (at ±√(Es/2)) and the Q-channel transmits another bit (at ±√(Es/2)). Since cos and sin are orthogonal:

• Noise on I-channel is independent of noise on Q-channel
• Detection of I-bit is independent of Q-bit

Step 2: Energy per bit

For QPSK: Es = 2·Eb (two bits per symbol)

Energy per channel: Es/2 = Eb

Each "mini-BPSK" stream has bit energy Eb.

Step 3: Bit error probability

Since each bit is effectively a BPSK transmission:

$$P_b^{QPSK} = Q\left(\sqrt{\frac{2E_b}{N_0}}\right)$$

This is identical to BPSK!

Symbol Error Rate:

A symbol error occurs when either the I-bit or Q-bit (or both) is detected incorrectly:

$$P_s = 1 - (1-P_b)^2 = 2P_b - P_b^2 \approx 2P_b \text{ (for small } P_b\text{)}$$

Comparison Table:

Offset QPSK (OQPSK), also known as Staggered QPSK (SQPSK), is a variant of QPSK designed to address a critical implementation problem: the 180° phase transitions that stress non-linear power amplifiers.

The 180° Phase Transition Problem:

In conventional QPSK, when both bits change simultaneously (e.g., 00→11), the phase changes by 180°. This instantaneous phase reversal causes:

• The signal envelope to pass through zero (for rectangular pulses)
• Spectral regrowth (sidelobe regeneration) after non-linear amplification
• Reduced power amplifier efficiency (must back off from saturation)

OQPSK Solution:

Offset the Q-channel by half a symbol period (T_s/2) relative to the I-channel:

$$s_{OQPSK}(t) = I_k \cdot p(t - kT_s)\cos(2\pi f_c t) - Q_k \cdot p(t - kT_s - T_s/2)\sin(2\pi f_c t)$$

Now, I and Q bits never change at the same instant. Maximum phase change per transition is 90° instead of 180°.

OQPSK Applications:

• CDMA systems (IS-95, CDMA2000): OQPSK reduces out-of-band emissions
• Satellite communications: Power efficiency is paramount
• OFDM subcarrier modulation: Some implementations use OQPSK
• Deep-space missions: Power-limited scenarios

Mathematical Identity:

Despite the timing offset, OQPSK has identical spectral shape to QPSK (for the same pulse shape). The timing offset only affects the time-domain envelope, not the frequency-domain power spectral density.

π/4-QPSK (also written as π/4-DQPSK when differential encoding is used) is a variant that alternates between two QPSK constellations rotated 45° apart. It combines advantages of both QPSK and OQPSK while enabling easier differential detection.

Operating Principle:

Instead of choosing from a fixed 4-point constellation, π/4-QPSK alternates between:

• Odd symbols: Standard QPSK at {45°, 135°, 225°, 315°}
• Even symbols: Rotated QPSK at {0°, 90°, 180°, 270°}

Data is encoded in the phase change (±45°, ±135°) from the previous symbol's phase rather than the absolute phase.

Phase Transition Table:

π/4-QPSK Properties:

1. Maximum phase transition: 135° (not 180° as in QPSK, not 90° as in OQPSK)
2. Differential operation natural: Phase changes encode data directly
3. 8-point signal space: But only 4 transitions are valid from any point
4. Coherent or non-coherent detection: Differential encoding enables simpler receivers

BER Performance:

• Coherent detection: Same as QPSK (Pb = Q(√(2Eb/N0)))
• Differential detection: Approximately 2-3 dB penalty vs. coherent

Applications:

• DECT (Digital Enhanced Cordless Telecommunications): European cordless phone standard
• TETRA: Terrestrial trunked radio
• IS-54/IS-136 (TDMA): North American digital cellular
• PDC: Japanese digital cellular standard

Differential Phase Shift Keying encodes information in phase transitions rather than absolute phase states. This eliminates the need for carrier recovery at the receiver, simplifying implementation at the cost of slightly worse error performance.

DBPSK Encoding:

The transmitted phase depends on both the current bit and the previous phase:

$$\phi_k = \phi_{k-1} + \Delta\phi_k$$

Where:

• Δφ_k = 0 if current bit is '0'
• Δφ_k = 180° if current bit is '1'

DBPSK Decoding:

Compare the current symbol's phase to the previous symbol's phase:

• If |φ_k - φ_{k-1}| ≈ 0°: decide '0'
• If |φ_k - φ_{k-1}| ≈ 180°: decide '1'

No absolute phase reference is needed—only the phase difference matters.

DQPSK Encoding:

For differential QPSK, the phase change encodes the dibit:

Error Performance:

$$P_b^{DBPSK} \approx \frac{1}{2}e^{-E_b/N_0}$$ (exact for noncoherent detection)

$$P_b^{DQPSK} \approx Q\left(\sqrt{\frac{2E_b(1-1/\sqrt{2})}{N_0}}\right)$$ (approximate)

Comparison at BER = 10⁻⁵:

BPSK and QPSK are ubiquitous in modern communications, each finding its optimal application domain based on system constraints.

Implementation Considerations:

1. Carrier Frequency Offset Sensitivity:

• BPSK: Most robust (only distinguishing +/- I values)
• QPSK: Sensitive to carrier offset (I/Q rotation causes decision errors)
• Higher-order: Increasingly sensitive

2. Phase Noise Impact:

• BPSK: Tolerant (180° apart is hard to confuse)
• QPSK: Moderate (90° margins)
• Solution: Carrier tracking loops must be designed for expected phase noise

3. Timing Sensitivity:

• Both: Require symbol timing within ~10% of symbol period for low degradation
• Raised-cosine filtering: Makes systems more tolerant of timing error

4. Power Amplifier Considerations:

• BPSK with rectangular pulses: Constant envelope (ideal for non-linear PA)
• QPSK with filtering: Envelope variations require back-off
• OQPSK: Best of both worlds (filtered + near-constant envelope)

We've developed comprehensive understanding of BPSK and QPSK—the foundational modulation schemes upon which virtually all modern digital communications are built. Let's consolidate the key insights:

What's Next:

With BPSK and QPSK thoroughly understood, the next page explores 8-PSK—the first step into higher-order PSK. We'll see how 8-PSK achieves 3 bits per symbol at the cost of reduced noise margin, and why this trade-off becomes increasingly unfavorable as the modulation order rises, eventually leading to QAM for high spectral efficiency.