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Embedded SystemsFPGAElectronics

Physical Audio Signal Processing
    Artificial Reverberation
       FDN Reverberation
          Choice of Lossless Feedback Matrix
             Householder Reflections

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Householder Reflections

For completeness, this section derives the Householder reflection matrix from geometric considerations [451]. Let $ \mathbf{P}_{\underline{u}}$ denote the projection matrix which orthogonally projects vectors onto $ {\underline{u}}$, i.e.,

$\displaystyle \mathbf{P}_{\underline{u}}= \frac{\underline{u}\,\underline{u}^T}... ...frac{\underline{u}\,\underline{u}^T}{\left\Vert\,\underline{u}\,\right\Vert^2} $

and

$\displaystyle \mathbf{P}_{\underline{u}}\, \underline{x}= \underline{u}\,\frac{... ...<\underline{u},\underline{x}\right>}{\left\Vert\,\underline{u}\,\right\Vert^2} $

specifically projects $ \underline{x}$ onto $ \underline{u}$. Since the projection is orthogonal, we have

$\displaystyle \left<\underline{x}-\mathbf{P}_{\underline{u}}\underline{x},\unde... ...}-\mathbf{P}_{\underline{u}})\underline{x},\underline{u}\right>=\underline{0}. $

We may interpret $ (\mathbf{I}-\mathbf{P}_{\underline{u}})\underline{x}$ as the difference vector between $ \underline{x}$ and $ \mathbf{P}_{\underline{u}}\underline{x}$, its orthogonal projection onto $ \underline{u}$, since

$\displaystyle (\mathbf{I}-\mathbf{P}_{\underline{u}})\underline{x}+ \mathbf{P}_{\underline{u}}\underline{x}= \underline{x} $

and we have $ (\mathbf{I}-\mathbf{P}_{\underline{u}})\underline{x}\perp \mathbf{P}_{\underline{u}}\underline{x}$ by definition of the orthogonal projection. Consequently, the projection onto $ \underline{u}$ minus this difference vector gives a reflection of the vector $ \underline{x}$ about $ \underline{u}$:

$\displaystyle \underline{y}= \mathbf{P}_{\underline{u}}\underline{x}- (\mathbf{... ...line{u}})\underline{x}= (2\mathbf{P}_{\underline{u}}- \mathbf{I})\underline{x} $

Thus, $ \underline{y}$ is obtained by reflecting $ \underline{x}$ about $ \underline{u}$--a so-called Householder reflection.

Previous: Householder Feedback Matrix
Next: Most General Lossless Feedback Matrices

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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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