A Survey on the Design of Lowpass Elliptic Filters

Transcription

1 British Journal of Alied Science & Technology 43: , 4 SCIENCEDOMAIN international A Survey on the Design of Lowass Ellitic Filters Athanasios I. Margaris Deartment of Alied Informatics, University of Macedonia, Thessaloniki, Macedonia, 574 6, Review Article Greece. Received: 3 Aril 4 Acceted: 6 June 4 Published: 3 June 4 Abstract The objective of this review aer, is the resentation of the basic features of the well known class of ellitic filters. Even though this is not a new subject, the theory of the ellitic filters found in most books is restricted only to a few resulting equations, due to the great comlexity associated with the Jacobian ellitic functions. The asects of the ellitic filters described in this aer include the detailed estimation of the minimum filter order, the construction of the filter transfer function via the identification of its oles and zeros in the comlex lane, as well as the alication of the resulting design rocedure for the construction of an ellitic filter that meets rescribed secifications. Keywords: Ellitic filters, transfer function, oles and zeros, frequency resonse Mathematics Subject Classification: A69 Introduction to the Theory of Filters Analog and digital filters are one of the most imortant aradigms of continuous and discrete time systems with great theoretical and ractical value. At it is well known from the fundamental system theory ]], their time behavior is described by the imulse resonse ht namely the system resonse to the delta or Dirac function δt, while in the domain of the comlex frequency, this behavior is described by the Lalace transform of their imulse resonse Hs = L{ht} = + hte st dt where s = σ + jω is the comlex frequency. The function Hs is known as transfer function. If the region of convergence of the Lalace transform includes the imaginary axis, the frequency resonse of the filter is defined as Hjω = + *Corresonding author: amarg@uom.gr hte jωt dt

2 British Journal of Alied Science & Technology 43, , 4 and therefore, it is the continuous time Fourier transform of the imulse resonse ht. The frequency resonse of a filter can be exressed in olar form as Hjω = Hjω e j Hjω where the function Hjω is known as amlitude resonse and the function Hjω is known as hase resonse. The heart of the filter design theory is the convolution theorem, according to which the convolution oeration in the time domain corresonds to a multilication in the frequency domain. It is well known that the inut xt, the outut yt and the imulse resonse ht of each continuous time system, are related via a convolution in the form yt = ht xt. If the Fourier transforms of these signals are Yjω, Hjω and X jω resectively, the convolution theorem says that j Hjω+ X jω] Yjω = HjωX jω = Hjω X jω e Therefore, if we want to eliminate the frequency comonent ω of the inut signal according to Fourier analysis each signal is the suerosition of an infinite number of frequency comonents whose frequencies are integral multiles of its fundamental frequency we just have to set Hjω =. If we want to erform this elimination for an entire frequency region we just have to design the filter accordingly. According to the osition of the frequency region to be eliminated in the frequency sectrum, a filter can be characterizes as low ass, high ass, band ass and band sto filter. In the case of the discrete time domain, the situation is exactly the same but there are two differences: a the Z transform is used instead of the Lalace transform with the corresonding discrete time Fourier transform to be defined if the region of convergence of the Z transform contains the unit circle in the comlex lane and b the amlitude and hase resonse of the filter are defined u to an uer frequency of ω max = π, since this is the maximum frequency associated with a discrete signal xn] 3]4]5]. The amlitude resonse of an ideal filter is characterized by an absolutely constant amlitude resonse with a value of unity in the assband, as well as a discontinuity namely, a ste-like behavior in the cut-off frequencies a single frequency for low ass and high ass filters and a air of frequencies for band ass and band sto filters. However, these filters are not causal and therefore, not realizable. Instead, in real alications a lot of different ideal filter aroximations are used, that are characterized by riles in the assband and/or in stoband as well as a transition band whose width is inverse roortional to the filter order. This order is defined as the degree of the olynomial Ds in the denominator of the rational filter transfer function whose general form is Hs = Ns/Ds in the circuit level, the filter order is defined as the number of the energy storage elements contained in the filter circuit. The roots of the olynomials Ns and Ds are known as the zeros and the oles of the filter transfer function and their identification allows the comlete descrition of the filter s behavior. There are four fundamental analog filter rototyes, the Butterworth aroximation characterized by the absence of riles in the assband as well in the stoband, the Chebyshev I filter aroximation whose magnitude resonse is characterized by riles only in the stoband, the Chebyshev II aroximation whose magnitude resonse is characterized by riles only in the assband and the ellitic or Cauer aroximation that leads to rile behavior in the assband as well in the stoband 6]. To describe the secification of a low ass or a high ass filter design, there are four different arameters that have to be defined, namely the maximum assband loss A, the minimum stoband loss A s, the assband edge ω and the stoband edge. On the other hand, the secifications for a band ass or a band sto filter need more arameters, even though they can be imlemented via a serial combination of a low ass and a high ass filter. The arameters A and A s exressed in db units decibels, are related to the rile amlitudes δ s and δ in the assband and the stoband via the equations 7] A = log δ + δ and A s = log δ s + δ The ratio k = ω / is known as selectivity factor, while, two additional arameters that are used frequently in the filter design are defined as ε = A / and A = As/. 38

3 British Journal of Alied Science & Technology 43, , 4 The above filter characteristics are also valid for the digital filters that can be either FIR finite imulse resonse or IIR infinite imulse resonse. In the fist case the filters are mainly designed via the alication of an aroriate window function 8] and the frequency samling method 3], while for the IIR case the filter can be derived by their analog rototyes using the techniques of the imulse invariance and the bilinear transformation 3]. In most cases the filter is designed to work as a low ass filter and if we need another filter tye such as a high ass, a band ass or a band sto filter, it can be derived by the analog low ass rototye via selected frequency transformations see for examle 9] for the case of digital filters. The toics covered in this aer are restricted only to the design of analog lowass ellitic filters. Ellitic Filters Ellitic filters also known as Cauer filters are secial tyes of analog and digital filters characterized by assband riles of equal amlitude, as Chebyshev I filters as well as stoband riles of equal amlitude as Chebyshev II filters, while, at the same time offer a very narrow transition band; however, their assband is associated with a maximal nonlinearity, regarding their hase resonse. The assband as well as the stoband rile amlitude can be adjusted indeendently and the filter reduces to Chebyshev I or Chebyshev II filter if one of them tends to zero note that if both amlitudes reach a zero value, the ellitic filter reduces to a Butterworth filter. Ellitic filters give the smallest filter order with resect to the other filter tyes for the same values of the filter design arameters. The riles of the amlitude resonse of the ellitic filters in the assband and the stoband for even and odd filter order are shown in Figure while the equation describing this resonse has the form Hjω = + ε J N ω,, ω, ε s, ε where J N ω,, ω, ε s, ε is the Jacobian ellitic function of order N defined as 7] ] J N ω,, ω, ε s, ε = sn κsn ω, ω + qk, ε ε s ω In the above equation, the function snx, k is the Jacobian ellitic sine function with ellitic modulus k, the symbols ω and describe the assband and the stoband edge frequencies, while the arameter κ deends on the filter order N and the values of the ellitic integrals associated with the filter design. Regarding the filter order N, it deends in turn on the arameters ω s and ε s and its estimation is discussed in Section.. On the other hand, the arameters ε and ε s are the assband and the stoband rile amlitudes, q is an auxiliary constant allowing the unified descrition of the odd q = and even q = order filters, while K is the comlete ellitic integral of first kind of the ellitic module ε ε s. It is convenient to define the ellitic modules k = ε ε s and k = ω / as well as the arameters ξ = κsn ω/ω, ω / and ζ = ω/ω, since, in this way, we can exress the Jacobian ellitic function in the simler form R N ω,, ω, ε s, ε = snκsn ζ, k + qk, k ] = snξ + qk, k As with the other filter tyes, the ellitic filters can be built using assive as well active circuit elements. An examle circuit of each one of these aroaches, is shown in Figure. By adoting the usual convention and alying the basic filter theory, the arameter A is defined as A = log + ε db and therefore we have ε = A/. On the other hand, the stoband is characterized by equiriles of amlitude ε, with the minimun amlitude to be exressed as /ε ε s = /k. Therefore, in comlete accordance with the Chebyshev II filters, we can write the equation A s = ε s + ε s = + /ε s 383

4 British Journal of Alied Science & Technology 43, , Passband riles of the amlitude resonse 6 Amlitude N= Frequency rad/s Stoband riles of the amlitude resonse Amlitude Frequency rad/s Figure : The assband and stoband riles in the amlitude resonse of the rototye lowass ellitic filter for arameter values A = db and A s = 3 db and for filter orders N =. or, in db units, A s = log + ε s which in turn, gives ε s = A s / Therefore, for secific values of the A and A s arameters we can use the above relations to estimate the values of ε s and ε associated with the amlitude resonse of the ellitic filter. The grahical interretation of these two arameters for the case of ellitic filters, is shown in Figure 6 see also the subsection.5 for a descrition of the loss characteristic associated with these arameters]. The last comment in this introductory section is associated with the filter transfer function. At is well known from the literature, there are two main categories of ideal filter aroximations, namely the olynomial and the rational aroximations. In both cases, the filter gain for a filter aroximation of order N has the form Gω = Hjω = H + γ P N ω where γ is the filter design arameter. Regarding the function P N ω, it is a olynomial for olynomial aroximations for examle, the olynomial P N ω = ω N for the Butterworth aroximation and the Chebyshev olynomials for the Chebyshev aroximations, and a rational function for rational aroximations, in the form P ω = Nω Dω = α N ω N + α N ω N + α N ω N + + α ω + α ω + α β M ω M + β M ω M + β M ω M + + β ω + β ω + β 384

5 British Journal of Alied Science & Technology 43, , 4 Inut Outut Passive toology - + Active toology Figure : A tyical assive and active third order ellitic analog filter. leading to the gain function Gω = H + γ N ω D ω = H Dω D ω + γ N ω It should be noted that the olynomial Dω must be an even function of frequency, since, otherwise the well known roerty D = of the odd functions, would result to the vanishing of the gain function for ω =, an unaccetable roerty for low ass filters. The rational aroximations are characterized by the resence of zeros in the gain function, a fact that it is comatible with the fundamental theorem of Paley and Wiener. It can be roven that these zeros have the form z m = ±jω m with each air to contribute to the numerator of the rational transfer function, the term s z m s zm = s + jω m s jω m = s + ωm. Ellitic filters are rational filter aroximations and therefore they are characterized by the whole set of the above roerties. The detailed construction of their rational transfer function, as well as the analytic form of this function for the case of the odd and even filter orders, are resented in Section.4. Since the filter order N affects the number of oles and zeros, namely the number of terms aearing in the numerator and the denominator of the transfer function, it is clear that this function is strongly deended of the filter order N, a fact that also holds for all the filters aroximations.. Estimating the minimum filter order To identify the minimum filter order that meets the rescribed secifications, we start from the squared amlitude filter resonse at the frequency ω = ω Hjω = + ε JN ω,, ω, ε s, ε = + ε 385

6 British Journal of Alied Science & Technology 43, , 4 that allows the descrition of the Jacobian ellitic function as or equivalently, as J N ω,, ω, ε s, ε = sn ξ ω=ω + qk, k = snξ ω=ω + qk, k = ± If we take into account the well known roerty snm + K, k] = ± we can write that ξ ω=ω + qk = m + K κsn ω, ω = m + qk ω or equivalently κ = m + q m + q K sn K =, ω /] sn K = m + q, k K since, from the defining equation of the Jacobian inverse ellitic function sn sin φ, k = φ dϑ k sin ϑ we get sn, k = sn sin ] π π/ dϑ, k = k sin ϑ = Kk = K On the other hand, at the frequency ω =, the amlitude resonse function is written as Hj = ] = + ε sn ωs κsn, ω + qk, ε ε s ω A s = + ε s and if we comare the radicand in the second and the forth exressions, we easily see that ] ε sn κsn ωs, ω + qk, ε ε s = ω ε s or equivalently sn κsn ωs, ω ] + qk, ε ε s = = ω ε s ε k Performing a comarison between the last equation and the exression snm + K + jk, k ] = k we have κsn ωs, ω + qk = κsn, k + qk = m + K + jk ω k and therefore sn k, k = m + q K + j K κ κ = K + j K κ 386

7 British Journal of Alied Science & Technology 43, , 4 Finally, using the equation sn k, k the resulting exression is = K + jk to get K + jk = K + j K κ κ = m + q K K = K K The last equation is an imortant one in the ellitic filter design, since it rovides a relationshi between the arameters ω,, ε and ε s, as a consequence of the fact that the ellitic integrals K and K deend on the rile amlitudes ε and ε s, while the ellitic integrals K and K deend on the frequencies ω and. Note the aearance of the term m + q in the last exression: its value is an even number if q = and an odd number if q =, and in fact, it describes the filter order N. Therefore, we can set N = m + q to get κ = N K K = K K an exression, that allows as to exress the minimum filter order associated with the values of the four filter design arameters as N = K K K K In most cases, the value estimated by this equation is a decimal number that has to be rounded to the next greater integer. Even though the above relation allows the estimation of the minimum order of an ellitic filter, it is not commonly used since it involves the estimation of ellitic integrals which is not a trivial task note, however that there is an efficient way of estimating these integrals based on the arithmetic - geometric mean ]. In ractical alications, the value of this arameter is estimated by a more simle exression that will be constructed later in this section. The ractical imortance of the above relation is that it defines a restriction regarding the design secifications, in the form f N, ω / = f A, A s that has to be satisfied by the four design filter arameters. Therefore, if we know the values of those arameters, the filter order that ensures the validity of the above relation is estimated as follows: Starting from the exression that relates the Jacobian ellitic sine function with ϑu, q, ] u snu, k] = ϑ K, qk ] = 4 qk n q nn+ k sin n= k u k ϑ 4 K, qk + n q n k cos n= n + πu K m πu K and since for u = K this function is equal to snk, k = and it holds that sin n + πu ] = sin n + π ] = cos n πu = cos nπ = n K K u=k we can write the exression 4 { } qk / { q nn+ k + k n= n= u=k q n k } = ] ] 387

8 British Journal of Alied Science & Technology 43, , 4 or equivalently k = 4 { } / { }] q k q nn+ k + q n k = 4 q k n= n= ] +q k +q 6 k +q k + +qk +q 4 k +q 9 k + where qk = ex πk /K. Using the simlifications k k that generally hold, we have K /K and qk and therefore the enclosed in square brackets and raised to the second ower quantity, can be aroximated by unity. Therefore we can set k 4 qk and get the exression k 6qk = 6 ex πk /K = 6 ex πnk /K = 6 ex πk /K ] N = 6q N k where we used the fact that N = K K /K K. Substituting this exression in the equation k = ε ε s = A / As/ = D we get κ = A / A s/ = D = 6qN k For given values of the design arameters ε s, ε s, A and A s, the minimum filter order can be estimated by solving the above equation with resect to N. In this case, we can easily verify that N = log6d log/qk ] where D = A s/ A / It is interesting to note, that if we know the filter order N as well as the values of the A and k arameters, the value of the A s that ensures that the constraints of the design are satisfied, can be estimated from the above equation as A / A s = log + 6q N k. Poles, zeros and min-max frequencies of the Jacobian ellitic function By studying the form of the ower resonse function Hjω it can be easily noted that it gets its maximum value for frequencies such that J N ω,, ω, ε s, ε = and its minimum value for frequencies such that JN ω,, ω, ε s, ε = J N assumes values in, ]. To identify the frequency values that maximize the function Hjω the roots of the equation sn κsn ω, ω ] + qk, ε ε s = ω have to be estimated, a task, that can be easily erformed since the function snu, k is zeroed in the values u = mk. Therefore, we have κsn ω, ω + qk = mk ω 388

9 British Journal of Alied Science & Technology 43, , 4 or equivalently sn ω, ω m qk m qk = = ω κ N since κ = NK /K. The solution of the above equation with resect to the frequency ω is based on the equivalence sn x, κ = α x = snα, κ and leads to the result m ωm max qk = ω sn, ω ] N for values m =,,,..., N / for odd order N and m =,,,..., N/ for even order N. This equation allows the identification of frequencies that maximize the squared magnitude of the frequency resonse function in the assband. On the other hand, the frequencies that minimize the above function, are estimated as the roots of the equation ] sn κsn ω, ω + qk, ε ε s = ω or equivalently ] sn κsn ω, ω + qk, ε ε s = ± ω To roceed, we use the roerty snu, k = ± where u = m + K, and working in the same way we get ] m + ωm min qk = ω sn, ω N where m =,,,..., N 3/ for odd filter orders N and m =,,,..., N / for even filter orders N. This exression allows the estimation of frequencies that minimize the squared magnitude of the frequency resonse function in the assband. Let us roceed now to the estimation of the cutoff frequency ω c in terms of the values of the arameters ω,, ε and ε s. As it is well known, the cutoff frequency ω = ω c corresonds to an attenuation equal to A = 3 db. Therefore, we have Hjω c = Hjω c max = or equivalently Hjω c = ωc + ε sn κsn, ω ] = + qk, ε ε s ω A direct comarison allows us to write that ε sn κsn ωc, ω ] + qk, ε ε s = ω leading to the result sn ωc, ω = ω sn /ε, ε ε s qk NK ] K 389

10 British Journal of Alied Science & Technology 43, , 4 Re sn - z, k ]/K.8 k = k = z Jm sn - z, k ]/K k =.7 k = z Figure 3: Variation of the real and the imaginary art of the comlex function sn z, k for ellitic module values k =.,.3,.5,.7. where we used again the fact that k = NK /K. The function sn z, k k = ω / is a comlex one, for z = ω c/ω >, with its imaginary art { } { ] } I sn ωc, ω sn /ε, ε ε s = I K ω NK to vanish for ω c = ω, or equivalently for z = this is not true for z <, or equivalently for ω c < ω as it can be seed from Figure 3 that shows the variation of the real and the imaginary art of the function sn z, k for values k =.,.3,.5,.7. Noting that the real art of this function is equal to K, we can write our initial function in the form sn ωc, ω { ] } sn /ε, ε ε s qk = K + ji K ω NK In this case, the solution with resect the frequency ω c will give the desired cutoff frequency as { ]} sn /ε, ε ε s qk ω c = ω sn K + ji NK Finally, let us estimate the ole frequencies of the Jacobian function J N ω,, ω, ε s, ε that send it to infinity. Using the fact that the ellitic sine function vanishes every K, or mathematically, snmk + jk, k = ±, we can write that Considering the exression κsn ω ω, ω + qk = mk + jk sn k, k = K + jk 39

11 British Journal of Alied Science & Technology 43, , 4 where k = ω /, this equality is satisfied by the frequency ω = /ω. On the other hand, for frequencies ω > ω this function gets a value of x+jk where x < K and therefore, for frequencies ω > ω we have kx+jk +qk = mk +jk. Using the k value as it is exressed by its defining equation and solving with resect to the x arameter, it is found that x = m qk N and therefore sn ω ω, ω = m qk N The oles of the Jacobian function ω are therefore given by the exression m ωm qk = ω sn + jk, ω ] = N snm qk /N, ω / ] = ω ωm max + jk that is derived using the roerty snu + jk, k] = /ksnu, k]. In the above exressions, the m arameter gets the values m =,,..., N / for odd filter orders and m =,,..., N/ for even filter orders. Note, that the Jacobian function associated with the odd order ellitic filters has an extra ole in the infinity. The defining equations of the oles and zeros of the Jacobian ellitic function, reveal an inverse roortionality relation between them. An imortant consequence of this roerty, is that the existence of equiriles in the assband, imoses the existence of equiriles in the stoband, too..3 Rationality of the Jacobian ellitic function After the identification of the oles ω m = ωm and the zeros ωz m = ωm max of the ellitic function J N this function in the literature is known as the Chebyshev rational function, it can be exressed as ] J N = Dω q L ω ωz m ] = Dω L q ω ω m ] L ω ωz m ] L ω ω ωs ] ωz m where the value of L is equal to N/ for even orders and equal to N / for odd orders, while the scaling factor D, has the form N/ ω m N/ ω ω ] s = ωz m 4 ω m z D = ω m ω] = ω ω ωz m ] N / ω N / N / N / ω s ω m z ] ω m z ω ω m z ] for even and odd filter orders resectively. This values ensure that in the assband edge ω = ω as well as at the frequency ω = the magnitude of the function J N is equal to unity. It can be roven that this function is charecterized by equirriles in the interval, +] as well as in the intervals, ] and +, + and its lot for odd and even filter order is shown in Figure 4. The frequecy ω in the exression of the J N for odd filter orders is due to the fact that the degree of the numerator olynomial for odd filter orders is equal to N and therefore a first degree olynomial term has to be added to get a olynomial of a degree of N. 39

12 British Journal of Alied Science & Technology 43, , 4 fx fx x x N even N odd Figure 4: The Chebyshev rational function for odd and even orders. It is interesting to note, that in ractical roblems the oles and zeros of the ellitic functions are not estimated from the above relations but via the next exressions, where the ellitic sinus function is exressed in terms of the functions ϑ u, q, ϑ 4 u, q and qk ] ] m q m q ω m ω m z = ω ϑ k = αk = ω ϑ k = αk N, qk = ω ϑ m q k ϑ 4 N, qk { n ex n= + n= m q N +j K N, ex m q ϑ 4 N, ex π M, k M, k π M, k M, k π M, k M, k { n ex π M, ] k n M, k, qk K m q ϑ 4 N +j K, qk K n= + { n ex n= = ω ϑ k π M, k M, k m q N +j K ] ] nn+ sin n + π m q N, ex K m q ϑ 4 N +j K, ex K ] nn+ sin { n ex π M, ] k n M, k In the above equations, the auxiliary function αk is defined as 4 ω ex π M, k M, k αk = k n+π cos cos nπ m q N ]} ]} π M, ] k M, k π M, ] k M, k nπ ]} m q N +j K K ]} m q N +j K K 39

13 British Journal of Alied Science & Technology 43, , 4 These relations do not include ellitic functions and allow the easy calculation of the oles and zeros of the ellitic function J N, since the arithmetic - geometric mean of two numbers Mx, y can be estimated quickly and easily, and furthermore, the functions ϑ u, q, ϑ 4 u, q and qk are estimated via their defining equations with the accurate estimation of their value to require the calculation of a few terms only..4 Poles and zeros of the transfer function To identify the oles and zeros of the transfer function let us start by the function ] HsH s = Hjω ω=s/j = + ε R N s/j,, ω, ε s, ε = = + ε D q+ s q { L s + ωz m } s + ω m L s + ω m ] L L s + ω m ] + ε D q+ s q s + ω z ] whose zeros, being the roots of the algebraic equation s + ω m =, are estimated as z m m = ±jω m qk jω sn + jk N, ω ] = ±jω m qk ω sn, ω ] = ±j ω ω m = ± jω z Ω m N where Ω m = m qk sn, ω ] N The above equation is valid for odd as well as even filter orders, namely for values q = and q = resectively. On the other hand, the oles of the transfer function are the roots of the equation + ε R N s/j,, ω, ε s, ε = or equivalently s sn κsn, ω ] + qk, ε ε s = j jω ε Since the Jacobian ellitic sine function is doubly eriodic with a real eriod of 4mK, we have s sn κsn, ω ] + q + 4mK, ε ε s = j jω ε 393

14 British Journal of Alied Science & Technology 43, , 4 ole x x x x zero Figure 5: Pole-zero lot for the fourth order lowass ellitic filter. and therefore, the oles of the filter transfer function can be estimated as snj/ε, ε ε s q + 4mK m = jω sn, ω ] k for values m =,,..., N. The ole zero lot for the fourth order lowass ellitic filter is shown in Figure Pole and zeros aroximations The ole defining equation can be simlified and exressed in terms of trigonometric and hyerbolic functions that are much simler; to do this, the simlification k = ε ε s can be alied, that generally holds in ractice. Using the results snu, = sin u and K = π/ we have κ k = = N K K = πn K and the ole defining equation gets the form πn s sn sn, ω + q π ] πn s K jω ω, = sin sn, ω + q π ] = j s K jω ε At this oint we distinguish between two cases: 394

15 British Journal of Alied Science & Technology 43, , 4 N is odd In this case we have q = and the last equation gets the form πn s sin sn, ω ] = j K jω ε or equivalently πn s j sin sn, ω ] = K jω ε From the mathematical analysis we know that j sin u = sinhju and therefore we have sinh j πn s sn, ω ] = K jω ε or j πn s sn, ω = sinh ε = sinh K jω ε Using the identity sinh x = lnx + x +, the right art of the equation is exressed as sinh + ε = ln + ε ε ε + = ln + ε ε = ln + + ε ε and since ε = A/ we get the exression sinh ε Therefore we have = ln + + A / A / = ln = ln A / + A/ = ln A/ + A/ j πn s sn, ω = K jω ln A/ + A/ + A / A / + A/ or equivalently, s sn, ω = j K jω Nπ ln A/ + A/ 395

16 British Journal of Alied Science & Technology 43, , 4 and solving with resect to s, s = jω sn j K Nπ ln A/ + A/, ω ] 4 qk n q nn+ k sin jn+ K = jω ωs ω n= + 4 qk = ω n q n k cos n= n q nn+ k sinh n= + n q n k cosh n= ] Nπ π ln A/ + A/ πnj K ] = Nπ ln A/ + A/ n + K ] N ln A/ + A/ n K ] = N ln A/ + A/ where we used the identities sinju = j sinh u and cosju = cosh u. It is clear that the s value derived above, is not a comlex but a real number. Therefore, the odd order ellitic low ass filters have a real ole s = that does not aear to the even order filters, even though this ole aears in the ole equations of the last filter tye. On the other hand, to identify the remaining oles of the transfer function, the eriodicity of the Jacobian ellitic sine has to be used; this roerty is described as snu + 4K, k = snu, k and results to a fundamental eriod of snαu, k equal to T = 4K/α. In this case, the multilicative coefficient of u = sn s/jω, ω / is α = NK /K = Nπ/K, a fact that can be easily verified noting that K = π/. Therefore, the eriod of the ellitic function that aears in the ole defining equation is T = 4K /α = π/nπ/k = 4K /N, allowing us to say that if the exression z = j K Nπ ln A/ + A / is a solution of this equation, then the same is true for the quantities z m = z + 4mK s, where z sn, ω N jω In other words, we can write that s sn, ω jω = j K Nπ ln A/ + A / + 4K N m and therefore, the oles of the filter transfer function can be estimated as m = σ m + jω m = jω sn j K Nπ ln A/ + A/ + 4K N m, ω for the values m =,,,..., N as it can easily be verified by solving the above equation with resect to s. The value m = is associated with the real ole, while the other values 396

17 British Journal of Alied Science & Technology 43, , 4 of m are used to estimate the remaining N oles. Noting that snu+k, k = snu, k and snu + 4K, k = snu, k, we have m = σ m + jω m = jω m sn j K Nπ ln A/ + A/ ± K N m, ω for the values m =,,..., N /. To identify the ellitic function in the above equation, we can use the sum roerty of the Jacobian ellitic sinus snu + υ, k = for arameter values snu, kcnυ, kdnυ, k ± snυ, kcnu, kdnu, k k sn u, ksn υ, k u = j K Nπ ln A/ + A /, υ = K N m, k = k = ω Starting from the estimation of the denominator which is much more simler and using the equation we get = jω sn j K Nπ ln A/ + A/, ω k sn u, ksn υ, k = k sn j K Nπ ln A/ + A/, ω sn K N m, ω ] where + ω ω s ω sn K N m, ω = + 4 Ω m = K sn N m, ω ] = qk n= ωs ω + sn K N m, ω n q nn+ k sin n= n q n k cos = = + Ω m n + πm N nπm for the values m =,,..., N /. To estimate the exressions in the numerator we combine the roerties sn u, k + cn u, k = and k sn u, k + dn u, k = to construct the auxiliary function cnu, kdnu, k = sn u, k] k sn u, k] N ] 397

18 British Journal of Alied Science & Technology 43, , 4 Therefore we have sn j K Nπ ln A/ + A/, ω K cn N m, ω sn j K Nπ ln A/ + A/, ω K sn N m, ω K dn N m, ω = ] k sn K N m, ω ] and using the defining equation of the arameters and Ω m we get sn j K Nπ ln A/ + A/, ω K cn N m, ω K dn N m, ω = ω jω Ω m ωsω m = V m jω where V m = ω Ω m ω sω m, m =,,..., N For the second exression in the numerator we have K sn N m, ω cn j K Nπ ln A/ + A/, ω dn j K Nπ ln A/ + A/, ω K = sn N m, ω sn ω j K s Nπ ln A/ + A/, ω k sn j K Nπ ln A/ + A/, ω Using the exressions of and Ω m we get K sn N m, ω cn j K Nπ ln A/ + A /, ω = Ω m + ω + ω s dn j K Nπ ln A/ + A/, ω = Ω m W where W = + ω + ωs Substituting the above exressions in the defining equation of the oles m, it gets the form m = σ m + jω m = jω m /jω V m ± Ω m V + Ω m = m V m ± jω m Ω m W + Ω m = = m V m ± jω Ω m W + Ω m 398

19 British Journal of Alied Science & Technology 43, , 4 for the values m =,,..., N/. This is the final result. Therefore, the transfer function has a negative real ole contributing the term s +, as well as N / airs of comlex conjugate oles σ m ± jω m each one of them contributes the term s m s m = s m + ms + m m = = s σ m s + σm + ωm s + V m s+ V m +ω Ω m W + Ω m + Ω m The transfer function is also characterized by comlex conjugate airs of zeros in the locations z m = ±jω /Ω m m =,,..., N / with each air to contribute the term s z m s z m = jω + jω = s + ω ω m ω m Ω m Based on the above results, the transfer function of the low ass ellitic filter of odd order is exressed as Hs= H s + N / { s + ω Ω m ] / s + V m + s+ V m +ω Ω m W ] } Ω m + Ω m The constant H can be estimated from the normalization requirement R N ω, ω,, ε, ε s = for odd filter order N, leading to the value H =. Therefore, H = H N / and the constant H is defined as N / H = ω Ω m V m + ω Ω m W = + Ω m V m + ω Ω m W + Ω m ω Ω m N even In this case we have q =, and therefore, πn s sin sn, ω + π ] K jω = j ε Working in the same way we get the exression s sn, ω = j K jω Nπ ln A/ + A/ K N 399

20 British Journal of Alied Science & Technology 43, , 4 Since the ellitic function involved in the ole equation has a fundamental eriod of 4K /N we have z m = z ± m N K and therefore, the ole equation of the filter transfer function is m = σ m + jω m = jω m sn j K Nπ ln A/ + A/ ± m N K, ω In this case, we have m =,,..., N/ and the filter function does not have a single real ole but only a number of airs of comlex conjugate oles. The subsequent analysis is the same as reviously and the final result is where m = ±σ m + jω m σ m + jω m = ± V m + jω m Ω m W ] + Ω m The arameters V m and W have already been defined, while Ω m is defined as Ω m = m sn N K, ω ] Therefore, the transfer function for even order ellitic filters has the form { ] N/ / Hs = H s + V m s + ω Ω m + s + V m + ω Ω m W Ω m + Ω m with the estimation of the H constant to rely on the requirement G = A/, as in the case of the Chebyshev I filters. Therefore we have N / H = H and the value of H is estimated as N / H =.5A ω Ω m V m + ω Ω m W =.5A + Ω m V m + ω Ω m W + Ω m Having identified the transfer function for odd as well as even order ellitic filters, we can easily identify the central frequency ω c ω and the quality factor Q as ωc m V m + ω Ω m W = + and Q m = ω Ω m W + Ω m V m ω Ω m ] } 33

21 British Journal of Alied Science & Technology 43, , 4 Αω Α s Α ω ω Figure 6: Loss characteristics for fifth order lowass filter..5 The loss characteristics The loss characteristics of the low ass ellitic filter has the form ] Aω = log + ε RN s/j,, ω, ε s, ε = log { + ε sn ω κsn, ω ]} + qk, ε ε s ω A curve for this function and for a 5th order lowass filter is shown in Figure 6..6 Phase characteristics and time resonse function At it is well known from the literature, the resonse of an LTI system to the elementary exonential signal xt = Ae jω t has the form yt = Ae jω t { Hjω e j Hjω } = A Hjω ex { j ω t + Hjω } and therefore the filter increases the hase of the inut signal by an amount of Hjω, namely, by the hase of the filter transfer function estimated for the inut signal frequency ω = ω. In the ideal case, the hase resonse of the filter is equal to zero, and no hase distortion is caused to the signal; however, these ideal filters are not realizable and in the real alications they are aroximated by the well known analog filter aroximations Butterworth, Chebyshev I and II and ellitic or Cauer aroximation. Another interesting case, is associated with the linear hase filters whose hase resonse has the simle form Hjω = αω where α = d Hjω]/dω is the constant grou delay of those filters. It can be easily roven, that in this case the inut signal is subjected a constant time delay for all frequencies, with no further distortion. In ractice, the most efficient and commonly used 33

22 British Journal of Alied Science & Technology 43, , 4 filters of this tye, are the Bessel - Thomson filters that are characterized by a maximally flat grou delay for the frequency ω =. The above described facts hold for a single frequency ω = ω, while, in the general case of an arbitrary inut signal, they are alied to each one of the single frequency comonents that form its sectrum. The hase resonse of the rototye lowass ellitic filter for cutoff frequency ω c =, arameter values A = db, A s = 3 db and filter orders N = to is lotted in Figure 7. The defining equation of this function can be derived from the general frequency resonse equations for even and odd filter orders N/ N/ { } Hjω = H k jω= tan β k ω β k β k ω tan α k ω α k α k ω k= Hjω = tan αk ω α k k= + N / k= { } tan β k ω tan α k ω β k β k ω α k α k ω for the arameters α =, α =, α =, β =, β = β =, α k = β k =, β k = and leading to the results β k = ω Ω, α k = V k + ω Ω k W k + Ω k, α k = V k + Ω k N/ Hjω = k= V k + Ω k ω V k + ω Ω k W + N even Ω k ω N / ω Hjω = tan k= V k + Ω k ω V k + ω Ω k W + N odd Ω k ω The set of arameters α i and β i aearing in the defining equations, are the coefficients of the olynomials of the denominator and the numerator resectively of the rational filter transfer fraction Hs = β + β s + β s + + β M s M α + α s + α s + + α N s N The differentiation of the above equations with resect to ω allows us to derive the grou delay as the negative sloe of the hase resonse. For the case of the ellitic filters this delay for even and odd filter orders is exressed as N/ { V k V k + ω Ω k W ]} + k= Ω k + + ω Ω k τω = { V k + ω Ω k W ] } N even + ω Ω k τω = N / k= { V k V k + ω Ω k W ]} + + ω Ω k + Ω k { V k + ω Ω k W + ω Ω k ] } + +ω N odd 33

23 British Journal of Alied Science & Technology 43, , Phase resonse Phase deg N= Frequency rad/s Figure 7: The hase resonse of the rototye lowass ellitic filter for cutoff frequency ω c =, arameter values A = db, A s = 3 db and filter orders N =. The imulse and the ste resonse of tyical examles of ellitic filters are loted in Figure 8 and their equations have the well known exonential form characterizes the other ideal filter aroximations. 3 The Design Procedure of Ellitic Filters After the analytical descrition of the fundamental roerties of the ellitic filters, let us now summarize the filter design rocedure for the low ass rototye ellitic filter. This rocedure is comosed of the following stes:. The filter design arameters are identified, namely, the maximum assband loss A, the minimum stoband loss A s, the assband edge ω and the stoband edge.. The ellitic modulus k = ε ε s = A/ A s/ = where D = As/ D A/ is estimated as well as the modulus k = ω / that gives the value of the filter selectivity. 3. The arameter values k = k and k = k as well as the ellitic integrals K, K, K and K are estimated using the arithmetic - geometric mean. 4. The minimum filter order is estimated as N = log6d log/qk 333

24 British Journal of Alied Science & Technology 43, , 4 Imulse resonse.4 N= Amlitude Time seconds Ste resonse N=.8 4 Amlitude Timeseconds Figure 8: The imulse resonse and the ste resonse of the rototye ellitic filter for cutoff frequency ω c =, arameter values A = db, A s = 3 db and filter orders N =. or alternatively by the relation N = K K K K and rounded to the next integer number. 5. The arameters 4 qk = ω W = + ω n q nn+ k sinh n= + + ω s n q n k cosh n= n + K ] N ln A/ + A/ ] n K N ln A/ + A/ 334

25 British Journal of Alied Science & Technology 43, , 4 and Ω m = 4 qk ωs ω n q nn+ k sin n= + n + πµ N nπµ n q n k cos N n= are estimated for the values { m N odd µ = m / N even where m = {,,..., N / N odd,,..., N/ N even leading to the estimation of the arameters V m = ωω m ωsω m α m = ω Ω m β m = σ V m + σ Ω m γ m = V m + Ω m W + Ω m 6. In the last ste, the filter transfer function is estimated as where and Hs = H Ds L s + α m s + β m s + γ m { N / N odd L = N/ N even { s + N odd Ds = N even H = L γ m α m A / L γ m α m N odd N even and the amlitude and hase resonse are derived and lotted. The alication of the above rocedure for the design of a filter with rescribed secifications is shown in the next examle. ] 335

26 British Journal of Alied Science & Technology 43, , 4 3. A low ass ellitic filter design examle To illustrate the above described ellitic filter design rocedure, let us design an ellitic filter with selectivity k = ω / =.95 and attenuation levels A =.3 db and A s = 6 db. We start with the identification of the minimum filter order via the equation N = log6d/ log/qk ] The value of the D arameter is estimated as D = A s / A / = 6/.3/ = = On the other hand, to estimate the function qk = q.95 we can aly two alternative techniques, and since this review is actually a tutorial on ellitic filters, let us resent both of them. The first technique uses the defining equation of this function, namely the equation qk = ex πk /K. To estimate the ellitic integrals we can use the arithmetic geometric mean, a comutationally simle and very efficient method that rovides the results K = Kk = K.95 =.59 and Therefore, K = Kk = K k = K.349 =.6337 qk = q.95 = ex π K = ex K.59 = ex =.4636 The second method is the rior estimation of the auxiliary arameter q = k + =.349 k + =.447 = and then the comutation of the required value via the aroximation qk =q.95=q +q 5 +5q 9 = =.4636 Even though both methods reach the same result, the second one is clearly more referable, since it does not use ellitic integrals. Therefore, the minimal filter order is equal to N = log6d log = = log/qk ] log/ = and since this value is a decimal number as haens in almost cases it has to be rounded to the next available integer leading thus to the value N =. Since the rescribed secifications do not include the frequencies ω and but only the selectivity factor k =.95, we can set ω = k =.95 = and furthermore 336

27 British Journal of Alied Science & Technology 43, , 4 = / k = /.95 =.5978 to construct a normalized filter with a cutoff frequency ω c = ω = this is a commonly used aroach. Therefore, j K Nπ ln A/ + A/ = j ln.3/ +.3/ = j and the arameter is estimated as = jω sn j K Nπ ln A/ + A/, ω ] = j snj.33465,.75 At this oint we have to estimate the Jacobian ellitic sinus function, and this is not a trivial rocedure. One way to do it, is to develo a user defined software function to comute this value from its series exansion in terms of the θ function. However, since this is not a crucial task in this resentation, we simly use the SN,CN,DN]=elliju,k MATLAB function ]3] that gets as inuts the arameters u and k and returns the values of the ellitic functions sn, cn and du. The roblem is that this MATLAB routine has been designed to work with real arguments, but in this situation we want to estimate the ellitic sinus of an imaginary quantity. To overcome this limitation we can use the identity snju, m = jscu, m = j snu, m cnu, m where u =.33465, m = k =.95 =.95 and m = k = k = m =.95 =.975 Therefore we have sn.33465,.975 =.3785 and cn.33465,.975 =.94478, reaching the result which in turn gives snj.33465,.975 = j = j = jω j = j j = In the final ste we use this value to estimate the W arameter as W = + ω + ωs = =.4634 At this stage of the design rocedure, we have in our disosal all of the required information for the estimation of the filter transfer function and lets start with the identification of its zero values. The Ω m arameters are estimated as Ω m = m sn N K, ω Using the ellij MATLAB function we get ] = sn.59m,.75],,, 3, 4, 5 Ω =.4764, Ω = , Ω 3 =.86897, Ω 4 =.95993, Ω 5 = and therefore, the zeros of the transfer function are the following: 337

28 British Journal of Alied Science & Technology 43, , 4 z, = ± j j = ± Ω.4764 = ± j z 3,4 = ± j j = ± Ω = ±.54596j z 5,6 = ± j j = ± Ω = ±.5857j z 7,8 = ± j j = ± Ω = ±.476j z 9, ± j j = ± Ω = ±.996j On the other hand, to identify the oles of the transfer function, we need the values V m = ωω m ωsω m for m =,, 3, 4, 5 that are estimated as V =.9386, V =.5784, V 3 =.4895, V 4 =.6, V 5 =.867 leading, in turn, to the results, =.3565 ±.75998j 3,4 = ±.68999j 5,6 =.75 ±.8954j 7,8 =.873 ±.96793j 9, =.863 ±.97688j The normalization constant H can be identified from its defining equation for even filter order and it is found equal to H =.97. To construct the filter transfer function we have to build the contribution of each ole and each zero and the results are the following: Zeros z, contribute to the numerator the term s Zeros z 3,4 contribute to the numerator the term s Zeros z 5,6 contribute to the numerator the term s Zeros z 7,8 contribute to the numerator the term s Zeros z 9, contribute to the numerator the term s Poles, contribute to the denominator the term s s Poles 3,4 contribute to the denominator the term s s Poles 5,6 contribute to the denominator the term s +.54s +.84 Poles 7,8 contribute to the denominator the term s s Poles 9, contribute to the denominator the term s +.76s After the identification of all those terms, we can construct the olynomial of the numerator by multilying the first five terms associated with the five zeros, while, the denominator olynomial can be constructed in a similar way by multilying the five terms associated with the five oles. The degree of both olynomials is equal to N =. The numerator olynomial contains only even owers of s, while the denominator olynomial contains all the owers of s between and. To comlete the design, the numerator olynomial must 338

29 British Journal of Alied Science & Technology 43, , Amlitude db Phase deg Frequency rad/s Figure 9: The amlitude and the hase resonse of the designed examle filter. be multilied by the normalization constant H. The final result exressed with an accuracy of three decimal digits has the form.97s +.7s s s s Hs= s +.s s s s 6 +4.s s s s +.9s+.64 The amlitude and hase resonse of this filter are lotted in Figure 9. However, it is clear that these results are only aroximative, since we used the aroximated exressions. If we construct the same filter using the aroriate MATLAB functions the results are z, = ±4.7687j, =.3785 ±.8554j z 3,4 = ±.58355j 3,4 =.959 ±.75739j z 5,6 = ±.9565j 5,6 =.8534 ±.963j z 7,8 = ±.8457j 7,8 =.36 ± j z 9, = ±.5635j 9, =.785 ±.47j and H = , and as it can be easily seen, our solution is associated with a good recision. In ractice and in real alications, the ellitic filters are designed using secialized software ackages due to the comlications associated with their mathematical foundations. 4 Conclusions The objective of this review aer was the detailed and rigorous mathematical resentation of the main asects, roerties and features of the low ass analog ellitic filter aroximation. 339

30 British Journal of Alied Science & Technology 43, , 4 These features include the minimum filter order that meets the rescribed secifications, the oles and zeros of the Chebyshev rational function, the filter transfer function, the amlitude and the hase resonse, the grou delay, as well as the imulse and ste resonses. An aendix describing the main features and roerties of the ellitic and theta functions is also included for the sake of comleteness. Cometing Interests The author declares that no cometing interests exist. References ] Oenheim AV, Willsky AS, Nawab SH. Signals and Systems, nd International Edition, Prentice Hall Signal Processing Series, ISBN ; 997. ] Lathi BP. Linear Systems & Signals. Oxford University Press, nd Ed., ISBN ; 5. 3] Proakis JP, Manolakis DK. Digital Signal Processing, Prentice Hall, 4 th Ed., ISBN ; 6. 4] Antoniou A,Digital Signal Processing, McGraw Hill, ISBN , 5. 5] Oenheim AV, Schafer RW, Buck JR. Discrete Time Signal Processing, nd Edition, Prentice Hall, ISBN ; ] Smith SW. The Scientist and Engineer s Designer Guide to Digital Signal Processing, California Technical Publishing, ISBN ; ] Paarmann L. Design and Analysis of Analog Filters - A Signal Processing Persective, Sringer, st Edition, ISBN ;. 8] Harris F. On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform, Proceedings of IEEE, Volume 66,.5-83, January ] Constantinides AG. Sectral Transformations for Digital Filters, Proceedinds of IEE. 97;78:-5. ] Dimooulos HG. Analog Electronic Filters: Theory, Design and Synthesis, st Edition, Sringer, ISBN ;. ] Abramowitz M, Stegun IA. Handbook of Mathematical Functions, Dover, ISBN ; 97. ] Wanhammar L. Analog Filters Using Matlab, Sringer Verlag, ISBN ; 9. 3] The Mathworks. Filter Design Toolbox User s Guide, Version. Release 3, July. 33

31 British Journal of Alied Science & Technology 43, , 4 Aendix - Ellitic Integrals and Related Functions Since the ellitic functions and the related stuff are not very common and they are rarely used in ractice, this aendix introduces the interested reader to the most imortant asects and roerties of them.. Ellitic Integrals Ellitic integrals were introduced by Wallis and Newton and they were used in a systematic way by Legendre for the study of classical mechanical roblems such as the single endulum. They are considered as generalizations of the inverse trigonometric functions, and they are defined as fx = αx + βx γx + δxζx In the above equation, the right-hand functions are olynomial of x, with the highest ower in ζx to have a value of 3 or 4. These functions deend on two variables, one of them is associated with the uer limit of integration. There are three kinds of ellitic integrals, namely, first, second, and third kind, each one of them is characterized as comlete or incomlete. The incomlete integral of first kind is defined as F φ, k = sin φ dt t k t where k and φ. The k arameter is known as ellitic module, and it can be defined alternatively in terms of the arameter m = k or the angle α = sin k. Performing the change of variable t = sin ϑ we have that dt = cos ϑdϑ = t dϑ and the integral gets the form F φ, k = φ dϑ k sin ϑ where φ π/. Regarding the comlete ellitic integral of first kind it is associated with the uer integration limit value φ = π/ and therefore we have F k = dt π/ t k t = dϑ k sin ϑ On the other hand, the incomlete and comlete ellitic integral of second kind are sin φ k t Eφ, k = φ t dt = k sin ϑ and Ek = k t π/ t dt = k sin ϑ while, the corresonding exressions for the incomlete and comlete ellitic integrals of third kind have the form sin φ dt φ Πφ, η, k = + ηt t k t = dϑ + η sin ϑ k sin ϑ Πη, k = dt + ηt t k t = π/ dϑ + η sin ϑ k sin ϑ 33

32 British Journal of Alied Science & Technology 43, , 4 In the literature, the comlete ellitic integrals of first and second kind are denoted as K and E, resectively, and therefore we have, F φ = π, k = F sin φ =, k = KK = K E φ = π, k = Esin φ =, k = EK = E Defining the arameter k = k and erforming the substitution υ = tan ϑ we define the socalled comlementary form for these integrals F φ = π, k = F sin φ =, k = KK = K E φ = π, k = Esin φ =, k = EK = E. Ellitic Functions The ellitic functions are closely related to the ellitic integrals and in fact they are associated with the inversion of those integrals. In this aendix the focus is given to a secific class of ellitic functions, namely the Jacobian ellitic functions, since they are used for the design of ellitic filters. These functions are defined via the inversion of the incomlete ellitic integral of first kind. Exressing this integral in the form u = F φ, k = φ dϑ k sin ϑ the uer limit of integration is known as Jacobian amlitude and it is denoted as amu, k. Therefore, we have φu, k = F u, k = amu, k. Based on this fact, we can define the three most well known ellitic functions as snu, k = sin φu, k = sinamu, k] cnu, k = cos φu, k = cosamu, k] dnu, k = k sin φu, k = k sin amu, k] These functions are generalizations of the well known trigonometric and hyerbolic functions for the arameter values k = and k =, since, based on their defining equations we can easily see that snu, = sin u, cnu, = cos u, dnu, =, cdu, = cos u, and furthermore, snu, = tanh u, cnu, = sechu, dnu, = sechu, cdu, =. Furthermore, using the auxiliary arameter k = k, it can be easily roven via their defining equations, that sn u, k + cn u, k =, k sn u, k + dn u, k = dn u, k k cn u, k = k k sn u, k + cn u, k = dn u, k sn u, k = cd u, k k cd u, k The ellitic functions used in the design of the ellitic filters, are the even function snu, k and the odd function cdu, k. These functions are related via the exressions cdu, k = snu + K, k = snk u, k 33