# scipy.signal.lfilter¶

scipy.signal.lfilter(b, a, x, axis=-1, zi=None)[source]

Filter data along one-dimension with an IIR or FIR filter.

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 Notes).

Parameters: b : array_like The numerator coefficient vector in a 1-D sequence. a : array_like 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 : array_like An N-dimensional input array. axis : int 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 is -1. zi : array_like, optional 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))-1. If zi is None or is not given then initial rest is assumed. See lfiltic for more information. y : array The output of the digital filter. zf : array, optional If zi is None, this is not returned, otherwise, zf holds the final filter delay values.

Notes

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


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