# Butterworth-Filter

Butterworth filters are continuous frequency filters, which are designed so that the frequency response below the cut-off frequency ωg as long as possible is horizontal. Only shortly before this cut-off frequency is the transfer function of slimming and go to the decrease in gain of 20 dB per decade of frequency n · (n is the order of the Butterworth filter ). The simplest form of the Butterworth filter 1st order constitutes the RC element

The attenuation in the cut-off frequency is approximately 3 dB, that is, a signal with the cut-off frequency is attenuated to the times of the original signal. Butterworth filters have both in the passband and stopband a uniform ( smooth ) shape of the transfer function.

Was named the Butterworth filter after the British physicist Stephen Butterworth, who described this type of filter for the first time.

## Transfer function

This results in a call to the transfer function:

With

Through coefficient comparison with the general transfer function of the coefficients of the Butterworth filter result.

## Coefficients

Bring to the transfer function in the normalized form ():

Are obtained for the coefficients and the following relations:

Order n of the filter just:

Order n of the filter odd:

## Properties

The Butterworth - filter has the following properties:

- Monotonic amplitude response in the passband as well as in the stopband
- Fast kink at the cutoff frequency improves with the order
- Significant overshoot in the step response deteriorates with the order
- The phase curve has a small non-linearity
- Relative frequency-dependent group delay
- Large implementation costs for the high -order

## Filter realization

The Butterworth filter with a given transfer function can be realized in the following form:

The k-th element is given by:

In the digital signal processing Butterworth filter ( recursive filter structure ) can be realized by selecting the appropriate filter coefficients in the IIR filters. The cascading two Butterworth filter of order n gives a Linkwitz -Riley filter 2nd order.

## Normalized Butterworth polynomials

The Butterworth polynomials are usually written as complex conjugate poles s1 and sn. The polynomials are normalized = 1 in addition to the factor? C. Thus, the normalized Butterworth polynomials have the following form:

Down to 4 decimal digits they are: