# Butterworthfilter data filter

I am currently doing a signal analysis and filtering using Matlab. The filter I am using is Butterworthfilter model to filter scattered data (vector data).

I am using Matlab as follows:

function [y]=fButterworthfilter(input_data,f_cut)
rng default;
[b,a] = butter(8,f_cut/(200/2));
y=filtfilt(b,a,input_data); % zero-phase filtering


I want to see how can I do the Butterworthfilter filtering using Mathematica. I tried ButterworthFilterModel in the help but the input is a function not a vector data.

Let us say I have the following noisy data:

data1 = Table[Exp[-i^2], {i, -5, 5, .01}];
noise = RandomReal /@ RandomReal[{-0.3, 1}, Length[data]];
data2 = data1+noise;


How can I do Butterworthfilter filtering for this data? Thanks

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Butterworth is a continuous-time recursive filter, so it needs to be translated to discrete form and then applied to the data using

 RecurrenceFilter[ToDiscreteTimeModel[ButterworthFilterModel[5], 1], data]


For example, with data={1, 2, 3, 4, 3, 2, 1, -1}', the output is

 {0.00649005, 0.0585661, 0.254294, 0.718228, 1.49564, 2.44071, 3.20403, 3.38901}

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Thank you fro you Replay, I got your idea but I am still not getting same answer that I am getting from Matlab. I don'e understand how to place the filtering inputs to produce same output as I am getting from. I will paste another Question and I would appreciate it if you look at it. –  Algohi Apr 21 '14 at 3:16
The RecurrenceFilter here is like Matlab's function filt. If you want the same answer as filtfilt gives, you will have to do what filtfilt does: filter forwards and then backwards. –  bill s Apr 21 '14 at 12:30
Thanks bill, i have done what you said, forwards and backwards filler as follows: Reverse[ RecurrenceFilter[ ToDiscreteTimeModel[ButterworthFilterModel[{c, a}], b], Reverse[ RecurrenceFilter[ ToDiscreteTimeModel[ButterworthFilterModel[{8, 0.04}], 3.2], data2]]]] is there any better way to do the backward and forward option? –  Algohi Apr 21 '14 at 20:32
I showed what I think is an imitation of what Matlab is doing in this answer: mathematica.stackexchange.com/a/46527/1783 If you don't want to write it all out each time, you can make a function (and call it filtfilt`!) –  bill s Apr 21 '14 at 21:11
Thank you. it works with me good. –  Algohi Apr 21 '14 at 22:07