319
CHAPTER
19
Recursive Filters
Recursive filters are an efficient way of achieving a long impulse response, without having to
perform a long convolution.
They execute very rapidly, but have less performance and
flexibility
than other digital filters.
Recursive filters are also called
Infinite Impulse Response
(IIR) filters,
since their impulse responses are composed of decaying exponentials.
This distinguishes them
from digital filters carried out by convolution, called
Finite Impulse Response
(FIR) filters.
This
chapter is an introduction to how recursive filters operate, and how simple members of the family
can be designed.
Chapters 20, 26 and 33 present more sophisticated design methods.
The Recursive Method
To start the discussion of recursive filters, imagine that you need to extract
information from some signal,
.
Your need is so great that you hire an old
x
[ ]
mathematics professor to process the data for you.
The professor's task is to
filter
to produce
, which hopefully contains the information you are
x
[ ]
y
[ ]
interested in.
The professor begins his work of calculating each point in
y
[ ]
according to some algorithm that is locked tightly in his overdeveloped brain.
Part way through the task, a most unfortunate event occurs.
The professor
begins to babble about analytic singularities and fractional transforms, and
other demons from a mathematician's nightmare.
It is clear that the professor
has lost his mind. You watch with anxiety as the professor, and your algorithm,
are taken away by several men in white coats.
You frantically review the professor's notes to find the algorithm he was
using.
You find that he had completed the calculation of points
through
y
[0]
, and was about to start on point
.
As shown in Fig. 191, we will
y
[27]
y
[28]
let the variable,
n
, represent the point that is currently being calculated.
This
means that
is sample 28 in the output signal,
is sample 27,
y
[
n
]
y
[
n
&
1]
is sample 26, etc.
Likewise,
is point 28 in the input signal,
y
[
n
&
2]
x
[
n
]
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y
[
n
]
’
a
0
x
[
n
]
%
a
1
x
[
n
&
1]
%
a
2
x
[
n
&
2]
%
a
3
x
[
n
&
3]
%
&
y
[
n
]
’
a
0
x
[
n
]
%
a
1
x
[
n
&
1]
%
a
2
x
[
n
&
2]
%
a
3
x
[
n
&
3]
%
&
%
b
1
y
[
n
&
1]
%
b
2
y
[
n
&
2]
%
b
3
y
[
n
&
3]
%
&
EQUATION 191
The recursion equation. In this equation,
is
x
[ ]
the input signal,
is the output signal, and the
y
[ ]
a
's and
b
's are coefficients.
is point 27, etc.
To understand the algorithm being used, we ask
x
[
n
&
1]
ourselves:
"What information was available to the professor to calculate
,
y
[
n
]
the sample currently being worked on?"
The most obvious source of information is the
input signal
, that is, the values:
.
The professor could have been multiplying each point
x
[
n
],
x
[
n
&
1],
x
[
n
&
2],
&
in the input signal by a coefficient, and adding the products together:
You should recognize that this is nothing more than simple convolution, with
the coefficients:
, forming the convolution kernel.
If this was all the
a
0
,
a
1
,
a
2
,
&
professor was doing, there wouldn't be much need for this story, or this chapter.
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 Summer '09
 Digital Signal Processing, Lowpass filter, recursive filters

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