Phase Modulation and PSK Techniques

Phase Shift Keying (PSK) stands as one of the most important modulation techniques in the history of digital communications. From the deep-space probes transmitting data across billions of kilometers to the Wi-Fi router in your home, PSK and its variants form the foundation upon which reliable, efficient digital communication is built.

In this page, we transition from the theoretical foundations of phase modulation to the precise technical specifications of PSK. We will dissect the mathematical formulation, analyze spectral characteristics, quantify error performance, and explore the hardware architectures that bring PSK to life in real-world systems.

M-ary Phase Shift Keying (M-PSK) is a digital modulation scheme where the carrier phase takes one of M distinct values, each representing a unique symbol from an alphabet of M symbols.

The General M-PSK Signal:

$$s_i(t) = \sqrt{\frac{2E_s}{T_s}} \cos\left(2\pi f_c t + \frac{2\pi(i-1)}{M} + \phi_0\right), \quad 0 \leq t \leq T_s$$

For $i = 1, 2, 3, \ldots, M$

Where:

• E_s = Energy per symbol (Joules)
• T_s = Symbol duration (seconds)
• f_c = Carrier frequency (Hz)
• M = Number of phase states (modulation order)
• φ_0 = Initial phase offset (often set to 0 or π/M for Gray coding alignment)

The amplitude coefficient $\sqrt{2E_s/T_s}$ normalizes the signal so that integrating $s_i^2(t)$ over one symbol period yields exactly E_s.

Phase Constellation:

The M phases in M-PSK are equally spaced around the unit circle:

$$\phi_i = \frac{2\pi(i-1)}{M} + \phi_0, \quad i = 1, 2, \ldots, M$$

The angular separation between adjacent phases is:

$$\Delta\phi = \frac{2\pi}{M} = \frac{360°}{M}$$

Critical Observation: As M increases, adjacent symbols become closer together in phase. This reduces the decision region for each symbol, making the system more susceptible to noise-induced errors.

The signal space representation of M-PSK provides geometric insight into modulation performance. By representing each PSK signal as a point in a two-dimensional space, we can visualize detection, analyze noise effects, and calculate error probabilities geometrically.

Orthonormal Basis Functions:

Any M-PSK signal can be expressed as a linear combination of two orthonormal basis functions:

$$\psi_1(t) = \sqrt{\frac{2}{T_s}} \cos(2\pi f_c t), \quad 0 \leq t \leq T_s$$

$$\psi_2(t) = -\sqrt{\frac{2}{T_s}} \sin(2\pi f_c t), \quad 0 \leq t \leq T_s$$

The signal for symbol i can then be written as:

$$s_i(t) = s_{i1} \cdot \psi_1(t) + s_{i2} \cdot \psi_2(t)$$

Where:

• $s_{i1} = \sqrt{E_s} \cos\left(\frac{2\pi(i-1)}{M}\right)$ (I coordinate)
• $s_{i2} = \sqrt{E_s} \sin\left(\frac{2\pi(i-1)}{M}\right)$ (Q coordinate)

Minimum Euclidean Distance:

The minimum distance (d_min) between constellation points is the critical parameter determining error performance. For adjacent symbols in M-PSK:

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

Derivation: Consider two adjacent symbols at angles θ = 0 and θ = 2π/M. Their coordinates are:

• Symbol 1: (√Es, 0)
• Symbol 2: (√Es·cos(2π/M), √Es·sin(2π/M))

Using the Euclidean distance formula and trigonometric identity: $$d = \sqrt{E_s(1-\cos(2\pi/M))^2 + E_s\sin^2(2\pi/M)} = 2\sqrt{E_s}\sin(\pi/M)$$

Key Insight: BPSK and QPSK have the same d_min normalized by bit energy, hence identical BER curves. Higher-order PSK has progressively smaller d_min, leading to worse BER at the same E_b/N_0.

Understanding the spectral properties of PSK is essential for spectrum-efficient system design. The spectrum of a PSK signal depends on the symbol rate, pulse shape, and statistical properties of the data.

Power Spectral Density (PSD):

For rectangular pulses (instantaneous phase transitions), the baseband PSD of M-PSK is:

$$S(f) = E_s T_s \left(\frac{\sin(\pi f T_s)}{\pi f T_s}\right)^2 = E_s T_s \cdot \text{sinc}^2(fT_s)$$

This sinc² shape has:

• Main lobe width: 2/T_s Hz (first nulls at f = ±1/T_s)
• Sidelobe structure: Decaying sidelobes at 20 dB/decade
• 99% power bandwidth: Approximately 10/T_s Hz
• First sidelobe level: -13 dB below main lobe peak

The Symbol Rate Connection:

If R_s = 1/T_s is the symbol rate and R_b is the bit rate:

• For M-PSK: R_s = R_b / log₂(M)
• Main lobe bandwidth: 2R_s = 2R_b / log₂(M)

Spectral Efficiency:

Spectral efficiency η is defined as the ratio of bit rate to bandwidth:

$$\eta = \frac{R_b}{B} \quad \text{(bits/second/Hz)}$$

For M-PSK with Nyquist bandwidth (B = R_s = 1/T_s):

$$\eta = \log_2(M) \quad \text{bits/s/Hz}$$

Practical Bandwidth Definitions:

Different bandwidth measures are used depending on context:

• Null-to-null bandwidth: 2R_s (99% power for rectangular pulses)
• 3 dB bandwidth: R_s (half-power points)
• Occupied bandwidth (99% power): ~10R_s for rectangular pulses; ~R_s for raised-cosine filtered

In practical systems, rectangular pulses (instantaneous phase transitions) are unacceptable because their sinc² spectrum has slowly decaying sidelobes that cause adjacent channel interference. Pulse shaping smoothly transitions between symbols, compacting the spectrum.

The Nyquist Criterion:

For zero inter-symbol interference (ISI), the transmitted pulse p(t) must satisfy:

$$\sum_{k=-\infty}^{\infty} P(f - k/T_s) = T_s$$

This means the frequency-domain pulse, when aliased (folded every R_s Hz), must be perfectly flat. The simplest pulse satisfying this is the sinc function, but it's impractical (infinite duration, sharp filter edges).

Raised-Cosine Spectrum:

$$H_{RC}(f) = \begin{cases} T_s & |f| \leq \frac{1-\alpha}{2T_s} \ \frac{T_s}{2}\left[1 - \sin\left(\frac{\pi T_s}{\alpha}\left(|f| - \frac{1}{2T_s}\right)\right)\right] & \frac{1-\alpha}{2T_s} < |f| \leq \frac{1+\alpha}{2T_s} \ 0 & |f| > \frac{1+\alpha}{2T_s} \end{cases}$$

Bandwidth with Raised-Cosine:

$$B = \frac{(1+\alpha)}{2T_s} = \frac{R_s(1+\alpha)}{2} = \frac{R_b(1+\alpha)}{2\log_2(M)}$$

Root-Raised-Cosine (RRC):

In practice, the raised-cosine response is split between transmitter and receiver (each implements √H_RC). This matched filtering maximizes SNR while ensuring zero ISI:

$$H_{TX}(f) \cdot H_{RX}(f) = H_{RC}(f) \quad \Rightarrow \quad H_{TX} = H_{RX} = \sqrt{H_{RC}}$$

The bit error rate (BER) is the fundamental performance metric for digital communication systems. For M-PSK, BER depends on the modulation order, signal-to-noise ratio, and detection method.

Symbol Error Probability for Coherent M-PSK:

For large M (M ≥ 4), the symbol error probability is approximately:

$$P_s \approx 2Q\left(\sqrt{2E_s/N_0} \cdot \sin(\pi/M)\right)$$

Where Q(x) is the Q-function (complementary cumulative distribution of standard normal).

Bit Error Probability:

Assuming Gray coding, adjacent symbols differ by only one bit. Most symbol errors occur between adjacent symbols, so:

$$P_b \approx \frac{P_s}{\log_2(M)} = \frac{1}{k} \cdot 2Q\left(\sqrt{2kE_b/N_0} \cdot \sin(\pi/M)\right)$$

Where k = log₂(M) is bits per symbol.

Differential PSK (DPSK) Performance:

Differential detection trades implementation simplicity for error performance:

BER Curves Interpretation:

A BER vs. E_b/N_0 (dB) plot reveals:

• All curves have similar "waterfall" shape with steep descent at high SNR
• BPSK and QPSK are identical (both require 2 dB less d²_min/N₀ per doubling of M)
• Curves shift right as M increases (more power needed for same BER)
• At low SNR, curves converge (all systems approach 50% error rate)
• At very high SNR, curves approach zero asymptotically

Implementing a PSK modulator requires converting digital data into precisely phase-shifted carrier signals. Modern implementations use I/Q modulators—the universal architecture for generating any digital modulation.

The I/Q Modulator Block Diagram:

Data Bits → [Symbol Mapper] → (I_k, Q_k) → [Pulse Shaper] → I(t), Q(t)
                                                              ↓       ↓
                                               cos(2πf_c·t) ←[×]     [×]→ -sin(2πf_c·t)
                                                              ↓       ↓
                                                             [    +    ] → s(t) (RF output)

1. Symbol Mapper:

Converts k-bit groups (for 2^k-PSK) into constellation coordinates:

• Input: k bits (e.g., "01" for QPSK)
• Output: (I_k, Q_k) coordinate pair (e.g., (0, +1) for phase = 90°)
• Uses lookup table (ROM) in hardware implementations

2. Pulse Shaping Filter:

Converts discrete symbol values to continuous waveforms:

$$I(t) = \sum_k I_k \cdot p(t - kT_s)$$ $$Q(t) = \sum_k Q_k \cdot p(t - kT_s)$$

Where p(t) is the pulse shape (typically root-raised-cosine). Implementation:

• Upsample symbols to DAC rate (e.g., 8× or 16× oversampling)
• Apply FIR filter with RRC coefficients
• Digital-to-analog conversion

3. Quadrature Upconversion:

The fundamental operation:

$$s(t) = I(t) \cdot \cos(2\pi f_c t) - Q(t) \cdot \sin(2\pi f_c t)$$

This produces the bandpass PSK signal. Implementation considerations:

• Local oscillator (LO) must be stable and low-phase-noise
• I and Q paths must be precisely 90° apart (quadrature accuracy)
• Mixer linearity affects spurious emissions
• DAC resolution affects signal quality (8-14 bits typical)

The PSK demodulator recovers the transmitted bits from the received RF signal. This involves several challenging tasks: frequency and phase synchronization, timing recovery, matched filtering, and symbol detection.

Coherent Demodulator Block Diagram:

RF in →[BPF]→[×]→[LPF]→[Matched Filter]→[Sample @ kTs]→ I_k
                    ↑
              cos(2πf_c·t + θ̂)
                    ↑
              [Carrier Recovery]

RF in →[BPF]→[×]→[LPF]→[Matched Filter]→[Sample @ kTs]→ Q_k
                    ↑
             -sin(2πf_c·t + θ̂)

(I_k, Q_k) → [Symbol Demapper] → bits out
        ↑
    [Timing Recovery]

Critical Subsystems:

1. Carrier Recovery:

Reconstructs the reference carrier phase. Techniques:

• Costas Loop (BPSK/QPSK): Decision-directed PLL
• M-th Power Loop: Removes data modulation by raising to M-th power
• Pilot-aided: Uses transmitted reference symbols

2. Matched Filtering:

Maximizes SNR at the sampling instant:

• Filter impulse response: h(t) = p(T_s - t) (time-reversed pulse)
• For RRC transmit filter: matched filter is also RRC
• Combined response: raised-cosine (zero ISI)

3. Symbol Timing Recovery:

Determines optimal sampling instants:

• Tracks symbol clock (frequency and phase)
• Adjusts sample timing to peak of matched filter output
• Must handle clock offset between TX and RX oscillators

4. Symbol Detection:

Maps received (I_k, Q_k) to nearest constellation point:

• For M-PSK: compute θ = atan2(Q_k, I_k); quantize to nearest allowed phase
• Equivalent: find decision region containing (I_k, Q_k)
• For BPSK: simple threshold (I_k > 0 → '1', else '0')

We've developed a comprehensive understanding of Phase Shift Keying from mathematical foundations through practical implementation. Let's consolidate the key concepts:

What's Next:

With the general PSK framework established, the next page focuses on BPSK and QPSK—the two most widely deployed PSK schemes. We'll examine their specific implementations, performance in practical channels, and the reasons these schemes dominate applications from satellite links to 5G cellular systems.