Application and Productive Examples¶

Application-level examples showing how to use lmlib on realistic signal-processing tasks, grouped by topic. Each thumbnail links to a full page with the plot(s), console output, any data files, and the complete source code.

Detection¶

Advanced event detection with ALSSMs.

	Rectangular Pulse Detection with Baseline [ex111.0] This example demonstrates the detection of a rectangular pulses of known duration in a noisy signal with baseline interferences.
	Multi-Channel Spike Detection [ex112.0] Demonstrates a spike detection algorithm that uses autonomous linear state space models together with exponentially decaying windows. The input is a multi-channel signal containing multiple spikes (sinusoidal cycles with decaying amplitude) with additive white Gaussian noise and a polynomial baseline. The algorithm fits a spike model and a...
	ECG Shape Detection [ex113.0] Detects a known reference shape in a single-channel electrocardiogram (ECG) signal using a template-matching approach based on ALSSMs. A reference template is extracted from the signal at a known location (K_REF). A polynomial ALSSM fits the template and every candidate window in the signal; the Log-Cost Ratio (LCR)...

Filtering¶

ALSSM-based signal filters.

	Symmetric Moving Average Filters with ALSSMs [ex121.0] Applies a CompositeCost with a two-sided symmetric window as a symmetric moving average filter of length L=100. A degree-0 polynomial ALSSM (i.e. a constant model) is combined with a forward left segment and a backward right segment of equal length. The resulting filter is equivalent to a finite-impulse-response...
	Symmetric and Non-Symmetric Polynomial Filters with Pascal Basis [ex122.0] Applies CompositeCost instances with AlssmPoly of degrees 0 through 3 to a rectangular test signal. Two filter configurations are shown for each polynomial degree: • Symmetric filter — a forward left window and a backward right window of equal size, yielding a zero-phase (non-causal) smoother. • Left (causal)...
	Symmetric and Non-Symmetric Polynomial Filters with Meixner Basis [ex122.1] Applies CompositeCost instances with AlssmPolyMeixner of degrees 0 through 7 to a rectangular test signal. The Meixner basis is orthogonal under the exponential (geometric) window weight, giving a well-conditioned Gram matrix for semi-infinite windows. This makes it the preferred polynomial basis for recursive least-squares filtering with exponential windows,...
	Multi-Channel Symmetric Signal Filter [ex123.0] Applies a CompositeCost with a symmetric two-sided window as a symmetric linear filter to a multi-channel signal. A degree-5 polynomial ALSSM is fitted over equal-length left and right windows. The same model is applied in parallel to all channels using the multi-channel (MC) output form of the ALSSM....
	Multi-Scale Polynomial (Savitzky-Golay) Smoothing: Causal vs Centered [ex125.0] Smooths a noisy single-channel EEG signal at a dyadic family of window lengths L \in {10, 20, 40, 80, 160, 320} with polynomial ALSSMs, and shows the two standard windowing choices side by side: • Causal (left) — a one-sided (left-sided) BACKWARD window (a = 0, b =...
	Symmetric and Non-Symmetric 2-D Polynomial Filters with ALSSMs [ex126.0] 2-D extension of the 1-D polynomial smoother shown in ex122.0. Instead of a single CompositeCost, a separable NDCompositeCost is built from one polynomial CompositeCost per image axis and applied to a noisy 2-D signal (an image) in a single fit call. At every pixel the filter fits a...

Polynomials Calculus¶

Calculus and cost functions with polynomials.

	Squared error between two polynomials [ex211.0] Closed form computation of the squared error between two time-continuous polynomials.
	Approximating a Polynomial by a Polynomial of Lower Degree [ex701.0] Fits a polynomial of lower order to a polynomial of higher order by minimizing the squared error over the given window. This example is published in [Wildhaber2020].

Event Detection with Two-Sided Line Models¶

Onset and peak detection with Two-Sided Line Models (TSLMs). Examples published in "Onset Detection of Pulse-Shaped Bioelectrical Signals Using Linear State Space Models" [Waldmann2022].

	(Edu) Edge Detection [ex401.0] This example is published in [Waldmann2022] as Example 1. This basic example illustrates the detection of edges using a Two-Sided Line Model (TSLM), weighting the two options of a "Continuous" versus a "Straight" line in a LCR term. Education Examples (Edu): This is an illustrative example to demonstrate...
	Edge Detection [ex401.1] Example published in [Waldmann2022] as Example 1. Illustrates edge detection using a Two-Sided Line Model (TSLM) that scores each sample by the Log-Cost Ratio (LCR) between a Continuous constraint (shared offset and matching slopes on both sides) and a Straight constraint (single global line). Peaks in the LCR...
	Notch Detection in ABP Signal [ex402.0] Example published in [Waldmann2022] as Example 2. Arterial blood pressure (ABP) signals typically show a dicrotic notch in the descending slope, marking the end of the systolic cycle. This example detects these notches using a Two-Sided Line Model (TSLM) that compares a Continuous line fit (no slope change)...
	ECG P, T Wave Onset and Peak Detection [ex403.0] Example published in [Waldmann2022] as Example 3. A common task in cardiology is to accurately measure the onsets and peaks of P and T waves in electrocardiography (ECG) signals. This example demonstrates a Two-Sided Line Model (TSLM) approach for extracting these fiducial points. Two separate TSLM cost functions...
	Overview of Two-Sided Line Model (TSLM) Applications [ex404.0] Compares multiple Two-Sided Line Model (TSLM) constraint types on a piecewise-linear test signal; cf. [Waldmann2022]. For each pair of hypothesis constraints (H0, H1), the Log-Cost Ratio (LCR) is computed and plotted. The constraint pairs tested are: • Left Horizontal vs. Horizontal • Continuous vs. Straight • Peak vs....

Convolution¶

Correlation (or convolution) and matched filters. The method was published in "Windowed State Space Filters for Peak Interference Suppression in Neural Spike Sorting" [Baeriswyl2022].

	Correlation/Convolution in Low-Dimensional ALSSM Feature Space [ex501.0] Performs correlation (or convolution) of a multi-channel signal (blue) with a given reference template (red) in a low-dimensional ALSSM feature space. Each signal window is projected onto the Legendre polynomial basis via an AlssmPolyLegendre and the resulting compact state-vector coefficients are used as features. Template matching is then...
	Matched Filter in Low-Dimensional ALSSM Feature Space [ex502.0] Implements a matched filter using ALSSM signal approximations (polynomial basis) to significantly increase processing speed compared to a direct sample-domain matched filter. A reference template is extracted from a 9-channel ECG signal and projected onto the Legendre polynomial basis via AlssmPolyLegendre. The matched filter response is computed as...
	Convolution and Correlation in Low-Dimensional ALSSM Feature Space [ex503.0] Computes the convolution, cross-correlation, and auto-correlation of a signal y with a reference filter h directly in a low-dimensional ALSSM feature space instead of in the original sample space. Both the signal and the filter are projected onto a polynomial basis through an AlssmPolyJordan model that is fitted...
	Correlation with a Triangular Pulse Template in Low-Dimensional ALSSM Cross-correlates a noisy sawtooth signal (gray) with a pulse template (black) in a low-dimensional ALSSM feature space. Both the multi-channel signal and the reference template are projected onto a polynomial basis through an AlssmPolyJordan model. The template's state vector is extracted and correlated against the signal with RLSAlssm.convolve,...

Shift and Time Dilation Estimation¶

Shift and Time Dilation Estimation with localized polynomial signal models. Examples published in "Signal Analysis Using Local Polynomial Approximations" [Wildhaber2020].

Audio Signal Shift Estimation [ex702.0]
Example 2 as published in [Wildhaber2020]. Equation references in the code (e.g. # Eq. 6.52) refer to equations in [Wildhaber2019]. Top plot: Two-channel acoustic signal from the left (L) and right (R) ear with an unknown interaural time delay (ITD) to be estimated. Middle plot: Local time-delay estimates...

Time and Amplitude Scaling [ex703.0]
Estimates both a time-scaling (dilation) factor and an amplitude-scaling factor between two locally polynomial signals. This example is published in [Wildhaber2020]. The polynomial coefficients of the reference and the candidate windows are related by the dilation and amplitude operators, enabling a closed-form least-squares estimator via the polynomial calculus.

N Dimensional Signal Processing¶

Processing of N-dimensional signals. Examples published in "Multi-Resolution Autonomous Linear State Space Filters for N-Dimensional Signals" [Baeriswyl2025].

	Text Recognition [ex801.0] Demonstrates 2-D ALSSM-based text recognition using an NDCompositeCost over a pixel image. A reference letter image is used as a template. The 2-D ALSSM filter projects each local image patch onto a separable polynomial basis (one per image dimension) and computes the Log-Cost Ratio (LCR) between the best-fit...
	Multi-Scale Water-Drop Detection with 2-D ALSSM / LCR [ex802.0] Round water drops in a microscope image are detected as local maxima of a Log-Cost Ratio (LCR) between two competing 2-D ALSSM hypotheses, evaluated at several scales. For each scale an isotropic 1-D model -- an offset plus two cosine modes, AlssmPoly(0) + AlssmSin + AlssmSin -- is...
	Separable ALSSM Gaussian Kernel via 2-D NDCompositeCost [ex803.0] A 2-D Gaussian smoothing kernel is approximated by a separable Autonomous Linear State-Space Model (ALSSM). Instead of building the kernel by hand, the approximation is obtained directly from a 2-D recursive least-squares fit: two identical one-dimensional CompositeCost terms (a two-sided, damped-polynomial model) are wrapped in an NDCompositeCost and...
No Plot	Multi-Channel ECG Shape Detection with a 2-D ALSSM / Cost-Ratio [ex804.0] Heartbeat shapes are detected in a multi-lead ECG by fitting a separable 2-D Autonomous Linear State-Space Model (ALSSM) over time (axis 0) and channel (axis 1) and scoring every time index with a cost ratio CR = \dfrac{J_0}{J_1} where J_0 is the squared error of a flat (zero)...