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

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 “Algorithm”).


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.


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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