Linear-Signal Analysis: Foundations and Applications

• Linear-signal analysis is the study of signals and systems using linear operators, focusing on LTI models, impulse responses, and transform-domain methods.
• It provides a unifying operator-theoretic framework that links time-domain convolution with frequency-domain representations and algebraic formulations.
• Applications span from analog circuit verification and magnetic hyperthermia to high-dimensional inference and adaptive time–frequency analysis in non-stationary signals.

Linear-signal analysis denotes the study of signals and systems through linear operators, linear time-invariant models, linear descriptor systems, and linearized or multilinear local approximations of more complex dynamics. Its central objects are impulse responses, transfer functions, convolution laws, Fourier- and Laplace-domain representations, and covariance- or Jacobian-based surrogates that preserve superposition or local differential structure. In this broad sense, the term covers continuous and discrete SISO LTI systems, algebraic signal models $(\mathcal{A},M,\varphi)$, linear response to arbitrary periodic excitation, Laplace-based analog-circuit analysis, high-dimensional linear inverse problems, and small-signal stability models for networked power systems (Mendez, 2022), [0612077], (Landi et al., 2012, Rashid et al., 2020, Kaufmann et al., 18 Oct 2025).

1. Foundational formalism

The classical core of linear-signal analysis is the LTI input–output map. For continuous time,

$$y(t)=(h*u)(t)=\int_{-\infty}^{\infty} h(\tau)\,u(t-\tau)\,d\tau,$$

and for discrete time,

$$y[k]=(h*u)[k]=\sum_{l=-\infty}^{\infty} h[l]\,u[k-l].$$

Causality imposes $h(t)=0$ for $t<0$ and $h[k]=0$ for $k<0$, while BIBO stability requires absolute integrability or summability of the impulse response (Mendez, 2022).

The same framework yields the transform-domain description. Complex exponentials are eigenfunctions of LTI systems, so the Laplace- and $z$-domain relations reduce to

$Y(s)=H(s)U(s),\qquad Y(z)=H(z)U(z),$

with $H(s)$ and $$y(t)=(h*u)(t)=\int_{-\infty}^{\infty} h(\tau)\,u(t-\tau)\,d\tau,$$0 the transfer functions. This is the operational basis for frequency response, pole–zero analysis, and the equivalence between convolution in time and multiplication in transform space (Mendez, 2022).

An algebraic formulation makes this structure explicit by defining a linear signal model as a triple $$y(t)=(h*u)(t)=\int_{-\infty}^{\infty} h(\tau)\,u(t-\tau)\,d\tau,$$1, where the filter space is cast as an algebra $$y(t)=(h*u)(t)=\int_{-\infty}^{\infty} h(\tau)\,u(t-\tau)\,d\tau,$$2, the signal space as a module $$y(t)=(h*u)(t)=\int_{-\infty}^{\infty} h(\tau)\,u(t-\tau)\,d\tau,$$3, and $$y(t)=(h*u)(t)=\int_{-\infty}^{\infty} h(\tau)\,u(t-\tau)\,d\tau,$$4 generalizes the $$y(t)=(h*u)(t)=\int_{-\infty}^{\infty} h(\tau)\,u(t-\tau)\,d\tau,$$5-transform to bijective linear mappings from a vector space of signal samples into $$y(t)=(h*u)(t)=\int_{-\infty}^{\infty} h(\tau)\,u(t-\tau)\,d\tau,$$6. In that view, the shift operator is the generator of the algebra of filters, and once the shift is chosen, notions of filtering, spectrum, and Fourier transform follow systematically [0612077]. This suggests that linear-signal analysis is not a single technique but a unifying operator-theoretic framework.

2. Periodic excitation, transfer functions, and spectral shaping

A central extension of the LTI viewpoint is linear response to arbitrary periodic forcing. In Kubo’s LRT, a weak periodic input $$y(t)=(h*u)(t)=\int_{-\infty}^{\infty} h(\tau)\,u(t-\tau)\,d\tau,$$7 drives an observable $$y(t)=(h*u)(t)=\int_{-\infty}^{\infty} h(\tau)\,u(t-\tau)\,d\tau,$$8 through the causal convolution

$$y(t)=(h*u)(t)=\int_{-\infty}^{\infty} h(\tau)\,u(t-\tau)\,d\tau,$$9

with response kernel $$y[k]=(h*u)[k]=\sum_{l=-\infty}^{\infty} h[l]\,u[k-l].$$0 and susceptibility $$y[k]=(h*u)[k]=\sum_{l=-\infty}^{\infty} h[l]\,u[k-l].$$1. Expanding $$y[k]=(h*u)[k]=\sum_{l=-\infty}^{\infty} h[l]\,u[k-l].$$2 in a Fourier series yields

$$y[k]=(h*u)[k]=\sum_{l=-\infty}^{\infty} h[l]\,u[k-l].$$3

so each harmonic is weighted only by its Fourier coefficient and the susceptibility at that harmonic (Landi et al., 2012).

For dissipative systems with a single relaxation time $$y[k]=(h*u)[k]=\sum_{l=-\infty}^{\infty} h[l]\,u[k-l].$$4, the paper introduces the dimensionless efficiency

$$y[k]=(h*u)[k]=\sum_{l=-\infty}^{\infty} h[l]\,u[k-l].$$5

with $$y[k]=(h*u)[k]=\sum_{l=-\infty}^{\infty} h[l]\,u[k-l].$$6, and derives

$$y[k]=(h*u)[k]=\sum_{l=-\infty}^{\infty} h[l]\,u[k-l].$$7

This separates waveform information from system dynamics and makes waveform design a spectral-allocation problem. Closed forms are obtained for several signals, including the harmonic wave $$y[k]=(h*u)[k]=\sum_{l=-\infty}^{\infty} h[l]\,u[k-l].$$8, the square wave $$y[k]=(h*u)[k]=\sum_{l=-\infty}^{\infty} h[l]\,u[k-l].$$9, the sawtooth wave $h(t)=0$0, and the $h(t)=0$1-pulse limit $h(t)=0$2 (Landi et al., 2012). This suggests that, under fixed input-energy normalization, changing only the signal shape can change dissipation substantially.

Linear-signal analysis also clarifies ostensibly nonlinear demodulation schemes. The Carré 4-step phase-shifting algorithm can be decomposed into a tunable linear 4-step PSA and a nonlinear phase-step estimator. The linear part has impulse response

$h(t)=0$3

and magnitude-relevant transfer function

$h(t)=0$4

Its zeros are at $h(t)=0$5, $h(t)=0$6, and $h(t)=0$7, and the SNR gain

$h(t)=0$8

attains its maximum value $h(t)=0$9 at $t<0$0 (Servin et al., 2012).

A related frequency-domain perspective appears in double-integrator design. The linear double integrator

$t<0$1

has transfer functions

$t<0$2

and tends to $t<0$3 as $t<0$4, so $t<0$5 tracks the signal, $t<0$6 approximates $t<0$7, and $t<0$8 approximates $t<0$9. The nonlinear counterpart is analyzed by a frequency-sweep method, showing that both integrators estimate low-frequency content accurately and suppress high-frequency noise, while the nonlinear integrator exhibits better estimation performance and stronger robustness (Wang, 2015).

3. Formal and computational realizations

In analog circuit analysis, linear-signal methods typically begin with Kirchhoff laws, proceed to an ODE, apply the Laplace transform, and solve for a rational transfer function

$h[k]=0$0

The FASiM framework automates this pipeline for Simulink models of linear analog circuits by extracting transfer-function coefficients from MATLAB, translating them into HOL Light lists, constructing the corresponding ODE with diff_eq_n_order, and proving ODE $h[k]=0$1 transfer-function equivalence in higher-order logic (Rashid et al., 2020).

Its workflow is explicitly

$h[k]=0$2

and the automation uses the tactics DIFF_EQ_2_TRANS_FUN_TAC and TRANS_FUN_2_DIFF_EQ_TAC (Rashid et al., 2020). For the Sallen–Key low-pass example, the formally verified transfer function is

$h[k]=0$3

under differentiability, zero-initial-condition, and Laplace-existence assumptions (Rashid et al., 2020).

This formal branch of linear-signal analysis makes explicit assumptions that are often left implicit in engineering calculations: linearity, time invariance, exponential order, zero initial conditions, and nonzero denominators. A plausible implication is that, in high-assurance settings, the traditional distinction between “analysis” and “verification” becomes part of the subject itself rather than a post-processing step.

4. High-dimensional inference, quantization, and statistical linear models

Modern linear-signal analysis extends beyond deterministic transforms to high-dimensional inference in random linear systems. In generalized linear mixing,

$h[k]=0$4

relaxed BP yields scalar sufficient statistics

$h[k]=0$5

and reconstruction under any additive error metric

$h[k]=0$6

reduces to coordinatewise Bayes decisions

$h[k]=0$7

This decoupling supports MMSE, MMAE, support recovery error, and large-$h[k]=0$8 $h[k]=0$9-type losses within the same linear mixing framework (Tan et al., 2013).

Quantized linear channels preserve the same structure but add a nonlinear observation layer. For the linear vector channel

$k<0$0

followed by scalar quantization, replica analysis in the large-system limit yields a bit-error formula

$k<0$1

where $k<0$2 is determined by coupled saddle-point equations. In the noiseless case $k<0$3, the optimal detector admits a perfect-detection solution $k<0$4, $k<0$5, hence $k<0$6, for any $k<0$7 and $k<0$8. In the noisy case, detection ability decreases monotonically as the quantization step size increases (0805.3406).

A complementary line studies recovery from $k<0$9-bit quantized Gaussian measurements while deliberately ignoring the quantizer nonlinearity at reconstruction time. The canonical linear estimator uses

$z$0

and satisfies

$z$1

Under a fixed bit budget $z$2, the paper concludes that $z$3 is optimal for estimating the unit vector corresponding to the signal, while $z$4 is optimal for estimating the direction and the norm; Lloyd–Max quantization is optimal with respect to $z$5-estimation error (Slawski et al., 2016).

In 1-bit MU-MISO downlink, Bussgang decomposition restores an effective linear model: $z$6 so the received signal becomes a linear effective channel plus colored quantization distortion. The achievable sum-rate lower bound is then expressed through an SQINR matrix, and the analysis indicates the merit of reconsidering traditional signal processing techniques, specifically transmit covariance matrices whose rank exceeds the channel rank. The resulting linear precoder design supports higher-order modulation schemes under 1-bit DACs (Candido et al., 2018).

Linear diagnostics remain meaningful even when the underlying parameter vector is not. In the high-dimensional model

$z$7

with $z$8, the paper proposes consistent, asymptotically normal estimators of residual variance $z$9, $Y(s)=H(s)U(s),\qquad Y(z)=H(z)U(z),$0-signal strength $Y(s)=H(s)U(s),\qquad Y(z)=H(z)U(z),$1, and signal-to-noise ratio, without sparsity assumptions and even when $Y(s)=H(s)U(s),\qquad Y(z)=H(z)U(z),$2 and the signal itself is non-estimable (Dicker, 2012). This broadens linear-signal analysis from system characterization to high-dimensional statistical identifiability.

5. Adaptive time–frequency analysis and direct component separation

For non-stationary multicomponent signals,

$Y(s)=H(s)U(s),\qquad Y(z)=H(z)U(z),$3

adaptive time–frequency methods retain linear analysis operators while making the window or wavelet parameters time-varying. The adaptive STFT uses

$Y(s)=H(s)U(s),\qquad Y(z)=H(z)U(z),$4

which remains linear in $Y(s)=H(s)U(s),\qquad Y(z)=H(z)U(z),$5 for fixed $Y(s)=H(s)U(s),\qquad Y(z)=H(z)U(z),$6 (Chui et al., 2020).

Under a sinusoidal local approximation,

$Y(s)=H(s)U(s),\qquad Y(z)=H(z)U(z),$7

the adaptive STFT expands as

$Y(s)=H(s)U(s),\qquad Y(z)=H(z)U(z),$8

Under a linear chirp local approximation,

$Y(s)=H(s)U(s),\qquad Y(z)=H(z)U(z),$9

the analogous expansion is

$H(s)$0

with $H(s)$1 the Fourier transform of a chirped window (Chui et al., 2020).

This leads to direct signal separation operation: extract a ridge $H(s)$2 by maximizing $H(s)$3 in the component’s TF zone, then reconstruct by evaluating the transform on that ridge. For the sinusoidal model,

$H(s)$4

whereas the linear-chirp correction gives

$H(s)$5

For Gaussian windows, the practical real-signal formula becomes

$H(s)$6

and explicit error bounds are obtained for both IF estimation and component recovery (Chui et al., 2020).

An adaptive continuous wavelet-like transform develops the same idea in a time–scale setting: $H(s)$7 Its sinusoidal and linear-chirp local models lead to analogous ridge-based IF estimators and direct reconstruction formulas, including the chirp-corrected form

$H(s)$8

The method is presented as an alternative to SST: SST estimates IF and then reconstructs by a definite integral along the estimated IF curve, whereas SSO and the adaptive CWLT avoid the second step and recover the component directly from the transform value on the ridge (Chui et al., 2020).

6. Applications, scope, and limits

Linear-signal analysis is applied across markedly different physical domains. In magnetic hyperthermia, the LRT framework predicts waveform-dependent heating efficiencies and shows that discontinuous waveforms and narrow pulses can outperform harmonic driving under equal input-energy normalization; the same formalism is then connected to magnetic Langevin or Fokker–Planck modeling of nanoparticles used as local heating centers in oncological treatments (Landi et al., 2012).

In analog electronics, the common collector amplifier is analyzed with an Early equivalent transistor model that preserves enough nonlinearity to estimate average voltage gain, total harmonic distortion, and average input and output resistances. The results confirm the importance of the trade-off of current gain and show that sub-optimal performance can occur when base and emitter resistances are not properly chosen; they also report that THD tends to be larger for PNP devices than NPN counterparts with the same average current gain (Costa, 2018).

In power systems, small-signal stability is formulated directly from implicit multilinear models. After suitable transformations of variables, including representations of trigonometric terms, the multilinear DAE is linearized into a descriptor model, and equilibrium stability is determined from generalized eigenvalues of the pair $H(s)$9. On a 3-bus network example, time-domain simulations of the implicit multilinear model agree with those of the nonlinear model, and the generalized eigenvalues agree with those of the linearized nonlinear model; the decomposed tensor representation also allows faster linearization compared to conventional methods in MATLAB Simulink (Kaufmann et al., 18 Oct 2025).

The literature also places clear validity conditions on these methods. LRT requires linearity, small driving amplitude, and near-equilibrium conditions; ideal discontinuous waveforms require infinite harmonic content, and their nonzero $$y(t)=(h*u)(t)=\int_{-\infty}^{\infty} h(\tau)\,u(t-\tau)\,d\tau,$$00 is explicitly identified as an artifact of ideal discontinuity (Landi et al., 2012). Formal Laplace-domain verification assumes linearity, time invariance, ideal components, and Laplace existence conditions (Rashid et al., 2020). Linear recovery from quantized measurements is competitive in moderate- and high-noise settings but can fall short of more sophisticated competitors in a low-noise setting (Slawski et al., 2016). This suggests that linear-signal analysis is most powerful when its approximation regime is made explicit: it is neither restricted to exactly linear physics nor universally exact once nonlinearity, finite bandwidth, quantization, or model mismatch dominate.