Transfer Functions of Second-Order Digital Filters with Two Equal Second-Order Modes Shunsuke Yamaki, Masahide Abe, and Masayuki Kawamata Department of Electronic Engineering, Graduate School of Engineering, Tohoku University, Japan E-mail:
[email protected] Tel: +81-22-795-7095 d
Abstract— This paper clarifies the class of second-order digital filters with two second-order modes equal. We consider three cases for second-order digital filters: complex conjugate poles, distinct real poles, and multiple real poles. We derive a general expression of the transfer function of second-order digital filters with two second-order modes equal. Furthermore, we show that the general expression is obtained by a frequency transformation on a first-order prototype FIR digital filter.
u(n)
b
Fig. 1.
II. P RELIMINARIES A. State-Space Digital Filters Consider a stable, controllable, and observable N th-order state-space digital filter described by x(n + 1) = Ax(n) + bu(n) y(n) = cx(n) + du(n) 978-1-4673-0219-7/12/$31.00 ©2012 IEEE
c
Block diagram of a state-space digital filter.
where x(n) ∈ RN ×1 is a state-vector, u(n) ∈ R is a scalar input, y(n) ∈ R is a scalar output, and A ∈ RN ×N , b ∈ RN ×1 , c ∈ R1×N , d ∈ R are coefficient matrices. The block diagram of the state-space digital filter (A, b, c, d) is shown in Fig. 1. The transfer function H(z) is described in terms of the coefficient matrices (A, b, c, d) as H(z) = c(zI − A)−1 b + d.
(3)
B. L2 -Sensitivity and Second-Order Modes The L2 -sensitivity is one of the measurements which evaluate coefficient quantization effects of digital filters. Ref. [6] defines the L2 -sensitivity of the filter H(z) with respect to the realization (A, b, c, d) by the general controllability Gramian K i and the general observability Gramian W i such as ∂H(z) 2 ∂H(z) 2 ∂H(z) 2 + + S(A, b, c) = ∂b ∂c ∂A 2 2 2 = tr(W 0 )tr(K 0 ) + tr(W 0 ) + tr(K 0 ) ∞ tr(W i )tr(K i ). (4) +2 i=1
The general Gramians K i and W i are expressed as 1 i Ki = A K 0 + K 0 (AT )i (5) 2 1 Wi = W 0 Ai + (AT )i W 0 (6) 2 where the controllability Gramian K 0 and the observability Gramian W 0 are solutions to the following Lyapunov equations: K 0 = AK 0 AT + bbT W 0 = AT W 0 A + cT c.
(1) (2)
y(n)
x(n)
A
I. I NTRODUCTION Second-order modes play quite important roles in synthesis of high-accuracy digital filter structures. In Refs. [1] and [2], minimum unit noise gain is expressed in terms of secondorder modes. In Ref. [3], coefficient sensitivity derived from a statistical approach is described by using second-order modes. Recently, the L2 -sensitivity minimization problems, in which second-order modes are significantly important, have been widely investigated [4]–[7]. L2 -sensitivity minimization problem is quite important in synthesis of high-accuracy digital filter structures. Since the L2 -sensitivity minimization of digital filters is a nonlinear problem, it is quite difficult to solve this problem by analytical method. In terms of computational complexity, it has been strongly desired to derive analytical solutions to the L2 -sensitivity minimization problems. Our group previously proved that the L2 -sensitivity minimization problem can be solved analytically if the second-order modes are all equal [8]. However, the result does not give the transfer functions of the digital filters with all second-order modes equal. In this paper, we clarify the class of second-order digital filters with all second-order modes equal. Second-order digital filters are classified into three cases: complex conjugate poles, distinct real poles, and multiple real poles. We derive a general expression of the transfer function of digital filters with all second-order modes equal. Furthermore, we show that the general expression is obtained by a frequency transformation on a first-order prototype FIR digital filter.
z −1
(7) (8)
The controllability Gramian K 0 and the observability Gramian W 0 are positive definite symmetric, and the eigenvalues
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θi2 (i = 1, · · · , N ) of the matrix product K 0 W 0 are all positive. The second-order modes θi ’s of the digital filter H(z) are defined by [3] (9) θi = eigenvalues of K 0 W 0 (i = 1, · · · , N ).
2 q2 =
q2
Let T be a nonsingular N × N real matrix. If a coordinate ¯ (n) = T −1 x(n) is applied to a transformation defined by x filter realization (A, b, c, d), we obtain a new realization which has the following coefficient matrices
(K i , W i ) = (T
K iT
−T
0 −1
multiple real poles distinct real poles
−2
(10)
−2
and the following general Gramians −1
T
, T W iT )
Fig. 2.
−1
S(P ) = tr(W 0 P )tr(K 0 P ) + tr(W 0 P ) + tr(K 0 P ) ∞ tr(W i P )tr(K i P −1 ) (13) +2 i=1
1
2
Stability triangle of second-order digital filters.
The balanced realization is the filter realization of which controllability Gramian K 0 and the observability Gramian W 0 are equal and diagonal such as K 0 = W 0 = Θ(= diag(θ1 , · · · , θN )).
(16)
Theorem 1 gives a sufficient condition for the L2 -sensitivity minimization problem to be solved analytically. However, it does not give the transfer functions of the digital filters with all second-order modes equal. III. S ECOND -O RDER D IGITAL F ILTERS WITH T WO E QUAL S ECOND -O RDER M ODES
where P = T T T . D. L2 -Sensitivity Minimization Problem The L2 -sensitivity minimization problem is formulated as follows: For an initial digital filter (A, b, c, d) with a given transfer function H(z), minimize the L2 -sensitivity S(P ) with respect to P , where P is an arbitrary positive definite symmetric matrix. The L2 -sensitivity S(P ) has the unique global minimum, achieved by P opt satisfying ∂S(P ) = 0. (14) ∂P P =P opt For this problem, many iterative algorithms has been proposed, which require many calculations to derive the minimum L2 sensitivity realization [5], [6]. E. Digital Filters with All Second-Order Modes Equal Our group previously proved that the L2 -sensitivity minimization problem can be solved analytically if the secondorder modes are all equal [8]. Theorem 1: If the second-order modes θi (i = 1, · · · , N ) of a digital filter H(z) are all equal, then (Aopt , bopt , copt , dopt ) = (Ab , bb , cb , db )
0 q1
(12)
which shows that K 0 W 0 has the same eigenvalues of K 0 W 0 . Thus, the second order modes, which are the square roots of the eigenvalues of K 0 W 0 , are invariant under coordinate transformation. The L2 -sensitivity of the transformed filter (T −1 AT , T −1 b, cT , d) can be expressed in terms of the infinite summation of general Gramians as −1
−1
(11)
respectively. From Eqs. (11), we have K 0 W 0 = T −1 K 0 W 0 T
complex conjugate poles
1
C. Coordinate Transformation
¯ c ¯ = (T −1 AT , T −1 b, cT , d) ¯ b, ¯, d) (A,
1 2 q 4 1
(15)
that is, the minimum L2 -sensitivity realization is equal to the balanced realization. 2
In this section, we derive the general expression of the transfer functions of second-order digital filters with two second-order modes equal. A. Second-Order Digital Filters Consider a stable second-order IIR digital filter given by H(z) =
p0 + p1 z −1 + p2 z −2 . 1 + q1 z −1 + q2 z −2
(17)
It is well known that the second-order digital filter H(z) is stable, if and only if q1 and q2 remain within the stability triangle described by |q2 | < 1, |q1 | < 1 + q2 .
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Fig. 2 shows the stability triangle. For stable second-order digital filters given by (17), the locations of the poles depend on the filter coefficients q1 and q2 as follows: ⎧ 2 ⎪ ⎨complex conjugate if q1 − 4q2 < 0. 2 Two poles are real and distinct if q1 − 4q2 > 0. (19) ⎪ ⎩ 2 real and multiple if q1 − 4q2 = 0. B. Explicit Expressions of Second-Order Modes [9], [10] Our group has derived explicit expressions of second-order modes for second-order digital filters with complex conjugate poles [9] and with real poles [10].
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ρ
1) Complex conjugate poles: Consider second-order digital filters with complex conjugate poles as follows: α∗ α + +d (20) z − λ z − λ∗ where λ = λr + jλi is a complex pole, and α is a complex scalar. We define the scalar parameters P , Q, and R as follows: H1 (z) =
P =
α |α| , R + jQ = . 1 − |λ|2 1 − λ2
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2) Distinct real poles: Consider second-order digital filters with distinct real poles as follows: α1 α2 H2 (z) = + + d (λ1 = λ2 ) (23) z − λ1 z − λ2 where (λ1 , λ2 ) are real poles, and (α1 , α2 ) are real scalars. We define the scalar parameters P1 , P2 , and P12 as follows: |α1 α2 | |α1 | |α2 | P1 = , P = , P = (24) 2 12 1 − λ21 1 − λ22 1 − λ1 λ 2 where P1 > 0, P2 > 0, and P12 > 0. Second-order modes depend on σ1 = sign(α1 ) and σ2 = sign(α2 ) as follows: (θ1 , θ2 ) = ⎧ ⎪ 2 (σ = σ ) ⎨ 1 (P1 + P2 ) ± 1 (P1 − P2 )2 + 4P12 1 2 2 2 (25) 1 1 ⎪ 2 ± (P − P ) (σ = σ ). ⎩ (P1 + P2 )2 − 4P12 1 2 1 2 2 2 3) Multiple real poles: Consider second-order digital filters with multiple real poles as follows: β1 β2 + +d z − λ0 (z − λ0 )2
ρ
u(n)
β12 1 + λ20 λ0 1 + σβ + |β | (27) 1 2 4|β2 | 1 − λ20 (1 − λ20 )2 (1 − λ0 )3 1 (28) Q2 = |β2 | 1 − λ20 1 λ0 1 Q12 = σβ1 + |β2 | (29) 2 1 − λ20 (1 − λ20 )2 where σ = sign(β2 ). Second-order modes are given by (θ1 , θ2 ) = Q1 Q2 ± Q12 . (30)
Fig. 3. Block diagrams of (a)prototype digital filter HP (z) and (b)transformed digital filter H(z).
where (λ1 , λ2 ) are poles of H(z), θ is a nonzero arbitrary real scalar, and ρ is an arbitrary real scalar. 2 Furthermore, it is remarkable that the transfer function H(z) in Eq. (31) is obtained by the frequency transformation on a first-order FIR prototype filter. Remark 1: A second-order digital filter H(z) in (31) is obtained by the frequency transformation such as H(z) = HP (z)|z−1 ←HAP (z)
(32)
where HP (z) = θz −1 + ρ is a first-order prototype FIR digital filter and HAP (z) is a second-order all-pass filter given by (λ1 z − 1)(λ2 z − 1) . (z − λ1 )(z − λ2 )
(33)
2 Block diagrams of the prototype digital filter HP (z) and transformed digital filter H(z) are shown in Fig. 3. The proof of Theorem 2 is given as follows. 1) Complex conjugate poles: For second-order digital filters with complex conjugate poles, H(z) in Eq. (31) can be rewritten by partial fraction expansion such as
1 − (λ∗ )2 1 − λ2 − + d1 (34) H1 (z) = jc1 z−λ z − λ∗ where c1 and d1 are real scalars given by
C. General Expressions of Transfer Functions of SecondOrder Digital Filters with Two Equal Second-Order Modes This subsection derives the general expression of transfer functions of second-order digital filters with two equal secondorder modes. Theorem 2: Transfer functions of second-order digital filters with two second-order modes equal can be expressed as
y(n)
(b) Second-order transformed digital filter H(z)
HAP (z) =
Q1 =
(λ1 z − 1)(λ2 z − 1) +ρ (z − λ1 )(z − λ2 )
θ HAP (z)
(26)
where λ0 is a multiple real pole, and (β1 , β2 ) are real scalars. We define the scalar parameters Q1 , Q2 , and Q12 as follows:
H(z) = θ
y(n)
(a) First-order prototype digital filter HP (z) (21)
Using these parameters, second-order modes are given by (22) (θ1 , θ2 ) = P 2 − Q2 ± R.
H3 (z) =
z
θ
−1
c1 = −θ
1 − |λ|2 , d1 = θ|λ|2 + ρ. 2λi
(35)
If two second-order modes are equal, that is, θ1 = θ2 , we have R = 0, which yields α = jc (36) 1 − λ2 or equivalently,
(31)
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α = jc(1 − λ2 )
(37)
where c is an arbitrary real scalar. Substituting Eq. (37) into Eq. (20) yields
1 − (λ∗ )2 1 − λ2 − + d. (38) H1 (z) = jc z−λ z − λ∗ Letting c = c1 and d = d1 in Eq. (38) shows that the transfer functions of second-order digital filters with complex conjugate poles of which second-order modes are equal can be expressed by Eq. (31). 2) Distinct Real Poles: For second-order digital filters with distinct real poles, H(z) in Eq. (31) can be rewritten by partial fraction expansion such as
1 − λ21 1 − λ22 + d2 − (39) H2 (z) = c2 z − λ1 z − λ2 where c2 and d2 are real scalars given by c2 = θ
1 − λ1 λ2 , d2 = θλ1 λ2 + ρ. λ1 − λ2
(ii) The case of σ1 = −σ2 : If two second-order modes are equal, that is, θ1 = θ2 , we have P1 = P2 , which yields
Since σ1 = −σ2 , we have α2 α1 =− 1 − λ21 1 − λ22
(41)
(42)
or equivalently, α1 = c(1 − λ21 ), α2 = −c(1 − λ22 )
(43)
where c is an arbitrary real scalar. Substituting Eqs. (43) into Eq. (23) yields
1 − λ21 1 − λ22 + d. (44) − H2 (z) = c z − λ1 z − λ2 Letting c = c2 and d = d2 in Eq. (44) shows that the transfer functions of second-order digital filters of which second-order modes are equal can be expressed by Eq. (31). 3) Multiple Real Poles: For second-order digital filters with multiple real poles, H(z) in Eq. (31) can be rewritten by partial fraction expansion such as
2λ0 1 − λ20 + d3 − (45) H3 (z) = c3 z − λ0 (z − λ0 )2 where c3 and d3 are real scalars given by c3 = −θ(1 − λ20 ), d3 = θλ20 + ρ.
1 λ0 1 β1 + β2 =0 2 1 − λ20 (1 − λ20 )2 or equivalently, β1 = 2cλ0 , β2 = c(1 − λ20 )
(49)
where c is an arbitrary real scalar. Substituting Eqs. (49) into Eq. (26) yields
2λ0 1 − λ20 H3 (z) = c +d (50) − z − λ0 (z − λ0 )2 Letting c = c3 and d = d3 in Eq. (50) shows that the transfer functions of second-order digital filters of which second-order modes are equal can be expressed by Eq. (31).
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This paper has derived the transfer functions of second-order digital filters with two second-order modes equal. We consider three cases for second-order digital filters: complex conjugate poles, distinct real poles, and multiple real poles. A general expression of the transfer function has been proposed for digital filters with all second-order modes equal. Furthermore, we show that the general expression is obtained by a frequency transformation on a first-order prototype FIR digital filter. R EFERENCES [1] C. T. Mullis and R. A. Roberts, “Synthesis of minimum roundoff noise fixed point digital filters,” IEEE Trans. Circuits Syst., vol. CAS-23, no. 9, pp. 551–562, Sept. 1976. [2] S. Y. Hwang, “Minimum uncorrelated unit noise in state-space digital filtering,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP25, no. 4, pp. 273–281, Aug. 1977. [3] M. Kawamata and T. Higuchi, “A unified approach to the optimal synthesis of fixed-point state-space digital filters,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-33, no. 4, pp. 911–920, Aug. 1985. [4] M. Gevers and G. Li, Parametrizations in Control, Estimation and Filtering Problems. Springer-Verlag, 1993, ch. 5. [5] W.-Y. Yan and J. B. Moore, “On L2 -sensitivity minimization of linear state-space systems,” IEEE Trans. Circuits Syst. I Fundamental theory and applications, vol. 39, no. 8, pp. 641–648, Aug. 1992. [6] T. Hinamoto, S. Yokoyama, T. Inoue, W. Zeng, and W.-S. Lu, “Analysis and minimization of L2 -sensitivity for linear systems and twodimensional state-space filters using general controllability and observability gramians,” IEEE Trans. Circuits Syst., vol. CAS-49, no. 9, pp. 1279–1289, Sept. 2002. [7] S. Yamaki, M. Abe, and M. Kawamata, “On the absence of limit cycles in state-space digital filters with minimum L2 -sensitivity,” IEEE Trans. Circuits Syst. II, vol. 55, no. 1, pp. 46–50, Jan. 2008. [8] ——, “Analytical synthesis of minimum L2 -sensitivity realizations of all-pass digital filters,” in Proc. IEEE International Symposium on Circuits and Systems, Paris, France, May 2010, pp. 729–732. [9] H. Matsukawa and M. Kawamata, “Design of variable digital filters based on state-space realizations,” IEICE Trans. Fundamentals of Electronics, Communications and Computer Sciences, vol. E84-A, no. 8, pp. 1822–1830, Aug. 2001. [10] S. Yamaki, M. Abe, and M. Kawamata, “Explicit expressions of balanced realizations of second-order digital filters with real poles,” IEEE Signal Processing Letters, vol. 55, pp. 465–468, 2008.
If two second-order modes are equal, that is, θ1 = θ2 , we have Q12 = 0, which yields 1 λ0 1 σβ1 + |β2 | = 0. 2 2 1 − λ0 (1 − λ20 )2
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IV. C ONCLUSION (40)
(i) The case of σ1 = σ2 : If two second-order modes are 2 = 0, equal, that is, θ1 = θ2 , we have (P1 − P2 )2 + 4P12 which does not hold since P1 > 0, P2 > 0, and P12 > 0. In this case, it is impossible for second-order modes to be equal.
|α1 | |α2 | = . 2 1 − λ1 1 − λ22
Since σ = sign(β2 ), we have
(47)
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