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).


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.


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