Information about Basics of Analogue Filters

3F3 – Digital Signal Processing (DSP), January 2009, lecture slides 6a, Dr Elena Punskaya, Cambridge University Engineering Department

Analogue Filters • Specified in a manner similar to digital filters (although frequencies are specified in the Ω domain (in rad/s)) ωp – pass-band edge frequency ωs – stop-band edge frequency δp – pass-band ripple ds – stop-band attenuation • The pass-band magnitude response is usually required to be in the range [1-dp, 1] – matter of convenience, can be adjusted to make the pass-band ripple symmetrical with respect to • Expressed as Ap= -20log10(1-δp) dB As= -20log10δs dB 2

Analogue Filter Parameters √( ) 10 0.1Ap -1 • The discrimination factor 10 0.1As -1 The selectivity factor ωp/ωs • • 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/decade p,q – numerator and denominator degrees (not defined for a digital filter as the frequency of interest is in the range from [–π,π] 3

Analogue Filter Prototypes Analogue 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 • Elliptic 4

Butterworth Filter Maximally flat frequency response near W=0 5

Nth-order Butterworth Filter An Nth-order lowpass Butterworth filter has transfer function H(s) satisfying This 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=3 N=4 Imag(s)= ω Imag(s)= ω X X X X ωc ωc X X i.e. at Re(s) Re(s) X X X X X X X X as illustrated on the right for N = 3 and N = 4: 6

Butterworth Filter Poles Clearly, if λi is a root of H(s), then - λi is a root of H(-s). For 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: H ( jω )H (− jω ) = H ( jω ) = 1 2 1 + (ω ω C ) (*) 2N Butterworth filters are known as quot;maximally flatquot; 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. 7 7

Chebyshev Filter Chebyshev – equiripple response in pass-band (up to ωc), monotonically decreasing in stop-band 8

Chebyshev Filter Chebyshev filters are characterised by the frequency response: where Tn(Ω) are so-called Chebyshev polynomials. 9 9

Elliptic Filter Equiripple in both pass-band and sto-band 10

Elliptic Filter Elliptic filters allow for equiripple in both pass and stop-bands. They are governed by a similar form: Where E(Ω) is a particular ratio of polynomials. 11

Other Types of Analogue Filter Other filter types include Bessel filters, which are almost linear phase. In general, there is a wide range of closed form analogue filters. • Some are all-pole; others have zeros. • Some have monotonic responses; some equiripple. • Each involve different degrees of flexibility and trade-offs in specifying transition bandwidth, ripple amplitude in pass- band/stop-band and phase linearity. For a given bandedge frequency, ripple specification, and filter order, narrower transition bandwidth can be traded off against worse phase linearity 12

Thank you! 13

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