Markov Parameters

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Index: Introduction to Digital Filters with Audio Applications

Introduction to Digital Filters with Audio Applications

State Space Filters

Response from Initial Conditions

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A transfer function matrix is simply a matrix of transfer functions. That is, each element of the matrix is a scalar transfer function. A matrix transfer function accommodates multiple inputs and multiple outputs (MIMO) in a linear, time-invariant (LTI) system, while each entry of the matrix relates a single-input to a single-output (SISO). — Click for https://ccrma.stanford.edu/~jos/filters/Transfer_Function_State_Space.html

A signal is typically a real-valued function of time. A discrete-time signal is typically a real-valued function of discrete time, and is therefore a time-ordered sequence of real numbers. — Click for http://ccrma.stanford.edu/~jos/filters/Definition_Signal.html

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The impulse signal is the shortest pulse signal. For discrete time (digital) systems, the impulse is a 1 followed by zeros. In continuous time, the impulse is a narrow, unit-area pulse (ideally infinitely narrow). The impulse is typically denoted by the Greek delta symbol δ and is often called the 'Dirac delta function', after Paul Dirac who pioneered its use (in addition to being a lead contributor in the development of Quantum Mechanics). — Click for https://ccrma.stanford.edu/~jos/filters/Impulse_Response_Representation.html

An initial value problem consists of a differential equation and an initial condition. The solution to an initial value problem is a function y(t) solving the differential equation and satisfying the initial condition y(t₀)=y₀. — Click for http://en.wikipedia.org/wiki/Initial_value_problem

The impulse response of a system is its output signal in response to the impulse signal. For discrete time (digital) systems, the impulse is a 1 followed by zeros. In continuous time, the impulse is a narrow, unit-area pulse (ideally infinitely narrow). — Click for https://ccrma.stanford.edu/~jos/filters/Impulse_Response_Representation.html

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Index: Introduction to Digital Filters with Audio Applications

Introduction to Digital Filters with Audio Applications

State Space Filters

Response from Initial Conditions

Markov Parameters

The Markov parameter sequence for a state-space model is a kind of matrix impulse response that easily found by direct calculation using Eq.(G.1):

$\begin{eqnarray*} \mathbf{h}(0) &=& C {\underline{x}}(0) + D\,\underline{\delta}(0) = D\\ [5pt] {\underline{x}}(1) &=& A\, {\underline{x}}(0) + B\,\underline{\delta}(0) = B\\ \mathbf{h}(1) &=& C B\\ [5pt] {\underline{x}}(2) &=& A\,{\underline{x}}(1) + B\,\underline{\delta}(1) = AB\\ \mathbf{h}(2) &=& C A B\\ [5pt] {\underline{x}}(3) &=& A\,{\underline{x}}(2) + B\,\underline{\delta}(2) = A^2B\\ \mathbf{h}(3) &=& C A^2 B\\ [5pt] &\vdots&\\ \mathbf{h}(n) &=& C A^{n-1} B, \quad n>0 \end{eqnarray*}$

Note that we have assumed ${\underline{x}}(0)=0$ (zero initial state or zero initial conditions). The notation $\underline{\delta}(n)$ denotes a $q\times q$ matrix having $\delta (n)$ along the diagonal and zeros elsewhere.^G.2 Since the system input is a $q\times 1$ vector, we may regard $\underline{\delta}(n)$ as a sequence of successive input vectors, each providing an impulse at one of the input components.

The impulse response of the state-space model can be summarized as

$\displaystyle \fbox{$\displaystyle \mathbf{h}(n) = \left\{\begin{array}{ll} D, & n=0 \\ [5pt] CA^{n-1}B, & n>0 \\ \end{array} \right.$}$

(G.2)

The impulse response terms for $n\geq 0$ are known as the Markov parameters of the state-space model.

Note that each ``sample'' of the impulse response $\mathbf{h}(n)$ is a $p\times q$ matrix.^G.3 Therefore, it is not a possible output signal, except when . A better name might be ``impulse-matrix response''. It can be viewed as a sequence of outputs, each $p\times 1$ . In §G.4 below, we'll see that $\mathbf{h}(n)$ is the inverse z transform of the matrix transfer-function of the system.

Given an arbitrary input signal $\underline{u}(n)$ (and zero intial conditions ${\underline{x}}(0)=0$ ), the output signal is given by the convolution of the input signal with the impulse response:

$\displaystyle \underline{y}_u(n) = (\mathbf{h}\ast \underline{u})(n) = \left\{\begin{array}{ll} D\underline{u}(0), & n=0 \\ [5pt] \sum_{m=1}^nCA^{m-1}B\underline{u}(n-m), & n>0 \\ \end{array} \right. \protect$

(G.3)

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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
Copyright © 2024-09-03 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

Markov Parameters

``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition) Copyright © 2024-09-03 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
Copyright © 2024-09-03 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University