State Space Filters

An important representation for discrete-time linear systems is the state-space formulation

where is the length state vector at discrete time , is a vector of inputs, and the output vector. is the state transition matrix,^G.1and it determines the dynamics of the system (its poles or resonant modes).

The state-space representation is especially powerful for multi-input, multi-output (MIMO) linear systems, and also for time-varying linear systems (in which case any or all of the matrices in Eq.(G.1) may have time subscripts ) [37]. State-space models are also used extensively in the field of control systems [28].

An example of a Single-Input, Single-Ouput (SISO) state-space model appears in §F.6.

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Subsections

• Markov Parameters
• Response from Initial Conditions
• Complete Response
• Transfer Function of a State Space Filter
  • Example State Space Filter Transfer Function
  
  
• Transposition of a State Space Filter
• Poles of a State Space Filter
• Difference Equations to State Space
  • Converting to State-Space Form by Hand
  • Signal Flow Graph to State Space Filter
  • Controllability and Observability
  • A Short-Cut to Controller Canonical Form
  • Matlab Direct-Form to State-Space Conversion
  • State Space Simulation in Matlab
  • Other Relevant Matlab Functions
  • Matlab State-Space Filter Conversion Example
  
  
• Similarity Transformations
• Modal Representation
  • Diagonalizing a State-Space Model
  • Finding the Eigenvalues of A in Practice
  • Example of State-Space Diagonalization
  • Properties of the Modal Representation
  
  
• Repeated Poles
  • Jordan Canonical Form
  
  
• Digital Waveguide Oscillator Example
  • Finding the Eigenstructure of A
  • Choice of Output Signal and Initial Conditions
  
  
• References
• State Space Problems

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.