On the Unique Optimality of Chirp Coding in Communications

1. Introduction

In communications engineering, chirp modulation is widely recognised as an effective and, under certain channel conditions, near-optimal approach for the transmission of information through noisy and dynamically varying channels. Its ability to provide favourable time–frequency localisation, processing gain, and resilience to frequency shifts makes chirp-based signalling particularly attractive for applications in which signals are exposed to interference, Doppler effects, multi-path propagation, and scattering. Nevertheless, an important theoretical question remains: ‘is chirp modulation uniquely optimal, or can alternative modulation schemes (some yet to be invented) achieve superior performance under equivalent channel and power constraints?’

This paper addresses this question by developing a rigorous mathematical framework for evaluating the optimality of chirp-based signalling in noise-dominated communication environments. In particular, it seeks to establish whether chirp coding can be characterised as a uniquely optimal modulation strategy, rather than merely one of several approximately equivalent approaches. The analysis formalises and extends an insight that has been recognised qualitatively within the communications engineering community, namely that chirp signals possess a combination of properties that can provide substantial advantages in challenging propagation environments.

Drawing upon concepts from information theory, functional analysis, and time–frequency analysis, the paper examines the fundamental performance characteristics of chirp-based signalling. The analysis considers the relationship between signal structure, channel capacity, noise tolerance, frequency selectivity, and propagation-induced distortions. Particular attention is given to the extent to which chirp signals maintain their distinguishability and recoverability when subjected to Doppler frequency shifts, multipath propagation, and multiple-scattering effects.

Within the proposed mathematical framework, we demonstrate that chirp-based signalling can achieve simultaneous optimality across several relevant performance dimensions, including information-theoretic efficiency, robustness to Doppler-induced frequency shifts, and resilience to multi-path and multiple-scattering environments**. Rather than evaluating modulation performance solely in terms of conventional measures such as bit-error rate or signal-to-noise ratio, the analysis considers a broader set of criteria that capture the behaviour of signals within realistic time-varying channels. This provides a more comprehensive basis for comparing chirp modulation with conventional approaches.

Analytical comparisons are subsequently developed between chirp-based signalling and established modulation techniques, including conventional sinusoidal and digitally modulated waveforms. These comparisons illustrate how the time-varying instantaneous frequency of a chirp can provide advantageous signal diversity and improved separability in the time–frequency domain. In particular, the analysis highlights the potential benefits of chirp signals in environments where frequency offsets, temporal dispersion, and multiple propagation paths would otherwise degrade communication performance.

The theoretical results have potentially significant implications for the design of next-generation wireless communication systems, radar and sensing technologies, and low-power Internet of Things (IoT) networks. In wireless communications, chirp-based schemes may offer advantages in environments characterised by mobility, interference, and rapidly changing propagation conditions. In radar and sensing applications, their time–frequency characteristics can support simultaneous ranging and velocity estimation, while in low-power IoT systems, their robustness and processing-gain properties may contribute to reliable communication over constrained or noisy links.

The findings may also have implications for the scalability and implementation efficiency of future communication architectures. If the mathematical optimality properties established here remain valid across a sufficiently broad class of practical channel models, chirp-based signalling could provide a common waveform framework capable of supporting diverse communication and sensing requirements. Such a framework could potentially simplify system design while improving reliability in increasingly heterogeneous wireless environments.

In summary, this paper argues that chirp-based modulation possesses a fundamental combination of information-theoretic, time–frequency, and propagation-resilience properties that distinguish it from any other (conventional or otherwise) modulation strategies. Under the assumptions and channel conditions defined in the analysis, the results suggest that chirp signalling constitutes a uniquely optimal framework for digital communication in noise-dominated and dynamically varying environments. Consequently, alternative modulation strategies may be regarded as comparatively suboptimal whenever the relevant performance criteria include not only spectral or power efficiency, but also robustness to Doppler shifts, multi-path propagation, scattering, and other realistic channel impairments.

2. Context: The Chirp Function Revisited

The chirp function and the chirplet transform are well-established mathematical concepts that have been extensively studied and applied across a broad range of information, communications, and signal-processing engineering applications. Their development can be traced to foundational work during the 1950s and 1960s, including the invention and patenting of chirp-pulse-based communication systems (e.g., [1,2]). Since this early work, chirp-based techniques have evolved considerably and are now important tools in signal analysis, digital communications, radar, sonar, imaging, remote sensing, and related areas of applied physics and engineering.

For unit amplitude, a linear frequency-modulated (LFM) chirp may be represented as a complex-valued function (signal) $s\left(t\right)$ say, of time t, in the form

$$s\left(t\right)=exp\left(i\alpha t^2\right),$$

where $\alpha$ is a constant commonly referred to as the ‘chirp rate’ or ‘chirp parameter’. The parameter $\alpha$ determines the rate at which the instantaneous frequency changes with time and has dimensions of ${\mathrm{Time}}^{-2}$ when the phase is expressed in radians. An equivalent representation may be obtained using the complex conjugate,

$$s^{*}\left(t\right)=exp(-i\alpha t^2).$$

The two forms correspond to opposite directions of a frequency sweep and are mathematically equivalent under complex conjugation.

The function $exp(\pm i\alpha t^2)$ is characterised by a quadratic phase function

$$\theta \left(t\right)=\alpha t^2,$$

which produces a linear variation in instantaneous angular frequency. This is because differentiating the phase with respect to time gives

$${\theta }^{\prime }\left(t\right)\equiv {\displaystyle \frac{d\theta \left(t\right)}{dt}}=2\alpha t.$$

Thus, the instantaneous angular frequency varies linearly with time. This linear relationship between frequency and time is the defining characteristic of an LFM chirp.

The above formulation represents the simplest form of a chirp, but the concept can readily be generalised to accommodate more complex frequency laws. For example, a quadratic chirp may be defined by

$${\theta }^{\prime }\left(t\right)=\alpha t^2,$$

whereas an exponential chirp may be described by a frequency law of the form

$${\theta }^{\prime }\left(t\right)={\alpha }^t,$$

subject to the appropriate restrictions on $\alpha$. More generally, a chirp may be represented as

$$s\left(t\right)=A\left(t\right)exp\left[i\theta \left(t\right)\right],$$

where $A\left(t\right)$ represents a potentially time-varying amplitude and $\theta \left(t\right)$ is a differentiable phase function. The instantaneous angular frequency is then given by ${\theta }^{\prime }\left(t\right)$. This general formulation reflects the flexibility of chirp-based representations and allows them to model a wide range of non-stationary signals whose spectral content evolves with time.

Linear frequency-modulated chirps represent a recurring theme across many areas of physics, mathematics, and communications engineering. In telecommunications and radio communications, LFM signals are widely employed as spread-spectrum and waveform-design techniques, in which the transmitted energy is distributed over a prescribed frequency range. Such frequency spreading can provide significant processing gain and can improve resilience to noise, interference, frequency-selective propagation, and other channel impairments. Chirp-based signalling has consequently been incorporated into a range of wireless communication technologies and standards, including systems defined by the Institute of Electrical and Electronics Engineers (IEEE) [3].

The usefulness of chirp waveforms extends well beyond conventional communications. In radar and sonar systems, for example, LFM pulses permit a relatively long-duration transmitted waveform to be compressed at the receiver, thereby combining high transmitted energy with fine range resolution. The associated time–frequency structure also provides useful information concerning the motion and propagation characteristics of targets. Similar principles have been exploited in medical imaging, remote sensing, geophysical exploration, spectroscopy, and other measurement systems in which signals undergo significant temporal or spectral variation. These applications are closely related to the fundamental properties of chirp signals, particularly their structured evolution in the time–frequency plane.

In practical applications, chirp functions are generally defined over a finite time interval, thereby producing a time-limited or compactly supported waveform. For a two-sided pulse, this interval may be written as

$$t\in [-T/2,T/2],$$

where T denotes the pulse duration, while a single-sided pulse may be defined over

$$t\in [0,T].$$

Applying a temporal window to an otherwise continuous chirp produces a finite-duration signal with well-defined temporal boundaries. Such windowed chirp functions are commonly referred to as ‘chirplets’. More generally, a chirplet may be understood as a localised segment of a chirp waveform obtained by applying a windowing function to a chirp, where the window may be continuous, discontinuous, compactly supported, or otherwise appropriately defined.

The concept of a chirplet is particularly important in signal and image processing because it provides a means of representing localised oscillatory structures whose frequency changes with time. This makes chirplets useful for analysing non-stationary signals that cannot be adequately represented by stationary Fourier methods alone. The chirplet transform extends these ideas by providing a flexible time–frequency representation in which both temporal localisation and frequency modulation can be incorporated into the analysis [4]. In this respect, the chirplet transform can be viewed as a generalisation of more conventional time–frequency techniques, offering additional degrees of freedom for analysing signals with evolving spectral characteristics.

The broad range of applications associated with chirp functions and chirplet transforms arises from their distinctive mathematical and physical properties. These include their structured time–frequency behaviour, well-defined autocorrelation characteristics, sensitivity to frequency and time shifts, and favourable interaction with Fourier transformation. These properties are revisited throughout this paper because they provide the mathematical basis for understanding why chirp-based representations can be particularly effective in communication and sensing systems.

It is also important to recognise that chirp functions and chirplet transforms are not merely convenient constructs developed for signal-processing applications. Chirp-like functions arise naturally in physics and applied mathematics, where they occur as components of solutions to a variety of models involving partial differential equations, wave propagation, dispersion, and diffusion. In such contexts, quadratic-phase functions emerge naturally from the mathematical description of wave propagation and the evolution of wave packets. This observation provides an additional motivation for examining the theoretical properties of chirp functions independently of their engineering applications. In particular, the role of quadratic-phase functions and chirp-like solutions within physical models is considered in the early sections of this paper.

Within this broader mathematical and physical context, the present work revisits several fundamental properties of chirp functions. In particular, we investigate their behaviour under the Fourier transform and establish conditions under which a chirp possesses a distinctive conjugate Fourier eigenfunction. We further examine the proposition that, within an appropriately defined class of constant-amplitude phase-only functions, the chirp is distinguished by the self-characteristic nature of its Fourier-transform behaviour. This observation provides a potentially important mathematical connection between the structural properties of chirps and their usefulness in signal representation.

When considered together with the well-known autocorrelation properties of chirp waveforms, these Fourier-domain characteristics provide further insight into the distinctive nature of chirp-based representations. In particular, the combination of time–frequency structure, autocorrelation behaviour, and Fourier-domain properties raises the possibility that chirps possess a form of mathematical uniqueness that may be relevant to communication and information-extraction problems. Establishing the precise scope and assumptions of this uniqueness is therefore an important objective of the present work.

The central proposition investigated in this paper is that these mathematical properties may have consequences for the transmission and recovery of information in noisy communication environments. Rather than considering chirps solely as one modulation option among many, the paper examines whether their combined properties can provide a more fundamental basis for optimal signal representation. This question is particularly relevant when communication takes place in environments affected simultaneously by additive noise, frequency shifts, multi-path propagation, and other channel distortions.

The remainder of this paper is organised as follows. Section 3 provides a concise overview of established modulation methods and introduces the conceptual framework required to understand chirplet modulation in the context of digital communications. Section 4 then formulates the essential requirements for extracting information from additive noise within a linear convolutional communication model. Particular attention is given to the mathematical conditions required for reliable information recovery and to the relationship between signal structure and channel characteristics.

Section 5 applies this framework to a complex chirp function and develops the principal mathematical arguments supporting the proposed optimality of chirp-based coding. The analysis draws upon the results established earlier in the paper and is supplemented by the mathematical developments presented in Appendix A and Appendix B. In particular, Appendix B contains the principal proof concerning the proposed uniqueness of chirp coding and represents the central mathematical contribution of this work. The distinction between mathematical uniqueness, optimality under specified constraints, and practical engineering performance is also considered where appropriate.

Finally, Section 6 discusses the implications of the results and considers their relevance to communications engineering and related applications. Particular attention is given to the potential implications for wireless communications, radar, sensing, and low-power communication systems. The section also identifies the assumptions and limitations of the analysis and outlines areas in which further theoretical and experimental investigation may be appropriate.

3. Overview of Modulation Methods

Modulation is a fundamental process in digital communication systems, enabling efficient transmission of information over physical channels. This introductory section briefly compares amplitude, frequency, phase, and chirplet modulation techniques in terms of performance, robustness, and efficiency. Additionally, it situates these methods within the broader context of modern modulation schemes.

Modulation enables the transmission of information by embedding signals onto carrier waves. Common techniques include amplitude modulation (AM), frequency modulation (FM), and phase modulation (PM), which underpin digital schemes such as Amplitude Shift Keying (ASK), Frequency Shift Keying (FSK), and Phase Shift Keying (PSK) [5]. More recently, chirplet modulation has emerged for non-stationary and time-varying channels. This paper compares these approaches in terms of performance and application.

3.1. Amplitude Modulation (AM)

Amplitude modulation varies the amplitude of a carrier signal:

$$s\left(t\right)=[A_c+m\left(t\right)]cos\left(2\pi f_{c}t\right)$$

where $A_c$ is the carrier amplitude, $m\left(t\right)$ is the message signal, and $f_c$ is the carrier frequency.

In digital systems, AM corresponds to ASK [6]. It is simple to implement [7] but highly susceptible to noise [8]. Additionally, it is power inefficient due to carrier energy consumption [9].

3.2. Frequency Modulation (FM)

Frequency modulation varies the carrier frequency:

$$s\left(t\right)=A_{c}cos\left(2\pi f_{c}t+2\pi k_f\int m\left(t\right)dt\right)$$

where $k_f$ is the frequency sensitivity. FM (or FSK in digital systems) is more robust to noise [8] and supports efficient amplification [7]. However, it requires larger bandwidth as described by Carson’s rule [9].

3.3. Phase Modulation (PM)

Phase modulation encodes information in phase variations:

$$s\left(t\right)=A_{c}cos\left[2\pi f_{c}t+k_{p}m\left(t\right)\right]$$

where $k_p$ is the phase sensitivity. PM is widely used in digital systems such as PSK [6]. It provides high spectral efficiency and robustness [14]. However, it requires coherent detection and phase synchronization [8].

3.4. Chirplet Modulation

Chirplet modulation extends chirp signals:

$$s\left(t\right)=Aexp\left[i\left(2\pi f_{0}t+\alpha t^2\right)\right]$$

where $f_0$ is the initial frequency and $\alpha$ is the chirp rate. Chirplet modulation is effective in time-varying environments and robust to Doppler and multipath effects [11]. It also offers high time-frequency resolution [13], but at the cost of increased computational complexity [4].

3.5. Other Modulation Methods

Modern communication systems employ additional techniques such as Orthogonal Frequency Division Multiplexing (OFDM), which improves spectral efficiency and mitigates multi-path interference [5]. Quadrature Amplitude Modulation (QAM) combines amplitude and phase modulation for high data rates [6]. Spread spectrum methods such as Direct Sequence Spread Spectrum (DSSS) and Frequency Hopping Spread Spectrum (FHSS) improve robustness and security, while adaptive modulation dynamically adjusts parameters based on channel conditions [5,14].

3.6. Comparative Discussion

The comparison reveals key trade-offs. FM and PM provide better noise immunity than AM [8], while PM achieves superior spectral efficiency [6]. AM is the simplest to implement, whereas chirplet modulation is computationally complex [4]. Constant-envelope schemes such as FM and PM improve power efficiency [9], while chirplet modulation excels in dynamic environments [11].

AM, FM, and PM form the foundation of digital communication systems, each offering trade-offs in complexity, efficiency, and robustness. Chirplet modulation provides enhanced performance in non-stationary environments. Modern techniques such as Orthogonal Frequency Division Multiplexing (OFDM) and Quadrature Amplitude Modulation (QAM) extend these principles to meet the demands of high-speed communication systems.

Instead of sending a data stream over a single, wide radio channel, OFDM splits the channel into many smaller, closely spaced sub-channels (called sub-carriers). The sub-carriers are arranged so they do not interfere with each other, even though they overlap. This makes the transmission highly efficient and resistant to interference (like echoes or signals bouncing off buildings). QAM mixes two carrier waves of the same frequency that are out of phase with each other by 90 degrees (called Quadrature). By changing both the amplitude (strength) and the phase (timing) of these waves, QAM can represent multiple bits of data at the exact same time. Both OFDM and QAM are fundamental digital communication technologies used to transmit large amounts of data wirelessly. They are frequently used together in modern technologies like Wi-Fi, 4G/5G cellular networks, and digital TV [12].

4. Communications Theory: On the Principles of Developing an Optimal Solution

Assuming that a signal is corrupted by additive noise, consider a model for the observed signal $s\left(t\right)$, as a function of time t, given by

$$s\left(t\right)=f\left(t\right)+n\left(t\right),$$

where $f\left(t\right)$ denotes the ‘information function’ and $n\left(t\right)$ represents the additive ‘noise function’, typically modelled as a stochastic process. The noise function characterises the distortion introduced to the information signal as it propagates through a communications channel. The fundamental problem is therefore to recover $f\left(t\right)$ from the observed signal $s\left(t\right)$ in the presence of $n\left(t\right)$, which corresponds to the general problem of extracting information from noise.

In general, exact recovery of $f\left(t\right)$ from $s\left(t\right)$ is not possible unless $n\left(t\right)$ is known for all values of t. Such a situation arises, for example, in cryptographic systems, where the noise function is deliberately generated to encrypt the information function. In this context, the noise (or ‘cipher’) is typically produced using a deterministic algorithm that generates the same sequence, subject to a given key. Provided the key is known, the noise can be reconstructed and removed, allowing the information function to be recovered exactly.

However, in most communications engineering applications, the noise function is not known explicitly. Instead, only certain statistical properties of the noise may be available, such as its probability distribution, mean, variance, or power spectral density. In such cases, Bayesian analysis or statistical estimation techniques may be applied, which require probabilistic models for both the information function $f\left(t\right)$ and the noise function $n\left(t\right)$. While these approaches can yield useful estimates of the information signal, they generally provide only approximate solutions rather than exact recovery.

This distinction is important in communications engineering, where the objective is typically to recover the transmitted information exactly, particularly in digital communication systems. Consequently, methods that rely solely on statistical estimation are often insufficient, motivating the need for alternative strategies that enable robust and exact information extraction in the presence of noise.

4.1. Transformation of the Information Function

Suppose that, prior to transmission, a transform is applied to the information function $f\left(t\right)$. The transmission model may then be written as

$$s\left(t\right)=\mathcal{T}\left[f\right(t\left)\right]+n\left(t\right),$$

where $\mathcal{T}$ denotes a transform operator. In the context of communications engineering, this operator may be loosely associated with a modulation method of the type briefly discussed in Section II, where the information signal is mapped into an alternative representation that is more suitable for transmission through a noisy channel.

The key question is therefore: which transform $\mathcal{T}$ provides optimal transmission of recoverable information in the presence of noise? In particular, we seek a transform that maximises the ability to recover the original information function $f\left(t\right)$ from the received signal $s\left(t\right)$, despite the presence of additive noise.

Ideally, the solution to this problem is to employ a transform for which the original information function can be recovered exactly through application of the inverse transform, i.e.,

$$f\left(t\right)={\mathcal{T}}^{-1}\left[s\left(t\right)\right].$$

Substituting for $s\left(t\right)$ yields

$$f\left(t\right)={\mathcal{T}}^{-1}\left[\mathcal{T}\left[f\right(t\left)\right]+n\left(t\right)\right].$$

For exact recovery of the information function, this expression requires the transform to satisfy the following conditions:

$${\mathcal{T}}^{-1}\mathcal{T}\left[f\left(t\right)\right]=f\left(t\right),{\mathcal{T}}^{-1}\left[n\left(t\right)\right]=0.$$

The first condition ensures that the transform $\mathcal{T}$ is invertible with respect to the information function, so that no information is lost during the transformation process. The second condition requires that the inverse transform suppresses or eliminates the noise component. Together, these conditions define an idealised transform that preserves the information signal while rejecting noise.

In practice, these requirements are highly restrictive, since noise is generally broadband and overlaps spectrally with the transformed information signal. Consequently, exact separation of signal and noise is rarely achievable using conventional transforms. This observation motivates the search for transform-based modulation methods that possess intrinsic robustness to noise, thereby enabling reliable recovery of the information function under realistic communication conditions.

4.2. Convolutional Transformation

In communications engineering, convolution is used to describe how a transmitted signal is modified by the response of a communication channel to an impulse, typically modelled as a delta function. This framework enables engineers to analyse important channel effects such as distortion, delay, dispersion, and interference. In this context, the received signal can be modelled as the convolution of the transmitted signal with the channel impulse response, together with an additive noise component.

For a convolution-based transform, the transmission model may be written as

$$s\left(t\right)=p\left(t\right)\otimes f\left(t\right)+n\left(t\right),$$

where ⊗ denotes convolution in time, defined by

$$p\left(t\right)\otimes f\left(t\right)\equiv {\displaystyle \underset{-\infty }{\overset{\infty }{\int }}}p(t-\tau )f\left(\tau \right)d\tau ,$$

and where $p\left(t\right)$ denotes the impulse response function (IRF) of the communication channel or, more generally, the transform kernel applied to the information function.

It is often convenient to analyse this model in the frequency domain. By application of the Convolution Theorem, convolution in the time domain corresponds to multiplication in the frequency domain, yielding

$$S\left(\omega \right)=P\left(\omega \right)F\left(\omega \right)+N\left(\omega \right),$$

where $\omega$ denotes angular frequency, and $S\left(\omega \right)$, $P\left(\omega \right)$, $F\left(\omega \right)$, and $N\left(\omega \right)$ represent the Fourier transforms of the corresponding time-domain functions. For example, the Fourier transform (denoted via the Fourier transform operator $\mathcal{F}$) of the information function is given by

$$F\left(\omega \right)=\mathcal{F}\left[f\left(t\right)\right]\equiv {\displaystyle \underset{-\infty }{\overset{\infty }{\int }}}f\left(t\right)exp(-i\omega t)dt.$$

Within this convolutional transform framework, the optimal recovery of the information function requires the removal of the effect of the impulse response function. One approach is to multiply both sides of the frequency-domain model by the complex conjugate $P^{*}\left(\omega \right)$, giving

$$P^{*}\left(\omega \right)S\left(\omega \right)={\left|P\left(\omega \right)\right|}^{2}F\left(\omega \right)+P^{*}\left(\omega \right)N\left(\omega \right).$$

For exact recovery of the information function, this expression must reduce to $F\left(\omega \right)$ which is achieved if and only if the following conditions are satisfied:

$${\left|P\left(\omega \right)\right|}^2=1,P^{*}\left(\omega \right)N\left(\omega \right)=0.$$

The first condition requires that the magnitude of the frequency response be unity for all frequencies, implying that the transform preserves the amplitude of the information signal. The second condition requires that the transformed noise component be eliminated, or at least minimised, after multiplication by the complex conjugate.

One way to guarantee that ${\left|P\left(\omega \right)\right|}^2=1$ is to select an impulse response function whose Fourier transform is phase-only, i.e.,

$$P\left(\omega \right)=exp\left[i\phi \right(\omega \left)\right],$$

where $\phi \left(\omega \right)$ denotes the phase spectrum. In this case, the magnitude of $P\left(\omega \right)$ is unity for all frequencies, satisfying the first optimality condition.

This observation provides a key insight into the design of optimal transform-based communication schemes. In particular, phase-only transforms preserve signal energy while enabling selective manipulation of phase structure. As will be discussed in the following section, the chirp function naturally satisfies these requirements and therefore provides a strong candidate for achieving optimal performance in noisy communication environments.

4.3. Application of the Chirp Function

In the context of the discussion presented in the previous section, consider the case in which the impulse response function (IRF) is given by the chirp function

$$p\left(t\right)=exp\left(i\alpha t^2\right),$$

where $\alpha$ is a real-valued chirp rate. Under this assumption, we analyse the recovery of the information signal by correlating the received signal with the complex conjugate of the impulse response function. This yields

$$p^{*}\left(t\right)\odot s\left(t\right)=p^{*}\left(t\right)\odot \left(p\left(t\right)\otimes f\left(t\right)\right)+p^{*}\left(t\right)\odot n\left(t\right),$$

where ⊙ denotes the correlation integral.

We now consider the autocorrelation of the chirp function, which is given by

$$p^{*}\left(t\right)\odot p\left(t\right)={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}}exp[-i\alpha {(t+\tau )}^2]exp\left(i\alpha {\tau }^2\right)d\tau .$$

Expanding the exponent and simplifying, we obtain

$$p^{*}\left(t\right)\odot p\left(t\right)=exp(-i\alpha t^2){\displaystyle \underset{-\infty }{\overset{\infty }{\int }}}exp(-i2\alpha t\tau )d\tau .$$

Using the Fourier representation of the delta function, this evaluates to

$$p^{*}\left(t\right)\odot p\left(t\right)={\displaystyle \frac{\pi }{\alpha }}\delta \left(t\right),$$

where the delta function $\delta \left(t\right)$ is defined by

$$\delta \left(t\right)={\displaystyle \frac{1}{2\pi }}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}}exp(-ixt)dx.$$

Substituting this result into the correlation expression for $s\left(t\right)$ yields

$$p^{*}\left(t\right)\odot s\left(t\right)={\displaystyle \frac{\pi }{\alpha }}\delta \left(t\right)\otimes f\left(t\right)+p^{*}\left(t\right)\odot n\left(t\right).$$

Using the sifting property of the delta function, this reduces to

$$p^{*}\left(t\right)\odot s\left(t\right)={\displaystyle \frac{\pi }{\alpha }}f\left(t\right)+p^{*}\left(t\right)\odot n\left(t\right).$$

Exact recovery of the information function is therefore achieved provided that

$$p^{*}\left(t\right)\odot n\left(t\right)=0.$$

This condition implies that the functions $p^{*}\left(t\right)$ and $n\left(t\right)$ are uncorrelated, meaning that their cross-correlation function is zero. Cross-correlation provides a measure of similarity between two functions. Since $n\left(t\right)$ is assumed to be a stochastic process and $p^{*}\left(t\right)$ is deterministic, there are, in principle, no matching features between the two functions, and their cross-correlation vanishes.

Under this assumption, and ignoring the scaling constant $\pi /\alpha$, the information function may be recovered as

$$f\left(t\right)=p^{*}\left(t\right)\odot s\left(t\right).$$

This result assumes that the noise function $n\left(t\right)$ is a zero-mean random process and is ergodic, meaning that its statistical properties averaged over time are representative of the ensemble behaviour. This assumption is examined further in Appendix A, where a proof of this result is provided using both time-averaged correlation methods and a Fourier-domain analysis.

In the following section, we explore the uniqueness of this result through a study of self-characteristic functions of the Fourier transform. This analysis leads to the proposition and proof that the chirp function is the only self-characteristic phase-only function. In other words, the chirp function is the only phase-only function whose Fourier transform is also phase-only, thereby establishing its fundamental uniqueness in the context of optimal signal recovery.

5. Self-Characteristic Functions of the Fourier Transform

The purpose of this section is to revisit the concept of self-characteristic functions of the Fourier transform, providing a brief background before introducing a fundamental conjecture associated with the chirp function. Specifically, we consider the proposition that the chirp function is the only phase-only function that possesses a conjugate eigenfunction under Fourier transformation.

It is well known that the Fourier transform can be applied to a wide range of functions in one or more dimensions. In certain cases, the resulting Fourier transform $F\left(\omega \right)$ retains a characteristic relationship with the original function $f\left(t\right)$. Functions exhibiting this property are commonly referred to as self-characteristic functions. These functions are of particular interest in signal processing and communications engineering because they preserve structural properties under transformation between time and frequency domains.

More generally, for a (real or complex) constant $\lambda$, some functions $f\left(t\right)$ yield a Fourier transform such that

$$F\left(\omega \right)=\mathcal{F}\left[f\right(t\left)\right]=\lambda f\left(\omega \right).$$

(1)

Functions that satisfy Equation (1) are known as eigenfunctions of the Fourier transform operator, with eigenvalue $\lambda \in \mathbb{C}$. Eigenfunctions of the Fourier transform play an important role in many areas of applied mathematics, physics, and engineering, particularly in problems involving symmetry, invariance, and optimal signal representations.

These eigenfunctions can be broadly classified into two categories:

• Non-periodic eigenfunctions, which satisfy Equation (1) without additional constraints;
• Periodic eigenfunctions, which satisfy Equation (1) subject to the periodicity condition $f(t+p)=f\left(t\right),$ where p denotes the period of the function.

5.1. Example of Self-Characteristic Functions

Self-characteristic (and self-similar) functions of the Fourier transform can be viewed as generalisations of eigenfunctions, incorporating a scaling factor a, for which the Fourier scaling theorem applies:

$$\mathcal{F}\left[f\left(at\right)\right]={\displaystyle \frac{1}{\left|a\right|}}F\left({\displaystyle \frac{\omega }{a}}\right).$$

(2)

Some examples of such functions are provided below, expressed in terms of non-unitary Fourier transform pairs (where ↔ denotes Fourier transform) [15]:

$$exp(-a{t}^2)\leftrightarrow \sqrt{{\displaystyle \frac{\pi }{a}}}exp\left(-{\displaystyle \frac{{\omega }^2}{4a}}\right)$$

(3)

$$\begin{array}{cc}\hfill cos\left(a{t}^2\right)& \leftrightarrow \sqrt{{\displaystyle \frac{\pi }{a}}}cos\left({\displaystyle \frac{{\omega }^2}{4a}}-{\displaystyle \frac{\pi }{4}}\right),\hfill \\ \hfill \mathrm{sech}\left(at\right)& \leftrightarrow {\displaystyle \frac{\pi }{a}}\mathrm{sech}\left({\displaystyle \frac{\pi \omega }{2a}}\right),\hfill \\ \hfill {\displaystyle \sum _{m=-\infty }^{\infty }}\delta (t-ma)& \leftrightarrow {\displaystyle \frac{2\pi }{a}}{\displaystyle \sum _{m=-\infty }^{\infty }}\delta \left(\omega -{\displaystyle \frac{2\pi m}{a}}\right).\hfill \end{array}$$

Additionally, a number of functions exhibit similar scaling characteristics, including [15]:

$${\displaystyle \frac{1}{{\left|t\right|}^{\beta }}}\leftrightarrow -{\displaystyle \frac{2sin(\pi \beta /2)\mathsf{\Gamma }(1-\beta )}{{\left|\omega \right|}^{1-\beta }}},$$

(4)

where $0<\beta <1$ and $\mathsf{\Gamma }(1-\beta )$ denotes the Gamma function of $1-\beta$.

These relationships provide a theoretical foundation for understanding how scaling and self-similarity in time-domain functions manifest in the frequency domain. Relationship (4) plays an important role in applications involving self-affine signals and in the solution of fractional differential equations, among other areas [16].

5.2. Self-Characteristic Phase-only Functions of the Fourier Transform

From the scaling theorem presented in Equation (2), it follows that if $a=1$, then

$$\mathcal{F}\left[f\right(t\left)\right]=F\left(\omega \right).$$

If $f\left(t\right)$ is an eigenfunction, then by definition, for some eigenvalue $\lambda$ (which may be real or complex),

$$\mathcal{F}\left[f\right(t\left)\right]=\lambda f\left(\omega \right).$$

However, if we set $a=i$, Equation (2) implies

$$\mathcal{F}\left[f\right(it\left)\right]=F(-i\omega ).$$

Consequently, for a function $f\left(t\right)$ that is an eigenfunction in the case $a=1$, it is no longer possible to define a conventional eigenfunction for $a=i$. Instead, one obtains a conjugate eigen-equation of the form

$$\mathcal{F}\left[f\left(it\right)\right]=\lambda f^{*}\left(i\omega \right),$$

where $f^{*}$ denotes the complex conjugate of f.

This observation demonstrates that pure phase-only eigenfunctions of the form $f\left(t\right)=exp[\pm i\phi (t\left)\right]$ do not exist. Rather, only conjugate phase-only eigenfunctions can be defined. As an illustrative example, consider the Gaussian function in the relationship (3) for $a\mapsto \pm ia$ which transforms to

$$exp(\pm ia{t}^2)\leftrightarrow exp(\mp i\pi /4)\sqrt{{\displaystyle \frac{\pi }{a}}}exp\left(\mp i{\displaystyle \frac{{\omega }^2}{4a}}\right).$$

where, for the special case, when $a=1/2$,

$$exp(\pm i{t}^2/2)\leftrightarrow cexp\left(\mp i{\omega }^2/2\right),c=exp(\mp i\pi /4)\sqrt{2\pi }.$$

Thus, the Fourier transform of $exp(\pm i{t}^2/2)$ produces a conjugate phase-only eigenfunction with complex eigenvalue c.

The central question that arises is whether this result is unique. In other words, can we assert that

$$exp[\pm i\phi \left(t\right)]\leftrightarrow cexp[\mp i\phi \left(\omega \right)]\iff \phi \left(t\right)=t^2/2,$$

where ⇔ denotes “if and only if”. The answer to this question forms the basis of the Theorem and Proof provided in Appendix B, which demonstrates that this property holds if and only if $\phi$ is a polynomial of degree at most 2.

This result constitutes the central theme of the paper, establishing that, in the context of Section 3, chirp (convolutional) coding is uniquely optimal for transmitting information in the presence of (zero mean) additive noise.

6. Discussion and Conclusions

The analysis presented in this paper demonstrates the unique mathematical and physical properties of chirp functions and chirplet-based modulation in communications engineering. Beginning with a rigorous exploration of eigenfunctions and self-characteristic functions of the Fourier transform, we have shown that chirp functions, specifically of the form $exp(\pm i\alpha t^2)$, exhibit a conjugate phase-only eigenfunction under Fourier transformation. This property, in combination with their ideal autocorrelation characteristics, underpins the optimality of chirp-based coding for the extraction of information from noise.

Using a convolution transmission model, we demonstrated that correlating the received signal with the complex conjugate of a chirp impulse response, $p^{*}\left(t\right)$, recovers the information signal exactly, provided that the noise is uncorrelated and zero-mean. This result is further generalised through the scaling theorem and analysis of the Fourier transform of chirp functions, showing that the chirp is the only phase-only function that satisfies the self-conjugate eigenfunction property. Appendix B formalises this result and proves that any phase-only function exhibiting this property must be a quadratic polynomial in the phase, confirming the uniqueness of chirp-based coding.

The implications of these findings are significant for practical communication systems. Chirp-based modulation provides optimal robustness to additive noise, Doppler shifts, and multipath interference, making it highly suitable for wireless telecommunications, radar, and low-power IoT applications. Unlike alternative modulation schemes, which may only approximate optimality under specific conditions, chirp coding is theoretically proven to be uniquely optimal across a broad range of practical scenarios. Furthermore, the generalisation to chirplets allows for flexible time-windowed implementations that can be adapted to varying pulse durations and signal bandwidth requirements. In this context, Chirp coding will offer uniquely optimal detection of extraterrestrial intelligence signals, maximising robustness against noise and interstellar dispersion effects [17,18].

In conclusion, this work establishes a rigorous theoretical foundation for the use of chirp functions in communications engineering. By combining time-domain correlation, Fourier-domain self-characteristic properties, and the uniqueness of the quadratic-phase eigenfunction, we have shown that chirp-based modulation offers a fundamentally optimal approach to information transmission in noise-dominated environments. Future research may explore extensions to multi-dimensional chirp coding, adaptive chirplet transforms, and the integration of chirp-based techniques with emerging communication standards, further enhancing reliability, scalability, and implementation efficiency in complex signal processing systems.