THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING THE SYNTHESIS OF LINEAR FILTERS WITH REAL OR IMAGINARY TRANSFER FUNCTIONS *~ Elnmer G..,Gilbert Joseph eO-terman September, 1959: IP-384

I X- 6 AS ""IV -~ ~~~~~~"Ill

INTRODUCTION The synthesis of linear filters with symmetrical (even-function) or antisymmetrical (odd-function) impulse responses has both theoretical interest and practical value. Such filters which can be called time-symmetric or time-antisymmetric, following the terminology of Storey and Grierson,l are characterized by transfer functions which are purely real or purely imaginary. Alternate names, proposed and. employed in this paper, are real or imaginary filters. Applications of real or imaginary filters include compensating systems which must have no phase distortion, such as those often used in the field of acoustics and. in the. compensation of signals generated by picture scanning devices.2,3 In the past little work of a general nature has been done. Although Kalmann,4Wiener and Lee,5,6 and others 7,8 consider delay line transversal filters which have real or imaginary transfer functions, their emphasis is on transfer functions periodic in frequency. An excellent paper by Levy 9 describes some of the properties and the usefulness of real and imaginary transfer functions;however, it limits synthesis to a lumped element delay line filter with many experimentally adjusted reflections. Storey and Grierson 1 have used a tape recorder scheme which involves storing the input signal on magnetic tape and processing the signal through reverse playback. Since these past contributions are restricted to limited types of transfer functions and physical realizations, their significance has not generally been appreciated. It is the purpose of this paper to extend previous work by developing fully the conditions under which real and imaginary filters can be realized and by utilizing an orthogonal function approach for solving the approximation problem. Filter realizations presented in this paper do require a delay element. And although it is true. that real or imaginary filters can be realized with arbitrarily small error by means of conventional lumped-element filters, it is felt that the filters presented here can often be more simply and economically realized, particularly when phase accuracy requirements are high. STATEMENT OF THE PROBLEM The problem of synthesis is to realize physically and as closely as practicable a prescribed mathematical operation on an input function of time. For a linear, time invariant system the prescribed operation may be expressed by the superposition integral 00 fo(t) = h(T)fi(t - T)dT (1) where t is time, fi is the input function, fo is the output function, and h(t) is the response of the prescribed system to the unit impulse uo(t) occurring at t =O.

-2The operation may also be expressed in terms of the Fourier transform of fi, fo and h by* Fo(n) = H(ja) Fi(ja)' (2) H(jOY) is the sinusoidal response function or the transfer unction The Forier transform is used rather than the Laplace transform becaue imulse reaporses h(t) will be considered whch are non-zero for negative t. Synthesis is assumed complete when a physical system is determined which has an impulse response h*(t) which approximates h(t) sufficiently well for a given purposeX An approximation generally results beause the physial system imposes the realizability condition h*(t) =0 for t <01 and possible practical eonditions, such as being realized by a liped-element system (H(JO) ratonal in J) In the present case ether HR or HI (both real functions) in H HR + JH () is zerol To approximate as closely as possible such an HI we will develop the onditions under which it is possible to realize an H* = H*R + j (4) (H*R and H*I are real functions) where either H* = 0 or H*R = 0 i*e*. where the realized filter is either real or imaginary, CONDITIONS FOR REALIZABII1TY From the Fourier transform we have H*R f h*(t) cos (t dt.00 H*I: - | h*(t) sin Et dr. (5) Thbus it is clear that a symmetric impulse response, hi*(t) h*(-t), (6) yields a real filter (H*I 0 ) and an antisymmetri impulse response h*(t) = -h*(t) (7) yields an imaginary filter (H*R = 0)''Tihe uer'ease-letter functions denote the transforis of the lower-case-letter funetions; i.e*. H(j<) f 0 e-J:'4 h(t) dt. It is tacitly assumed that the transforms exist when employed:

-3Sinae realizability requires h*(t) = 0 for t < 0, it is clear from (6) and (7) that realizable real and imaginary filters require h*(t) = 0 exeept at t - O* The only functions which satisfy this condition are thbe unit impulse uo and its derivatives,ul, u2,,,, But the impulse derivatives have no value as approximating functions and are not realizable by physical systems having finite gain at infinite frequency4 Hence real and imaginary filters cannot be realized:directly. By introducing a time delay T dertain real and imaginary filters can be realized~ The time delay allows h*(t) to be symmetia or antisymmetic about t 3 T instead of t. Realizability then means that h*(t) = O, t < O, t >2T (8) h*(t + T) = h*(-t + T)' 0 < t < T (9) (a delayed real filter) h*(t + T) h +T) =h(- < t < T (10) (a delayed imaginary filter) Fortunately in most practial applications the added delay T does not limit the usefulness of real and imaginary filterss Sinee lumped-element systems have impulse responses of the form N t h*(t)= Zntkn e n + Co(t) t >0 (11) n=l 00 O, t < (kn = positive integer, c = real, imaginary, or complex number) and have infinite memory, it is obvious that they cannot satisfy (8) which states that h*(t) has a finite memory of 2T, Only through the introduction of distributed elements may (8) and (9) or (8) and (10) be satisfied. PHYSICAL REALIZATION Figure shows how a singe delay-element transfer function e-JT and two lumped-element transfer funcstions Hl(jI) and H2 (J), may be used. to realize H*(jF), For t < 2T the impulse response is simply h(t); for t > 2T the impulse response is'h(t) -h2(t - 2T), To make h*(t) = 0 for t > 2T.it is necessary to ae the responses of the two paths cancel for t > 2T, ie to mak h1(t) = h2 (t - 2T) for t > 2T, Thu h*(t) =, t < 0 = hl(t), 0 < t < 2T 5= 0, t > 2T (12)

and h2.(t) = O, t < 0 = hl(t + 2T), t > 0 (13) Sinee hi must symmetric or antisymmetric about t = T in (0O 2T) it is clear that hi can have no terms with an exponential funetion of time as a factor, That is, the terms in hi must be of the following t.ype: Type of funtQions Form of realization 1r t, t2, - tk k integrators cos Ml t sin 01 t. acos'.2 tj sin.2 t- undamped filter with poles at "*~E - + -U1 + -2cos O l t ~sin M t, t aos a t, t sin ic t$, --- undamped filter with (k + 1) Ckos t! tk sin ^ t, order poles at = + tj In every ease the poles of H1 (and also the poles of H2 since by (13) the poles of H2 must also be the poles of H1) are on the imaginary axis in the aomplex p plane, In a later section approximating series based on the first two types of terms will be described, Sinee passive eiruit realization of H1 or H2 demands lossless induators and eapacitors, active circuitry is preferred for high accuracy realization, Aetually, active circuit realization is mandatory wiere H1 or H2 are voltages ratios and have poles at p = 0, Other fo for h*(t) can be obtained if more delay elements are introduced. Figure 2 shows an example, Here h*(t) = hT(t) + h2T(t) (14) where hT(t) = t < t > T h2T(t) =, t < T t > 2T (15) and a delayed real filter) - orh2T(t + T) -= _l4~(t + T)e0 < t < T (17) (a delayed imaginary filter)

-5The functions hT and h2T are realized as h*(t) was realized in Figure l* In addition to requiring more than one delay element, the realization possesses practical' limitations, If either hT or h2T contains a term with a decaying exponential factor, then by the symmetry conditions (16) or (17) the other contains a term with a aQoresponding growing exponential factor which leads to an active unstable lumped element filter in-the overall realization, Since the matching of gains in the unstable lupped element filters cannot be exact, it is impossible to make Ih*(t)l < o as t -, oo When the lumped element filters in hT and h2T have all their poles on the imaginary axis in the p plane a gain matching problem is still presents though it is not nearly so severe, Further remarks on accuracy of realization are given in the section titled "Illustrative Examples." Many other types of responses are available from filters which contain both lumped elements and delay elements. Various authors have discussed limited classes of such filterslO0ll For the realization of real and imaginary filters it does not seem worth-while to use more than a single delay element unless several delay elements will allow exact or nearly exact synthesis of the prescribed impulse response (at least within the time delay T). Certainly, when h(t) consists of sections of rather simple functions this is the case, Another example occurs when N h*(t)l an u(t - tn) (18) tn = real number an = real number Such an h*(t) can be realized with a tapped delay line and a suming amplifier and leads to the periodic transfer functions described by Kalmannm4 Wiener and Lee,5,6 White and Ruvin,7 and Urkowitz,8 THE APPROXIMATION PROBLEM Approximation involves the representation of a prescribed system function (h(t) or H(jO)) in terms of a physically realizable system function (h*(t) or H*(JcD)), If the physical system is lumped, then H*(j) is rational in jt; if the physical system has:.adeLayed real or imaginary transfer function, then h*(t) and H*(QJ) assume the forms discussed in the previous section, Of the many approximating techniques available, this paper will present a minimum integral-square-error approach based on orthogonal approximating functions, rthogonal function approximation has the following advantages ) t iegral-square-error is minimized and easily computed. 2) the approximation may be computed readily in either the time or frequency domains, 3) approximation error tends to be small throughout the interval of expansion (as opposed to a Taylor expansion where the approximation tends to be good only at one point), 4) orthogonal functions suitable for realization of real and imaginary filters are well known and tabulated, 5) additional terms may be added to an approximating series to further reduce error without forcing a new calculation of the terms already determined, Orthogonal functions have been employed by many authors for the synthesis of lumped-element systems.12135,14,15 However, because of the objectives of this paper, different types of apoximat4'ng functions must be considered here, Whereas

-6the orthogonal time functions which appear in the series for h*(t) for a lumpedelement filter consist of exponential functions defined in the time interval (Oo) the functions used in real or imaginary filter approximation mst be orthogonal in (0, 2T) and. be symmetric or antisyzmletrie about t =-T, ORTHOGONAL APPROXIMATING FUNCTIONS Legendre Polynomials When h*(t) is a polynomial in t, as is the ease when a real or imaginary filter is realized by a delay line and a number of integrators, Legendre polynomials are employed to form an orthogonal set o!(t), 02(t)..*- * Since Legendre plynomials are orthogonal in (-, 1) it is necessary to time scale the prescribed response so h(t) is satisfactorily represented in (-1 1 ). Once the approximation has been completed, h*(t) which is in (-1, 1) my be delayed and time scaled to (0, 2T) for physical realization. A later example will illustrate how these delaying and scaling steps are earried out Following the usual definition of the Legendre polynomials the 0n are given by 16 On O It1 > 1,n I= 0 1, 2, * 1, -1 < t,< 1l n = O 1 dn(t2_ l)n = 2nn, dtn t -l <t < 1, n = 1, 2, 3, (19) This is know as Rodrigues'formula. Several of the 0n are tabulated and plotted in Table I. If higher order functions are-required they can be obtained most conveniently from the recursion relation 16 nn= (2n - 1) t On-1 -(n 1) 0n-2 (20) The approximating series assumes the form h*(t) = Cnn(t) (21) n:O where cn=(n ) 1 hdt = (n+ *) f hOdt. (22)'-1 _ -00, 1 bcuete12 The factor (n +') ocurs in (22), because the n are not normalized, i.e.~ 0 dt = (n + )- 1 Since N is finite (filter complexity grows directly with N) the integral.-square error, 2 E = j(h-h*)t t= i h dt-' —1 (23) -.o~ -~ n=O'..n+.

-7is in general not zero, Approximation accuracy rleuirements set an upper bound on E which in turn establishes the minimum N, From Table I it is seen that the 0n are symmetric functions for even n and antisymmetri functions for odd n. Thus the even terms are used for a real filter approximation and the odd terms for an imaginary filter approximation, This aan also be seen from (22) which gives en = 0 for n odd when h is a symetric function and Cn = 0 for n even when h is an antisymmetria function, In the frequency domain (21), (22) and (23) become respectively N H*(J.) n= Cnen(j), (24) n-=O -*oo.. 1cn (n+ ) f Hen d~ (25) -00 and 2 N E = / IHI- E (26) ^ ~''~ ~60Q ^ ^_ ~*1 n=O n+ where 6n(J)) is the Fourier transform of Qn(t) and. n8 is the complex conjugate of 6n, Expr.essions (25) and (26) are derived from (22) and (23) by application of Parseval s theorem, 17 The formulas for e0 and. e are -e = ef-j o at. fl ej'. dt 2s) (27) an-d 40' ~C1 dt 1 e-b cos' sin.' S S - J e 1 1 dt f e't dt = 2 [ — — ] (28) -* -l Although the higher order functions can be obtained in the same way, a much more rapid procedure is to se the recursion formula en = 2n 1 n-1 + 2 (29) which is developed in the Appendix* The n are also tabulated and plotted.in Table I. As should be expected., the 8n are real for even n and imaginary for odd n,

-8Table I shows that the approximation H* contains terms of the type sin. ej -e" (:)n 2(.)n and cos0 e" + e( )n 2 (j@)n When h*(t) is delayed by T -1 to the interval (0, 2) for realizstion (time scaling to (0, 2T) will not be considered here) functions of the type 1l e 2(j)n and + e 2(^' are obtained. Thus H* asumes the form N N H* E= 9' + e, -I (30) n=O ((n n=O (j)n where the q;h and r;n are determined from the cn and. the expressions for the en. Figure 3 shows the realization* Sine and Cosine Funations in the Time Domain If a real or imginary filter is realized with a single delay line and an undamped lumped element filter, h* consists of sections of sine and cosine functions and the most obvious orthogonal series is a Fourier series in the time domain,* For aonvenienee we will consider a Fourier series with the interval of expansion (-.1 1), Then the previous remarks which were made concerning time scale still hold. *It is possible to generate other orthogonal series from sine and cosine functions. For example the sine and cosine. functions would not have to be harmonic.

-9Since the period of expansion is 2 the Fourier series can be written as a N h* = + (an cos mnt + bn sin nnt)i - < t < 1 n=l 5 0, Itl > I (3) Because two types of approximating functions are involved for each n (sin nnt and cos nt-) the functions oC(t) = Cos inti -1< t < 1 = 0, lItl> (32) n(t) sin nt -1 < t < 1 = O Itl> (33) are defined. The Fourier series (31) then becomes a N N h~(t) Co % a + b h(t) = ~ i na n — bn (34) 2 n=l n=1 where the coeffiients are given by.00, an f h an dt = h cos Itt (35) and bn 5 h n dt = fl h sin gnt dt (36) For a real filter bn = and for an imaginary filter an F 0. Thus the aC form real filter approximations and. the Bn form imaginary filter approximations. In the frequency domain the real-filter funtons are given by the Fourier transform of the a An e- n dt = e cos nnt dt J_ Jl)n (ej e-j) ( )2 + (n)2 _a. ((-1) sin (37) (.-'~,2 + (: ()2

-10Similarly for the imaginary-filter funtions 00 1 P Bn e 3 dt = e sin nt dt n t -00 _ n(-1)n (eJ - et^~) (jp+)2 + (gn)2 =j2 )n sjir (38) () (2 + (tn)2 Table II summarizes these results and shows plots of the fuctions an An and Bn,* The approximation coeffieients can be evaluated in the frequenc domain by the following equations (obtained by aplying Parseval' s theorem to (35) and (36) 00 aJ' H A d (39) n 2 H Jdn (40) The integral-squae-error is 2 N +1 2 E 1J h2 _. ai2 2 i 1 Ia2 22 2t- b =i I -_ h1.(a +bI HI2dm. (a +bn n n C Ca 2 n'n 2 n=J.- n=l (41) When h* is delayed to the interval (0, 2) for realization the functions An and.Bn become A - (-l)l (1- e-j) (42) (*be2 + ( n)2 *A brief tabulation of the An and Bn and also the n is given in Report S58 of the Radiation Laboratory, MIT,, August 1945.

-IIand Bn e.J = n(-1) (1 2e) (4)) (jwc~o)2 + (on)2 Thus H* assumes the form H* (1 - e- )H (44) and may be realized as shown in Figure 4, For a real filter -a N H, +^ Z.(4'5) 3 2J n=l (-j)2 + (nn)2 for an imaginary filter H bnrin(-l) H' ~ ~;, (46) 3 n:l ( Qi)2 + (nn)2 ao H cean be realized as a. voltage ratio with lossless passive circuits except for the term., 2J ILLUSTRATIVE EXAMPLES For the first example a real, ideal low pass filter is approximated in the time interval (-l, 1) by sine and cosine functions, If the ideal filter has a Cutoff fre'quency -, then the transfer function is: H HR =1 - -< = o,1 (47) and -h si t (48) The coefficients an (the bn 0 ) are determined most easily from (35) with h given by (48): ~ ~i.,~~a.i sint a=J-os tt d2 A.t 2 o tdt 1- t

-121 c ntdt,1 1 sin(- + +n)t si I II sn( + n)t dt + I ain.-. dt Ott at [ [SQi + Tn) + Si( - iin)J, (49) Si(x) = Jy ~ dy the sine integral function for which tables are available, The integial-s.quare error is given by (41) as o a2 N 2 E= o-. an (50),it 2 n=l A measure of the fractional -;proximation error is the relative mean square error E (51) rel 7o. J h-dt which in this ase is 2 a N re l 1- ( - + ) (52) Since Erel varies with..: (the an depend on ) it is natural to ask what outoff frequeney yields the bet- approximation Table III gives some results for N = 3145,5 For N = 3 (four non-zero terms in h*) the smallest E e1 is obtained for e = 3*5,5 Figure 5 gives the an and shows h*(t) and H*(J). for"" =- 3*,5 and N = 3* It is not surprising that "%= 3=,5n gives the best approximation sinae 3*5-. 3, the highest frequency component in h*(t) for N 5 3, Inasmuch as the c = 3*55 design gives the smallest Ere it should serve as the prototype for other cutoff frequencies Thus, where a cutoff frequenqy of 35.T yields T 1 a cutoff frequency of % yields T 3*5X& The transfer function is 3_5___ 3,* [5(c 2 -.9957 H*= 2( ~ )sin,^ 5( ti + 2 (1)2 22 525 ((2+5f))2 ^ - (l~ and. an be realized in the form of Figure 4 with an added delay T -. c

-13Proceeding in a similar fashion the real, ideal low pass filter can be approximated by Legendre polynomials. The cn are given by (22) with h given by (48) and 0 by Table I. Of course, cn 0 for odd n. Equation (22) involves integrals of the form tk sin u t dt (k = -1, 1 3, 5*. ) which are readily evaluated* From (23) and 0 c (51) Z n 1 2 Ere! = i - an Z (54) c n=o 1 + " n 2 In this case the smallest Erel is obtained for m = 2r when N = 6 (four non-zero terms in h*). Figure 6 gives the cn and shows h (t) and H*(jd) for e = 2i and N = 6. Note that Erel = 0506 which is somewhat higher than obtained for the same number of terms in the aosine funetion representation. Using the above design for anN = 6 prototype yields T = 2 and the transfer function * = sin [-,1988 + 115.6 (2) - 5,95 + 12.(), )7] + aos 1 [- 2.747 ()2 + 1,148 (2<1)4 - 12,140. (j -)] (55) 2v Realization in the form of Figure 3 requires an-additional delayT = - The last example is a sine and cosine function approximation of the imaginary, ideal low pass filter characteristic H= jHI +J 0, <-: < -j, <Q- < O - o0f - = ~0to 11} > (56) and cos t - 1 h (57) nt

-14The coefficients bn (the an = 0) are given from (36) and (57) by 1 Cos t -1 b = 2 o - i sin int dt n lo.it =. [Si(tn + c) - 2Si(nn) + Si(n - c)] (58) and the relative mean square error by N Er l — Z b2 (59) Erel = 1 -.- n=l In this case, Erel is near its smallest value at e, = 3x when N = 4 (four non-zero C terms in h*) where Erel =.119 The relatively large Erei is probably due to the discontinuity in H at. = 0 in addition to those at f% = ^. Figure 7 gives the bn and shows h*(t) and H*i(jO) for = 315 and N 4 Both of the realizations in Figure 3 and Figure 4 have parallel signal paths, one of which is direat and a second which is through a delay line* Gain matching in the two paths must be exact or cancellation of hl(t) and h2(t - 2T) will not occur for t > 2T. In the case of the Legendre realization any difference between hl(t) and h2(t - 2T) grows as tN which causes a large error in the realized frequency response at zero frequency; in the case of sines or cosines realization the difference between h1(t) and h2i(t - 2T) oscillates with small amp...itude at frequencies Jtn which causes undesiied resonances in the realized frequeney response. A practical expedient for reducing the ariticalness in gain matching is to shift the singularities of H* slightly to the left in complex p plane (by replacing j'in H*(j) by j) + ao). This causes h(t) to become multiplied by e-aot, a damping factor which is effective in reducing the error for large t, Thp shift ao should be made as small as possible consistent with the obtainable accuracy in gain matching as the shift causes H* to depart from the desired real or imaginary characteristics. CONCLUSIONS In this paper the synthesis of real and imaginary transfer functions has been described. Realization is only possible within a time lag T, and requires a delay element filter with a finite memory of 2T, The approximation problem has been solved in both the frequency domain and time domain by the introduction of two sets of orthogpnal approximating functions, one based on Legendre polynomials in the time domin and the other on sines and cosines in the time domain, It appears that the latter approximating functions have the following realization adiantages: 1) realization is simple in form, consisting of (1 - e-J2T) and H3 in cascade and 2) at most, only one pole is required at p = 0O

-15APPENDIX Development of Recursion Formula for Legendre Function en(jd) From (19) n(j0) = f e J n dt -00 2n e dt dtn.t 2n1 n',1 d5(t _ l~n _a(Dt; (A - 1) Integration by parts of (A - 1) gives (making the derivative of e-it appear in the new integrand) nj1) = {dn- (t - 1)n } 1 ne (j n n,-nWl I2 n dat - dn-1 (t2 _1) e dt, (A 2) -.1 dtn-i dn-l (t2_ l)n but ~~n-l = O for t 1 l1 Hence dt JIl n-1 et _ ln _ )( n -1 d (t 1 e dt Repeating the integration by parts a total of n times yields en f) (t - e d-J at (A ) 2n'n -1 Integration by parts of (A - 4) gives (imaking the integral of eJt appear in the new integrand) e { {(t2 _ 1)fl } 1 + f (t -l)n t e3l i dt n n n)- 1J (n-l) t- n (j )n- ^1 (t2 _ l)n-1 t e-' dt, n > (A )

Finally, a similar integration by parts of (A- 5) gives (jn-l (t2)n-t 11 +e-J1% i 2(n-l) l t2 1)n-2 2 t dt ((t2 - 1)n-1 t~ ( - l) t e dt 2 n 2n-l (n-l)! (-) -l j) -1 + 1 l (t2 1)n-1 -j4dt (j)n-2 l (t 2 )n-2 t2 -jt t + jca l(~t- 1)( 1 + ^~ ^f (t^ -i e adt f -i t 2n-2(n-2)-1 l +(j)n-2 I:t2 l)n l eji dat, n >1 (- 6) 2n-1 (n-l) -1 Ading ar snubtraeting - j 1 (t2 1)n-2 (t2 - 1) e dt, 22 (n-2)j -I at, t Ie (t2 nl)n-2 de dt ( )n-2 2n2 (d)2 1(2n-2 t 2 jt + f ~u... ~. / (t2-l) (1 -e )ed dt 2nl (n-l) - + 2 j (2 - 1)n2 e 2 at (AJ-7) 2 (n-2) -1 n-2 2(n- 1) 1 2n — 1 e J = e )1 + )l n-22 21 e2 An- n This is the desiredt result n-2 2 (n-2); -1 (t 1~t O + O O + O (A - 8d'S 2 ( n-l) 8n-1 2 nI2 This-is the' desired res u l t(

-17REFERENCES 1. Time Symmetric Filters, L, R, 0, Storey and J. K. Grierson. Electronic Engineering, vol. 30, No. 568 and No. 569, pp. 586-592 (Pt. 1) and pp. 648-6i5 (Pt. II). 2. Apeture Compensation for Television-Cameras, R. JC. Dennison, RCA Review} December, 1953, pp. 569-585, 3. Aperture Corrective Systems, Joseph Otterman. IRE Transactionson Instrumentation, vol. 1-8, No, 1, March, 1959, pp. 8-19. 4. Traisversal Filters, Heinz E, Kallmann. Proc ]RE, vol. 28, July, 1940, pp. 302-308. 5. Electrical Networks System, Y, W. Lee and N. Wiener. Patent 2,128,257, filed July 7, 1936. 6. Electrical Networks System, N. Wiener and Y. W. Lee. Patent Number 2,124,599 filed July 1 18 936. 7. Recent Advances in the Synthesis of Comb Filters, W, D, White and A. E. Ruvin. 1957 IRE Convention Record, vol. 5, Part 2, pp. 186-199. 8. Analysis and Synthesis of Delay Line Periodic Filters, Harry Urkowitz. IRE Transactions on Circuit Theory, vol. CT-4, June, 1957, pp. 41-53. 9. The Impulse Response of Electrical Networks, with Special Reference to the Use of Artificial Lines in Network Design, M. Levy Journ,. IEE, vol. 90, Part III, No. 12, December, 1943, pp. 153-164.. 10. Filters for Sampled Signals, J. Brogan. Proc. of the Symposium on Information Networks, Polytechnic Institute of Brooklyn,$ April, 1954 pp 71-8. 11, The Continuous Delay-Line Synthesizer as a System Analogue, J. H. Westcott. IEE Monograph No. 176M, May, 1956. 12. Synthesis of Electrical Networks by Means of Fourier Transforms of Laguerrets Functions, Y. W. Lee, Journ. Math. and Physics, vol. II 1932, pp. 83-113. 13, Transient Synthesis in the Time Domain, W. H. Kautz. IRE Transactions on Circuit Theory, vol. CT-1, September, 1954 pp. 29-39. 14. Time Domain Network Synthesis for an Analog Computer Setup, Joseph Otterman. Proceedings of National Simulation Conference, Dallas, Texas, January 1956, pp. 24.1-24. 15. Linear System Approximation by Differential Analyzer Simulation of Orthonormal Approximating Functions, Elmer G. Gilbert. IRE.Transactions on Electronic Computers vol. EC-8, No. 2, June, 1959, pp, 204-209. 16. Methods of Mathematical Physics (book), Courant and Hilbert. Interscience Publishers, New York, N. Y., vol. 1, 19531 17. Introduction to the Theory of Fourier Integrals (book), E. E, E9i.hmarsh. Oxford, 1937.

TABLE I TIME DOMAIN FREQUENCY DOMAIN______ 10 1. 80 ^ ~ ^ J ^ L ^\0 -I i.. I I t @0 2 sin go T 1 2 "______________^u 1 I' -SnV\.tv I ff\.j2,r -37r @1 i ~1lL.81: ~~ t @.~i(-2,'' 2 sin 2 cosa w 02 6 sin' w 6cosa' 2 sinLa'.b^^^ ~-~~~~~~~~~9 x 1) +-^^^^ +^ ^^ 2 2' 3 o o n I'R\ Z 3r ~ \~~ t a' a at -2 3 - 683 2II~' "^h~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i 833\ / (5~ t 3 @ f ji(t 300sin w 30 cos w 12sin c 2 cosm \, -2'' -w \ ^ 30 rus P g cm2' ~,,/" ^^ y\ ~^ ^~^~~~~~83 (5t -3) ~3( W4 -~^T —~,r*~T~ +^~ j - WJ O ~~~~~~~~~~~~~~~~~~~~~Ai 94 @- 210 sina 210cos a 4 C05 ~4 r-3 /^ y \^ ~I\y\ ~ t 94 (35430t2L+3) 90 sin 20cos + 2 sin wa t ++ -2-r -T o 7r 2 v a' 80 ~5 I ^-1890 sin w f 1890 cos w j ce 5 3-ir _ r 2r 3w 85^~~~ ^ ^5- (6~3t5-70t3+ 15) 3 8\j5\\ ~ 8. (6t7t l)+ 840sin. 210cos 30Ssina w 2co0s -2t 4 ~'3 a2 J 86 @6k -20790sin a 20790cos0 9450sina i 6 Ia'7 a6,8 5 -4 -3,w -2w -v w 2 r 3w 4w 866 ( 31t-315+105t-5) 2520cos w 420sinc w42cosw 2 sinc NJ I t 6 ++ -7 7 270270 sin a 270270 cos a'7 7~~~~~~~~~~~~~~~~~~w - -~~~~~~~~~~~~~~-w-v-v13 -^^v ^ ^ I ~~ 12.4740 sina' 34650 cos co 30in -' 3 ~ l7~-^(429t7-693t+315It-35,t) -~ ~5~6. sin~..... 34w.7' 16 V E,5 ~ 4 a r 32 4a' I yv i" VI 756 cog wc 56 sin w 2 cos w I_~2

Table II TIME DOMAIN FREQUENCY DOMAIN _ ao A0o.:j ~~~ 7~r'^J-2~A T.,, ~3,(t)~S.NC t BA (w) +j l 27rS1N Cu a' r2-w 2 A2 \ f Q-I \/ \/i T 4^2-2 -3\ z, 2./ SIN B / 3(t)=SIN 2 rt 2 4r2 -.'1 -~:3;r-2 /I"~.//t..r'3~r"': A A2 A' twSN w \ \ \, I a2(t) COS 27rt A2(W) 4r2-w2 3 I t'P(t) =OSIN6t A- ( 7=r SINw. SIN 3771 B3 (Cu) \ 9~2~2 -Wil3w23- ( W) w 2w2w Cu 2w SIN. a3(t) OS 37rt A3 (W) Z -' - I' { B4 (t) =SIN aw't z(W):-j:-= 16 t'sa -2nrz`-w) " Z2\ /3f W t~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~'''

-20TABLE III Erel for Cosine Function Approximation of Real Low Pass Filter N r1T 2 r 31 3. 5rf 4 w 4. 5wr 5 r 61 7T3.0978.0511.0363.0307.0869.2176.3125 4.0976.0506.0345.0287.0275.0230.0702.2600 5.0975;.0504.0340.0286.0260.0221.0222.0589.2226

-21~INP-~UT~'.. +P OUTPUT 1..' j.' i ew (j2T) _______H 2 (J "- _1!' Figure 1. Single Delay Element Realization of H*(jw). h t) O T 2T t Figure 2. An h*(t) Utilizing More Than One Delay Element-.

q. INPUT i +I OUTPUT I1 ~ ~ ~ ~ ~ q 2 Nx __ 2 -' ~ ^ ro. Figure 3. Realization of Real or Imaginary Filter Based on Legendre Function Approximation.

INPUT OUTPUTH3(jw) Figure Ie Realization of Real or Imaginary Filter Based' on Fourier Series Approximation.

-24h*(t) 3.5 3.0 2.0 1.0 -1.08-6 -4 -2 246 8 1.0 t n. - -0.5 1 Figure 5. H* and h* for Cosine Function Approximation of Real Low Pass Filter.

-25. h*(t) 2.0.5-5 0.5 \ 1.0-.8 -.6 -.4 2 2.4.6.8 1.0 \.5 0.5 -47r -37r -27T -7r 0. T 2r. 3r 4ir Figure 6. H* and h* for Legendre Function Approximation of Real Low Pass Filter.

-26h*(t) 2.0 1.5 1.0 -I'.0-8 -.6 -.4 -2. 4 8.0 t o5 1.5 2.0 Hf(w) 1.00 75 25.50.75 1.025 * Figure 7. H* and h* for Sine Function Approximation of Imaginary Low Pass Filter.