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State Space Representation

State Space Representation

The frequency-domain technique, commonly used in analyzing and designing feedback control systems, is effective for linear, time-invariant systems. However, it falls short when dealing with nonlinear, time-varying, and multiple-input multiple-output systems. The time-domain or state-space approach addresses these limitations by utilizing state variables to construct simultaneous, first-order differential equations, known as state equations, for an nth-order system.
Consider an RLC circuit, a...

State Space to Transfer Function

State Space to Transfer Function

The conversion of state-space representation to a transfer function is a fundamental process in system analysis. It provides a method for transitioning from a time-domain description to a frequency-domain representation, which is crucial for simplifying the analysis and design of control systems.
The transformation process begins with the state-space representation, characterized by the state equation and the output equation. These equations are typically represented as:

Bandpass Sampling

Bandpass Sampling

In signal processing, bandpass sampling is an effective technique for sampling signals that have most of their energy concentrated within a narrow frequency band. This type of signal is known as a bandpass signal. The key principle of bandpass sampling involves sampling the signal at a rate that is greater than twice the signal's bandwidth to prevent aliasing.
A bandpass signal has a spectrum with a lower frequency limit, denoted as ω1, and an upper frequency limit, denoted as ω2. The spectrum...

IR Spectrum Peak Splitting: Symmetric vs Asymmetric Vibrations

IR Spectrum Peak Splitting: Symmetric vs Asymmetric Vibrations

Identical bonds within a polyatomic group can stretch symmetrically (in-phase) or asymmetrically (out-of-phase). Similar to hydrogen bonding, these vibrations also influence the shape of the IR peak. Generally, asymmetric stretching frequencies are higher than symmetric stretching frequencies. For example, primary amines exhibit two distinct IR peaks between 3300–3500 cm−1 corresponding to the symmetric and asymmetric N-H stretching, while secondary amines exhibit a single stretching vibration...

Transfer Function to State Space

Transfer Function to State Space

State-space representation is a powerful tool for simulating physical systems on digital computers, necessitating the conversion of the transfer function into state-space form. Consider an nth-order linear differential equation with constant coefficients, like those encountered in an RLC circuit. The state variables are selected as the output and its n−1 derivatives. Differentiating these variables and substituting them back into the original equation produces the state equations.
In an RLC...

¹H NMR Signal Multiplicity: Splitting Patterns

¹H NMR Signal Multiplicity: Splitting Patterns

When protons A and X are coupled, their nuclear spin energy levels are slightly modified. This is because the energy required to excite proton A to a spin state parallel to proton X is slightly different from the energy required for it to become anti-parallel to spin X. Consequently, there are two possible excitation frequencies for A (A1 and A2), depending on the spin state of X, and vice versa. The mutual nature of coupling implies that the difference between frequencies A1 and A2, indicated...

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Related Experiment Video

Updated: Jun 18, 2026

Spatial Separation of Molecular Conformers and Clusters

Spatial Separation of Molecular Conformers and Clusters

Published on: January 9, 2014

Signal space separation beamformer.

Jiri Vrba¹, Samu Taulu, Jukka Nenonen

¹Elekta Oy, Helsinki, Finland. jvrba@shaw.ca

Brain Topography

|November 28, 2009

Summary

This summary is machine-generated.

The novel Signal Space Separation (SSS) beamformer matches or exceeds conventional beamformer spatial resolution, especially for deeper sources. This computationally efficient method offers improved brain noise analysis.

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Last Updated: Jun 18, 2026

Spatial Separation of Molecular Conformers and Clusters

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Area of Science:

Biomedical Engineering
Neuroimaging
Signal Processing

Background:

Conventional beamformers are widely used for source localization in neuroimaging.
Existing methods face limitations in spatial resolution and computational efficiency, particularly with complex noise.
Signal Space Separation (SSS) offers a potential alternative for improving signal extraction.

Purpose of the Study:

To develop and evaluate a combined Signal Space Separation and beamformer (SSS beamformer).
To compare the spatial resolution and computational efficiency of the SSS beamformer against conventional beamformers.
To investigate the impact of noise covariance matrix properties on SSS beamformer performance.

Main Methods:

Simulated brain noise was used to test the SSS beamformer's performance.
Spatial resolution was assessed for different source depths and beamformer output metrics.
The effect of diagonalizing the sensor noise covariance matrix in the SSS basis was analyzed.

Main Results:

The SSS beamformer performs comparably to or better than conventional beamformers, contingent on sufficient expansion order.
For power-normalized outputs, SSS beamformer spatial resolution is superior for deeper sources.
SSS beamformers demonstrate greater computational efficiency than conventional approaches.

Conclusions:

The SSS beamformer is a viable and potentially superior alternative for source localization in neuroimaging.
Diagonalizing the noise covariance matrix further enhances the spatial resolution of SSS beamformers.
The SSS beamformer offers improved performance and efficiency for analyzing brain activity.