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Chapters

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Embedded SystemsFPGAElectronics

Introduction to Digital Filters
    Recursive Digital Filter Design
       Filter Design by Minimizing the L2 Equation-Error Norm
          Equation Error Formulation

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Equation Error Formulation

The equation error is defined (in the frequency domain) as

$\displaystyle E_{\mbox{ee}}(e^{j\omega}) \isdef \hat{A}(e^{j\omega})H(e^{j\omega}) - \hat{B}(e^{j\omega}) $

By comparison, the more natural frequency-domain error is the so-called output error:

$\displaystyle E_{\mbox{oe}}(e^{j\omega}) \isdef H(e^{j\omega}) - \frac{\hat{B}(e^{j\omega})}{\hat{A}(e^{j\omega})} $

The names of these errors make the most sense in the time domain. Let $ x(n)$ and $ y(n)$ denote the filter input and output, respectively, at time $ n$. Then the equation error is the error in the difference equation:

\begin{eqnarray*} e_{\mbox{ee}}(n) &=& y(n) + \hat{a}_1 y(n-1) + \cdots + \hat{a... ...0 x(n) - \hat{b}_1 x(n-1) - \cdots - \hat{b}_{{n}_b}x(n-{{n}_b}) \end{eqnarray*}

while the output error is the difference between the ideal and approximate filter outputs:

\begin{eqnarray*} e_{\mbox{oe}}(n) &=& y(n) - \hat{y}(n) \\ \hat{y}(n) &=& \ha... ...}_1 \hat{y}(n-1) - \cdots - \hat{a}_{{{n}_a}} \hat{y}(n-{{n}_a}) \end{eqnarray*}

Denote the $ L2$ norm of the equation error by

$\displaystyle J_E(\hat{\theta}) \isdef \left\Vert\,\hat{A}(e^{j\omega})H(e^{j\omega}) - \hat{B}(e^{j\omega})\,\right\Vert _2,$ (I.11)

where $ \hat{\theta}^T = [\hat{b}_0,\hat{b}_1,\ldots,\hat{b}_{{n}_b}, \hat{a}_1,\ldots, \hat{a}_{{n}_a}]$ is the vector of unknown filter coefficients. Then the problem is to minimize this norm with respect to $ \hat{\theta}$. What makes the equation-error so easy to minimize is that it is linear in the parameters. In the time-domain form, it is clear that the equation error is linear in the unknowns $ \hat{a}_i,\hat{b}_i$. When the error is linear in the parameters, the sum of squared errors is a quadratic form which can be minimized using one iteration of Newton's method. In other words, minimizing the $ L2$ norm of any error which is linear in the parameters results in a set of linear equations to solve. In the case of the equation-error minimization at hand, we will obtain $ {{n}_b}+{{n}_a}+1$ linear equations in as many unknowns.

Note that (I.11) can be expressed as

$\displaystyle J_E(\hat{\theta}) = \left\Vert\,\left\vert\hat{A}(e^{j\omega})\ri... ...ot\left\vert H(e^{j\omega}) - \hat{H}(e^{j\omega})\right\vert\,\right\Vert _2. $

Thus, the equation-error can be interpreted as a weighted output error in which the frequency weighting function on the unit circle is given by $ \vert\hat{A}(e^{j\omega})\vert$. Thus, the weighting function is determined by the filter poles, and the error is weighted less near the poles. Since the poles of a good filter-design tend toward regions of high spectral energy, or toward ``irregularities'' in the spectrum, it is evident that the equation-error criterion assigns less importance to the most prominent or structured spectral regions. On the other hand, far away from the roots of $ \hat{A}(z)$, good fits to both phase and magnitude can be expected. The weighting effect can be eliminated through use of the Steiglitz-McBride algorithm [45,78] which iteratively solves the weighted equation-error solution, using the canceling weight function from the previous iteration. When it converges (which is typical in practice), it must converge to the output error minimizer.

Previous: Filter Design by Minimizing the L2 Equation-Error Norm
Next: Error Weighting and Frequency Warping

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About the Author: 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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