Design of Infinite Impulse Response (IIR) filters with almost linear phase characteristics via Hankel-norm approximation

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1 Design of Infinite Impulse Response IIR filters with almost linear phase characteristics via Hankel-norm approximation GD Halikias, IM Jaimoukha and E Milonidis Abstract We propose a method for designing Infinite Impulse Response IIR filters with specified magnitude frequency-response tolerances and approximately linear phase characteristics The design method consists of two steps First, a Finite Impulse Response FIR filter with linear phase is designed using standard optimisation techniques eg linear programming In the second stage, the designed FIR filter is approximated by a stable IIR filter using a Hankel-norm approximation method This is based on a recent technique for approximating discrete-time descriptor systems and requires only standard linear algebraic routines, while avoiding altogether the solution of two matrix Lyapunov equations which is computationally expensive A detailed solution is developed in state-space form, using only the Markov parameters of the FIR filter Further, a-priori bounds on the magnitude and phase errors are obtained which may be used to select the lowest-order IIR filter order which satisfies the specified design tolerances The efectiveness of the method is illustrated with a numerical example Keywords: FIR filters, linear phase, Hankel-norm approximation, descriptor systems 2 Introduction Finite Impulse Response FIR filters are widely used in many digital signal processing applications, especially in the field of communications, because they can exhibit linear phase characteristics Unfortunately, in many cases the order of such filters is prohibitively high for practical implementation In general, the number of delay elements and multipliers needed for an FIR design tends to be much higher compared to similar Infinite Impulse Response IIR implementations This is especially true for filters with sharp cut-off characteristics which have a long impulse response [6, 9] It is therefore natural to ask whether a linear-phase FIR filter Hz can be approximated by a loworder IIR filter, without degrading significantly its magnitude and phase characteristics In this paper we apply Hankel-norm approximation techniques to matrix FIR filters Our technique is based on a recent result involving approximations of discrete-time descriptor systems [, 2] This allows us to treat systems with poles at the origin and has been applied successfully in the context of mixed H /H 2 optimisation problems [5] Our results apply both to the γ-suboptimal and the strictly optimal Control Engineering Research Centre, School of Engineering and Mathematical Sciences, City University, Northampton Square, London ECV HB, UK Department of Electrical and Electronic Engineering, Imperial College of Science, Technology and Medicine, Exhibition Road, London SW7 2BT

2 problem, although in the later case the resulting state-space realisation is non-minimal Using an allpass matrix dilation technique, a state-space parametrisation of the complete family of solutions can be obtained, in the form of a linear fractional map of the ball of unstable contractions [, 2, 4] This can be subsequently reformulated using only the matrix Markov parameters of Hz When the results are specialised to the scalar case, magnitude and phase error bounds of the approximation error can be obtained in terms of the Hankel singular values of Hz, which allows the a-priori determination of the IIR filter order satisfying specified magnitude ripple specifications and an acceptable phase deviation from linearity The effectiveness of the method is illustrated via a numerical example, involving the approximation of a high-order linear-phase FIR filter, designed using linear programming techniques [8] Finally, extensions and further applications of the method are briefly discussed 3 Notation Most of the notation used is standard and is reproduced here for completeness D denotes the open unit disc D = {ζ C : ζ < }, with D and D its closure and boundary respectively L 2 D denotes the Hilbert space of all matrix-valued functions F defined on the unit circle such that 2π trace[f e jθ F e jθ ]dθ < where denotes the complex conjugate transpose of a matrix The corresponding inner product of two L 2 D functions F and G of compatible dimensions is given as: < F, G >= 2π trace[f e jθ Ge jθ ]dθ 2π H 2 D and H 2 D are the closed subspaces of L 2 consisting of all functions analytic in C \ D and D, respectively L D is the space of all uniformly-bounded matrix-valued functions in D, ie all functions defined on the unit circle whose norm: F = sup σ[f e jθ ] θ [,2π is finite Here σ denotes the largest singular value of a matrix H D and H D denote the closed subspaces of L D consisting of all functions analytic in C \ D and D, respectively, while H,k D is the set of all functions in L D with no more than k poles in D Spaces of real-rational functions will be indicated by the suffix R before the corresponding space symbol The unit ball of H D is the set BH D = {U H D : U } If G L D, then the Hankel operator with symbol G is defined as: Γ G : L 2 D H 2 D, Γ G = Π + M G H 2 D in which M G denotes the multiplication operator M G : L 2 D L 2 D, M G f = Gf and Π + denotes the orthogonal projection Π + : L 2 D H 2 D If H L D has dimensions p +p 2 m +m 2 and U RL D has dimensions m 2 p 2, we define the lower linear fractional map of H and U as F l H, U = H + H 2 UI H 22 U H 2, provided the indicated inverse exists If U is a set of m 2 p 2 matrix functions, then F l H, U denotes the set {F l H, U : U U} 2

3 4 FIR filters with linear phase response The fact that FIR filters can have a linear phase response has been used extensively in many digital signal processing applications A filter with a nonlinear phase characteristic causes distortion to its input signal, since the various frequency components in the signal will be delayed, in general, by amounts which are not proportional to their frequency, thus altering their mutual harmonic relationships A distortion of this form is undesirable in many practical digital signal processing applications such as music, video, data transmission and biomedicine and can be avoided by using filters with linear phase over the frequency range of interest [9] Suppose that the unit pulse response of an FIR filter is given by {h, h, h2,, hn} For this filter to have a linear phase response, say θω = αω for α constant, it must have an impulse response with positive symmetry, ie hn = hn n, where n varies as n =,, 2,, N 2 for N odd and n =,, 2,, N 2 for N even Such a filter has identical phase and group delay and these are independent of frequency, ie T p = θω ω = T g = dθω dω An FIR filter whose impulse response has negative symmetry, ie hn = hn n, has an affine phase-response characteristic of the form θω = β αω with α and β constant and its group delay T g = α is independent of ω The simple relationship between the impulse response of an FIR filter and its frequency response has been exploited to design filters of this type via a variety of optimisation methods [3], [7], [8], [] One of the earliest and simplest approaches is [8], in which a linear programming procedure is proposed for designing linear phase FIR filters with specified magnitude-response characteristics low-pass, high-pass, etc As an example, consider the filter Hz = h + hz + + hp z p + hpz p + hp + z p+ + + h2pz 2p in which positive symmetry of the impulse response is enforced around the central coefficient hp by setting h = h2p, h = h2p,, hp = hp + The frequency response of the filter may be written as: He jω = e jωp Mω = α where Mω is a real linear function of the filter s coefficients: Mω = 2 cospω 2 cosp ω h h hp Suppose now that we want to satisfy the following specifications: a δ He jω + δ for all frequencies in the pass-band ω [, ω p ], b He jω δ for all frequencies in the stopband ω [ω s, π] where ω p < ω s, and c minimise the ripple δ subject to constraints a and b Specifications a and b can enforced by discretising the pass-band and stop-band frequency intervals using n frequencies, say, ie = ω < ω 2 < < ω n = ω p 3

4 and ω s = ω n+ < ω n+2 < < ω 2n = π and enforcing the specifications via the inequalities: δ Mω i + δ, i =, 2,, n and δ Mω i δ, i = n +, n + 2,, 2n 2 The optimisation problem: min δ subject to and 2, can now be formulated as a linear programme in the standard form: min c x st Ax b where x = h h hp δ, c =, b is a 4n-dimensional vector and A is a 4n p + 2- dimensional matrix This can be solved efficiently using standard techniques, eg the simplex algorithm or interior point methods 5 Hankel-norm approximation of FIR filters In this section we propose a Hankel-norm method for approximating FIR filters via lower-order IIR filters The method is based on a recent result involving model-reduction of descriptor discrete-time systems [], [2] The main advantage of Hankel-norm approximation methods over other model-reduction techniques is that they offer tight bounds on the infinity-norm of the approximation error; in particular, it is shown in [4] how to construct k-th order approximations Xz RH D of Hz RH D with degxz = k < deghz = n, such that: n Hz + Xz σ i Γ H 3 i=k+ where σ i Γ H denotes the i-th singular value of Γ H, indexed in non-increasing order of magnitude This inequality can be used to determine a-priori the order k of the low-order system Xz which satisfies magnitude error specifications Theorem 2 below parametrises all solutions Xz to the Hankel-norm approximation problem Γ H + Γ X γ, in which Hz is the matrix FIR filter Hz = H + H z + + H n z n and Xz is a matrix IIR filter of degree degxz k The parametrisation is given in descriptor form and hence applies both in the sub-optimal case σ k+ Γ H < γ < σ k Γ H and the optimal case γ = σ k+ Γ H Before stating this theorem, however, the following preliminary result is needed: Theorem : Let Hz = Hz H = H z + H 2 z H n z n with H i R p l for i =, 2,, n Then: 4

5 The singular values of Γ H, σ i Γ H indexed in decreasing order of magnitude are the singular values of the Hankel matrix: H H 2 H n H n H 2 H 3 H n H 3 H 4 R = H n H n H n 2 A state-space realisation of Hz is given by Hz = CzI A B with: I p A = I p R np np I p B = H n H n H 2 H R np l C = I p R p np Further, the realisation is output balanced, ie the unique solution of the Lyapunov equation Q = A QA + C C is given by Q = I np 3 The controllability grammian of the realisation in part 2, ie the unique solution of the Lyapunov equation P = AP A + BB, can be factored as P = R 2 R 2 where: H n H R 2 = n H n Equivalently P can be block-partitioned as: and σ 2 i Γ H = λ i P for each i H H 2 H n mini,j P ij j=,2,n i=,2,n where P ij = H n i+k H n j+k k= Proof: Straightforward and hence omitted The following Theorem gives a complete parametrization of all k th order γ-suboptimal Hankelnorm approximations in descriptor form All optimal approximations can also be obtained by setting γ = σ k+ Γ H Theorem 2: Let Hz be the matrix FIR filter defined in Theorem Let γ satisfy σ k+ Γ H γ < σ k Γ H Then all Xz H,k D such that Hz + Xz γ are generated via the lower linear fractional transformation: X = {F l S a, U : γu BH D} 5

6 where: with and E S = B S = S a = D S + C S ze S A S B S I np I np γ 2 I np P, A S = X A CX P, C S = B B C CX B γi p D s = γi l where A, B, C and P are defined in Theorem, and X = I np +A, A = P γ 2 XA X, B = γxx C and C = γb X Proof: Minor adaptation of result in [], [2] The proof of Theorem 2 follows readily by specialising the results of [] and [2] to our case, using the state-space description in Theorem Note that the generator of all Hankel norm approximations S a z is given in descriptor form; this can be converted into a state-space form, if required, using a standard procedure In particular, it is always possible to obtain a state-space realisation of S a z of order 2np RankA ; if A is non-singular, we obtain a state-space description of order np Note also, that although Theorem 2 is still valid in the optimal case γ = σ k+ Γ H, the resulting realisation is non-minimal In this case, a minimal realisation can be obtained in closed form via a singular perturbation argument see [2] for details It is clear from Theorem 2 that the solution of Lyapunov equations is completely avoided in our framework; in fact the generator S a z is completely defined by the Markov parameters of Hz and γ We make this dependence explicit in Theorem 4 below, which derives a state-space realization of the generator of all γ-suboptimal Hankel-norm approximations We first reformulate the parametrization of all Hankel-norm approximations in a state-space rather than descriptor setting: Theorem 3: Let Hz be the matrix FIR filter defined in Theorem Let γ satisfy σ k+ Γ H γ < σ k Γ H and assume that A is non-singular Then all Xz H,k D such that Hz+Xz γ are generated via the lower linear fractional transformation: where: with X = {F l H a, U : γu BH D} H a = D H + C H zi A H B H A H = I pn γ 2 I pn P A X, B H = γ 2 I n P A B B C H = CX P C A X, D H = CX B CX P A B γi CX P A B γi C A B C A B 6

7 where all variables are defined in Theorem and Theorem 2 Proof: Follows by the standard procedure of transforming descriptor realisations to state-space form The following Theorem shows that in the state-space realization given in Theorem 3 has a special structure In particular, the corresponding A matrix of the realization is in companion form, which allows us to obtain an analytic expression for the transfer function of the so-called central approximation corresponding to Uz = For simplicity the results are stated only for the scalar case Theorem 4: Let all variables be defined as in Theorems -3 above, set p = m = and assume that A is non-singular and that σ k+ Γ h < γ < σ k+ Γ h sub-optimal case Then the realization of H a z = A H, B H, C H, D H defined in Theorem 3 is of the form: a a 2 a 3 a n a n b b 2 h n A H =, B H = h n 2 h C H = c c 2 c 3 c n c n, D H = d 22 where hz = h z + h 2 z h n z n and in which a i = γ2 θ i + θ j γ 2 i =, 2,, n, a n = γ2 θ n θ γ 2 θ γ 2 n γ b = h n + γ 2 θ i h n i+, b 2 = θ γ 2 θ θ j = i= n i ˆPij, j =, 2,, n, i= ˆP = P γ 2 I n In particular, the transfer function of the central approximation Uz = is given as: Xz = h z n + h 2 h a z n h n h a n 2 z + b h a n z n a z n a 2 z n 2 a n z a n Proof: First note that since σ k+ Γ h < γ < σ k+ Γ h, P γ 2 I n is nonsingular It is also easy to see that: X =, X = 7 n+ n n+

8 Thus, P γ 2 I n A = γ 2 XA X I n = γ 2 n := uv and hence, using the matrix inversion lemma, A = P γ 2 I n uv = P γ 2 I n + λp γ 2 I n uv P γ 2 I n where we have defined Setting λ = v P γ 2 I n u = v ˆP u θ = θ θ 2 θ n = v P γ 2 I n shows that and hence A H = I n γ 2 I n P A X = γ 2 I n P A = I n λuθ λγ 2 θ + θ 2 λγ 2 θ 2 + θ 3 λγ 2 θ n + θ n λγ 2 θ n which is of the required form on noting that λ = Next consider B H Using the fact that γ 2 I n P A = θ u = γ 2 = θ γ 2 n i= i ˆP i + λγ 2 θ λγ 2 θ 2 λγ 2 θ n λγ 2 θ n the first column of B H can be written as: γ 2 I n P A B = h n + γ2 γ 2 θ n i= θ ih n i+ h n h 2 h 8

9 Similarly, since B = γxx C = γ n = γ the second column of B H can be written as: + λγ 2 θ λγ 2 θ 2 λγ 2 θ n λγ 2 θ n γ 2 I n P A B = γ = γ γ 2 θ as required Next consider the first row of C H ; this is given by: CX P A X = CX P P γ 2 XA X X = CX P γ 2 XA X + γ 2 XA X P γ 2 XA X X = C γ 2 CA X P γ 2 XA X X = C since CA = Finally, consider D H Using a similar argument to the one above, its, element is: Similarly, D H, = CX B CX P P γ 2 XA X B = CX B CX B γ 2 CA X P γ 2 XA X B = D H,2 = γ γcx P P γ 2 XA X XX C = γ γcx C γ 3 CA X P γ 2 XA X XX C = γ γcc = Finally consider the 2, element of D H, D H,2 = γ C A B We first show that A B is a zero column vector, except from its first element which is equal to h n Write A = P γ 2 XA X = x X 2 where x is a column vector and set where φ is a scalar Then: A B = φ φ 2 φ x + X 2 φ 2 = B It is easy that the first column of XA X is zero, and hence x is the first column of P which is given by x = h n h n h h n 9

10 Thus x is a multiple of B, from which it follows that φ = h n and φ 2 =, so that A B = h n Thus, D H,2 = γ γb X A B = γ γh n h n h n h n n+ = The transfer function of the central approximation follows by routine calculations since A H companion form is in In the remaining part of this section we derive a bound on the phase of the approximation error Note first, that in the scalar case, the solution to the Hankel-norm approximation problem is, in general, unique only in the optimal case The so-called central solution is obtained by setting Uz = Having obtained the central approximation, one next needs to remove its anti-stable component The extraction of the stable projection can be performed either by factoring the denominator polynomial of Xz, pz = z n a z n a 2 z n 2 a n z a n which is guaranteed to have k roots strictly inside the unit circle and n k roots strictly outside the unit circle Alternatively, the stable projection can be extracted from the state-space realization of Xz by transforming the system to block-schur form using an appropriate orthogonal state-space transformation and ordering the eigenvalues of A H in ascending order of magnitude; decoupling of the stable and anti-stable parts then requires the solution of a matrix Sylvester equation Reference [4] shows that having solved the k-th order Hankel-norm approximation problem, it is always possible to choose a constant term x so that the approximation error satisfies the bound: n hz + xz + x σ i Γ h 4 i=k+ Since this bound applies uniformly in frequency, it can be used to give an immediate bound on the approximation phase error: Theorem 5: Let ˆxz = xz + x be a k-th order optimal Hankel norm approximation of the scalar FIR filter hz such that 4 holds Then, if φω = arghe jω + argˆxe jω, we have that: sinφω n i=k+ σ iγ h he jω 5 for every ω [, π In particular, if the frequency interval [ω ω 2 ] lies in the filter s passband in which δ he jω + δ, then n i=k+ sinφω σ iγ h 6 δ for every ω [ω ω 2 ]

11 Proof: Straightforward and hence omitted The phase error bound given in Theorem 5 can be used to select the minimum order of approximation k consistent with a worst-case phase-error specification; If hz is a linear-phase FIR filter, then the RHS of 5 or 6 quantifies the deviation in the phase of the IIR approximation of hz from linearity If the magnitude tolerances resulting from the optimization procedure are tight, the specifications need to be tightened to account for the approximation error 6 Example In this section some of the results presented in the paper are illustrated by means of a computer example First, a linear phase FIR filter of order n = 2 was designed using the linear programming procedure outlined in section 3 The design specifications were defined as: ω p = rad/sample, ω s = 5 rads/sample and the frequency intervals [, ω p ] and [ω s, π] were each discretised using 5 equally-spaced frequencies The linear programme was then set up and solved using Matlab s function linprogm The minimum ripple was obtained as δ = 232; the first optimal impulse response coefficients of the resulting filter hz are tabulated below the last coefficients are symmetric and are not included: i hi i hi i hi Table : Impulse response coefficients The magnitude frequency response of the filter is shown in Figure below Note that the response in the passband and the stopband lies within the desired bounds ± δ Next the Hankel-norm approximation method outlined in Theorem 2 was applied to the FIR filter hz The Hankel singular values of hz are shown in Table 2 Parameter γ was chosen as γ = 3 resulting in a 7-th order sub-optimal Hankel-norm approximation The seven stable poles of the approximant were obtained as: 7467, 284 ± 852j, 4977 ± 6347j and 6886 ± 3363j i σ i i σ i i σ i i σ i Table 2: Hankel singular values

12 4 Optimal FIR filter and bounds 2 8 Gain angular frequency rads/sample Figure : Magnitude frequency response The magnitude response of hz and its IIR approximation is shown in Figure 2 It can be seen that the IIR filter has a slightly larger ripple in the pass-band Figure 3 shows the impulse responses of the two filters 4 FIR/IIR magnitude response 2 Magnitude linear angular frequency rads/sample Figure 2: FIR/IIR magnitude frequency response Finally, Figure 4 shows the phase error in the passband ω rad/sample arising from the approximation ie the deviation of the phase of the IIR filter from linearity, while Figure 5 shows the pole/zero pattern of the IIR filter there is an additional zero at z = 5652 not shown in the figure 2

13 4 FIR and IIR impulse response Amplitude index Figure 3: FIR/IIR impulse response 7 Conclusions In this work we have presented a systematic way for designing low-order IIR filters with approximately linear phase response characteristics The method relies on two steps: a First a linear-phase FIR filter is designed using linear programming or an alternative optimisation technique; b In step 2, the FIR filter is approximated by a low-order IIR filter using Hankel-norm approximation methods The method is computationally efficient and relies on the specialization of recent results for approximating discrete-time descriptor systems A-priori bounds are obtained for the magnitude and phase approximation error which can assist the designer to choose an approximation order consistent with the design specifications A low-order example has illustrated the effectiveness of the method There are a number of issues related to the proposed technique that we intend to pursue in the future These include: Derivation of tighter error bounds than those applying in general for the Hankel-norm approach This seems possible given the special structure of FIR filters all poles lie at the origin Investigation of the sensitivity properties of the proposed design under finite-precision implementation Generalisation of the method to other types of approximation, eg relative-error and general frequency weighted approximations Extension of the method to more general application domains in the fields of Signal Processing, Systems Identification and Control References [] M Al-Husari, IM Jaimoukha and DJN Limebeer 993, A descriptor approach for the solution of the one-block discrete distance problem, Proc IFAC World Congress, Sydney, Australia 3

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15 5 Pole/zero plot 5 Imaginary axis Real axis Figure 5: Pole/zero pattern of IIR filter 5