# 0.12 Convolution algorithms  (Page 3/7)

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## Block signal processing

In this section the usual convolution and recursion that implements FIR and IIR discrete-time filters are reformulated in terms of vectors andmatrices. Because the same data is partitioned and grouped in a variety of ways, it is important to have a consistent notation in order to beclear. The ${n}^{th}$ element of a data sequence is expressed $h\left(n\right)$ or, in some cases to simplify, ${h}_{n}$ . A block or finite length column vector is denoted ${\underline{h}}_{n}$ with $n$ indicating the ${n}^{th}$ block or section of a longer vector. A matrix, square or rectangular, is indicatedby an upper case letter such as $H$ with a subscript if appropriate.

## Block convolution

The operation of a finite impulse response (FIR) filter is described by a finite convolution as

$y\left(n\right)=\sum _{k=0}^{L-1}h\left(k\right)\phantom{\rule{0.166667em}{0ex}}x\left(n-k\right)$

where $x\left(n\right)$ is causal, $h\left(n\right)$ is causal and of length $L$ , and the time index $n$ goes from zero to infinity or some large value. With a change of index variables this becomes

$y\left(n\right)=\sum h\left(n-k\right)\phantom{\rule{0.166667em}{0ex}}x\left(k\right)$

which can be expressed as a matrix operation by

$\left[\begin{array}{c}{y}_{0}\\ {y}_{1}\\ {y}_{2}\\ ⋮\end{array}\right]=\left[\begin{array}{ccccc}{h}_{0}& 0& 0& \cdots & 0\\ {h}_{1}& {h}_{0}& 0& & \\ {h}_{2}& {h}_{1}& {h}_{0}& & \\ ⋮& & & & ⋮\end{array}\right]\left[\begin{array}{c}{x}_{0}\\ {x}_{1}\\ {x}_{2}\\ ⋮\end{array}\right].$

The $H$ matrix of impulse response values is partitioned into $N$ by $N$ square sub matrices and the $X$ and $Y$ vectors are partitioned into length- $N$ blocks or sections. This is illustrated for $N=3$ by

${H}_{0}=\left[\begin{array}{ccc}{h}_{0}& 0& 0\\ {h}_{1}& {h}_{0}& 0\\ {h}_{2}& {h}_{1}& {h}_{0}\end{array}\right]\phantom{\rule{72.26999pt}{0ex}}{H}_{1}=\left[\begin{array}{ccc}{h}_{3}& {h}_{2}& {h}_{1}\\ {h}_{4}& {h}_{3}& {h}_{2}\\ {h}_{5}& {h}_{4}& {h}_{3}\end{array}\right]\phantom{\rule{36.135pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\text{etc.}\phantom{\rule{4.pt}{0ex}}$
${\underline{x}}_{0}=\left[\begin{array}{c}{x}_{0}\\ {x}_{1}\\ {x}_{2}\end{array}\right]\phantom{\rule{72.26999pt}{0ex}}{\underline{x}}_{1}=\left[\begin{array}{c}{x}_{3}\\ {x}_{4}\\ {x}_{5}\end{array}\right]\phantom{\rule{72.26999pt}{0ex}}{\underline{y}}_{0}=\left[\begin{array}{c}{y}_{0}\\ {y}_{1}\\ {y}_{2}\end{array}\right]\phantom{\rule{36.135pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\text{etc.}\phantom{\rule{4.pt}{0ex}}$

Substituting these definitions into [link] gives

$\left[\begin{array}{c}{\underline{y}}_{0}\\ {\underline{y}}_{1}\\ {\underline{y}}_{2}\\ ⋮\end{array}\right]=\left[\begin{array}{ccccc}{H}_{0}& 0& 0& \cdots & 0\\ {H}_{1}& {H}_{0}& 0& & \\ {H}_{2}& {H}_{1}& {H}_{0}& & \\ ⋮& & & & ⋮\end{array}\right]\left[\begin{array}{c}{\underline{x}}_{0}\\ {\underline{x}}_{1}\\ {\underline{x}}_{2}\\ ⋮\end{array}\right]$

The general expression for the ${n}^{th}$ output block is

${\underline{y}}_{n}=\sum _{k=0}^{n}{H}_{n-k}\phantom{\rule{0.166667em}{0ex}}{\underline{x}}_{k}$

which is a vector or block convolution. Since the matrix-vector multiplication within the block convolution is itself a convolution, [link] is a sort of convolution of convolutions and the finite length matrix-vector multiplication can be carried out using the FFT or otherfast convolution methods.

The equation for one output block can be written as the product

${\underline{y}}_{2}=\left[{H}_{2},{H}_{1},{H}_{0}\right]\left[\begin{array}{c}{\underline{x}}_{0}\\ {\underline{x}}_{1}\\ {\underline{x}}_{2}\end{array}\right]$

and the effects of one input block can be written

$\left[\begin{array}{c}{H}_{0}\\ {H}_{1}\\ {H}_{2}\end{array}\right]{\underline{x}}_{1}=\left[\begin{array}{c}{\underline{y}}_{0}\\ {\underline{y}}_{1}\\ {\underline{y}}_{2}\end{array}\right].$

These are generalize statements of overlap save and overlap add [link] , [link] . The block length can be longer, shorter, or equal to the filter length.

## Block recursion

Although less well-known, IIR filters can also be implemented with block processing [link] , [link] , [link] , [link] , [link] . The block form of an IIR filter is developed in much the same way as for the block convolutionimplementation of the FIR filter. The general constant coefficient difference equation which describes an IIR filter with recursivecoefficients ${a}_{l}$ , convolution coefficients ${b}_{k}$ , input signal $x\left(n\right)$ , and output signal $y\left(n\right)$ is given by

$y\left(n\right)=\sum _{l=1}^{N-1}{a}_{l}\phantom{\rule{0.166667em}{0ex}}{y}_{n-l}+\sum _{k=0}^{M-1}{b}_{k}\phantom{\rule{0.166667em}{0ex}}{x}_{n-k}$

using both functional notation and subscripts, depending on which is easier and clearer. The impulse response $h\left(n\right)$ is

$h\left(n\right)=\sum _{l=1}^{N-1}{a}_{l}\phantom{\rule{0.166667em}{0ex}}h\left(n-l\right)+\sum _{k=0}^{M-1}{b}_{k}\phantom{\rule{0.166667em}{0ex}}\delta \left(n-k\right)$

which can be written in matrix operator form

$\left[\begin{array}{ccccc}1& 0& 0& \cdots & 0\\ {a}_{1}& 1& 0& & \\ {a}_{2}& {a}_{1}& 1& & \\ {a}_{3}& {a}_{2}& {a}_{1}& & \\ 0& {a}_{3}& {a}_{2}& & \\ ⋮& & & & ⋮\end{array}\right]\left[\begin{array}{c}{h}_{0}\\ {h}_{1}\\ {h}_{2}\\ {h}_{3}\\ {h}_{4}\\ ⋮\end{array}\right]=\left[\begin{array}{c}{b}_{0}\\ {b}_{1}\\ {b}_{2}\\ {b}_{3}\\ 0\\ ⋮\end{array}\right]$

In terms of $N$ by $N$ submatrices and length- $N$ blocks, this becomes

$\left[\begin{array}{ccccc}{A}_{0}& 0& 0& \cdots & 0\\ {A}_{1}& {A}_{0}& 0& & \\ 0& {A}_{1}& {A}_{0}& & \\ ⋮& & & & ⋮\end{array}\right]\left[\begin{array}{c}{\underline{h}}_{0}\\ {\underline{h}}_{1}\\ {\underline{h}}_{2}\\ ⋮\end{array}\right]=\left[\begin{array}{c}{\underline{b}}_{0}\\ {\underline{b}}_{1}\\ 0\\ ⋮\end{array}\right]$

From this formulation, a block recursive equation can be written that will generate the impulse response block by block.

${A}_{0}\phantom{\rule{0.166667em}{0ex}}{\underline{h}}_{n}+{A}_{1}\phantom{\rule{0.166667em}{0ex}}{\underline{h}}_{n-1}=0\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}n\ge 2$
${\underline{h}}_{n}=-{A}_{0}^{-1}{A}_{1}\phantom{\rule{0.166667em}{0ex}}{\underline{h}}_{n-1}=K\phantom{\rule{0.166667em}{0ex}}{\underline{h}}_{n-1}\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}n\ge 2$

with initial conditions given by

${\underline{h}}_{1}=-{A}_{0}^{-1}{A}_{1}{A}_{0}^{-1}\phantom{\rule{0.166667em}{0ex}}{\underline{b}}_{0}+{A}_{0}^{-1}\phantom{\rule{0.166667em}{0ex}}{\underline{b}}_{1}$

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Source:  OpenStax, Fast fourier transforms. OpenStax CNX. Nov 18, 2012 Download for free at http://cnx.org/content/col10550/1.22
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