Basics of Analogue Filters

Basics of Analogue FiltersElena Punskayawww-sigproc.eng.cam.ac.uk/~op205Some material adapted from courses by Prof. Simon Godsill, Dr. Arnaud Doucet, Dr. Malcolm Macleod and Prof. Peter Rayner1Analogue Filters• Specified in a manner similar to digital filters (although frequencies are specified in the Ω domain (in rad/s))ω – pass-band edge frequencypω – stop-band edge frequencysδ – pass-band ripplepd – stop-band attenuations• The pass-band magnitude response is usually required to be in the range [1-d , 1] – matter of convenience, can be adjusted to pmake the pass-band ripple symmetrical with respect to • Expressed asA = -20log (1-δ ) dBp10pA = -20log δdBs10 s2Analogue Filter Parameters10 0.1Ap -1• The discrimination factor √( )10 0.1As -1• The selectivity factor ω /ωps• The -3dB cutoff frequency – at which the magnitude response of the filter is 1/√2 of its nominal value at the bass band• The asymptotic attenuation at high frequencies 20(p-q) dB/decadep,q – numerator and denominator degrees(not defined for a digital filter as the frequency of interest is in the range from [–π,π]3Analogue Filter PrototypesAnalogue designs exist for all the standard filter types (lowpass, highpass, bandpass, bandstop). The common approach is to define a standard lowpass filter, and to use standard analogue-analogue transformations from lowpass to the other types, prior to performing the bilinear transform. It is also possible to transform from lowpass to other filter types directly in the digital domain, but we do not study these transformations here.Important families of analogue filter (lowpass) responses are described in this section, including:•Butterworth•Chebyshev•Elliptic4Butterworth FilterMaximally flat frequency response near W=05Nth-order Butterworth FilterAn Nth-order lowpass Butterworth filter has transfer function H(s) satisfyingThis has unit gain at zero frequency (s = j0), and a gain of -3dB ( = √0.5 ) at s = jΩc.The poles of H(s)H(-s) are solutions of N=3N=4Imag(s)= ωImag(s)= ωXXXXωcXωXci.e. at XXRe(s)Re(s)XXXXXXas illustrated on the right for N = 3 and N = 4:6Butterworth Filter PolesClearly, if λ is a root of H(s), then - λ is a root of H(-s). iiFor a stable filter, the poles of H(s) must be those roots lying in the left half-plane,.The frequency magnitude response is obtained as:21H ( jω )H (− jω ) = H ( jω ) =(*)1 + (ω ω )2NCButterworth filters are known as "maximally flat" because the first 2N-1 derivatives of (*) w.r.t. ω are 0 at ω = 0.Matlab routine BUTTER designs digital Butterworth filters (using the bilinear transform):[B,A] = BUTTER(N,Wn) designs an Nth order lowpass digital Butterworth filter and returns the filter coefficients in length N+1 vectors B and A. The cut-off frequency Wn must be 0.0 < Wn < 1.0, with 1.0 corresponding to half the sample rate.77Chebyshev FilterChebyshev – equiripple response in pass-band (up to ω ), cmonotonically decreasing in stop-band8Chebyshev FilterChebyshev filters are characterised by the frequency response:where T (Ω) are so-called Chebyshev polynomials.n99Elliptic FilterEquiripple in both pass-band and sto-band10Document Outline

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