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:

RC circuit Bode plot Decibel Sallen–Key topology Group delay and phase delay Linkwitz–Riley filter Elliptic filter
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