# matlab, filter function

Travis Oliphant oliphant at ee.byu.edu
Tue Oct 17 16:55:54 CDT 2006

```Kenny Ortmann wrote:

>just looking for some help, most of the time you guys are good with matlab
>code, i am trying to use the filter function under this setting
>
>y = filter(b,a,X) filters the data in vector X with the filter described by
>numerator coefficient vector b and denominator coefficient vector a. If a(1)
>is not equal to 1, filter normalizes the filter coefficients by a(1). If
>a(1) equals 0, filter returns an error.
>
>

There is scipy.signal.lfilter which implements this algorithm.  It's doc
string is

lfilter(b, a, x, axis=-1, zi=None)
Filter data along one-dimension with an IIR or FIR filter.

Description

Filter a data sequence, x, using a digital filter.  This works for
many
fundamental data types (including Object type).  The filter is a
direct
form II transposed implementation of the standard difference equation
(see "Algorithm").

Inputs:

b -- The numerator coefficient vector in a 1-D sequence.
a -- The denominator coefficient vector in a 1-D sequence.  If a[0]
is not 1, then both a and b are normalized by a[0].
x -- An N-dimensional input array.
axis -- The axis of the input data array along which to apply the
linear filter. The filter is applied to each subarray along
this axis (*Default* = -1)
zi -- Initial conditions for the filter delays.  It is a vector
(or array of vectors for an N-dimensional input) of length
max(len(a),len(b)).  If zi=None or is not given then initial

Outputs: (y, {zf})

y -- The output of the digital filter.
zf -- If zi is None, this is not returned, otherwise, zf holds the
final filter delay values.

Algorithm:

The filter function is implemented as a direct II transposed
structure.
This means that the filter implements

y[n] = b[0]*x[n] + b[1]*x[n-1] + ... + b[nb]*x[n-nb]
- a[1]*y[n-1] + ... + a[na]*y[n-na]

using the following difference equations:

y[m] = b[0]*x[m] + z[0,m-1]
z[0,m] = b[1]*x[m] + z[1,m-1] - a[1]*y[m]
...
z[n-3,m] = b[n-2]*x[m] + z[n-2,m-1] - a[n-2]*y[m]
z[n-2,m] = b[n-1]*x[m] - a[n-1]*y[m]

where m is the output sample number and n=max(len(a),len(b)) is the
model order.

The rational transfer function describing this filter in the
z-transform domain is
-1               -nb
b[0] + b[1]z  + ... + b[nb] z
Y(z) = ---------------------------------- X(z)
-1               -na
a[0] + a[1]z  + ... + a[na] z

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```