Analog Filters: Network Functions Franco Maloberti

Analog Filters: Network Functions Introduction Magnitude characteristic Network function Realizability Can be implemented with real-world components No poles in the right half-plane Instability: goes in the non-linear region of operation of the active or passive components Self destruct Franco Maloberti Analog Filters: Network Functions

Analog Filters: Network Functions General Procedure The approximation phase determines the magnitude characteristics This step determines the network function H(s) Assume that The procedure to obtain P(s) for a given A(w2) and that for obtaining Q(s) are the same Franco Maloberti Analog Filters: Network Functions

General Procedure (ii) P(s) is a polynomial with real coefficients Zeros of P(s) are real or conjugate pairs Zeros of P(-s) are the negative of the zeros of P(s) Zeros of A(w2) are Quadrant symmetry Franco Maloberti Analog Filters: Network Functions

General Procedure (iii) In A(w2) replace jw by -s2 Factor A(-s2) and determine zeros Split pair of real zeros and complex mirrored conjugate Example Four possible choices, but …. B(s) must be Hurwitz, for a the choice depends on minimum-phase requirements The polynomial A(s) [or B(s)] results Franco Maloberti Analog Filters: Network Functions

General Procedure (iv) EXAMPLE one NO Franco Maloberti Analog Filters: Network Functions

Analog Filters: Network Functions Use of Matlab % Specify coefficient vector % a=w^6+3*w^4+12*w^2+100 a=[1 0 3 0 12 0 -100] % Obtain zero roots b= roots(a) % Plot the zeros zplane(b) % Form the polynomial x1=input('first zero is # ') x2=input('second zero is # ') x3=input('third zero is # ') c= poly([b(x1) b(x2) b(x3)]) Franco Maloberti Analog Filters: Network Functions

Butterworth Network Functions Remember that therefore: The zeros of Q are obtained by Therefore Franco Maloberti Analog Filters: Network Functions

Butterworth NF with Matlab »ButterNet order of the filter 5 n = 5 a = 1 0 0 0 0 0 0 0 0 0 -1 b = -1.0000 -0.8090 + 0.5878i -0.8090 - 0.5878i -0.3090 + 0.9511i -0.3090 - 0.9511i 0.3090 + 0.9511i 0.3090 - 0.9511i 1.0000 0.8090 + 0.5878i 0.8090 - 0.5878i c = 1.0000 3.2361 5.2361 5.2361 3.2361 1.0000 Result with n=5 m-file clear all; n=input('order of the filter ') zerocoeff=2*n-1; lastcoeff=(-1)^n; a=[1 zeros(1,zerocoeff) lastcoeff] b=roots(a) c=poly([b(1:n)]) Franco Maloberti Analog Filters: Network Functions

Butterworth NF with Matlab (ii) BUTTAP Butterworth analog lowpass filter prototype. [Z,P,K] = BUTTAP(N) returns the zeros, poles, and gain for an N-th order normalized prototype Butterworth analog lowpass filter. The resulting filter has N poles around the unit circle in the left half plane, and no zeros. clear all; n=input('order of the filter ') [z p k] =buttap(n) zplane(p) c=poly(p) Franco Maloberti Analog Filters: Network Functions

Chebyshev Network Functions Remember that Therefore The zeros of Q are obtained by Let Franco Maloberti Analog Filters: Network Functions

Chebyshev Network Functions (ii) Equation Becomes Equating real and imaginary parts For a real v this is > 1 Franco Maloberti Analog Filters: Network Functions

Chebyshev Network Functions (iii) Remember that Therefore The real and the imaginary part of wk are such that Zeros lie on an ellipse. Franco Maloberti Analog Filters: Network Functions

Chebyshev NF with Matlab CHEB1AP Chebyshev type I analog lowpass filter prototype. [Z,P,K] = CHEB1AP(N,Rp) returns the zeros, poles, and gain of an N-th order normalized prototype type I Chebyshev analog lowpass filter with Rp decibels of ripple in the passband. Type I Chebyshev filters are maximally flat in the stopband. %CHEBYNET clear all; N=input('order of Chebyshev ') Rp=input('ripple in the pb (dB) ') [z,p,k]=cheb1ap(N,Rp) figure zplane(p) e=poly(p) k 0.1 dB Franco Maloberti Analog Filters: Network Functions

NF for Elliptic Filters Obtained without obtaining the prior magnitude characteristics Based on the use of the Conformal transformation Mapping of points in one complex plane onto another complex plain (angular relationships are preserved) Mapping of the entire s-plane onto a rectangle in the p-plane sn is the Jacobian elliptic sine function Derivation complex and out of the scope of the Course Design with the help of Matlab Franco Maloberti Analog Filters: Network Functions

Elliptic NF with Matlab ELLIPAP Elliptic analog lowpass filter prototype. [Z,P,K] = ELLIPAP(N,Rp,Rs) returns the zeros, poles, and gain of an N-th order normalized prototype elliptic analog lowpass filter with Rp decibels of ripple in the passband and a stopband Rs decibels down. %ElliptNet clear all; N=input('order of the Elliptic ') Rp=input('ripple in the pb (dB) ') Rs=input('stopband attenuation (dB) ') [z,p,k]=ellipap(N,Rp,Rs) figure zplane(z,p) num=poly(z) den=poly(p) k N=4 Rp=1dB Rs=25dB Franco Maloberti Analog Filters: Network Functions

Elliptic NF with Matlab (ii) [n1 n2]=size(num); [n3 n4]=size(den); xmax = input('what is the max plotted freq? '); npoints=500; w0=linspace(0,xmax,npoints); p1=0; for m=1:npoints w=w0(m); for j=1:n2 p1=p1+num(j)*(i*w)^(n2-j); end numer=abs(p1); for j=1:n4 p1=p1+den(j)*(i*w)^(n4-j); denom=abs(p1); H(m)=k*numer/denom; figure plot(w0,H) Estimate the Module response Franco Maloberti Analog Filters: Network Functions

Elliptic NF with Matlab (iii) »ElliptResp order of the Elliptic 4 N = 4 ripple in the pb (dB) 1 stopband attenuation (dB) 20 z = 0 - 2.0392i 0 + 2.0392i 0 - 1.1243i 0 + 1.1243i p = -0.4003 - 0.6509i -0.4003 + 0.6509i -0.0516 - 1.0036i -0.0516 + 1.0036i k = 0.1000 what is the max plotted freq? 10 Franco Maloberti Analog Filters: Network Functions

Bessel-Thomson Filter Function Useful when the phase response is important Video applications require a constant group delay in the pass band Design target: maximally flat delay Storch procedure Franco Maloberti Analog Filters: Network Functions

Bessel-Thomson Filter Function (ii) Find an approximation of in the form And set Approximations of Example Franco Maloberti Analog Filters: Network Functions

Analog Filters: Network Functions Delay Equalizer It is a filter cascaded to a filter able to achieve a given magnitude response for changing the phase response It does not disturb the magnitude response Made by all-pass filter The magnitude response is 1 since Moreover Franco Maloberti Analog Filters: Network Functions

Analog Filters: Network Functions Examples Franco Maloberti Analog Filters: Network Functions

Analog Filters: Network Functions Examples Franco Maloberti Analog Filters: Network Functions