Approximation in the time domain whentransfer-function poles are preassigned

I.M. Filanovsky and K.A. Stromsmoe, M.Sc, Ph.D.

Indexing terms: Linear network analysis, Poles and zeros, Time-domain synthesis, Transfer functions

Abstract: Parseval's theorem gives the possibility of connecting the approximation in the time domain in the senseof least squares with the corresponding approximation in the frequency domain. If the poles of the approximatingtransfer function are known or preassigned (as it can be in the case of active 7?C-synthesis) the best location of itszeros can be obtained with the help of the proposed Lagrange-wise ratio. The location of the transfer-functionpoles can be evaluated, for example, by the decomposition of hyperbolic functions into infinite products; thelocation of zeros is obtained as the numerator of the above-mentioned ratio at the second stage of the approxi-mation process. The example shows that the combination of these two steps in the frequency domain can give riseto very satisfactory time-domain approximation.

List of symbols

HOh*{t)Kit)

a unit impulse responsean impulse response approximating h it)the impulse response of n natural fre-quencies approximating h it)

fo(t),f\{t),. .. ,fm(t) = functions for the piece wise approxi-

H(s)H\s)

Fois),Fiis),. . . ,1

5, 5', 5"s,p, V,W,\t =x =6CO =

COo =

S i , S2 , • • • , Sj, . . .

n —Pl,P2,P3,P4 =

w l , w 2 , . . . , w h .

Pi =

a» =

//(/co),//*(/co) =

A0,Au- . -,Ar =

w ==

r =

mation of hit)a given transfer functionan approximating transfer functionthe approximating transfer function with npreassigned poles

Fm(s) = the Laplace transforms of thefunctions/0(0,/i(0. • • • »/m(0the approximation integral errorscomplex variablestime variablereal variableangular variable in the W-planeangular frequencyfrequency of normalisation

, sn = preassigned poles in the s-planetotal number of preassigned polespreassigned poles in p-plane. . , Wn = mappings of the preassignedpoles in W-planereal part (with negative sign) of a complexpole in the s-planeimaginary part of a complex pole in thes-planereal pole (with negative sign) in the s-planetransfer-function values on the imaginaryaxis of the s-planemappings of His) and H% (s) in the s-planepolynomial coefficientsconstant multiplierscomplex amplitudesreal amplitudespole multiplicity

r i , T2 , . . . , r m , rm + a = delay time intervalsW-2 =/ , *

M,N,mu,v =^ O i ^ i • • • Ki . . .

the unit ramp functioncurrent summation and multiplicationindicesfinal summation indicesfinal multiplication indices

Kn = the numerators in the decompositioninto a sum of simple ratios

1 Introduction

Paper 1120G, first received 16th July and in final form 16th October1980The authors are with the Department of Electrical Engineering,University of Alberta, Edmonton, Alberta, Canada T6G 2G7

IKEPROC, Vol. 128, Pt. G, No. 1, FEBRUARY 1981

In some cases in the time-domain synthesis, a given unit im-pulse response h (t) has the Laplace transform H(s) which is atranscendental function. Such H(s) cannot be realised as acircuit with lumped parameters. In such circumstances we tryto find a substituting transfer function H*(s) which is analgebraic ratio approximating H(s). This function is theLaplace transform of the function h*(t) approximating h (t).

We usually assume that'/i*(0 can contain only terms of thefollowing type:

(Ao +Alt+A2t2 + . . .+Art

r)esit (1)

where s,- = — (a,- + 7/3,), a,- > 0 and r — 0 if a,- = 0. Moreover, inmany cases we also assume that n natural frequencies s,- of thesystem are somehow known (at least approximately) or evenpreassigned as, for example, in the case of active /?C-synthesiswhere we can have restrictions on the pole ^-values.

Once st have been determined or estimated, ^£(0 (whichincludes n terms of the type of expr. 1 only) can be evaluateddirectly by one of the possible criteria [1]. For example,assume that all poles are simple and let

M

Kit) = 1

AT N

+ I (Ai-jBi)e-(f*'-ipi>t+ I (2)

where a,-, |3,-, a,- are real and nonnegative,^,-, Bh C{ are real and2M + N = n. As an approximation criterion, we can use

5 = T \h(t)-h*(t)\2dtJo

(3)

and try to find a set of Ah Bt and Q in such a way that thisintegral is minimised. If h(f) is given, this set can be evaluated[1, 2] either by analytical methods when possible, or bynumerical methods. This is connected with calculation of alarge amount of integrals in the time domain. After this, wefind the complete H*(s) whose n poles were preassigned.

The goal of this paper is to find H*(s) in the frequencydomain directly. The proposed method gives the best locationof zeros for //^(s) if its poles are preassigned or prescribed. Asa result, we can use methods for evaluation of poles which giverise (if they were used for calculation of zeros also) to poortime-domain approximations but assure the locations of thesepoles in the left half of the s-plane or its preassigned points.The subsequent correction of zeros is essential in that it canstrongly improve the time-domain approximation.

The fact that the approximation procedure corresponding

35

0143- 7089/81/010035 + 06 $1-50/0

to minimisation of 6 can be transferred into the frequencydomain has been alluded to in Reference 2 but the approxi-mation relationship of H(s) and H*(s), to these writers'recollection, has not appeared in the literature.

2 The frequency-domain approximation

According to Parseval's theorem [3],

\~\h(t)-h*(t)\2dt =

if the integral of the left side in this equality exists. This con-dition is satisfied for the wide class of unit impulse responseswhich do not include pulses at t = 0 and decay in the courseof time (more strictly, for h(t) and h*(t) which are squareintegrable).

Thus, we can approximate H(s) by an algebraic ratio H%(s)restricted to have all n poles at the preassigned points st andthe function H%(s) has to be chosen so as to minimise theintegral error

6' = (5)

Let us introduce the weight function 1/|1 + /w|2 which willhelp us simplify the integrand in the following, when thevariable transformation is introduced below. Then we willminimise the integral

5" = MM (6)

6" will be close to 5' if we assume that the approximationerror |#(/a>) — //*(/o>)|2 is concentrated in the range CJ< 1and the values \H(jco)\ and |//n(/co)| decrease asymptoticallyas 1/w2 at least for CJ > 1. This last assumption assumes that

-W

from the co-ordinate origin in the s-plane and neglect them,retaining the proper difference between the orders of thenumerator and denominator polynomials of H^ (s) which wewill use finally. According to the initial values theorem [2]this procedure changes the behaviour of hn(t) in the vicinityof t = 0 mostly.

We map the s-plane onto the W-plane in accordance withthe well-known [4] mapping

W =1 + s

(8)

This mapping transforms the right half of the .s-plane into theinterior of the unit circle and the /co-axis is mapped onto theunit circumference.! The mapping (expr. 8) is a one-to-onecorrespondence, i.e. for each point in the s-plane there is onlyone corresponding point in the W-plane and vice-versa. If weapply the mapping of eqn. 8 to eqn. 6, we find that one hasto minimise

6" = l- (9)

where G(W) and G*(W) are the mappings of H(s) and#*(s).If a given h (t) satisfies the condition

C\h(t)\dtJo

(10)

then H(s) has no /co-axis or right half-plane poles [1] andG(W) will by analytic for I W\ < 1. G*(W) can be representedas

G*(W) = a0 ...+anWn

(W-W1)(W-W2)...(W-Wn) 01)

where the Wt are mapping of the preassigned st and the at

remain to be determined.If each of the Wt are different, the function of best approxi-

mation minimising eqn. 9 is given [5] by the formula

- i)(w2w-1)... (wnw-1)(\_- wl)(\-w2)... (\-_wn)(W- WX)(W- W2).. . (W- Wn)(K- W)\(WlX- l)(W2X- I) .. . (Wn\-

h(t) and hn(t) will not include pulses at t = 0. The first con- where X = e'd (0 < 0 < 2TT) and bar means complex con-dition can be satisfied by the proper frequency normalisation. jugate. Using the residues theorem [4] we obtained [6] that

The choice of the normalisation frequency influences onlythe difference between 5" and 5' (some tries may be necessary

c a n be represented as a Lagrange-wise ratio (seeAppendix 7 also)

G*(W) = G(0)W- l)(W2W~ 1 ) . . . (WnW- 1)

- Wx ){W - W2) . . . (W - Wn)

- Wy Wt)(l - W2Wt) . . . (1 - WnWt)\-G(()){

iTt (W.-w-iXW! -

to keep it small) but does not change the form of the obtainedsolution. But due to the weight function, 6" can converge evenif

Urn | / / * ( / G J ) | = const (7)

, - W f ) - - - ( W / - i -

It is not difficult to see that

(13)

G*(0) = and

Such functions give rise to hn{t) which do not satisfy the con-ditions for application of Parseval's theorem. But the approxi-

i.e. the ratio G*(W) of the best approximation is the function

TThus, the obtained approximation results can be easily applied todi

,. c x- n*r \ Li • j i_ i u 2.x. L TThus, the obtained approximation results can be easily applied tomating functions Hn (s) obtamed below have the same number d i g i t a l f i l t e r s a l s o T h i s j u s t i f i e s and explains partially the introductionof zeros and poles. We can take some zeros which are farthest of this mapping

36 IEEPROC, Vol. 128, Pt. G, No. 1, FEBRUARY 1981

found by interpolation of G(W) at the origin and at the points1/ Wh the inverses in the_unit circle of the preassigned poles Wt.We can calculate G(l/Wt) if G(W) is analytic at these points.But this condition will be satisfied if H(s) is analytic in theright half of the s-plane. In all other respects, H(s) can bearbitrary, algebraic or transcendental.

The formula, eqn. 13, assumes that all the points W{ aredifferent. This is usually the case in the time-domain synthesis,especially if we preassign the pole locations. But if some of thepoints W( coalesce, G*(W) can be calculated also [7].

To obtain H*(s) we substitute eqn. 8 instead of W into

It is worth noticing that if the pole locations are notrestricted for G*(W), the solution for minimisation of 6 isgiven by the Pade approximant for G(W) [8].

3 Two-stage approximation procedure

The preliminary evaluation of the transfer-function polelocations can be made by any of the known [1] methods. Butmany of them do not guarantee the stability of an approxi-mating transfer function, and its poles can be found after aseries of tries.

For one relatively broad class of problems, the followingprocedure is useful. Assume that we are given the unit impulseresponse h(t) and it can be represented (Fig. 1) as

h{t) = (14)

where /o(0>/i(0 • • -/m(0 have the corresponding Laplacetransforms F0(s), F^ (s). . . Fm(s). Let each of these transformsbe a ratio with poles in the left half of the s-plane (i.e. they areanalytic in the right half-plane). In this case, the transferfunction of the network under synthesis is a transcendentalfunction of the form

(15)fe=O

Following eqn. 9, we multiply and divide H(s) by eSTm

(the choice of Tm + 1 depends on the particular case). Then

H(s) = fe = O

H*(s) can be used as an approximation for H(s) also. The mainadvantage of this truncated expansion is its physical realis-ability. It is not difficult to see that eqn. 18 has a Hurwitzpolynomial in the denominator for an arbitrary v. On theother hand, the calculation of its zeros does not subdue to anyoptimisation restrictions as far as Tm + 1 is chosen. As a result,the approximation in the time domain is poor for the chosenorder of the transfer function (an example is given below).

/fo(t)

\ f (t-r2)

Fig. 1 Approximation of a given transient response by the finiteseries of delayed functions

But we can take the poles of H*(s) as the poles of Hn(s)with n = 2v or n = 2v + 1 and calculate the zeros of //,?(s)using the procedure described in the previous Section. In thiscase, we do all calculations in the frequency domain, beingsure that we obtain the best (in the sense, with the adoptedassumptions, of minimisation of the integral, eqn. 3) approxi-mation in the time domain for chosen pole locations and theirnumber. The calculations show that such a two-stage approxi-mation procedure gives rise to an essential improvement forthe approximation of h(t).

4 Example and additional remarks

Assume that the required unit impulse response is

h(t) = H _ a ( 0 - 2 K - 2 ( f - i ) + l l - 2 ( f - l ) (19)

exp(sr

2 Fk (s) [cosh s {rk - (rm +! /2)} - sinh s {rk -(rm + l 12)}]

cosh (sTm +112) + sinh (srm + , /2)

Hyperbolic functions can be expanded into infinite products[10]

sinh x = x 1 +

(17)

cosh x = 11 1 +4JC 2

1 = 1 27T2(2/+l)27T

Taking the finite numbers u,v of the multipliers in the ex-pansions of the numerator and denominator, we obtain

(16)

T2 K 1-6

Fig. 2 Approximation of the triangular transient responsea h(t)b h*(t)c h*(t)

H*(s) =fe = 0 fe = (2A:+1)V

•s n (i +

n 1 +(2fc+l)V

4k'71IJi.

(2A: ) 2 TT :

08)

IEEPROC, Vol. 128, Pt. G,No.l, FEBRUARY 1981 37

i.e. it represents the triangular pulse (Fig. 2, curve a). TheLaplace transform of this impulse response is the trans-cendental transfer function

H(s) = (20)

We multiply and divide H(s) by e^2 and obtain

H(s) =

- cosh (s/2) - 12s2 (cosh s/2 + sinh s/2)

Substituting into eqn. 21 the truncated expansions

s i n h s / 2 ^

(21)

cosh s/2 — ^r 1 +9iT<

(22)

we obtain an algebraic ratio

0.05629 + 0.00057s2

H ~ 1 + 0.5s + 0.11258s2 + 0.01267s3 + 0.00114s4

0.5 (s2 +98.69604)

~ (s2 + 6.93183s + 16.73378)(s2 + 4.17147s + 53.37632)

(23)

which can be used as a physically realisable transfer functionapproximating H(s). But the unit impulse response

h*(t) = 0.54462 exp (-3.765920 x

cos (2.10266? -1.58285)

+ 0.09368 exp (-2.08574?) x

cos (6.99972f-3.79189) (24)

corresponding to this transfer function is a very poor approxi-mation for the given transient response (Fig. 2, curve b).

4.05386. The given transfer function //(s) becomes

1 - 2 exp ( - 2.02693p) + exp ( - 4.05386p)H(p) =

16.43378/?2(25)

We can easily check that this function is analytic in the righthalf of the p-plane. The poles of the future approximatingfunction H%{p) will be p , 2 =-0.85497 ±/0.51868 andP3,4 = -0.51450 ±/1.72668.'

Applying the transformation

W =l+P

to the function, eqn. 25, we obtain

(1 + W)2(coshV-l)G(W) =

32.86757(1 - W)2(cosh V+ sinh V)

(26)

(27)

where

V = 2.026931 + W

Poles of the function G%(W) are the mappings of plt 2 andp3> 4 in the W-plane. They are Wi<2= ±/3.57631 and W3> 4 =- 6 . 6 9 8 1 8 ±/1.07343. Now we calculate

G(0) = 0.04587

G hH = G f-i-l = 0.03784 •/0.03577

and

(Notice, that all these calculations can be done at corre-sponding points of the s-plane using a given transfer functionH(s).) Substituting all these results into eqn. 13, we find thealgebraic ratio

0.99068W4 + 3.50926W3 + 5.04634W2 + 3.48552W + 0.96205W* + 1.39632M/3 + XAA9269W2 + 17.S5916W + 20.97185

(28)

approximating the transcendental function, eqn. 28. To obtainthe transfer function / / | (s) we use eqn. 26 and do the de-normalisation. Finally, we obtain

2.50559.1O~4(s2 + 1.09342s + 143.31 U5)(s2 -67.36284s + 6138.45529)

(s2 + 6.93183s + 16.73378)(s2 + 4.17147s + 53.34632)(29)

Now we can do the approximation in two stages. Assumethat the previously obtained approximating function, eqn. 23,is used for evaluation of the pole locations only and we decidethat the location of its poles at the points

= -3.76592 ±/2.10166

and

s3>4 = -2.08574 ±/6.99972

is satisfactory for the future approximating function H%(s)with four poles. In this case, we have to determine only thezeros and constant multiplier for H%(s).

To simplify the calculations, we normalise the pole locations,introducing p = S/OJ0 where co0 = x/3^76592 + 2.102662 =

This transfer function has a finite value when s -> °°. But wenotice that eqn. 29 has one pair of zeros which are far apartin the right half of the s-plane. We can consider them aslocated at infinity and discard them, preserving their squaremodulus 6138.45529 as a multiplier (in Reference 2, thesimilar problem is solved by an accurate choice of the basisfor the system of orthogonal functions). Finally, we have

1.53805(s2 + 1.09342s + 143.31115(s2 + 6.93183s + 16.43378)(s2 + 4.17147s + 53.34632)

(30)

38 IEEPROC, Vol. 128, Pt. G, No. 1, FEBRUAR Y 1981

The unit pulse transient response h%(t) corresponding to thistransfer function is equal to

hZ(t) = 2.30533 exp(-3.465920 x

cos (2.102665-1.52982)

+ 0.46330 exp (-2.085740 x

cos (6.99972/+1.77607) (31)

and is shown in Fig. 2, curve c. As we can see, the correctionof the transfer function zeros which we have fulfilled at thesecond stage of the approximation process in the frequencydomain essentially improves the approximation in the timedomain.

5 Conclusions

The approximation in the time domain with minimisation ofthe integral square error can be reduced to the approximationproblem in the frequency domain. If the location of the poles'for the approximating transfer function is known, it is possibleto find the Lagrange-wise ratio which approximates the trans-formed given transfer function with minimal integral squareerror on the border of a unity circle in the W-plane. This ratiocan be calculated if a given transfer function (algebraic or

with preassigned poles in the analog and digital filter design', Pro-ceedings of IEEE International Symposium on circuits and systems,Houston, 1980, pp. 105-112

8 TEASDALE, R.D.: 'Time domain approximation by use of Pad6approximant', IRE Convention Record, 1953, 1, pp. 89-94

9 MATKHANOV, P.N.: 'Synthesis of the pulse shaping circuits byexpansion of entire functions in infinite products', Radiotekhnika,1960, 15, (10), pp. 43-46, 1960 (in Russian)

10 ABRAMOWITZ, M., and STEGUN, I.A.: 'Handbook of mathe*matical functions' (Dover Publications, Inc., New York, 1965)

7 Appendix

The function G(W) is analytic in the unity circle, and accordingto Cauchy's theorem [4],

-wd\ (32)

Then we obtain

G*(W) = G(W)-2-nj{W-Wx)...{W-Wn)

The integrand of the last expression has simple poles at X = W,X = 0 and X,- = 1/W,- (/= 1, 2 , . . . , « ) . Using the residuestheorem [4] we calculate that

= (WlW-l)(W2W-l)...(WnW-\)n( } (WW)(WW)(WW)

wt

(W-Wl)(W-W2)...(W-Wn)

i v-w-^-^-1(34)

transcendental) is analytic in the right half of the s-plane.With small modifications caused by the usual demands fortime-domain approximation, this ratio can be considered asminimising the integral square error on the /cj-axis, whichcorresponds, according to Parseval's theorem, to the minimis-ation of the integral square error in the time domain.

6 References

1 SU, K.L.: 'Time domain synthesis of linear networks' (Prentice Hall,Inc., Englewood Cliffs, New Jersey, 1971)

2 KAUTZ, W.H.: 'Transient synthesis in the time domain', IRETrans., 1954, CT-1, pp. 29-38

3 KORN, G.A., and KORN, T.M.: 'Mathematical handbook forscientists and engineers' (McGraw Hill, 1968)

4 GUILLEMIN, E.A.: 'The mathematics of circuit analysis' (JohnWiley & Sons, 1958)

5 WALSH, J.L.: 'On interpolation and approximation by rationalfunctions with preassigned poles', Trans. Am. Math. Soc, 1952,34, pp. 21-74

6 FILANOVSKY, I.M., and STROMSMOE, K.A.: 'Approximation oftransfer functions with high Q-factor poles by rational functionswith preassigned poles'. Proceedings of the 22nd Midwest Sym-posium on circuits and systems, Philadelphia, 1979, pp. 571-575

7 FILANOVSKY, I.M. and STROMSMOE, K.A.: 'On approximation

Kt =

We can rewrite

W (W-Wl)(W-W2)...(W-Wn)

= G(0)-G(0)

(W -

(W- Wt)(W- W2)... (W- Wn)

-W2)...(W-Wn)

W{WXW- 1)(W2W- l)...(WnW-

wlw2...wnw (35)

The expression in the braces of eqn. 35 can be represented asthe sum of simple ratios

Ki

w w- —(38)

We calculate that KQ = 0 and

w, 'I \wt

W1 1_

ft

(39)

IEEPROC, Vol. 128, Pt. G, No. 1, FEBRUARY 1981 39

Substituting eqns. 35 and 39 into eqn. 34, we obtain

cum - now

- W2 Wt) . . . (1 - WnWt) ( -G(O) + G ^

- Wt)(W2 ~ Wt) . . . - Wt) ...(Wn- Wt)(40)

Igor M. Filanovsky was born in Kirov,USSR, on the 8th April 1940. He re-ceived the equivalent of the M.S. degreein 1962, and the equivalent of the Ph.D.degree in 1968, both in electrical engin-eering, from V.I. Ulianov (Lenin) Instituteof Electrical Engineering, Leningrad.From 1967 to 1971 he was a researchassociate at All-Union Scientific ResearchInstitute of High Frequency Currents,Leningrad, and was involved in the fields

of applied and industrial electronics. From 1971 to 1974 hewas a research associate at 'Giricond' Scientific ResearchInstitute, Leningrad, and was involved in applied micro-electronics and circuit theory. In November 1976 he joinedthe Department of Electrical Engineering, University ofAlberta, Edmonton, Alberta, Canada. His current researchinterests are in the areas of active networks and approximationtheory.

A k

Keith A. Stromsmoe received the B.Sc.degree in engineering physics from theUniversity of Alberta, Edmonton, Canada,in 1958, the M.Sc. degree in electricalengineering from the University of NewBrunswick, Fredericton, New Brunswick,Canada, in 1963, and the Ph.D. degreefrom Purdue University, Lafayette,Indiana, USA in 1970. From 1958 to1962 he served as a TelecommunicationsOfficer in the Canadian Air Force. Since

1963 he has held research and teaching positions with theUniversity of Alberta, where he is presently an AssociateProfessor. In 1975 he spent a sabbatical leave at the Universityof California, Berkeley, where he held a position as ResearchAssociate. His current research interests are nonlinear circuitsand digital systems.

40 IEEPROC, Vol. 128, Pt. G, No. 1, FEBRUARY 1981