Shifted-Signal Formulation in Signal Processing

• Shifted-Signal Formulation is a framework that models signals using explicit shift operations on their arguments, domains, or indices, applicable to discrete, continuous, and multidimensional cases.
• It facilitates robust shift retrieval and sparse representation by leveraging cyclic operators, set-function shifts, and spectral decompositions across various practical applications.
• The formulation underpins efficient transform designs and signal processing strategies in communications, radar, and optimization, integrating algebraic and analytic techniques.

A shifted-signal formulation describes the algebraic, analytic, or algorithmic modeling of signals that are related via explicit shift operations on their arguments, domains, or indices. This framework appears across discrete, continuous, and multidimensional signal processing; it underpins canonical results in shift retrieval, sparse modeling, efficient transforms, OFDM system design, analytic signal theory, and the modern theory of set-function signal processing. The core concept is to operationalize shift—whether by permutation, modulation, coordinate translation, or set operation—and organize models, operations, and inference around this structure.

1. Mathematical Foundations of Shifted Signals

A foundational model for discrete signals employs the circular shift operator. For $N$-length vectors $x \in \mathbb{C}^N$, a shift $s$ is realized via $y[n] = x[(n-s) \bmod N]$, equivalently $y = D^s x$ with $D^s$ an $N \times N$ circulant permutation matrix, $(D^s)_{ij} = 1$ if $i \equiv j + s \pmod N$ and $0$ otherwise (Ohlsson et al., 2013). In multidimensional continuous contexts, shifted-signal decomposition indexes phase-shifted signal components $f_j(x)$ by binary multi-indices $j$, with kernels $$\alpha_j(x,\omega) = \prod_{\ell=1}^d \cos(\omega_\ell x_\ell - j_\ell \pi/2)$$ and associated mixed Hilbert transforms (Tsitsvero et al., 2017). For set-function signals $f:2^N \to \mathbb{R}$, the shift can be implemented by set union, difference, or symmetric difference: $T_X^\cup[f](A) = f(A \cup X)$, $T_X^-[f](A) = f(A \setminus X)$, $T_X^\Delta[f](A) = f(A \Delta X)$ (Püschel et al., 2020).

In OFDM systems, the time-domain sequence $x(n)$ is decomposed into $V$ subblocks $x_v(n)$, each cyclically shifted by $\tau_v^{(u)}$; the candidate signal is $$x^{(u)}(n) = \sum_{v=1}^V x_v((n-\tau_v^{(u)}) \bmod N)$$ (Kim, 2015).

2. Shift Retrieval and Compressive Models

The classical shift retrieval problem seeks to estimate the unknown shift $\ell$ relating signals $x$ and $y$ via $y = D^\ell x$. The standard estimator maximizes the cross-correlation: $$\ell^* = \arg\max_{0 \leq s < N} \langle y, D^s x \rangle$$, where $\langle \cdot, \cdot \rangle$ is the Hermitian inner product (Ohlsson et al., 2013).

In compressive shift retrieval, only compressed measurements $z = Ax$ and $w = Ay$ are available. Under matrix commutativity and scaling conditions,

• $A D^s = \overline{D}^s A$ for all $s$,
• $AA^H = \alpha I_m$ for some $\alpha \in \mathbb{C}$,

the shift may be recovered via $$\ell^* = \arg\max_{0 \leq s < N} \mathrm{Re} \langle z, \overline{D}^s w \rangle$$ (Ohlsson et al., 2013).

For partial Fourier measurements, with $A$ the partial DFT matrix selecting $k_1,\dots,k_m$ rows, only one non-trivial DFT coefficient can, under mild conditions, uniquely determine the shift. In scenarios with additive noise and well-separated compressed representations, recovery is robust, provided noise levels are moderate (Ohlsson et al., 2013).

3. Shifted Sparse Representations

Sparse decomposition involving shifted signals posits that an unknown shift $\tau \in \{0,1,\dots,n-1\}$ of a signal $s$ admits a sparse representation in an overcomplete dictionary $D$. The operator $S_\tau$ acts as $(S_\tau s)[n] = s[(n-\tau) \bmod n]$ (0809.3485). The optimization seeks

$$\min_{\tau, x} \frac{1}{2} \|Dx - S_\tau s\|_2^2 + \lambda \|x\|_0$$

where $\|x\|_0$ promotes sparsity. In the DFT domain, $S_\tau$ corresponds to modulation by a diagonal matrix of complex exponentials, $W(\tau)$.

A graduated non-convexity algorithm proceeds by minimizing a smoothed sparsity objective with decreasing smoothness parameter $\sigma$, promoting convergence to the joint optimal shift and sparse code. The procedure involves cyclic least-squares fitting for each integer shift, followed by descent on

$$H(x, \tau) = \lambda G(x, \tau) + (1-\lambda) F_\sigma(x)$$

where $G$ encodes DFT-domain data fidelity, and $F_\sigma$ smooths the $\ell_0$ penalty. Empirical verification yields high recovery accuracy for both shift and sparse support under Gaussian noise (0809.3485).

4. Shifted-Signal Models in Multidimensional and Set-Function Spaces

In multidimensional analytic signal theory, the shifted-signal formulation generalizes to $2^d$ phase-shifted signal components $f_j(x)$. The analytic signal in $d$ dimensions is constructed in a commutative Scheffers algebra $S_d$, via a hypercomplex Fourier transform

$$\mathcal{F}_H[f](\omega_1, ..., \omega_d) = \int_{\mathbb{R}^d} f(x) e^{-e_1 \omega_1 x_1} \cdots e^{-e_d \omega_d x_d} dx$$

with positive-frequency restriction yielding all phase-shifted components (Tsitsvero et al., 2017). The Paley-Wiener theorem ensures holomorphic extension in $d$ commuting upper half-planes and links signal analytic continuation to positive-frequency support.

For set-functions, three shift operators—union, difference, and symmetric difference—yield distinct convolution and Fourier transform laws. Each admits jointly diagonalized shift-invariant filters, and explicit spectral formulas:

Shift Model	Shift Operator	Fourier Transform Formula
Union	$T_X^\cup$	$$\widehat{f}^{(1)}(B) = \sum_{A: A \cap B = \emptyset} f(A)$$
Difference	$T_X^-$	$$\widehat{f}^{(3)}(B) = \sum_{A \supseteq B} (-1)^{|A \setminus B|}\,f(A)$$
Symmetric Diff	$T_X^\Delta$	$$\widehat{f}^{(5)}(B) = \sum_{A} (-1)^{|A \cap B|} f(A)$$

Frequency response and convolution theorems follow by spectral pointwise multiplication; explicit inversion formulas are provided in each case. Notably, the difference-model spectrum recovers (up to sign) multivariate mutual information for entropy set-functions (Püschel et al., 2020).

5. Applications in Communications, Signal Processing, and Optimization

In OFDM systems, cyclically shifted sequences (CSS) involve partitioning an $N$-point symbol into $V$ nonoverlapping subblocks, applying individual shifts $\tau_v^{(u)}$ to each, and summing to produce candidate signals. The selection of shift value sets is crucial: maximizing independence of peak-to-average power ratio (PAPR) among candidates improves transmission robustness (Kim, 2015).

Matched filtering for time-frequency shift estimation in communications employs algebraically structured waveforms. Given received signal $$R(t) = \sum_{j=1}^r \pi(\tau_j, \omega_j)[S_j](t) + W(t)$$, with $\pi$ the shift operator over $\mathbb{F}_p$, fast matched filter algorithms using Heisenberg (line) or Weil (peak) group-theoretic signals allow recovery of all $(\tau_j, \omega_j)$ in near-linear time $O(rp \log p)$ (Fish et al., 2011).

Key applications include:

• Multiuser detection (CDMA): assignment of algebraic "flag" waveforms to users enables separate shift retrieval.
• GPS: cross-correlation with known satellite signals recovers range and Doppler using fast algorithms.
• Radar: Doppler and delay estimation for multiple targets.

In set-function signal processing, shifted-signal formulations enable spectral compression and efficient sampling for combinatorial optimization and preference elicitation, leveraging frequency ordering indexed by set cardinality (Püschel et al., 2020).

6. Spectral Properties and Algebraic Structures

The spectral landscape arising from shifted-signal models is strongly determined by the algebraic properties of the underlying shift operators:

• In classical signal processing, circulant structure admits diagonalization by the DFT, and shifts correspond to frequency-domain phase modulation.
• For set-functions, the mutual commutativity of shift maps enables joint diagonalization, explicit spectral decompositions, and convolution-type laws.
• The necessity of commutative elliptic algebras for multidimensional analytic signals reflects the limitations of noncommutative (Clifford, Cayley-Dickson) constructions: in dimension $d>2$, only commutative models recover all $2^d$ phase-shifted components via frequency filtering (Tsitsvero et al., 2017).

The spectrum can often be indexed and ordered by subset cardinality (set-functions) or by frequency bins (DFT), with band-limitation, frequency response, and convolution mapped directly to shift-invariant operations.

7. Principles for Shift Selection and Model Design

Model performance and interpretability depend critically on the explicit choices of shift parameters and value sets:

• In OFDM CSS, selection criteria for shift values include distinctness of relative shifts across candidates and near-$N/2$ separation to minimize PAPR interdependence (Kim, 2015).
• In sparse representation, the shift parameter is estimated jointly with sparse support via descent on a smooth $\ell_0$ surrogate objective, graduated to nonconvexity (0809.3485).
• In set-function SP, shift types mediate model choice (coverage, difference, symmetric difference), each with unique spectral and filtering behavior (Püschel et al., 2020).

Shift value sets and the underlying algebraic, spectral structure determine feasibility and efficiency in practical retrieval, coding, optimization, and signal analysis tasks.