Analog-Digital Mapping Paradigm

• Analog-digital mapping is a framework that defines methods for converting continuous signals into discrete representations, crucial for signal processing, quantum computation, and cyber-physical systems.
• It integrates diverse methodologies—from Shannon/Nyquist sampling to neuromorphic and quantum-classical interfacing—using rigorous mathematical models and innovative device architectures.
• The paradigm systematically addresses challenges such as error propagation, quantization noise, and computational limits, thereby influencing hardware design and algorithmic co-optimization.

The analog-digital mapping paradigm encompasses a rich set of theoretical frameworks, device architectures, and algorithmic strategies for representing, converting, and processing signals or states across analog (continuous, physical, or high-dimensional) and digital (discrete, logical, or finite-alphabet) domains. This paradigm is foundational across signal processing, quantum computation, neural modeling, cyber-physical systems, and information theory. It governs not only practical procedures for analog-to-digital (A/D) and digital-to-analog (D/A) conversion––from Shannon/Nyquist-based sampling to neuromorphic and quantum-classical interface protocols––but also exposes deep limitations and opportunities as determined by the structure of the underlying mathematics, the physics of devices, and the computability of abstract representations.

1. Mathematical Structures and Mapping Principles

The formalization of analog-digital mapping begins with the mathematical models used to describe analog and digital objects. In classical signal processing, analog signals $x(t)$ are typically modeled as elements of function spaces (e.g., $L^2(\mathbb{R})$ or Paley–Wiener/Bernstein spaces of bandlimited functions), while digital signals are finite or countable sequences (e.g., $x[n]$), possibly quantized in amplitude. The canonical A/D pipeline consists of anti-aliasing filtering, uniform or non-uniform time sampling, and quantization to a finite alphabet. In this context, the mapping is rigorously described by an operator

$$T_{\rm AD}: f(t) \mapsto \{ Q_\Delta(f(nT+\tau)) : n \in \mathbb{Z} \},$$

where $Q_\Delta$ is a uniform quantizer and $T$ is the sampling period.

More advanced formulations, such as the lattice-function approach, model the output of A/D converters as lattice functions—continuous-time functions $f(t)$ constrained so that $f(nT+\tau) = m_n \Delta + \gamma$ for $n \in \mathbb{Z}$, $m_n \in \mathbb{Z}$, with both time and amplitude quantized. These are shown to correspond, via the Paley–Wiener theorem and Boas' theorem on integral-valued entire functions, to a class of analytic functions with strictly constrained spectral properties (Martínez-Nuevo et al., 2018).

In quantum settings, the mapping is formulated either as a transformation between quantum registers (digital: finite-dimensional qubits, analog: infinite-dimensional continuous-variable modes) or between quantum and classical descriptions, as in decoherence-driven digital-to-analog conversion (SaiToh, 2014), and via explicit AD/DA unitaries for hybrid quantum systems (Liu et al., 27 Aug 2024).

In computational and cyber-physical contexts, the mapping paradigm further encompasses digital twin representations, emphasizing the algorithmic and computability-theoretic properties of alternative digital encodings—such as sample series versus continuous-time series expansion representations of bandlimited functions (Boche et al., 2022).

2. Device-Level Realizations and Signal Conversion Strategies

A spectrum of device-level mapping strategies has been engineered for both classical and quantum platforms:

• Threshold-reliant arrays: Memristive A/D (and D/A) converters utilize arrays of bistable resistive devices, each encoding comparator logic and storage in a single physical entity. A two-stage pulsed voltage protocol determines which devices switch, directly mapping the analog input to a digital code without the need for separate latches; D/A conversion exploits the same hardware via simple current summing (Pershin et al., 2011).
• Unlimited dynamic range ADCs: The modulo-based “self-reset” ADC applies a folding function $$\sigma(x) = x-2V_{\rm ref} \lfloor \frac{x+V_{\rm ref}}{2V_{\rm ref}} \rfloor$$, ensuring every input sample falls within a bounded quantization window; digital recovery applies a cumulative sum, leading to an invertible and non-clipping mapping (Krishna et al., 2019).
• Deep neural network-enabled photonic front-ends: Here, ultrafast photonic sampling is followed by high-throughput, imperfect electronic quantization. Convolutional neural networks then nonlinearly invert device defects and channel mismatches, achieving high ENOB and bandwidth within purely data-driven, learnable mapping blocks (Xu et al., 2018).
• Analog in-memory FFTs: Structured transformations such as the Cooley–Tukey FFT are mapped recursively onto charge-trapping memory arrays (SONOS), where each array is programmed to execute small DFT/MVM steps, with digital twiddle operations between stages. The mapping is parameterized by physical array dimension $K$, device nonidealities, and peripheral circuit constraints (Xiao et al., 27 Sep 2024).

Each of these architectures embodies a distinct analog-digital mapping principle: fold-and-unfold, physically-coded-lattice-decoding, hierarchical algorithmic mapping (FFT factorization), or learnable/nonlinear inversion.

3. Error Propagation, Precision, and Performance Metrics

Analog-digital mappings are subject to error from multiple sources: quantization noise, device/programming errors, analog drift and IR drops (in memristive arrays and charge-trapping crossbars), nonlinearity and mismatch (in photonic front-ends), A/D conversion artifacts (aliasing, clipping), as well as stochastic and systematic errors in quantum conversions.

Key error and performance measures include:

• Cumulative error propagation in recursive mappings (FFT), modeled as additive across logarithmic-depth stages:

$$|\Delta X_k| \lesssim \sum_{\ell=1}^{\log_K N} [ \|x^{(\ell)}\|_1\,\delta_w^{(\ell)} + \delta_q^{(\ell)} + \epsilon_{IR}^{(\ell)} ],$$

with $x^{(\ell)}$ the intermediate vector at stage $\ell$, $\delta_w$ and $\delta_q$ the relative and quantization errors, and $\epsilon_{IR}$ the IR-drop systematic error (Xiao et al., 27 Sep 2024).

• Quantization-induced bandwidth inflation: No matter how fine the amplitude quantizer, any time-and-amplitude-sampled, bandlimited function must introduce a minimum out-of-band spectral component, with

$$\max\,{\rm supp}\,\mathcal{F}\{ Q(x) \}(\omega) \geq 0.8/T$$

regardless of $\Delta$ (Martínez-Nuevo et al., 2018).

• Signal-to-quantization-noise ratio (SQNR) and signal recovery fidelity, displaying nontrivial jumps as a function of digital mapping strategy (e.g., deep-learned vs. hardware-only ADC) (Xu et al., 2018).
• In quantum DAQC or hybrid analog-digital circuits, performance is measured by Frobenius norm error for target unitaries, resource scaling in number of analog blocks or circuit depth, and fidelities in the presence of device noise (Garcia-de-Andoin et al., 2023, Li et al., 8 Oct 2024, Kumar et al., 2 May 2024).

These metrics are used not only in target-agnostic acquisition, but also in task-driven contexts where estimation or inference is paramount, as formalized in task-based ADC approaches (Neuhaus et al., 2020).

4. Theoretical Foundations: Transformations, Computability, and Information Limits

Mapping between analog and digital domains is constrained not only by device architecture but fundamentally by mathematical and algorithmic structures. Key theoretical pillars include:

• Rate–distortion tradeoffs: Joint optimization of sampling and quantization, with minimal mean-squared error in reconstruction recovered by aligning sampler design with rate-distortion-preserving spectrum (water-filling over preserved subbands) (Kipnis et al., 2018).
• Impossibility results in computability: There exist digital twin representations (sampling-based vs. continuous-expansion) that, while analytically equivalent, do not support mutually efficient conversion. For bandlimited functions, the peak-to-average power ratio or $L^\infty$ norms are not computable from sample data alone, and bounded input–bounded output (BIBO) norms are similarly uncomputable in sample-only representations (Boche et al., 2022).
• Quantum resource lower bounds: In mapping n-qubit digital data into continuous-variable analog amplitudes (or analogizing the QFT), any such transfer requires exponential phase-space volume or runtime: minimal run-time scaling is exponential in $n$ due to the need to resolve $2^n$ distinguishable states, set by sampling-theoretic arguments (Liu et al., 27 Aug 2024, SaiToh, 2014).
• Noninvertibility and information loss: Neural mapping cascades (e.g., from analog sensory signals to digital attractor states in cortex) act as many-to-one mappings, erasing analog trajectory histories and thereby establishing a mathematically irreducible boundary on explanatory reduction (Ahissar et al., 20 Feb 2024).

The upshot is that analog–digital mappings are inherently constrained by the interplay of algebraic, analytic, and computational structure, setting both the power and the limitations of possible transformations.

5. Algorithmic Paradigms: Recursive, Task-Based, and Hybrid Mappings

A range of algorithmic frameworks has emerged to maximize efficiency, scalability, or task-specific performance:

• Recursive/Hierarchical Mappings: The Cooley–Tukey FFT maps large DFTs onto sequences of small MVMs or programmable arrays, leveraging divide-and-conquer to balance device and circuit non-idealities while preserving the desired $\mathcal{O}(N\log N)$ complexity (Xiao et al., 27 Sep 2024).
• Task-Based Mappings: When the system objective is not signal reconstruction but recovery of a functionally relevant parameter (the "task"), analog filtering, sampling, and quantization are jointly optimized to minimize estimation MSE under bit-rate and hardware constraints (Neuhaus et al., 2020).
• Quantum/Classical Hybrids: DAQC decomposes general quantum evolution into sequences of analog Hamiltonian blocks (e.g., Ising, XY) interleaved with single-qubit digital rotations, enabling more efficient simulation and circuit depth reduction at the code level (Parra-Rodriguez et al., 2018, Garcia-de-Andoin et al., 2023, Gonzalez-Raya et al., 2020, Martin, 19 Jan 2024).
• Learned and Non-Unitary Quantum Mappings: Deep learning or quantum ensemble-based protocols perform highly nonlinear A/D inversion; in decoherence-assisted quantum DAC, classical signal amplitudes emerge from parallelized channel actions (dephasing/depolarization) (SaiToh, 2014, Xu et al., 2018).
• Generic Linear Transform Extension: Structured transforms beyond FFT (DCT, Winograd convolution, wavelets, linear filterbanks) can be factorized and mapped to analog IMC substrates, combining analog acceleration for core computation with small digital stages for twiddle factors or index manipulations (Xiao et al., 27 Sep 2024).

These paradigms are typically tailored ("hardware-aware algorithm redesign") to the salient constraints of the underlying substrate––array size, error sources, bit-growth, and energy budgets––and can trade off among cost, error robustness, and speed.

6. Foundational and Philosophical Dimensions

The analog–digital mapping paradigm has deep implications beyond circuit design and algorithmics, influencing foundational questions in physics, neuroscience, and information theory:

• Layered emergence: There exists a conceptual hierarchy—from pure analog dynamics of non-commuting variables in trace dynamics, to emergent digital (quantum) laws by statistical averaging, to semi-digital (quantum on classical time), and eventual return to analog (classical) in the macroscopic limit. Transitions between these layers are controlled by system size, energy scales, and coupling strengths (Singh, 2011).
• Neural and cognitive dualities: Perceptual and cognitive processes can be analyzed as mappings between continuous, dynamical (brain–world, BW) representations and discrete, categorical (brain–brain, BB) codes, implemented via hierarchical comparator loops. The irreversibility and many–to–one nature of these mappings lead to practical and philosophical dualisms (mind–body, physical–mental) grounded in neural computation (Ahissar et al., 20 Feb 2024).
• Limits of digital twin fidelity: Not all digital representations (twins) of analog systems are equally computable or sufficient for reconstructing or predicting physical behavior. For systems whose critical constraints depend on non-computable properties (peak values, BIBO norms), sample-only twins are inadequate, motivating the search for representations that retain all relevant analog invariants (Boche et al., 2022).

These foundational perspectives inform both the selection of mapping strategy and the epistemological confidence in digital abstractions of physical systems, impacting system reliability, safety, and scope of representation.

7. Outlook: Extensibility and Emerging Directions

Current and emerging work extends the analog-digital mapping paradigm in several directions:

• Generalization to Structured Linear Operators: Any linear transformation with exploitable structure—circulant, block, separable/dimensional—can be parsed and mapped recursively or blockwise onto analog IMC plus lightweight digital control (Xiao et al., 27 Sep 2024).
• Reconfigurability and Programmability: Hardware arrays can be reprogrammed or partially addressed to implement DFTs or related transforms of arbitrary size, introducing flexibility in edge and embedded contexts (Xiao et al., 27 Sep 2024).
• Quantum-Classical-Analog Interfacing: Hybrid quantum information processing (CV-DV quantum processors, quantum signal processing) necessitates efficient and scalable analog-digital mappings, with resources and error scaling now an active area of complexity theory (Liu et al., 27 Aug 2024).
• Device/Algorithm Co-Design: Increased focus on co-optimizing physical substrate (e.g., crossbar size, charge-trap characteristics, noise floor) and algorithmic mapping (partitioning, radix choice, error correction) for end-to-end efficiency and reliability (Xiao et al., 27 Sep 2024, Neuhaus et al., 2020).
• Hybrid data-driven mapping: Deep-learning-enabled analog-digital mappings that learn to invert defects and nonlinearities open up new design spaces, especially as hardware-accelerated learning devices proliferate (Xu et al., 2018).

The analog–digital mapping paradigm is thus central and evolving across physical, algorithmic, and theoretical axes in computational science and engineering. Its progress will continue to inform the scalable design of edge, quantum, neuromorphic, and cyber-physical systems.