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Continuous phase modulation

Continuous phase modulation (CPM) is a class of constant-envelope digital modulation techniques in which the phase of the carrier signal varies continuously as a function of the modulating data symbols, ensuring no abrupt phase jumps and thereby minimizing spectral sidelobes. Introduced in seminal works in the early 1980s, CPM achieves high spectral and power efficiency by constraining the phase trajectory to be smooth, making it particularly suitable for bandwidth-limited and power-constrained communication systems. The general form of a CPM signal is given by s(t) = \sqrt{\frac{2E_s}{T}} \cos\left(2\pi f_c t + \phi(t, \mathbf{a})\right), where \phi(t, \mathbf{a}) = 2\pi h \sum_{i=-\infty}^{\infty} a_i q(t - iT), with h denoting the modulation index, a_i the M-ary data symbols, T the symbol duration, and q(t) the normalized phase response function derived from the frequency pulse shaping. CPM schemes are classified by their frequency pulse duration LT (full-response for L=1 or partial-response for L>1), common pulse shapes such as rectangular, raised cosine, or Gaussian, and the choice of h, which determines the maximum phase change per symbol. Notable variants include continuous phase frequency shift keying (CPFSK) with rectangular pulses, minimum shift keying (MSK) (h=0.5, rectangular pulse), and Gaussian minimum shift keying (GMSK), which applies Gaussian filtering for even tighter spectral containment. CPM's primary advantages stem from its constant envelope, which permits the use of efficient nonlinear power amplifiers without significant distortion or spectral regrowth, and its continuous phase, which yields a power spectral density with faster roll-off (e.g., f^{-4} for MSK) compared to discontinuous schemes like QPSK. These properties make CPM robust in fading channels and ideal for applications requiring high efficiency. In practice, GMSK with a modulation index of 0.5 and Gaussian pre-filtering (BT=0.3) is employed in the Global System for Mobile Communications (GSM) for its compact spectrum in the 900 MHz band. Additionally, CPM finds use in satellite and deep space communications, such as in NASA's systems, where constant envelope operation maximizes power efficiency with traveling-wave tube amplifiers. CPM is also used in emerging dual-function radar-communication waveforms for spectrum sharing.

Introduction

Definition and Principles

Continuous phase modulation (CPM) is a non-linear digital modulation scheme in which the phase of a constant-amplitude carrier signal is continuously varied according to the input data symbols, avoiding any abrupt discontinuities in the phase trajectory. This results in a transmitted signal with a constant envelope, which is ideal for efficient amplification. The technique was formalized through key contributions, such as the amplitude modulated pulse (AMP) representation introduced by Laurent in 1986, enabling precise construction of optimal CPM signals. In CPM, the modulating data symbols—typically from an M-ary alphabet—are mapped to incremental phase changes that accumulate over successive symbol periods, shaped by a frequency pulse to ensure smooth transitions. Rather than resetting the phase at each symbol boundary, the changes build upon the previous state, introducing inherent phase memory that influences the overall signal structure. This continuous accumulation distinguishes CPM from traditional phase modulations and contributes to its compact spectral occupancy. Compared to discontinuous methods like M-ary phase-shift keying (M-PSK) or quadrature phase-shift keying (QPSK), which suffer from sudden phase jumps causing significant spectral sidelobes and requiring linear amplification, CPM's phase continuity minimizes out-of-band emissions and supports non-linear amplifiers for higher power efficiency. These attributes yield superior spectral efficiency, enabling denser data packing in limited bandwidth, alongside robust performance in power-limited environments. CPM finds widespread application in wireless communications for bandwidth- and power-efficient data transmission, notably in standards like GSM via its Gaussian minimum shift keying variant.

Historical Development

Continuous phase modulation (CPM) originated in the late 1970s as an extension of traditional frequency modulation techniques, aiming to improve spectral efficiency in digital communications by ensuring a continuous phase trajectory that minimizes discontinuities and reduces out-of-band emissions. This development addressed the limitations of earlier non-continuous phase schemes, which suffered from poor spectral containment in bandwidth-constrained environments. The foundational work began with Tor Aulin's PhD dissertation in 1979 at Lund University, where he explored continuous phase signaling for enhanced performance in mobile systems. Key contributions came in 1981 through two seminal papers by Tomas Aulin and Carl-Erik Sundberg, published in IEEE Transactions on Communications. The first paper introduced full-response CPM, detailing its constant envelope properties that enable efficient power amplification, while the second extended the framework to partial-response CPM, offering trade-offs between bandwidth efficiency and intersymbol interference for practical implementations. These works emphasized CPM's advantages in maintaining a constant signal amplitude, which is crucial for nonlinear amplifiers in power-limited devices, and laid the groundwork for its analysis using phase trees and minimum distance metrics. In the 1980s, CPM gained adoption in mobile radio systems due to its spectral efficiency and robustness against fading, as demonstrated in studies applying partial-response variants to cellular networks for reduced adjacent-channel interference. By the 1990s, refinements led to the Gaussian minimum shift keying (GMSK) variant—a specific partial-response CPM scheme with a modulation index of 0.5 and Gaussian pulse shaping—which was selected for the Global System for Mobile Communications (GSM) standard launched in 1991, enabling widespread deployment in second-generation cellular networks. Motivations for this evolution were driven by spectrum scarcity in wireless bands and the need for power-efficient modulation in battery-powered mobile devices, where constant envelope signals allow operation near amplifier saturation without distortion. Milestones include CPM's integration into Bluetooth specifications in 1999, utilizing Gaussian frequency shift keying (GFSK)—a binary CPM form—for short-range wireless links, and its established use in satellite communications for telemetry and data transmission, leveraging multi-h CPM waveforms in military standards like MIL-STD-188-181B from the 1980s onward. Post-2000 developments focused on faster-than-Nyquist (FTN) extensions of CPM, which intentionally introduce controlled intersymbol interference to achieve higher data rates beyond Nyquist limits, with key advancements in receiver designs for practical deployment in bandwidth-limited channels.

Mathematical Description

General Signal Model

The general signal model for continuous phase modulation (CPM) describes a constant-envelope waveform where the information-bearing data sequence modulates the instantaneous phase of a carrier in a continuous manner. The transmitted signal can be expressed as s(t, \mathbf{a}) = \sqrt{\frac{2E_s}{T}} \cos\left(2\pi f_c t + \phi(t, \mathbf{a})\right), where E_s denotes the symbol energy, T is the symbol duration, f_c is the carrier frequency, and \phi(t, \mathbf{a}) is the time-varying phase function that depends on the data sequence \mathbf{a} = \{a_k\}. This form ensures a constant amplitude \sqrt{2E_s / T}, which is advantageous for power-efficient transmission in nonlinear amplifiers. The phase function \phi(t, \mathbf{a}) is derived from the principle of phase accumulation, where the instantaneous frequency deviation is shaped by a frequency pulse and scaled by the data symbols. Specifically, \phi(t, \mathbf{a}) = 2\pi h \sum_{k=-\infty}^{\infty} a_k q(t - kT), with h as the modulation index, a_k as the M-ary data symbols (typically from the set \{\pm 1, \pm 3, \dots, \pm (M-1)\}), and q(t) as the phase pulse shaping function. This summation represents the cumulative effect of past and present symbols on the current phase, introducing inherent memory into the signal. The model arises from integrating the instantaneous frequency f(t) = f_c + h \sum_k a_k g(t - kT), where g(t) is the frequency pulse, yielding the phase as the integral of $2\pi (f(t) - f_c). The phase pulse q(t) is normalized such that q(t) = \int_{-\infty}^t g(\tau) \, d\tau and q(\infty) = 1/2, ensuring that each symbol contributes a phase shift of \pi h a_k over its full duration in the full-response case, while maintaining overall phase continuity. Since g(t) and thus q(t) are continuous functions (typically smooth and time-limited or decaying), the phase \phi(t, \mathbf{a}) exhibits no discontinuities at symbol boundaries t = nT, distinguishing CPM from discontinuous phase modulations like phase-shift keying. This continuity minimizes spectral sidelobes and enhances bandwidth efficiency. For analytical purposes, particularly in receiver design and performance evaluation, the complex envelope representation is often employed: \tilde{s}(t, \mathbf{a}) = \sqrt{\frac{2E_s}{T}} \exp\left(j \phi(t, \mathbf{a})\right). This baseband equivalent facilitates vector-space analysis and correlation-based detection without altering the constant modulus property.

Phase Pulse and Modulation Index

In continuous phase modulation (CPM), the modulation index h quantifies the maximum phase shift per symbol relative to the carrier frequency and is typically selected as a rational number h = \frac{k}{q}, where k and q are coprime positive integers. This rational form ensures a finite number of phase states in the modulation trellis, facilitating efficient maximum-likelihood detection via the Viterbi algorithm. For binary CPM schemes, h = \frac{1}{2} is a standard choice, as realized in minimum shift keying (MSK), while multi-level CPM often employs smaller indices such as h = \frac{1}{4} or h = \frac{1}{3} to balance spectral efficiency and trellis complexity. The phase pulse q(t), which shapes the cumulative phase trajectory, is defined as the integral of the frequency pulse g(t): q(t) = \int_{-\infty}^{t} g(\tau) \, d\tau. The frequency pulse g(t) is nonnegative and has finite duration L T, where L is an integer representing the number of symbol periods over which it spans, causing q(t) to have support across L+1 symbols and asymptotically approach \frac{1}{2} for t \to \infty. Frequency pulse shapes are designed to control phase continuity and spectral occupancy. The rectangular pulse, used in full-response CPM (L=1), is g(t) = \frac{1}{2T} for $0 \leq t \leq T and zero otherwise, yielding a simple but spectrally broad signal. The raised-cosine pulse, which provides smoother transitions, is expressed as g(t) = \frac{1}{2T} \left[ 1 - \cos\left( \frac{2\pi t}{T} \right) \right], \quad |t| \leq \frac{T}{2}, and zero elsewhere, reducing out-of-band emissions compared to the rectangular shape. Gaussian pulses, applied in partial-response CPM for enhanced spectral confinement, feature a bell-shaped profile g(t) = \frac{1}{2T} \left[ \erf\left( \frac{t + \frac{LT}{2}}{\sqrt{2} \sigma T} \right) - \erf\left( \frac{t - \frac{LT}{2}}{\sqrt{2} \sigma T} \right) \right], where \sigma tunes the bandwidth-time product B T. All frequency pulses obey the normalization \int_{-\infty}^{\infty} g(t) \, dt = \frac{1}{2}, guaranteeing that each binary symbol induces a phase increment of \pm \pi h at the pulse's steady state. The modulation index h trades off bandwidth and error performance: larger h expands the signal's occupied bandwidth through increased frequency deviation but proportionally enlarges the squared minimum Euclidean distance d_{\min}^2 \propto h^2, thereby improving bit error rate under AWGN conditions. The phase pulse q(t) thereby governs the intersymbol interference and memory depth in the CPM waveform.

Core Properties

Phase Memory

In continuous phase modulation (CPM), phase memory arises from the cumulative effect of transmitted symbols on the signal phase, where the phase at any time t depends on the current symbol and the previous \mu symbols due to the finite support length LT of the phase pulse q(t), with \mu = L. This intersymbol dependence stems from the integral form of the phase expression, \phi(t) = 2\pi h \sum_{k=-\infty}^{n} a_k q(t - kT) \mod 2\pi, where only the most recent \mu + 1 terms contribute non-negligibly when q(t) = 0 for t < 0 and q(t) = 1/2 for t \geq LT. The memory length \mu equals L in standard CPM formulations, where L is an integer representing the pulse duration in symbol periods; for full-response schemes, L = 1 (\mu = 1), while partial-response variants use L > 1 (up to 5 in typical designs) to shape the spectrum. For rational modulation indices h = p/q, the number of states in the state-space representation via a trellis diagram is typically S = M^\mu \times q, where each state encodes the combination of the last \mu symbols and the phase state to fully determine the current phase trajectory; for cases where the phase states align with M, it simplifies to M^\mu. The phase memory introduces controlled intersymbol interference (ISI) that enhances spectral efficiency but necessitates joint sequence detection over multiple symbols, rather than independent symbol-by-symbol decisions as in memoryless modulations like PSK. For instance, in binary CPM (M = 2) with L = 1 (\mu = 1), the trellis has 2 states, enabling low-complexity detection; increasing L to 3 or 4 raises the state count to 8 or 16 (simplified), amplifying computational demands but yielding better power-bandwidth trade-offs through smoother phase transitions. Although the phase is defined modulo $2\pi, exact tracking requires retaining the full accumulated phase value across states to avoid ambiguity in the continuous waveform representation, distinguishing CPM from discontinuous phase schemes.

Phase Trajectory

In continuous phase modulation (CPM), the phase trajectory refers to the curve traced by the instantaneous phase \phi(t) modulo $2\pi over one or more symbol periods, where the trajectory starts and ends at multiples of $2\pi h, with h denoting the modulation index. This continuous path in the phase domain ensures that the signal maintains phase coherence across symbol transitions, distinguishing CPM from discontinuous phase modulations like phase-shift keying (PSK). The shape of the phase trajectory depends on the frequency pulse g(t). For rectangular pulses, as in continuous-phase frequency-shift keying (CPFSK), the trajectory is linear, resulting in constant frequency shifts during each symbol interval. In contrast, smoothed pulses such as raised-cosine produce curved trajectories that avoid abrupt phase changes, enhancing the signal's compactness in the frequency domain. For instance, a modulation index of h = 1/2 in minimum-shift keying (MSK), a special CPM case, yields linear phase ramps of \pm \pi/2 per symbol period, with full cycles spanning two symbol periods ($2T). The modulation index h, along with the pulse shape and memory length L, governs trade-offs in spectral occupancy and intersymbol interference. A phase tree diagram visualizes multiple possible phase trajectories originating from different symbol sequences, with branches merging at convergence points that facilitate calculations of minimum Euclidean distance for error performance analysis. These diagrams highlight how phase continuity propagates memory effects across symbols. Smoother phase trajectories in CPM significantly reduce out-of-band emissions relative to discontinuous modulations, improving spectral efficiency by confining more energy within the assigned bandwidth.

Classification and Variants

Full-Response CPM

Full-response continuous phase modulation (CPM) refers to a CPM scheme where the duration of the frequency pulse is exactly one symbol period, denoted as L = 1. In this configuration, the phase response function q(t) has finite support over [0, T), ensuring that the phase change induced by each symbol is confined to its own symbol interval without introducing intersymbol interference from previous symbols. This results in a signal where the phase continuity is maintained solely through the instantaneous frequency deviation within the current symbol period. The primary characteristic of full-response CPM is its minimal memory requirement, with the modulation memory \mu = 1, leading to a trellis structure with M states for an M-ary alphabet. This simplicity facilitates lower computational complexity in signal generation and processing compared to schemes with extended memory. A representative pulse shape is the rectangular frequency pulse defined as g(t) = \frac{1}{2T} for $0 \leq t \leq T and zero otherwise, which produces a linear phase ramp over the symbol duration, akin to continuous-phase frequency-shift keying (CPFSK) as a special case. In terms of performance, full-response CPM offers advantages in systems prioritizing low complexity, such as early digital mobile standards, but its spectral occupancy is broader than that of partial-response CPM due to the abrupt frequency transitions at symbol boundaries. For binary full-response CPM with modulation index h = 1/2, the transmitted signals corresponding to different symbols are orthogonal over the symbol period, enhancing detectability in additive white Gaussian noise channels. The minimum squared Euclidean distance, a key metric for error performance, is determined by the maximum phase difference over one symbol interval T; for the binary case with h = 1/2, this distance achieves d_{\min}^2 = 2, providing a baseline for power efficiency comparable to minimum-shift keying.

Partial-Response CPM

Partial-response continuous phase modulation (CPM) is characterized by a phase pulse duration that extends over multiple symbol intervals, with the pulse length parameter L > 1, typically taking values between 2 and 5. In this framework, the phase response function q(t) spans LT seconds, where T is the symbol duration, thereby inducing inherent intersymbol interference (ISI) as each symbol influences the phase across subsequent symbols. This controlled ISI contrasts with schemes where the pulse is confined to a single symbol period, enabling more gradual phase changes that enhance spectral properties. The design of partial-response CPM often employs frequency pulses such as raised-cosine shapes extended over LT to promote smooth transitions and minimize discontinuities in the phase trajectory. For example, with L=2, a raised-cosine pulse produces a duobinary-like response, where the phase evolution incorporates contributions from the current and previous symbol, balancing smoothness with modulation efficiency. These pulses are normalized such that q(\infty) = 1/2, ensuring the phase increment per symbol aligns with the modulation index h. Longer pulses in this category also contribute to trajectory smoothing, reducing abrupt shifts. A key advantage of partial-response CPM lies in its superior spectral containment, achieving narrower bandwidth than full-response counterparts for equivalent bit error rates, though this comes at the cost of higher modulation memory \mu = L. In some configurations, such as binary schemes with optimized pulse shapes, partial-response formats yield a larger minimum Euclidean distance, improving error performance under maximum-likelihood detection. An illustrative example is Gaussian minimum shift keying (GMSK), a binary partial-response CPM variant using a Gaussian frequency pulse with normalized bandwidth-time product BT = 0.3, which effectively spans L = 3 to $4 symbols due to the pulse's tail. This design delivers a compact power spectral density, making it suitable for bandwidth-constrained environments while maintaining constant envelope properties. Detection of partial-response CPM involves a trellis with increased state complexity, scaling as M^L for M-ary alphabets, reflecting the memory depth from the extended pulses. However, reduced-state sequence estimation techniques, such as those using decision feedback or per-survivor processing, can substantially lower the computational burden without significant performance loss.

Continuous-Phase Frequency-Shift Keying

Continuous-phase frequency-shift keying (CPFSK) is defined as a binary continuous phase modulation (CPM) scheme with M=2 symbols, employing a rectangular frequency pulse of duration L=1 symbol period T, which renders it equivalent to a continuous-phase variant of frequency-shift keying (FSK). In this modulation, binary data symbols a_k \in \{ \pm 1 \} select between two frequency deviations from the carrier, while ensuring phase continuity across symbol boundaries. The transmitted signal for CPFSK can be expressed as s(t) = A_c \cos\left(2\pi f_c t + 2\pi h \sum_k a_k \int_{-\infty}^t g(\tau - kT) \, d\tau \right), where A_c is the signal amplitude, f_c is the carrier frequency, h is the modulation index, and the frequency pulse g(t) = \frac{1}{2T} for $0 \leq t \leq T and zero otherwise. This rectangular pulse results in a piecewise constant instantaneous frequency, with the frequency deviation given by \Delta f = \frac{h}{2T}, representing the shift from the carrier for each binary symbol. A notable case occurs when h = 1/2, yielding minimum-shift keying (MSK), where the two possible signals are orthogonal over the symbol period T and achieve the minimum bandwidth among binary CPFSK schemes for coherent detection. The modulation index h governs the separation between the frequency tones; for non-coherent detection, h = 0.715 provides optimal performance, outperforming coherent detection of standard phase-shift keying in certain scenarios. CPFSK exhibits a constant envelope and absence of phase discontinuities, enhancing power amplifier efficiency in transmission systems. As a special case within the broader CPM framework, CPFSK features linear phase changes over each symbol interval due to the rectangular pulse, producing piecewise constant frequency trajectories that maintain overall phase continuity. This positions CPFSK as a foundational example of full-response CPM.

Detection and Performance

Demodulation Methods

The optimal demodulation of continuous phase modulation (CPM) signals is achieved through maximum-likelihood sequence estimation (MLSE), which accounts for the intersymbol interference introduced by the phase memory inherent in CPM. Due to the correlated nature of the phase trajectory across symbols, MLSE processes the received signal as a sequence estimation problem over a finite-state trellis, outperforming symbol-by-symbol detection methods that ignore memory effects. This approach exploits the convolutional structure of CPM, treating it as an uncoded modulation with built-in error correction properties similar to a rate-1 convolutional code. The Viterbi algorithm implements MLSE efficiently on the CPM trellis, where each state corresponds to the accumulated phase from the previous \mu symbols, with \mu denoting the memory length of the phase pulse. The number of trellis states is typically M^{\mu} for an M-ary alphabet, enabling branch metrics to be computed from phase differences between hypothesized transmitted sequences and the received signal matched to the correlator outputs. For full-response CPM schemes like continuous-phase frequency-shift keying (CPFSK), the trellis simplifies, reducing complexity while maintaining optimality under coherent reception. Suboptimal demodulation methods reduce computational complexity for high-memory or high-order CPM schemes, where full MLSE becomes impractical due to the exponential growth in states. Reduced-state sequence estimation (RSSE) partitions the trellis states and uses decision feedback to approximate branches, achieving near-optimal performance with significantly fewer states—often by a factor of the alphabet size M. Decision-feedback equalization variants further simplify processing by canceling estimated interference from prior decisions, suitable for partial-response CPM with longer memory. For non-coherent scenarios, frequency discriminator detection demodulates the instantaneous frequency without phase reference, performing adequately for binary schemes like minimum-shift keying (MSK) but with a SNR penalty of 1-2 dB compared to coherent MLSE. CPM demodulation requires accurate synchronization of carrier phase and symbol timing to align the receiver's reference with the transmitted signal, as phase errors degrade sequence estimation. Common techniques include non-data-aided phase-locked loops for carrier recovery and early-late timing estimators, often augmented by pilot symbols or known preamble sequences inserted periodically to aid acquisition and tracking in bursty transmissions. Performance is typically evaluated via bit error rate (BER) as a function of signal-to-noise ratio (SNR), where coherent MLSE yields superior results, with CPM exhibiting 2-4 dB coding gain over uncoded phase-shift keying at BER $10^{-5} due to its memory-induced diversity. Suboptimal methods like RSSE incur less than 0.5 dB degradation for moderate memory lengths, while non-coherent discriminator detection shows higher error floors but enables simpler, low-power receivers.

Spectral Characteristics and Efficiency

The power spectral density (PSD) of continuous phase modulation (CPM) signals exhibits a narrow main lobe and reduced sidelobes compared to discontinuous phase modulations, primarily due to the continuous phase trajectory that minimizes abrupt phase changes. For binary CPM, an approximation of the PSD is given by S(f) \propto \left| \int_{-\infty}^{\infty} g(t) e^{-j 2\pi f t} \, dt \right|^2, where g(t) is the phase pulse shaping function, highlighting how the smooth phase evolution contributes to compact spectral occupancy. This property stems from the inherent phase continuity, which confines more power within the main lobe while suppressing out-of-band emissions. CPM achieves higher spectral efficiency than traditional phase-shift keying (PSK) schemes, enabling more bits per hertz in bandwidth-constrained environments. For instance, Gaussian minimum-shift keying (GMSK), a partial-response CPM variant with modulation index h = 0.5, delivers approximately 1.35 bits/s/Hz in the Global System for Mobile Communications (GSM), supporting a 270.8 kbps data rate within a 200 kHz channel. These efficiencies arise from optimized pulse shapes that trade minimal intersymbol interference for tighter spectral containment, outperforming binary PSK's typical 1 bit/s/Hz limit. In practical systems, CPM's spectral advantages support diverse applications, including GSM for voice and data transmission using GMSK, Bluetooth for low-power short-range links also employing GMSK, satellite communications with continuous-phase frequency-shift keying (CPFSK) to fit bandlimited channels, and navigation signals leveraging partial-response CPM for enhanced anti-jamming resilience. The constant envelope nature of CPM further boosts power efficiency, permitting operation of nonlinear amplifiers in saturation with up to 6 dB power gain without significant distortion, though this involves trade-offs in modulation index h and pulse shape that balance bandwidth compression against increased intersymbol interference. Relative to quadrature PSK (QPSK), CPM demonstrates superior out-of-band radiation control; for example, the 99% power bandwidth for minimum-shift keying (MSK, a full-response CPM with h = 0.5) is approximately 1.2/T (where T is the symbol duration), versus approximately 3.2/T for unfiltered QPSK (normalized to symbol duration). This containment reduces adjacent channel interference, making CPM preferable for systems prioritizing bandwidth economy over peak data rates.