Butterworth Filters Two design parameters:order of the filter N,cutoff frequency we.The squared magnitude response of a Butterworth filter has the following form: 1 HBw(ex)2= 1+ tan(w/2)2N tan (we/2) The tan function arises here because of the bilinear transformation.In the analog case, the transfer function is in the form of 1/1+()2.See Appendix B.1 of OSB for analog Butterworth filter design techniques. The magnitude response of a Butterworth filter decreases monotonically with frequency. The order of the system can be estimated by examining the desired filter gain at the cut-off frequency. Integer round up may be required in estimating the order of the system.Specs are therefore often exceeded at the passband and stopband edges.In addition,the specs are often much exceeded in the stopband,with attenuation approaching zero (-oodB)as frequency approaches m.This is because all zeros of the system are located at z=-1. Since the gain diminishes quickly as frequency increases.It is possible that a lower order filter exists such that it satisfies the given specifications,but does not exceed them as greatly as the Butterworth design. The following sets of figures are examples of Butterworth filter design.For an additional example,see OSB Example 7.4. 3Butterworth Filters • Two design parameters: order of the filter N, cutoff frequency ωc. The squared magnitude response of a Butterworth filter has the following form: 1 2 |HBW (e = jω)| 1 + � tan (ω/2) �2N tan (ωc/2) The tan function arises here because of the bilinear transformation. In the analog case, the transfer function is in the form of 1/1 + ( Ω )2N . See Appendix B.1 of OSB for analog Ωc Butterworth filter design techniques. • The magnitude response of a Butterworth filter decreases monotonically with frequency. The order of the system can be estimated by examining the desired filter gain at the cut-off frequency. • Integer round up may be required in estimating the order of the system. Specs are therefore often exceeded at the passband and stopband edges. In addition, the specs are often much exceeded in the stopband, with attenuation approaching zero (−∞dB) as frequency approaches π. This is because all zeros of the system are located at z = −1. Since the gain diminishes quickly as frequency increases. It is possible that a lower order filter exists such that it satisfies the given specifications, but does not exceed them as greatly as the Butterworth design. • The following sets of figures are examples of Butterworth filter design. For an additional example, see OSB Example 7.4. 3