Sinusoidal Displacement Model

• The sinusoidal displacement model is a parametric framework that represents physical or temporal displacements using sine functions defined by amplitude, offset, phase, and frequency.
• It is applied in diverse areas such as 2D statistical estimation, PWM pulse optimization, and modeling disorder in nanocrystal superlattices with tailored symmetry and closed-form solutions.
• Its effectiveness relies on domain-specific approximations and optimization techniques, ensuring accurate parameter inference amidst noise and complex spatial or temporal variations.

A sinusoidal displacement model is a parametric construction in which position, displacement, or displacement-controlled placement is represented by sinusoidal functions. In the recent arXiv literature, the expression is not attached to a single canonical formalism. Instead, it appears in several domain-specific senses: a 2D sinusoidal displacement/intensity model with a constant offset for statistical parameter estimation (Hosseinbor et al., 2017), a displacement-factor parametrization of pulse locations in sinusoidally modulated PWM waveforms (Tyagi et al., 2020), and a collective longitudinal/transverse displacement-wave model for disorder in nanocrystal superlattices (Filippi et al., 13 Sep 2025). These usages share a periodic parametrization, but they differ substantially in what is being displaced, which parameters are unknown, and how inference or optimization is carried out.

1. Scope of the term in current technical usage

The literature uses “sinusoidal displacement” in both literal and extended senses. In some cases, the displaced quantity is an actual spatial coordinate field; in others, it is a positional degree of freedom that determines where a pulse or event is placed within a fixed interval.

Context	Modeled quantity	Representative parameters
2D estimation	Sinusoidal image/displacement/intensity with DC offset	$A,B,\phi,f_0,f_1$
PWM	Pulse placement inside a subinterval	$\alpha_l$
Superlattices	Collective positional disorder of nanocrystals	$A_l,\Lambda_l,A_t,\Lambda_t,\phi$
Driven mechanics	Prescribed sinusoidal motion or load	$H_0,\Delta H,\omega$ or $P_0,\omega,t_d$

In the 2D estimation setting, the model is an explicit sinusoid over a spatial grid with additive white Gaussian noise. In PWM, the sinusoid fixes pulse width, while a displacement factor determines the pulse position inside each switching subinterval. In nanocrystal superlattices, the model is a static collective displacement field with longitudinal and transverse components. In several adjacent mechanics papers, sinusoidal motion is prescribed as an external trajectory or forcing protocol rather than inferred as an unknown field (Hosseinbor et al., 2017, Tyagi et al., 2020, Filippi et al., 13 Sep 2025).

2. Statistical 2D sinusoidal displacement/intensity model with offset

A precise and fully specified sinusoidal displacement/intensity model appears in the estimation problem studied in "2D Sinusoidal Parameter Estimation with Offset Term" (Hosseinbor et al., 2017). The discrete signal is

$$f(x,y)=A\sin\!\big(2\pi(f_0 x+f_1 y)+\phi\big)+B,\qquad x,y=0,\dots,N-1,$$

with noisy observations

$$s(x,y)=f(x,y)+w(x,y), \qquad w(x,y)\sim\mathcal N(0,\sigma^2).$$

The parameter vector is

$\boldsymbol\theta=(A,\;B,\;\phi,\;f_0,\;f_1).$

Here $A$ is the sinusoid amplitude, $B$ is the offset term or DC level, $\alpha_l$0 is the phase, and $\alpha_l$1 are the spatial frequencies in the $\alpha_l$2- and $\alpha_l$3-directions. In the fingerprint interpretation reported in the paper, $\alpha_l$4 encode ridge spacing and orientation, $\alpha_l$5 captures ridge/furrow contrast, and $\alpha_l$6 captures the average grayscale level in a block.

Under i.i.d. Gaussian noise, maximum likelihood estimation is equivalent to least-squares minimization,

$\alpha_l$7

For fixed frequencies $\alpha_l$8, the sinusoid is linearized with

$\alpha_l$9

and the reparametrization

$A_l,\Lambda_l,A_t,\Lambda_t,\phi$0

With sampled sine and cosine basis vectors $A_l,\Lambda_l,A_t,\Lambda_t,\phi$1 and

$A_l,\Lambda_l,A_t,\Lambda_t,\phi$2

the model becomes $A_l,\Lambda_l,A_t,\Lambda_t,\phi$3, giving the closed-form LS/MLE solution

$A_l,\Lambda_l,A_t,\Lambda_t,\phi$4

After profiling out $A_l,\Lambda_l,A_t,\Lambda_t,\phi$5, the remaining frequency search is, under the large-$A_l,\Lambda_l,A_t,\Lambda_t,\phi$6 orthogonality approximations, equivalent to maximizing the 2D periodogram,

$A_l,\Lambda_l,A_t,\Lambda_t,\phi$7

where

$A_l,\Lambda_l,A_t,\Lambda_t,\phi$8

The resulting estimators are

$A_l,\Lambda_l,A_t,\Lambda_t,\phi$9

and

$H_0,\Delta H,\omega$0

The offset term is the distinguishing feature. It introduces an additional parameter into the Fisher information matrix and requires explicit separation of the DC component from the sinusoidal basis. Asymptotically, this is handled by the approximate orthogonality of the constant vector $H_0,\Delta H,\omega$1 to the sine and cosine vectors, so that $H_0,\Delta H,\omega$2 reduces to the sample mean.

The paper also derives Cramér–Rao lower bounds. Using large-$H_0,\Delta H,\omega$3 approximations,

$H_0,\Delta H,\omega$4

$H_0,\Delta H,\omega$5

$H_0,\Delta H,\omega$6

These bounds make explicit that frequency and phase identifiability improve with larger $H_0,\Delta H,\omega$7 and larger $H_0,\Delta H,\omega$8, while amplitude and offset bounds depend chiefly on noise variance and sample size (Hosseinbor et al., 2017).

3. Displacement-factor models in sinusoidal PWM

In power electronics, a sinusoidal displacement model is used in a different sense. "Optimal Time-Domain Sinusoidal Pulse Width Modulation Technique" introduces displacement factors that parameterize pulse locations within switching subintervals while keeping pulse widths determined by a sampled sinusoidal reference (Tyagi et al., 2020).

For the single-phase inverter, the center of the $H_0,\Delta H,\omega$9 subinterval is

$P_0,\omega,t_d$0

and for the desired voltage $P_0,\omega,t_d$1, the $P_0,\omega,t_d$2 pulse width is

$P_0,\omega,t_d$3

The widths are therefore fixed by sinusoidal modulation. The remaining degree of freedom is positional. The displacement factor $P_0,\omega,t_d$4 is the time from the beginning of the $P_0,\omega,t_d$5 subinterval to the rising edge of the pulse, with

$P_0,\omega,t_d$6

The switching instants are

$P_0,\omega,t_d$7

In conventional sinusoidal PWM, $P_0,\omega,t_d$8, so the pulses are centered. The model generalizes centered-pulse PWM into a family of admissible pulse placements indexed by $P_0,\omega,t_d$9.

Harmonic performance is optimized in the time domain. For the single-phase case, the inductor current depends on the switching times, hence on the displacement factors, and the paper minimizes the $$f(x,y)=A\sin\!\big(2\pi(f_0 x+f_1 y)+\phi\big)+B,\qquad x,y=0,\dots,N-1,$$0 tracking error

$$f(x,y)=A\sin\!\big(2\pi(f_0 x+f_1 y)+\phi\big)+B,\qquad x,y=0,\dots,N-1,$$1

with

$$f(x,y)=A\sin\!\big(2\pi(f_0 x+f_1 y)+\phi\big)+B,\qquad x,y=0,\dots,N-1,$$2

Symmetry is central. Half-wave symmetry,

$$f(x,y)=A\sin\!\big(2\pi(f_0 x+f_1 y)+\phi\big)+B,\qquad x,y=0,\dots,N-1,$$3

eliminates even harmonics, and quarter-wave symmetry,

$$f(x,y)=A\sin\!\big(2\pi(f_0 x+f_1 y)+\phi\big)+B,\qquad x,y=0,\dots,N-1,$$4

implies

$$f(x,y)=A\sin\!\big(2\pi(f_0 x+f_1 y)+\phi\big)+B,\qquad x,y=0,\dots,N-1,$$5

The number of independent variables is thereby reduced from $$f(x,y)=A\sin\!\big(2\pi(f_0 x+f_1 y)+\phi\big)+B,\qquad x,y=0,\dots,N-1,$$6 to $$f(x,y)=A\sin\!\big(2\pi(f_0 x+f_1 y)+\phi\big)+B,\qquad x,y=0,\dots,N-1,$$7.

The three-phase formulation is analogous. If there are $$f(x,y)=A\sin\!\big(2\pi(f_0 x+f_1 y)+\phi\big)+B,\qquad x,y=0,\dots,N-1,$$8 pulses in each $$f(x,y)=A\sin\!\big(2\pi(f_0 x+f_1 y)+\phi\big)+B,\qquad x,y=0,\dots,N-1,$$9 interval, then $$s(x,y)=f(x,y)+w(x,y), \qquad w(x,y)\sim\mathcal N(0,\sigma^2).$$0, with $$s(x,y)=f(x,y)+w(x,y), \qquad w(x,y)\sim\mathcal N(0,\sigma^2).$$1 odd, and the displacement factors satisfy

$$s(x,y)=f(x,y)+w(x,y), \qquad w(x,y)\sim\mathcal N(0,\sigma^2).$$2

Conventional SVPWM corresponds to $$s(x,y)=f(x,y)+w(x,y), \qquad w(x,y)\sim\mathcal N(0,\sigma^2).$$3. The optimization is performed numerically with the Interior point method in MATLAB. The practical conclusion reported in the paper is that pulse widths alone do not determine harmonic performance; pulse placement matters as well, and the optimal pattern is generally nonuniform rather than centered (Tyagi et al., 2020).

4. Collective sinusoidal displacement fields in nanocrystal superlattices

A more literal spatial-displacement model appears in "Sinusoidal Displacement Describes Disorder in CsPbBr$$s(x,y)=f(x,y)+w(x,y), \qquad w(x,y)\sim\mathcal N(0,\sigma^2).$$4 Nanocrystal Superlattices" (Filippi et al., 13 Sep 2025). Here the purpose is to explain anisotropic disorder that cannot be captured by a point-defect or one-dimensional cumulative-disorder picture.

The experimental problem is defined by a paradox in diffraction. In GISAXS, many samples show peak broadening that increases with diffraction order, indicating cumulative disorder. In GIWAXS, multilayer interference is often visible only along the axial $$s(x,y)=f(x,y)+w(x,y), \qquad w(x,y)\sim\mathcal N(0,\sigma^2).$$5 directions and absent along diagonal directions such as $$s(x,y)=f(x,y)+w(x,y), \qquad w(x,y)\sim\mathcal N(0,\sigma^2).$$6. The least soft sample, C8 with $$s(x,y)=f(x,y)+w(x,y), \qquad w(x,y)\sim\mathcal N(0,\sigma^2).$$7, is the exception: it shows clear multilayer modulation of the $$s(x,y)=f(x,y)+w(x,y), \qquad w(x,y)\sim\mathcal N(0,\sigma^2).$$8 reflection. Softness is defined as

$$s(x,y)=f(x,y)+w(x,y), \qquad w(x,y)\sim\mathcal N(0,\sigma^2).$$9

where $\boldsymbol\theta=(A,\;B,\;\phi,\;f_0,\;f_1).$0 is interparticle distance, $\boldsymbol\theta=(A,\;B,\;\phi,\;f_0,\;f_1).$1 is the nanocrystal lattice constant, and $\boldsymbol\theta=(A,\;B,\;\phi,\;f_0,\;f_1).$2 is the number of lattice planes through the nanocrystal thickness.

To rationalize these anisotropies, the nanocrystal coordinate is modeled as

$\boldsymbol\theta=(A,\;B,\;\phi,\;f_0,\;f_1).$3

The longitudinal component modulates interparticle distance along the propagation direction. In the paper’s schematic form,

$\boldsymbol\theta=(A,\;B,\;\phi,\;f_0,\;f_1).$4

where $\boldsymbol\theta=(A,\;B,\;\phi,\;f_0,\;f_1).$5 is the longitudinal amplitude, $\boldsymbol\theta=(A,\;B,\;\phi,\;f_0,\;f_1).$6 the longitudinal wavelength, and $\boldsymbol\theta=(A,\;B,\;\phi,\;f_0,\;f_1).$7 a random phase shift. The transverse component is another sinusoidal modulation, with amplitude $\boldsymbol\theta=(A,\;B,\;\phi,\;f_0,\;f_1).$8 and wavelength $\boldsymbol\theta=(A,\;B,\;\phi,\;f_0,\;f_1).$9, arranged so that the displacement is orthogonal to the propagation direction and acts as a shear-like positional modulation.

This two-mode decomposition separates effects that a simple paracrystal model conflates. Longitudinal fluctuations primarily affect peak broadening for axis-aligned stacking, while transverse fluctuations are the key to the loss of diagonal coherence. The paper’s geometric argument is that $A$0 planes depend on a single Cartesian translation, whereas $A$1 planes are sensitive to combinations of directions; consequently, a small transverse shear-like displacement can destroy $A$2 coherence much more efficiently than $A$3 coherence.

The simulations reported in the paper support this interpretation. Changing longitudinal parameters mainly affects $A$4 and $A$5 broadening. By contrast, changing transverse parameters controls whether the $A$6 reflection retains multilayer modulation. Increasing $A$7 from about 400 nm to 600 nm, or decreasing $A$8 from about 5 nm to 3 nm, restores more ordered diffraction and allows the $A$9 modulation to reappear. Altering longitudinal parameters alone, such as $B$0 from about 200 nm to 300 nm or $B$1 from about 0.3 nm to 0.2 nm, sharpens axial peaks but does not reproduce the diagonal-reflection behavior.

The softness dependence is central. Higher softness samples, with $B$2–0.7, support larger longitudinal and transverse fluctuations and lose diagonal coherence more readily. Intermediate softness samples preserve axial coherence while diagonal coherence remains weak or absent. Lower softness suppresses the displacement wave and leads to a more coherent superlattice. The C8 sample is the reported endpoint of this trend, with a fitted $B$3 stacking-disorder value $B$4 Å compatible with the axial values, indicating that diagonal coherence becomes comparable to axial coherence only in the stiffest sample (Filippi et al., 13 Sep 2025).

5. Prescribed sinusoidal motion and forcing in adjacent mechanics models

Several mechanics papers use sinusoidal motion or sinusoidal forcing in ways adjacent to, but not identical with, the displacement-field formulations above. In these works, the sinusoidal quantity is prescribed rather than inferred.

In the bouncing-ball problem, the limiter displacement in the standard model is explicitly sinusoidal,

$B$5

and the dynamics are represented by the impact-to-impact Poincaré map

$B$6

$B$7

Because sinusoidal table motion makes the impact equation transcendental, the paper also introduces a piecewise cubic periodic approximation $B$8 that preserves the qualitative bifurcation structure while making impact times analytically tractable. Fixed points, low-velocity and high-velocity cycles, grazing, and chattering are then analyzed in detail (Okniński et al., 2013).

In interface creep, the prescribed sinusoid is a drive rather than a spatial displacement. The field is

$B$9

and the positive-half-cycle average velocity is

$\alpha_l$00

A saddle-point/Laplace-type approximation shows that motion is dominated by the maximum field $\alpha_l$01, leading to

$\alpha_l$02

For short-range elasticity, the resulting behavior can appear nearly power-law with a material-dependent exponent; for long-range elasticity, the dependence remains essentially exponential with a square-root prefactor (Savolainen et al., 2022).

In railway-track dynamics, the prescribed sinusoid is a short-duration load pulse,

$\alpha_l$03

contrasted with the rectangular pulse

$\alpha_l$04

The track is modeled as an Euler–Bernoulli rail coupled to sleepers and ballast through springs and dampers, and the paper reports that rectangular pulses produce larger rail, sleeper, and substructure responses than sinusoidal pulses. The sinusoidal pulse is therefore proposed as a more realistic short-duration wheel-load representation (Touati et al., 2023).

These cases broaden the technical landscape of sinusoidal displacement modeling. A plausible implication is that, across nonlinear dynamics and structural mechanics, sinusoidal prescriptions are often adopted when analytic tractability, modal interpretation, or realism of transient loading is a design objective.

6. Adjacent sinusoidal models that are not displacement models

Several arXiv papers contain sinusoidal parameterizations but explicitly fall outside the displacement-model sense.

The speech-synthesis paper "RNN-based speech synthesis using a continuous sinusoidal model" states directly that it does not propose a displacement-based sinusoidal model. Its continuous sinusoidal model decomposes speech as

$\alpha_l$05

with a harmonic voiced component

$\alpha_l$06

where continuous F0 determines harmonic spacing and MVF determines the voiced/noise partition. This is a sinusoidal vocoder formulation, not a model of physical displacement (Al-Radhi et al., 2019).

In PINNs, the sinusoidal element is an input feature map,

$\alpha_l$07

introduced to increase input-gradient variability and to avoid deceptive local minima associated with nearly flat initial outputs. The model is labeled sf-PINN, and the central object is a neural representation of PDE solutions, not a displacement field as such (Wong et al., 2021).

"Sinusoidal Flow" likewise uses sinusoidal functions as the basis of an invertible residual transformation. The diagonal nonlinear component is built from the integral of

$\alpha_l$08

yielding a monotone autoregressive flow with exact Jacobian determinant and fixed-point inversion. Here “sinusoidal” refers to the basis of the normalizing flow rather than to any displaced physical coordinate (Wei, 2021).

Finally, "Simulation of residual oil displacement in a sinusoidal channel with the lattice Boltzmann method" concerns the displacement of an oil slug through a channel whose wall profile is sinusoidal. The sinusoidal element is the geometry of the confining channel, not the form of a displacement field. The study benchmarks critical capillary pressure, wettability effects, and resolution requirements in a controlled porous-media analogue (Otomo et al., 2016).

These distinctions are terminologically important. The presence of sine functions, sinusoidal geometry, or sinusoidal feature maps is not sufficient to make a formulation a sinusoidal displacement model in the stricter sense used in estimation, PWM pulse placement, or superlattice disorder.

7. Shared analytical themes, identifiability, and limitations

Despite their domain differences, the principal sinusoidal displacement formulations share several structural features.

First, they reduce complex spatial or temporal organization to a small set of parameters with direct physical meaning. In the 2D estimation problem, those parameters are amplitude, offset, phase, and two spatial frequencies. In PWM, the free variables are the displacement factors $\alpha_l$09, constrained by waveform symmetries. In nanocrystal superlattices, the disorder is decomposed into longitudinal and transverse amplitudes and wavelengths.

Second, each formulation relies on a mechanism that isolates the sinusoidal degrees of freedom from nuisance structure. In the 2D MLE, large-$\alpha_l$10 orthogonality between sine, cosine, and constant components permits closed-form estimation of $\alpha_l$11 after a frequency search. In PWM, half-wave and quarter-wave symmetry reduce the admissible space from $\alpha_l$12 or $\alpha_l$13 variables to $\alpha_l$14 or $\alpha_l$15. In the superlattice model, the separation into longitudinal and transverse modes resolves a diffraction anisotropy that a single scalar disorder parameter cannot capture.

Third, the models are effective only within explicit approximation regimes. The 2D estimator assumes additive white Gaussian noise, sufficiently large $\alpha_l$16, and frequencies away from degenerate points where orthogonality approximations fail (Hosseinbor et al., 2017). The PWM formulation assumes sinusoidally prescribed pulse widths and optimizes only the placement variables under symmetry and box constraints (Tyagi et al., 2020). The superlattice model is an interpretive diffraction model for correlated disorder, and its reported success depends on anisotropic coherence trends and softness-dependent fitting behavior rather than on a universal defect theory (Filippi et al., 13 Sep 2025).

A common misconception is that “sinusoidal displacement model” names a single transferable recipe. The literature instead supports a narrower conclusion: the phrase denotes a family of domain-specific models in which sinusoidal functions encode positional structure, placement freedom, or correlated coordinate perturbations. What carries across domains is not a unique equation, but a methodological pattern—periodic parameterization, physically interpretable amplitudes and phases, and a strong reliance on symmetry or asymptotics to keep inference and optimization tractable.