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`ts1 = filter(ts, numerator, denominator) ts1=filter(ts, numerator, denominator, index) `

` ts1 = filter(ts, numerator, denominator) `
applies the transfer function filter

` ts1=filter(ts, numerator, denominator, index) `
uses the optional

The time-series data must be uniformly sampled to use this filter.

The following function

y = filter(b,a,x)

creates filtered data

`y`by processing the data in vector`x`with the filter described by vectors`a`and`b`.The

`filter`function is a general tapped delay-line filter, described by the difference equation:*a*(1)*y*(*n*) =*b*(1)*x*(*n*) +*b*(2)*x*(*n*− 1) + ... +*b*(*n**b*)*x*(*n*−*n**b*+ 1) −*a*(2)*y*(*n*− 1) − ... −*a*(*N*_{a})*y*(*n*−*N*_{b}+ 1).Here,

*n*is the index of the current sample,*N*_{a}is the order of the polynomial described by vector`a`, and*N*_{b}is the order of the polynomial described by vector`b`. The output*y*(*n*) is a linear combination of current and previous inputs,*x*(*n*)*x*(*n*−1)..., and previous outputs,*y*(*n*− 1)*y*(*n*− 2)... .You use the discrete filter to shape the data by applying a transfer function to the input signal.

Depending on your objectives, the transfer function you choose might alter both the amplitude and the phase of the variations in the data at different frequencies to produce either a smoother or a rougher output.

In digital signal processing (DSP), it is customary to write transfer functions as rational expressions in

*z*^{−1}and to order the numerator and denominator terms in ascending powers of*z*^{−1}.Taking the z-transform of the difference equation

*a*(1)*y*(*n*) =*b*(1)*x*(*n*) +*b*(2)*x*(*n*−1) + ... +*b*(*n**b*)*x*(*n*−*n**b*+ 1) −*a*(2)*y*(*n*− 1) − ... −*a*(*n**a*)*y*(*n**a*+ 1),results in the transfer function

where

*Y*(*z*) is the z-transform of the filtered output*y*(*n*). The coefficients*b*and*a*are unchanged by the z-transform.

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