# 0.12 Convolution algorithms  (Page 3/7)

 Page 3 / 7

## 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

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

How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
Got questions? Join the online conversation and get instant answers!  By By By By Rhodes  By   By