Discrete Oscillator Transform (DOT)

• DOT is a family of unitary transforms that map discrete signals onto eigenfunctions of finite oscillator models using structure-preserving methods.
• It leverages orthogonal polynomials and group-theoretic constructs to ensure unitarity, exact spectral decomposition, and efficient algorithm implementation.
• DOT finds applications in signal processing, radar, communications, and optical systems by providing precise modeling of discrete harmonic oscillators.

The Discrete Oscillator Transform (DOT) is a family of structure-preserving, unitary transforms that map discrete signals onto orthogonal bases of eigenfunctions associated with finite-dimensional quantum harmonic oscillator models. These transforms emerge in finite quantum systems, information theory, signal processing, and mathematical physics, serving as the discrete analogues of the Hermite (oscillator) transform and the fractional Fourier transform. The kernels of various DOTs are typically realized by finite orthogonal polynomials or group-theoretic constructs, ensuring key properties such as unitarity, exact spectral decomposition, and, in some cases, closure under fast algorithms. DOTs provide a framework for diagonalizing discrete analogues of harmonic oscillator Hamiltonians, and their mathematical formulation leverages deep connections to representation theory, especially of $\mathfrak{su}(2)$, the Weil representation, and orthogonal polynomials.

1. Algebraic and Analytical Foundations

At the core of the DOT is the algebraic structure provided by finite analogues of the harmonic oscillator. In the su(2)/Kravchuk oscillator model, each fixed resolution $N$ corresponds to a $(N+1)$-dimensional Hilbert space equipped with canonical $\mathfrak{su}(2)$ generators,

$$J_3 = 2 H_I, \quad J_+ = \frac{1}{2} D^\dagger, \quad J_- = \frac{1}{2} D,$$

with commutation relations

$$[J_3, J_\pm] = \pm J_\pm, \quad [J_+, J_-] = 2 J_3,$$

and quadratic Casimir $\Omega = J_+J_- + J_-J_+ + 2 J_3^2 = N(N+2) I$ (May et al., 2023).

Energy ladder operators $D$, $D^\dagger$, and resolution ladder operators $B_1$, $B_1^\dagger$ permit transformations both within and between different-resolution oscillators. The eigenfunctions in the position basis $\{\psi_{x,N}\}$ are discrete analogues of the continuous Hermite functions and are most canonically realized as Kravchuk functions,

$$\alpha_{x,N}^{(n)} = \langle \psi_{x,N} | n,N \rangle = (-1)^n 2^{-N/2} \sqrt{\binom{N}{n} \binom{N}{\frac{x+N}{2}}} \quad {}_2F_1(-n, -\tfrac{x+N}{2}; -N; 2).$$

Alternatively, in finite field and modular arithmetic settings, oscillator eigenmodes are constructed via the Weil representation of $Sp(2, \mathbb{F}_p)$ and maximal tori, providing a group-theoretic basis diagonalizing key operators such as the discrete Fourier transform (DFT) (0808.1495, 0808.3214, 0902.0668).

2. Definition and Structure of the DOT

The DOT is a unitary transformation mapping a discrete signal in position space to its expansion in the orthogonal basis of eigenfunctions associated with a finite oscillator. For the $(N+1)$-dimensional $\mathfrak{su}(2)$ Kravchuk oscillator, the forward and inverse DOT are: $$F(n) = \sum_{x=-N}^{N} \alpha_{x,N}^{(n)} f(x), \quad f(x) = \sum_{n=0}^{N} \alpha_{x,N}^{(n)} F(n).$$ In matrix notation, DOT$_N$ is realized via the orthonormal matrix $A^{(N)}$ with

$$\mathbf{F} = \mathbf{A}^{(N)} \mathbf{f}, \quad \mathbf{f} = (\mathbf{A}^{(N)})^T \mathbf{F}.$$

For oscillator systems defined over finite fields, the DOT maps $f \in \mathbb{C}(\mathbb{F}_p)$ to mode coefficients $\widehat{f}(j) = \langle f, \varphi_j \rangle$ with respect to oscillator eigenfunctions $\varphi_j$ (0808.1495, Chauleur et al., 2022).

The fundamental properties of the DOT are:

• Unitarity: $A^{(N)}$ is real and orthonormal, so DOT preserves the $\ell^2$-norm.
• Invertibility: The transform is self-inverse up to transpose or conjugation, depending on the setting.
• Spectral decomposition: DOT diagonalizes the corresponding discrete Hamiltonian or DFT.

3. Connections to Finite Oscillator and Signal Models

DOTs arise in several contexts:

• su(2)/Kravchuk Oscillator: The DOT is canonically associated with finite, equally-spaced spectra of the DQHO, with kernels given by Kravchuk polynomials. This transform provides a structure-preserving analogue of the Hermite transform (May et al., 2023, Chauleur et al., 2022).
• Weil Representation and Tori: In settings over finite fields, the DOT arises as the map onto the canonical basis of eigenvectors for the DFT furnished by the Weil representation and its restriction to maximal tori (split or non-split) (0808.3214, 0902.0668). This approach supplies not only the transform but also the machinery for fast algorithms when $p \equiv 1 \pmod{4}$.
• Hahn and Hahn-deformed Oscillators: For models based on su(2)$_\alpha$ deformed algebras, the DOT generalizes via dual Hahn polynomials, yielding a discrete Hahn-Fourier transform with analogous properties (symmetry, unitarity, eigenfrequency structure) (Jafarov et al., 2011).
• Integrated Optical Implementations: In discrete quantum optics, the DOT is realized as the propagator of DQHO in an array of waveguides, directly implementing the discrete fractional Fourier transform (DFrFT) (Urzúa et al., 2024).

4. Computational Methods and Algorithms

The structure of the DOT admits efficient numerical implementations:

• Tridiagonalization and Exponentiation: In the $\mathfrak{su}(2)$ or Kravchuk oscillator, the DOT can be written in terms of the exponent of a tridiagonal matrix $A$, leveraging three-term recurrence relations. This allows computation in $O(N)$–$O(N^2)$ time depending on the numerical approach (Chauleur et al., 2022).
• Chirp-DFT Factorizations: For DOTs derived from the Weil representation, the matrix can be written as a composition of chirp multiplications, DFT, and (when $p \equiv 1 \pmod{4}$) a Mellin transform over multiplicative characters, enabling $O(p \log p)$ complexity (0902.0668, 0808.3281, 0808.3214).
• Dynamic Resolution Scaling: The resolution ladder operators enable changing the transform’s dimension by padding or truncating energy coefficients and reprojecting using the transform matrix of the new resolution, an operation unique to the su(2) formalism (May et al., 2023).

DOT Model	Basis Functions	Fast Algorithm
su(2)/Kravchuk	Kravchuk polynomials	Tridiagonal exp., $O(N)$–$O(N^2)$
Weil/Field	Chirps via Weil representation	Chirp-DFT-Mellin, $O(N \log N)$
Hahn-deformed	(Dual) Hahn polynomials	O($j^2$), no FFT variant
Optical/FrFT	Jacobi polynomials, FrFT kernel	On-chip, direct implementation

5. Relation to Classical Oscillator and Limit Cases

The DOT is designed as a finite, structure-preserving analogue of the continuous oscillator (Hermite) transform. Rigorous results show second-order convergence: $$\|\varphi_{n,h} - \psi_n\|_{L^2} = O(N^{-1+\epsilon})$$ for Kravchuk-based DOTs, where $\varphi_{n,h}$ is the $n$th Kravchuk function for grid parameter $h$, and $\psi_n$ is the $n$th Hermite function, with errors controlled for modes $n$ up to $O(\ln N)$ (Chauleur et al., 2022).

In the context of the DFT, the DOT diagonalizes the DFT operator, with the canonical oscillator modes corresponding to eigenvalues among $\{\pm1, \pm i\}$. The finite oscillator basis thus provides a granular replacement for continuous harmonic analysis in discrete settings (0808.3214, 0902.0668).

6. Applications and Statistical Properties

DOTs have central roles in applications where discrete symmetry, orthogonality, and spectral localization are essential:

• Discrete Radar: Oscillator signals constructed via the DOT have thumbtack ambiguity functions, enabling high-resolution radar implementations with minimal sidelobes (0808.1495, 0808.1417).
• CDMA Communications: Assignment of distinct oscillator waveforms to multiple users ensures that each user’s bit stream can be separated in the presence of many other transmitters, with cross-interference controlled by orthogonality properties (0808.1495, 0808.1417).
• Optical Phase-Space Analysis: On-chip implementations of the DOT in integrated waveguide arrays provide efficient tools for measuring discrete Wigner distributions and fractional spectral content (Urzúa et al., 2024).
• Pseudorandom Sequences: The near-Gaussian law of linear statistics derived from oscillator eigenvectors shows their quasi-random behavior, supporting secure sequence design (0808.1417).

The table below summarizes statistical properties for finite-field oscillator systems:

Property	Value	Reference
Orthonormality	$$\sum_x \varphi_{T,\chi}(x) \overline{\varphi_{T',\chi'}(x)} = \delta_{T,T'} \delta_{\chi,\chi'}$$	(0808.1495)
Ambiguity Bounds	$$|\mathcal{A}_{\varphi,\psi}(t,\omega)| \le 2/\sqrt{p}$$	(0808.1495)
PAPR	$\le 2$	(0808.1495)
Closure under DFT	DFT maps oscillator basis to oscillator basis (modulo phase)	(0808.1495)

7. Extensions and Variants

Significant variants generalize the core concepts of the DOT:

• Fractional and Generalized DOTs: The DOT framework naturally extends to discrete fractional Fourier transforms (DFrFT), where spectral powers of the DFT operator are constructed via polynomial or Vandermonde approaches (Moya-Cessa et al., 2016).
• Hahn/Associated Family Transforms: One-parameter deformations of su(2) yield DOTs whose kernels involve Hahn polynomials, generalizing the spectral, transformation, and analytic properties to broader classes of finite models (Jafarov et al., 2011).
• Resolution Ladder (Dynamic Scaling): The su(2)-based DOT uniquely possesses resolution ladder operators, allowing the dynamic scaling of basis sets and Hilbert space dimension with precise control (May et al., 2023).

A plausible implication is that further generalizations to quantum group or $q$-deformed algebras could yield new classes of DOTs suitable for discretized non-classical or non-equidistant settings, although explicit fast algorithms may not always be available.

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References:

• "Algebraic discrete quantum harmonic oscillator with dynamic resolution scaling" (May et al., 2023)
• "The finite harmonic oscillator and its applications to sequences, communication and radar" (0808.1495)
• "Discrete quantum harmonic oscillator and Kravchuk transform" (Chauleur et al., 2022)
• "On the diagonalization of the discrete Fourier transform" (0808.3281)
• "Application of the Weil representation: diagonalization of the discrete Fourier transform" (0902.0668)
• "Discrete fractional Fourier transform: Vandermonde approach" (Moya-Cessa et al., 2016)
• "The su(2)_α Hahn oscillator and a discrete Hahn-Fourier transform" (Jafarov et al., 2011)
• "Integrated optical wave analyzer using the discrete fractional Fourier transform" (Urzúa et al., 2024)
• "The finite harmonic oscillator and its associated sequences" (0808.1417)