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Introduction to Digital Filters
    State Space Filters
       Similarity Transformations

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Similarity Transformations

A similarity transformation is a linear change of coordinates. That is, the original $ N$-dimensional state vector $ {\underline{x}}(n)$ is recast in terms of a new coordinate basis. For any linear transformation of the coordinate basis, the transformed state vector $ \underline{{\tilde x}}(n)$ may be computed by means of a matrix multiply. Denoting the matrix of the desired one-to-one linear transformation by $ E$, we can express the change of coordinates as

$\displaystyle {\underline{x}}(n) \isdef E\underline{{\tilde x}}(n) $

or $ \underline{{\tilde x}}(n)=E^{-1}{\underline{x}}(n)$, if we prefer, since the inverse of a one-to-one linear transformation always exists.

Let's now apply the linear transformation $ E$ to the general $ N$-dimensional state-space description in Eq.$ \,$(G.1). Substituting $ {\underline{x}}(n) \isdeftext E\underline{{\tilde x}}(n)$ in Eq.$ \,$(G.1) gives

$\displaystyle E\underline{{\tilde x}}(n+1)$ $\displaystyle =$ $\displaystyle A \, E\underline{{\tilde x}}(n) + B \underline{u}(n)$  
$\displaystyle \underline{y}(n)$ $\displaystyle =$ $\displaystyle C E\underline{{\tilde x}}(n) + D\underline{u}(n)$ (G.17)

Premultiplying the first equation above by $ E^{-1}$, we have
$\displaystyle \underline{{\tilde x}}(n+1)$ $\displaystyle =$ $\displaystyle \left(E^{-1}A E\right) \underline{{\tilde x}}(n) + \left(E^{-1}B\right) \underline{u}(n)$  
$\displaystyle \underline{y}(n)$ $\displaystyle =$ $\displaystyle \left(C E\right) \underline{{\tilde x}}(n) + D\underline{u}(n).$ (G.18)

Defining
$\displaystyle \tilde{A}$ $\displaystyle =$ $\displaystyle E^{-1}A E$  
$\displaystyle {\tilde B}$ $\displaystyle =$ $\displaystyle E^{-1}B$  
$\displaystyle {\tilde C}$ $\displaystyle =$ $\displaystyle C E$  
$\displaystyle {\tilde D}$ $\displaystyle =$ $\displaystyle D$ (G.19)

we can write
$\displaystyle \underline{{\tilde x}}(n+1)$ $\displaystyle =$ $\displaystyle \tilde{A}\underline{{\tilde x}}(n) + {\tilde B}\underline{u}(n)$  
$\displaystyle \underline{y}(n)$ $\displaystyle =$ $\displaystyle {\tilde C}\underline{{\tilde x}}(n) + D\underline{u}(n)$ (G.20)

The transformed system describes the same system as in Eq.$ \,$(G.1) relative to new state-variable coordinates. To verify that it's really the same system, from an input/output point of view, let's look at the transfer function using Eq.$ \,$(G.5):

\begin{eqnarray*} {\tilde H}(z) &=& {\tilde D}+ {\tilde C}(zI - \tilde{A})^{-1}{... ...ht]^{-1} B\\ &=& D + C \left(zI - A\right)^{-1} B\\ &=& H(z) \end{eqnarray*}

Since the eigenvalues of $ A$ are the poles of the system, it follows that the eigenvalues of $ \tilde{A}=E^{-1}A E$ are the same. In other words, eigenvalues are unaffected by a similarity transformation. We can easily show this directly: Let $ \underline{e}$ denote an eigenvector of $ A$. Then by definition $ A\underline{e}=\lambda\underline{e}$, where $ \lambda$ is the eigenvalue corresponding to $ \underline{e}$. Define $ \underline{\tilde{e}}=E^{-1}\underline{e}$ as the transformed eigenvector. Then we have

$\displaystyle \tilde{A}\underline{\tilde{e}}= \tilde{A}(E^{-1}\underline{e}) = ... ...^{-1}A\underline{e}= E^{-1}\lambda\underline{e}= \lambda\underline{\tilde{e}}. $

Thus, the transformed eigenvector is an eigenvector of the transformed $ A$ matrix, and the eigenvalue is unchanged.

The transformed Markov parameters, $ {\tilde C}\tilde{A}^n {\tilde B}$, are obviously the same also since they are given by the inverse $ z$ transform of the transfer function $ {\tilde H}(z)$. However, it is also easy to show this also by direct calculation:

\begin{eqnarray*} {\tilde h}(n) &=& {\tilde C}\tilde{A}^n{\tilde B}= (CE)(E^{-1}... ...ilde B}= C(EE^{-1}) A(EE^{-1}) \cdots A(EE^{-1}) B)\\ &=& CA^nB \end{eqnarray*}

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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.


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