Orthogonal Frequency Square Representation

• Orthogonal Frequency Square Representation is a structured framework organizing data on 2D matrices that exhibit strong algebraic and combinatorial orthogonality.
• It underpins applications in space–time coding, OTFS modulation for wireless communications, and balanced experimental designs.
• Efficient computational methods like DSFT and OCCPT enable robust equalization, full channel diversity capture, and fast signal processing.

Orthogonal Frequency Square Representation is a structured mathematical framework that appears across coding theory, signal processing, wireless communication, and combinatorics. The unifying principle is the organization of data or signal components in a two-dimensional (typically square) matrix such that strong algebraic or combinatorial orthogonality holds. The term is instantiated in several disciplines: complex orthogonal designs for space–time codes, 2D basis designs in modern wireless signaling (OTFS, OSTF), harmonic and combinatorial decompositions (Fourier, Hadamard, frequency squares), and algebraic maximality constructions for experimental design.

1. Algebraic Structures and Square Complex Orthogonal Designs

The theory of square complex orthogonal designs (CODs) underpins much of the algebraic methodology in orthogonal frequency square representation (Li, 2012). A square COD is defined as an $n \times n$ matrix $\mathcal{O}_z$, each entry being a complex linear combination of indeterminates $z_i$ and their conjugates:

$$\mathcal{O}_z^{H} \mathcal{O}_z = \left(\sum_{i=1}^k |z_i|^2\right) I_n$$

The structure theorem for CODs imposes rigid divisibility: a square COD of size $[n, n, k]$ exists if and only if $2^{k-1}$ divides $n$, and all such CODs are, up to unitary equivalence, block-diagonal forms composed of canonical matrices $C_k$ and $C_k'$ defined by recursive relations. The underlying theory is rooted in representation theory; the algebraic condition is enforced by matrices generating a Clifford algebra, with anticommutation relations:

$G_i G_j = -G_j G_i, \quad G_i^H = -G_i$

This group-theoretic perspective uniquely determines the decomposition of any square COD and, by extension, canonical forms for square orthogonal matrix representations in coding theory.

2. Signal Processing: 2D Orthogonal Basis and OTFS Modulation

Modern wireless signaling leverages orthogonal frequency square representations through two-dimensional orthogonal bases in schemes such as Orthogonal Time Frequency Space (OTFS) signaling (Zemen et al., 2017, Nimr et al., 2018, Mohammed et al., 2023, Yang, 24 Jan 2024). In OTFS, data is mapped onto a 2D grid parameterized by delay and Doppler, using a symplectic finite Fourier transform (SFFT/DSFT):

$$x[m, q] = \frac{1}{\sqrt{MN}} \sum_{n=0}^{M-1} \sum_{p=0}^{N-1} d[n, p] e^{j2\pi\left(\frac{mn}{M} - \frac{qp}{N}\right)}$$

The orthogonal spreading ensures each transmitted symbol experiences full channel diversity, as opposed to the one-dimensional assignment in OFDM. For frequency square representation, the set of DSFT (or SFFT) basis functions forms a 2D orthogonal "square" facilitating diversity capture and efficient equalization.

OTFS can be realized within the GFDM framework by permuting the allocation of time-frequency resources. The permutation connects subsymbols and subcarriers, leading to per-symbol SNR uniformity (Nimr et al., 2018). With appropriately designed basis and equalization methods (notably MMSE and iterative interference cancellation), robust performance in high-Doppler mobility scenarios is observed, substantially outperforming OFDM by 3–5 dB in certain channel models (Zemen et al., 2017).

3. Combinatorial Orthogonality: Frequency Squares and Maximal Sets

Combinatorial orthogonal frequency square representations are formalized through the paper of frequency squares and their maximal mutually orthogonal sets (MOFS, maxMOFS) (Britz et al., 2019, Jedwab et al., 2020, Cavenagh et al., 2020, Bodkin et al., 2022, Rahim et al., 2022). An $n \times n$ frequency square of type $(n; d)$ has each symbol appearing $d$ times in each row and column from a multiset. In the binary case, $d = n/2$, all rows and columns have equal numbers of $0$ and $1$.

Orthogonality between two frequency squares $F$ and $F'$ requires that each ordered pair $(i,j)$ appears $n^2 / 4$ times:

$$\forall\, (i,j)\in\{0,1\}^2,\quad |\{ (r,c) : (F(r,c), F'(r,c)) = (i,j) \}| = \frac{n^2}{4}$$

For $k$ mutually orthogonal frequency squares (e.g., $k$-MOFS), the maximum cardinality is limited by combinatorial constraints, notably $k \leq (n-1)^2$ in the binary case (Britz et al., 2019). Existence of complete MOFS is tied to Hadamard matrices of order $n$; when such matrices exist, there are at least $2^{n^2/4-O(n\log n)}$ isomorphism classes of complete MOFS (Britz et al., 2019).

The representation via indicator {0,1} matrices formalizes frequency squares for algebraic analysis, proving classical bounds and maximality via block structure and parity/modular conditions (Jedwab et al., 2020, Bodkin et al., 2022). These conditions generalize earlier parity-based maximality proofs to broader modular settings.

4. Fourier and Harmonic Series: Orthogonal Functional Decomposition

From a functional perspective, orthogonal frequency square representations generalize harmonic decomposition into two dimensions (Aristidi, 2018). Standard 1D series include:

• Fourier series (complex exponentials, periodic boundary)
• Fourier–Legendre (orthogonal polynomials, bounded interval)
• Fourier–Bessel (circular/radial symmetry)
• Spherical Harmonic series (2D sphere)

The decomposition onto mutually orthogonal bases ensures that 2D signals (images, spectrograms, etc.) can be efficiently represented using appropriate products (or tensor squares) of orthogonal functions in each dimension. For square representation, one typically employs products of orthogonal bases adapted to the problem geometry.

5. Computational Realizations: Fast Algorithms and Channel Equalization

Fast realization of orthogonal frequency square transforms is typified by OCCPT and related algorithms (Shah et al., 2021). OCCPT employs pairs of cosine and sine basis sums (Complex Conjugate Pair Sums, CCPS) to span each frequency "square" (2D subspace):

$$x(n) = \sum_{p|N} \sum_{k \in \hat{U}_p} [\beta_{0,k} c^{(1)}_{p,k}(n) + \beta_{1,k} c^{(2)}_{p,k}(n)]$$

Here, each $(\beta_{0,k}, \beta_{1,k})$ pair spans a real-valued frequency square component (cosine/sine; amplitude/phase) for period $p$. FOCCPT, the fast OCCPT algorithm, achieves multiplicative complexity $N \log_2(N) - N +1$, lower than FFT for real inputs.

In OTFS systems, the channel matrix induced by practical (rectangular) waveforms preserves a block-circulant structure (Xu et al., 2019). This property allows low-complexity algorithms for Zero-Forcing (ZF) and MMSE equalization by reducing the full $NM \times NM$ matrix inversion to a series of $M \times M$ invertible blocks, solvable efficiently with FFT and sparse LU factorization.

6. Applications in Wireless Communication, Experimental Design, and Radar Sensing

Orthogonal frequency square representation has broad application impact:

• Wireless communication: OTFS leverages frequency squares in the delay–Doppler domain to achieve non-fading, uniformly predictable channels even under doubly selective conditions (Mohammed et al., 2023). The channel parameterization, waveform design (Zak transform), and equalization strategies are fundamentally 2D square-based.
• Experimental design and coding theory: Binary frequency squares and their maximal mutually orthogonal sets (MOFS) provide robust combinatorial designs for statistically balanced experiments, error-correcting codes, and equidistant permutation arrays. Existence of maximal sets is connected to Hadamard matrices, t–independent vector sets, and orthogonal arrays (Bodkin et al., 2022, Rahim et al., 2022).
• Radar sensing: The DD domain square arrangement in OTFS enables high-resolution estimation of both delay and Doppler, yielding precise radar images for applications in autonomous vehicles and joint communication/radar platforms (Mohammed et al., 2023).

7. Comparative Perspectives and Structural Constraints

While orthogonal frequency square representation in coding theory (square CODs, Clifford algebras) is uniquely constrained by group representation and anti-commutation relations, signal processing approaches (DFT/SFFT, OTFS, OCCPT) rest on orthogonal function decomposition and resource spread over 2D grids. In combinatorics, maximality and existence bounds derive from combinatorial enumeration, block structure, parity, and modular arithmetic. The distinction between algebraic/group-theoretic, functional/harmonic, and combinatorial paradigms is crucial when extending or adapting orthogonal frequency square representation to new domains.

Tables and summary formulas formalize key quantitative relationships:

Framework	Orthogonality Condition	Existence Criterion
Square COD (Li, 2012)	$$\mathcal{O}_z^H \mathcal{O}_z = (\sum |z_i|^2 ) I_n$$	$2^{k-1}$ divides $n$
Binary MOFS (Britz et al., 2019)	Each $(i,j)$ pair appears $n^2 / 4$ times	$k \leq (n-1)^2$; Hadamard
OTFS (Zemen et al., 2017)	DSFT/SFFT process yields 2D spreading of symbols	Complete basis; time/frequency
OCCPT (Shah et al., 2021)	Cosine/sine basis pairs are mutually orthogonal	Decomposition by period $p$

The landscape of orthogonal frequency square representation captures a spectrum of algebraic, analytic, and combinatorial methods for constructing 2D orthogonal grids suited for full diversity extraction, balanced design, and spectral decomposition, with each domain governed by its own existence constraints, computational methods, and application priorities.