Phase Modulation and PSK Techniques

The transition from QPSK to 8-Phase Shift Keying (8-PSK) represents a critical inflection point in digital modulation theory. By increasing from 4 to 8 phase states, we gain 50% more spectral efficiency (3 bits per symbol vs. 2). But this gain comes at a significant cost: the constellation points crowd closer together, demanding substantially more transmit power to maintain the same error rate.

8-PSK sits at a fascinating crossroads in modulation design. It's the last pure PSK scheme that's commonly deployed in practical systems; beyond 8-PSK, the power penalties become so severe that hybrid schemes like QAM (which modulate both phase and amplitude) become more attractive. Understanding 8-PSK deeply illuminates why modulation design is fundamentally about navigating the trade-off between spectral efficiency and power efficiency.

8-Phase Shift Keying (8-PSK) uses eight equally-spaced phase states to encode 3 bits per symbol, providing a spectral efficiency of 3 bits/s/Hz.

The 8-PSK Signal:

$$s_i(t) = \sqrt{\frac{2E_s}{T_s}} \cos\left(2\pi f_c t + \frac{(2i-1)\pi}{8}\right), \quad i = 1, 2, \ldots, 8$$

Alternatively, with phases at the natural positions {0°, 45°, 90°, 135°, 180°, 225°, 270°, 315°}:

$$s_i(t) = \sqrt{\frac{2E_s}{T_s}} \cos\left(2\pi f_c t + \frac{(i-1)\pi}{4}\right), \quad i = 1, 2, \ldots, 8$$

I/Q Representation:

$$s_i(t) = \sqrt{E_s}\cos\left(\frac{(i-1)\pi}{4}\right)\cdot\psi_1(t) + \sqrt{E_s}\sin\left(\frac{(i-1)\pi}{4}\right)\cdot\psi_2(t)$$

Where ψ_1(t) and ψ_2(t) are the orthonormal basis functions (cos and -sin).

Gray coding ensures that adjacent constellation points differ by only one bit. This is crucial for minimizing bit errors since most symbol errors result in detection of an adjacent symbol.

8-PSK Gray Code Assignment:

One standard Gray coding arrangement:

Verification: Check that adjacent entries differ by exactly one bit:

• 000 → 001: differs in bit 0 ✓
• 001 → 011: differs in bit 1 ✓
• 011 → 010: differs in bit 0 ✓
• 010 → 110: differs in bit 2 ✓
• ... and so on (including wrap-around 100 → 000: differs in bit 2 ✓)

Alternative Gray Code:

Another valid Gray coding (rotated by 22.5°):

This rotation places decision boundaries on the cardinal and diagonal axes, which can simplify implementation in some architectures.

Bit-to-Symbol Mapping Process:

1. Group incoming bits into tribits (3-bit groups)
2. Look up the tribit in the Gray code table
3. Map to corresponding phase angle
4. Generate I = √Es·cos(φ) and Q = √Es·sin(φ)

The minimum distance between constellation points is the fundamental parameter determining error performance. For 8-PSK, the reduced angular separation compared to QPSK leads to smaller minimum distance and higher error rates.

Minimum Distance Calculation:

For M-PSK, the minimum distance between adjacent points is:

$$d_{min} = 2\sqrt{E_s} \sin\left(\frac{\pi}{M}\right)$$

For 8-PSK (M = 8):

$$d_{min}^{8PSK} = 2\sqrt{E_s} \sin\left(\frac{\pi}{8}\right) = 2\sqrt{E_s} \times 0.3827 = 0.765\sqrt{E_s}$$

Comparison to QPSK:

For QPSK (M = 4): $$d_{min}^{QPSK} = 2\sqrt{E_s} \sin\left(\frac{\pi}{4}\right) = 2\sqrt{E_s} \times 0.707 = 1.414\sqrt{E_s}$$

Ratio: $$\frac{d_{min}^{8PSK}}{d_{min}^{QPSK}} = \frac{\sin(\pi/8)}{\sin(\pi/4)} = \frac{0.3827}{0.707} = 0.541$$

8-PSK's minimum distance is only 54.1% of QPSK's minimum distance for the same symbol energy.

Decision Regions:

The optimal detector divides the signal space into 8 decision regions, each a 45° "pie slice" centered on a constellation point:

Detection Algorithm:

1. Compute received angle: θ = atan2(Q_rx, I_rx)
2. Add 22.5° to center the regions on the boundaries
3. Divide by 45° and floor: index = floor((θ + 22.5°) / 45°) mod 8
4. Look up Gray code for the detected symbol index

The symbol and bit error probabilities for 8-PSK can be derived using the geometry of the decision regions and the Gaussian noise distribution.

Symbol Error Probability:

For M-PSK with coherent detection, the exact symbol error probability involves integration over the decision region. A close approximation (excellent for M ≥ 4 and moderate-to-high SNR):

$$P_s \approx 2Q\left(\sqrt{\frac{2E_s}{N_0}} \sin\left(\frac{\pi}{M}\right)\right) = 2Q\left(\frac{d_{min}}{\sqrt{2N_0}}\right)$$

For 8-PSK:

$$P_s^{8PSK} \approx 2Q\left(\sqrt{\frac{2E_s}{N_0}} \sin\left(\frac{\pi}{8}\right)\right) = 2Q\left(\sqrt{0.293 \cdot \frac{E_s}{N_0}}\right)$$

Since E_s = 3E_b for 8-PSK (3 bits per symbol):

$$P_s^{8PSK} \approx 2Q\left(\sqrt{0.879 \cdot \frac{E_b}{N_0}}\right)$$

Bit Error Probability (Gray-coded):

$$P_b^{8PSK} \approx \frac{2}{3}Q\left(\sqrt{0.879 \cdot \frac{E_b}{N_0}}\right) \approx \frac{2}{3}Q\left(0.937\sqrt{\frac{E_b}{N_0}}\right)$$

Numerical BER Values for 8-PSK:

Required E_b/N_0 for Target BER:

When designing an 8-symbol constellation, we have a choice: arrange 8 points on a circle (8-PSK) or allow amplitude variation (8-QAM). This comparison illuminates why QAM increasingly dominates at higher modulation orders.

8-QAM Constellation Options:

1. Rectangular 8-QAM: Remove 4 corners from 16-QAM; non-uniform power per symbol
2. Star 8-QAM: Two concentric 4-point squares; average power = geometric mean
3. Circular 8-QAM: Same as 8-PSK (constant envelope)

The ideal 8-point constellation (maximizing d_min for fixed average energy) is neither a circle nor a rectangle—it's a specific pattern that must be computed numerically. However, rectangular constellations are favored for their implementation simplicity.

The Crossover Point:

8-PSK remains competitive with 8-QAM because:

1. 8 points on a circle can still achieve reasonable spacing
2. The constant envelope benefit is significant in power-constrained systems
3. The QAM advantage at M=8 is relatively small (~0.5-1 dB)

However, at M=16 and beyond, QAM's geometric advantage becomes decisive:

Implementing 8-PSK requires careful attention to the additional complexity compared to QPSK, particularly in the symbol mapping and detection stages.

8-PSK Modulator Architecture:

                  ┌─────────────┐
Bit Stream ──────►│ Serial-to-  │──► (b₀, b₁, b₂)
                  │  Parallel   │
                  └─────────────┘
                        │
                        ▼
                  ┌─────────────┐
                  │   Symbol    │──► (I_k, Q_k)
                  │   Mapper    │      8 possible values
                  └─────────────┘
                        │
                        ▼
                  ┌─────────────┐
                  │   Pulse     │──► I(t), Q(t)
                  │   Shaping   │
                  └─────────────┘
                        │
                        ▼
                  ┌─────────────┐
                  │    I/Q      │──► RF Output
                  │ Upconverter │
                  └─────────────┘

Symbol Mapper Implementation:

The mapper is typically a lookup table (ROM) storing the 8 (I, Q) pairs:

Address (3 bits) → (I[N bits], Q[N bits])
000 → (1.000, 0.000)      // 0°
001 → (0.707, 0.707)      // 45°
011 → (0.000, 1.000)      // 90°
010 → (-0.707, 0.707)     // 135°
... (Gray coded)

8-PSK Demodulator Architecture:

RF Input ──► [I/Q Downconverter] ──► I_rx(t), Q_rx(t)
                    │
                    ▼
            [Matched Filter + Sample] ──► (I_k, Q_k)
                    │
                    ▼
            [Phase Detector] ──► θ_k = atan2(Q_k, I_k)
                    │
                    ▼
            [Quantize to 8 Levels] ──► symbol index
                    │
                    ▼
            [Gray Decode] ──► 3 bits
                    │
                    ▼
            [Parallel-to-Serial] ──► Bit Stream

Phase Detection Methods:

1. CORDIC algorithm: Iterative coordinate rotation; hardware-efficient
2. Lookup table: Store atan2 values in ROM indexed by quantized (I, Q)
3. Linear approximation: For rough phase, with refinement
4. Comparison: For each of 8 phases, compute correlation; select maximum

8-PSK finds its niche in systems where bandwidth is at a premium but power is not the dominant constraint. Its constant-envelope property also makes it attractive for systems with non-linear amplification.

Case Study: DVB-S2 Modulation Options

The DVB-S2 satellite standard illustrates adaptive modulation across PSK and QAM:

8-PSK in DVB-S2:

• Used with LDPC codes at rates 3/5, 2/3, 3/4, 5/6, 8/9, 9/10
• Typical application: live video contribution feeds where latency matters
• Power margin: approximately 2 dB above QPSK threshold
• Gain over QPSK: 50% more bits/Hz when power allows

When to Choose 8-PSK:

1. Bandwidth is expensive, power is not — Satellite transponder lease costs
2. Non-linear amplifier required — Constant envelope property
3. Upgrade existing links — Same bandwidth, 50% more throughput
4. Intermediate SNR conditions — Too good for QPSK, not enough for 16-QAM

We've comprehensively explored 8-PSK as the threshold modulation scheme that illustrates the fundamental trade-offs between spectral efficiency and power efficiency. Let's consolidate the key insights:

What's Next:

The final page of this module explores Phase Modulation Applications—a comprehensive survey of how PM and PSK techniques are deployed across the full spectrum of modern communication systems, from cellular networks to satellite links, from Wi-Fi to deep-space communications. We'll see how the principles developed in this module translate into real-world engineering decisions.