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 rest is assumed. SEE signal.lfiltic for more information.
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
- a[0]*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