Analog Filter Design. Part. 1: Introduction. P. Bruschi - Analog Filter Design 1

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1 Aalog Filter Desig Part. : Itroductio P. Bruschi - Aalog Filter Desig

2 Defiitio of Filter Electroic filters are liear circuits hose operatio is defied i the frequecy domai, i.e. they are itroduced to perform differet amplitude ad/or phase modificatios o differet frequecy compoets. Digital Filters (or umerical Filters) operates o digital (i.e. coded) sigals Aalog Filters operate o aalog sigals, i.e. sigals here the iformatio is directly tied to the ifiite set of values that a voltage or a curret may assume over a fiite iterval (rage). P. Bruschi - Aalog Filter Desig

Harmoic Aalysis Brief Filter History: The Basis BC: Apolloius of Perga theory of deferets ad epicycles (the basis of later ( AD) Ptolemaic system of astroomy), maybe aticipated by Babylo

3 Harmoic Aalysis Brief Filter History: The Basis BC: Apolloius of Perga theory of deferets ad epicycles (the basis of later ( AD) Ptolemaic system of astroomy), maybe aticipated by Babylo mathematicias ituitios 8 Joseph Fourier: Théorie Aalytique de la Chaleur. Preceded by Euler, d Alembert ad Beroulli orks o trigoometric iterpolatio. Electrical etork Aalysis 845 Gustav Robert Kirchhoff: Kirchhoff Circuit Las (KCL, KVL) Olivier Heaviside developed the Telegrapher's equatios ad coied the terms iductace, coductace, impedace etc. 893 Charles P. Steimetz: Complex Quatities ad Their Use i Electrical Egieerig I the same year, Arthur Keely itroduced the complex impedace cocept P. Bruschi - Aalog Filter Desig 3

Brief Filter History: Resoace Acoustic resoace as ko sice the ivetio of the first musical istrumets. Acoustical (i.e. mechaical) resoace allos selectig ad ehacig idividual frequecies.

4 Brief Filter History: Resoace Acoustic resoace as ko sice the ivetio of the first musical istrumets. Acoustical (i.e. mechaical) resoace allos selectig ad ehacig idividual frequecies. It suggested the first FDM (frequecy divisio multiplexig) applicatio to telegraph lies based o electrical resoace. Electrical resoace as first observed i the discharge trasiet of Leide Jars ot suitable for telephoe FDM, but sufficiet for Radio tuig 898 Sir Oliver Lodge: Sytoic tuig Marcoi RTX ith sytoic tuig P. Bruschi - Aalog Filter Desig 4

Brief filter history: The begiig of the electroics era 94 Joh Ambrose Flemig Valve (Vacuum Diode) 96 Lee De Forest Audio Vacuum Triode 9 First Triode-based Amplifiers ad

5 Brief filter history: The begiig of the electroics era 94 Joh Ambrose Flemig Valve (Vacuum Diode) 96 Lee De Forest Audio Vacuum Triode 9 First Triode-based Amplifiers ad Oscillators 9 Lee De Forest: cascaded amplifier stages 95 G. A. Campbell K.W. Wager ave filters first example of filter theory: Image Parameters Theory P. Bruschi - Aalog Filter Desig 5

Brief Filter History: Toards Moder Filter Theory 97 Edi Hoard Armstrog: First Superhetorodye Radio 9 Itroductio of the term feedback (referred to positive FB) 97 Harold Stephe Black: applicatio of

6 Brief Filter History: Toards Moder Filter Theory 97 Edi Hoard Armstrog: First Superhetorodye Radio 9 Itroductio of the term feedback (referred to positive FB) 97 Harold Stephe Black: applicatio of egative feedback to amplifiers 9-3 Diffusio of Frequecy Divisio Multiplexig for telephoe calls. 93 Stephe Butterorth itroduces maximally flat filters. (ith amplifier to separate stages) 95-4 Progressive itroductio of the etork Sytesys approach to filter desig. Primary cotributors as Wilhelm Cauer Vacuum tube era P. Bruschi - Aalog Filter Desig 6

7 The role of a filter ca be: Filter operatio Modify the magitude of differet frequecy compoets. These filters are by far the most commoly used. Modify the phase of differet frequecy compoets (i.e. to compesate for a uated phase respose of a filter of a amplifier) Real filters geerally chage both the phase ad magitude of a sigal P. Bruschi - Aalog Filter Desig 7

8 Aalog filters: Geeral properties Commo properties: Liearity Time Ivariace Stability (BIBO: Bouded Iput -> Bouded Output) Filters may be: Lumped / Distributed Active / Passive Cotiuous-time / Discrete-Time P. Bruschi - Aalog Filter Desig 8

9 Lumped elemet etorks Lumped elemet etorks are made up of compoets, hose state ad/or behavior is completely defied by a discrete umber of quatities. Lumped elemet etorks are simplificatios or real systems, hich are spatially distributed (i.e. the relevat quatities are give as fuctio of space variables, defied over a cotiuous domai). Lumped Distributed (e.g. trasmissio lie) P. Bruschi - Aalog Filter Desig 9

iductors, trasformers) Whe coected to exteral idepedet sources, the

10 Passive ad Active etorks (Liear) Passive etork Active etork Passive etork: Icludes oly passive compoets (resistors capacitors, iductors, trasformers) Whe coected to exteral idepedet sources, the et eergy flux ito the etork is alays positive P. Bruschi - Aalog Filter Desig

11 Cotiuous ad Discrete Time Cotiuous time: the sigal at all time values belogig to a cotiuous iterval are sigificat Discrete time: the sigal assumes sigificat values oly at time itervals that form a coutable (i.e. discrete) ad ordered set. The sigal is the a sequece of values s() A sequece ca be cosidered as the result of samplig a cotiuous-time sigal This relatioship is ot uivocal P. Bruschi - Aalog Filter Desig

12 Time ad Value Discretizatio Discrete magitude (digital) Cotiuous magitude (aalog) Discrete Time Cotiuous Time P. Bruschi - Aalog Filter Desig

13 Filter types accordig to the ideal magitude respose f c syoyms: Corer frequecy Cut-off frequecy f CL Corer Freq Lo f CH Corer Freq High P. Bruschi - Aalog Filter Desig 3

14 Real filters: the approximatio fuctio 3 db The Lo Pass filter is the referece for other types Frequecy are geeral give as agular frequecies () A Trasitio Bad (TB) is itroduced c ote: p is geerally differet from c (-3 db frequecy) P. Bruschi - Aalog Filter Desig 4

15 Approximatio parameters for high-pass, bad-pass, bad-stop filters P. Bruschi - Aalog Filter Desig 5

16 The approximatio problem for time-cotiuous aalog filters P. Bruschi - Aalog Filter Desig 6

17 Approximatio Problem The first step is reducig the ide variability i filter characteristics by desigig a referece filter from hich the actual filter ca be derived through trasformatios. The referece filter, H (j), is: Lo Pass ormalized Gai ormalizatio Characteristic frequecy ormalizatio H ( j) H ( j H ( j) ) H ( j ) H ( j ) H ( j ) ad H(j) deped o the actual filter H ( j) : P P P S S S P. Bruschi - Aalog Filter Desig 7

18 Approximatio Problem I order to classify the filters, it is coveiet to defie the fuctio k(j) as follos: K ( j) H ( j) H ( j) K ( j) Ideal Case: H ( j) i the PB i the SB K ( j) i the PB i the SB P. Bruschi - Aalog Filter Desig 8

19 Approximatio Problem Real Case: K ( j) a i the PB i the SB A P H ( ) j log log ( ) ( ) H j H j log K ( j ) A P log a Worst case: a Similarly: log S A P. Bruschi - Aalog Filter Desig 9 A P A S

20 Approximatio Problem We eed to fid a fuctio K(s) such that K(j) satisfies the coditios: K ( j) a for for P S Clearly: There are ifiite solutios to this mathematical problem Solutios should lead to a feasible H (s) Lumped elemet filter: H (s) should be a ratioal fuctios: ( s) H ( s) Where (s) ad D(s) are polyomial D( s) P. Bruschi - Aalog Filter Desig

21 otable cases Maximally Flat Magitude Filters (e.g. Butterorth Filters) Chebyshev Filters Iverse Chebyshev Filters Elliptical Filters Bessel Filters Geeral selectio criteria: The loer the polyomial order, the better the solutio Mootoic behavior i the PB ca be required Asymptotic behavior for >> S could be importat Phase respose: a liear respose is ofte required P. Bruschi - Aalog Filter Desig

22 Maximally Flat Magitude (MFM) Approximatio P. Bruschi - Aalog Filter Desig s s K ) ( j K j j K ) ( ) ( ) ( j K j H ) ( ) ( H j All derivatives up to the (-)th are zero for =.

23 Maximally Flat Magitude (MFM) Approximatio H j ( ) Sice H (s) is the Fourier trasform of a real sigal (the impulsive respose): H *(j)=h (-j). The: H ( j) H ( j) P. Bruschi - Aalog Filter Desig 3

24 Maximally Flat Magitude (MFM) Approximatio s j js H ( s) D( s) H ( s) H ( s) H ( s) H ( s) D( s) D( s) s Thus, H (s) is a all-poles fuctio: s The roots of D(s)D(-s) are the solutios of + (-s ) = P. Bruschi - Aalog Filter Desig 4

25 Maximally Flat Magitude (MFM) Approximatio j s s e s k exp j k k :iteger Radius: agle odd multiples of for eve : eve multiples of for odd P. Bruschi - Aalog Filter Desig 5

26 Maximally Flat Magitude (MFM) Approximatio eve odd s k H ( s) k ( s s k ) P. Bruschi - Aalog Filter Desig 6

27 Butterorth Filters The Butterorth filter is a MFM filter ith = It ca be sho that the case is idetical to = but ith simple frequecy trasformatio H j ( ) D (s) are the Butterorth polyomials o ormalized (actual) filter H ( j) P. Bruschi - Aalog Filter Desig 7

28 H j ( ) Butterorth Filters H ( j) 3dB C C Logarithmic Magitude P. Bruschi - Aalog Filter Desig 8

29 Filter Parameter Determiatio K ( j ) K ( j ) a a S P A P A S S P K ( j ) S P a A S A p log log S P P a P a P P C P a P. Bruschi - Aalog Filter Desig 9

30 Butterorth filter : example f P = khz f S = khz A P = db A S =6 db S P f f S P a A P.59 A S 965 a C P a f P.63 log log S P.94 P. Bruschi - Aalog Filter Desig 3

31 K j) C Chebyshev Filters ( C (): -th order Chebyshev polyomial C cos arccos cosh arccos h C x xc ( x) C C x C x x C x x ; P. Bruschi - Aalog Filter Desig 3

32 Chebyshev polyomials: properties C cos arccos cosh arccos h For << oscillates betee ad For =. For = : if odd, if eve For = : for every d For > icrease mootoically st Chebyshev polyomial have the highest leadig term coefficiet tha ay other Polyomial costraied to be less tha (i modulus) for betee ad C 6 3 rd P. Bruschi - Aalog Filter Desig 3

33 Chebyshev filters: Pass Bad Atteuatio K ( j) C H ( j) H ( j) C ( ) H ( j) P ( ) ( ) H j H j a A P e.g. A A P P db 3 db.5 P. Bruschi - Aalog Filter Desig 33

34 Chebyshev filters: Stop Bad Atteuatio K ( j S ) C cosh arccosh S S cosh mi arccosh S arccosh ( arccosh ( ) ) S A S A P P. Bruschi - Aalog Filter Desig 34

35 Iverse Chebyshev filters (or Chebyshev type II) P. Bruschi - Aalog Filter Desig 35 Target: Obtai mootoic behavior i the pass-bad (o ripple) ad ripple I the stop-bad ) ( C j K ) ( ) ( C j H ) ( j H

36 Elliptic (or Cauer) Filters K ( j) R ( j) R : elliptical ratioal fuctio : R () / D R () H R R pk zk M M M such that s / k k R / ( ) pk zk pk zk for for eve for odd P. Bruschi - Aalog Filter Desig 36

37 Phase, delay, group delay I order to maitai the shape of a geeric sigal, the folloig coditios must be respected: All the sigificat frequecy compoets of the sigal fall ito the filter pass-bad, hich should be as flat as possible: The filter phase respose i the pass-bad should be of the type: t R If this coditio is fulfilled, the iput sigal is simply delayed by time t R. I other ords, the group delay should be costat. The group delay is defied as: G d d P. Bruschi - Aalog Filter Desig 37

38 Bessel Filters Costat group delay is very importat i systems that has to hadle digital trasmissios, here sigal distortio may result i high BER (Bit Error Rate), or eve i urecoverable sigal. The Bessel filter is obtaied by cosiderig a all-pole fuctio: H s K Ds We start from a geeric polyomial D(s), substitute s=j ad tha calculate the phase of H by: ImD j arcta ReD j P. Bruschi - Aalog Filter Desig 38

39 Bessel Filters The Bessel Filters are derived by: Taylor s expasio of the arcta() fuctiois calculated obtaiig a polyomial approximatio of the phase; The first derivative of the phase approximatio is calculated The costat term of the derivative is set to (group delay=) Higher order terms are set to zero; this correspods to settig the derivatives of the group delay to zero. The umber of derivatives that ca be ulled depeds o the filter order. The result is a maximally flat group delay Ca be used to purposely itroduce a delay P. Bruschi - Aalog Filter Desig 39

40 : : 3 :... D D( s) D( s) D( s) s s 3 3s 6s D s D Bessel filters, examples Deomiators of various order for uity group delay s ; D 3; 5s K K 3 5; K 5 3dB (Bessel polyomials, recursive expressio) The Bessel filter is less selective tha a Butterorth filter of same order, but its phase respose is much more liear l() Approximate expressio for 3 (pass-bad group delay after frequecy scalig) P. Bruschi - Aalog Filter Desig 4

41 Summary of filter characteristics Butterorth: maximally flat i the pass-bad ad mootoic everyhere Chebyshev: More selective tha Butterorth (sharper trasitio), but ripple i the pass-bad (mootoic i the stop-bad) Iverse Chebyshev: Same selectivity tha Chebyshev, but ripple i the stop-bad (flat i the pass-bad). Magitude do ot decrease asymptotically i the stop-bad Elliptic: Best selectivity, but ripple i both the pass-bad ad stopbad. Magitude do ot decrease asymptotically i the stop-bad Bessel: The least selective of all other filters, but the best i terms of phase liearity (costat group-delay i the pass-bad) P. Bruschi - Aalog Filter Desig 4

42 Other cotiuous time filters Optimum L-filter (Papoulis) Obtais the best selectivity ith a mootoic respose. Compared ith a Butterorth of the same order filter it is sharper i the trasitio bad, but less flat (but still mootoic) i the pass-bad. All pass filters (phase equalizers). Their commo characteristic is that for each pole they have a zero ith opposite real part. As a result, they have RHP (right half plae) zeros ad their step respose is geerally preceded by a glitch i the opposite directio ith respect to the fial value. For step-like sigals, lo-pass phase equalizers (e.g. Bessel filters) are to be preferred. Filters based o Padè approximatios: The Padè approximatio is the best -order ratioal fuctio that approximate a arbitrary fuctio. It is used for the approximatio of the ideal delay: exp(-jt D ). The all-pass fuctios are a particular case of Padè approximatio. P. Bruschi - Aalog Filter Desig 4

43 Frequecy trasformatios P. Bruschi - Aalog Filter Desig 43 The aim of frequecy trasformatios are: Chage the characteristic frequecies ith respect to the ormalized case Chage the lo pass respose ito a high pass, bad-pass etc. s s s s B s s s s s B s All characteristic frequecies are multiplied by High pass Bad-Pass Bad Stop

44 Pass Bad trasformatio: meaig of B, P. Bruschi - Aalog Filter Desig 44 B j j j B j B B B s s B s for small variatios aroud, such that:

45 P. Bruschi - Aalog Filter Desig 45 B B B B Pass Bad trasformatio: meaig of B, B B : for B

Pass Bad trasformatio: meaig of B, Whe =, =. The, the respose of the pass-bad filter at is the D.C. value ( =) of the prototype lo pass filter. For variatios from, moves aay from the origi.

46 Pass Bad trasformatio: meaig of B, Whe =, =. The, the respose of the pass-bad filter at is the D.C. value ( =) of the prototype lo pass filter. For variatios from, moves aay from the origi. Whe <, is egative, so that H() is the complex cojugate of the values at > (see the phase diagram i the figure) The badidth B is the differece betee the frequecies ad, for hich the absolute value of the ormalized frequecy is uity. If the badidth B is much smaller tha frequecy (selective filter), tha ad are symmetrical ith respect to. P. Bruschi - Aalog Filter Desig 46

6.003 Homework #3 Solutions

6.003 Homework #3 Solutions 6.00 Homework # Solutios Problems. Complex umbers a. Evaluate the real ad imagiary parts of j j. π/ Real part = Imagiary part = 0 e Euler s formula says that j = e jπ/, so jπ/ j π/ j j = e = e. Thus the