SICE Annual Conference 2007 Sept. 17-20, 2007, Kagawa University, Japan
A New Efficient Matrix Spectral Factorization Algorithm Lasha Ephremidze1 Gigla Janashia and Edem Lagvilava2 1
Department of Mathematics, Tokai University, Shizuoka, Japan (Tel : +81-543-37-0140; E-mail:
[email protected]) 2 Razmadze Mathematical Institute, Tbilisi, Georgia (Tel : +995-32-334596; E-mail:
[email protected])
Abstract: An absolutely new method of matrix spectral factorization is proposed which leads to the most simple computational algorithm. A demo version of the software implementation is located at www.ncst.org.ge/MSF-algorithm. Keywords: Matrix spectral factorization algorithm.
1. INTRODUCTION
2. SOME BASIC FACTS FROM THE THEORY OF HARDY SPACES HP
Spectral factorization plays a prominent role in a wide range of fields in system theory and control engineering. In the scalar case, which arises in systems with single input and single output, the factorization problem is relatively easy and several classical methods exist to perform this task (see a survey paper [5]). The matrix spectral factorization, which arises in multi-dimensional systems, is significantly more difficult. Since Wiener’s original efforts ([6]) to create a sound computational method of such factorization, dozens of papers addressed the development of appropriate algorithms. Nevertheless, the problem was far from satisfactory solution.
Let D = {z ∈ C : |z| < 1} and T = ∂D. The Hardy space Hp = Hp (D), p > 0, is the set of analytic functions f (z), z ∈ D, such that 2π p |f (reiθ )|p dθ < ∞. f Hp = sup r 0, then there exist the boundary values f (z) = f (eiθ ) := lim f (reiθ ) r→1−
for almost all z = eiθ ∈ T. Furthermore f (eiθ ) ∈ Lp (T). The boundary values of f ∈ Hp , p > 0, cannot be 0 on a set of positive measure. Furthermore, log |f (eiθ )| ∈ L1 (T) which is equivalent in this case to 2π log |f (eiθ )| dθ > −∞.
It should be mentioned that though the branch of mathematics where the spectral factorization problem is posed is the theory of complex functions (namely, a familiarity with some introductory facts from the theory of Hardy spaces is required for a strict formulation of the problem in its general non-rational setting), none of the existing methods used this theory for solution. Mathematicians, including Wiener and his followers, were using methods of Functional Analysis, while engineers introduced a state-space model and reduced the problem to the solution of algebraic Riccati equation.
0
In particular, if the boundary values of two functions from Hp , p > 0, coincide (almost everywhere) then these functions are the same. For p ≥ 1, if f ∈ Hp , then the negative Fourier coefficients of f (eiθ ) are equal to 0, i.e. f (eiθ ) ∈ L+ p (T) := {f ∈ Lp (T) : cn (f ) = 0, n < 0}. Conversely, each f (eiθ ) ∈ L+ p (T) can be (uniquely) extended in D to some function from Hp . Thus Hp (D) and L+ p (T) can be naturally identified when p ≥ 1. − If f (z) ∈ L+ p (T), then f (z) ∈ Lp (T) := {f ∈ − Lp (T) : cn (f ) = 0, n > 0}. Obviously L+ P ∩ LP consists only of constant (a.e.) functions. Consequently, if f (z) ∈ L+ 1 (T) is real (a.e.), then f is constant (a.e.). An analytic function 2π iθ e +z 1 iθ log q(e Q(z) = c · exp ) dθ , (1) 2π 0 eiθ − z
A completely new approach to the matrix spectral factorization problem was developed by the last two authors in [2], where the two dimensional case is considered. In the present paper we extend the same method to arbitrary dimensional matrices. This is the first time that the theories of Complex Analysis and Hardy spaces are used for solution of the problem which turned out to be very effective. Namely, the decisive role of unitary matrix-functions in the factorization process is revealed which, by flexible manipulations, completely absorbs all the technical difficulties of the problem leaving very few and simple procedures for computation.
where |c| = 1 and a positive (a.e.) function q is such that log q ∈ L1 (T), is called outer. The class of outer functions from Hp will be denoted by HpO . For boundary values, we have
The algorithm can be applied to any matrix spectral densities which satisfy the necessary and sufficient PaleyWiener condition (see (4)) for the existence of factorization, though further simplifications in the calculation procedure are available in the rational case (see [1]).
|Q(eiθ )| = q(eiθ ) a.e. on T,
- 20 -
(2)
PR0001/07/0000-0020 ¥400 © 2007 SICE
and if boundary values of two outer functions coincide with absolute values a.e., then these functions differ from each other by a constant multiplier with absolute value 1. If a function I ∈ H∞ is such that 0 ≤ I(z) ≤ 1 on D, and |I(z)| = 1 a.e. on T, then it is called inner. Every function f ∈ Hp , p > 0, can be represented as a product f (z) = Q(z)I(z), where Q ∈ HPO and I is a inner function (see Riesz factorization Theorem, e.g., in [4], p. 105; I itself can be factorized as a Blaschke product and a singular inner function, but we do not need to consider this for current purposes). Obviously |f (z)| ≤ |Q(z)| for z ∈ D, and |f (z)| = |Q(z)| for a.a. z ∈ T. Thus, the outer functions are the ones that take maximal possible absolute values in D whenever absolute values on the boundary T are fixed. We will use the following generalization of Smirnov’s theorem (see [4], p. 109).
which is a core of Kolmogorov spectral factorization method. There is no analog of this formula in the matrix case, since in general eA+B = eA eB for noncommutative matrices A and B. This is the main reason that the approximate computation of the spectral factor (5) for a given matrix-function (3) is significantly more difficult. Our method does not contain any improvement in the scalar spectral factorization, but uses it to perform the matrix spectral factorization.
4. DESCRIPTION OF THE METHOD First we perform the lower-upper triangular factorization of S(z). S(z) = M (z)(M (z))∗ , where
Theorem A. Let f (z) = g(z)/h(z) where g ∈ Hp1 and h ∈ HpO2 . If the boundary values f (eiθ ) ∈ Lp (T) , then f ∈ Hp .
M = ϕm−1,1 ϕm1
3. FORMULATION OF THE PROBLEM Let
f1,1 (z) f2,1 (z) S(z) = . .. fm,1 (z)
f1,2 (z) f2,2 (z) .. .
··· ··· .. .
fm,2 (z) · · ·
f1,m (z) f2,m (z) , .. .
(3)
(4)
S(z) = χ+ (z)(χ+ (z))∗ (χ+ )T
for a.a. z ∈ T, where (χ ) = is the adjoint of χ+ . A matrix-function χ+ is analytic (i.e. it can be extended in D by ρk z k , |z| < 1,
(5)
where ρk are matrix-coefficients) with entries from H2 (we say that χ+ ∈ H2 in similar situations) and it has an O . outer determinant, det χ+ (z) ∈ H2/m + A spectral factor χ is unique up to a constant right unitary multiplier. With a suitable constraint on χ+ (0) we can ensure that the spectral factor χ+ is unique. In the scalar case, m = 1, a spectral factor can be explicitly written by the formula (see (1), (2)) χ (z) = exp
1 4π
0
2π
+ fm−1 ϕm,m−1
0 0 .. .
. (7) 0 + fm
(8)
(9)
then χ+ = M U, i.e. M U is a spectral factor of S. Proof: We have S = M U ·(M U )∗ since (6) holds and U is unitary. By virtue of (9), M (z)U (z) can be extended to a analytic matrix-function from H2 whose determinant + (z), z ∈ D, will be the outer function f1+ (z)f2+ (z) . . . fm because of Eq. (8) and the uniqueness property of the boundary values of functions from H2/m . We recurrently represent a unitary matrix-function U of Lemma 1 as a product U = U2 U3 . . . Um , where Uk , k = 2, 3, . . . m, are unitary matrix-functions with determinant 1 of a block matrix form 0 Uk (10) Uk = 0 Im−k
k=0
··· ···
M (z)U (z) ∈ L+ 2 (T), + ∗
+
ϕm−1,2 ϕm2
0 0 .. .
We pose the factorization problem in an equivalent form: Given a matrix-function M, find a unitary matrix-function U such that the product M U is a spectral factor of S. Lemma 1: If U (z) is a unitary matrix-function with determinant 1, i.e. if U (z)(U (z))∗ = Im and det U (z) = 1 for a.a. z ∈ T, such that
then it admits the spectral factorization, i.e.
∞
··· ··· .. .
+ det M (z) = f1+ (z)f2+ (z) . . . fm (z), |z| = 1.
z ∈ T, be a positive definite (a.e.) matrix-function with integrable entries, fkj ∈ L1 (T). If S satisfies the PaleyWiener condition
χ+ (z) =
0 f2+ .. .
We take fk+ , k = 1, 2, . . . , m, equal to a scalar spectral factor of the positive function det Sk (z)/ det Sk−1 (z), where S0 (z) = 1 and Sk (z) is the left upper k×k submatrix of S(z). Thus ϕkj ∈ L2 (T), 2 ≤ k ≤ m, 1 ≤ j < k and fk+ ∈ H2O in (7). Note that
fm,m (z)
log det S(z) ∈ L1 (T),
f1+ ϕ21 .. .
(6)
such that the k × k left upper submatrix of the product Mk := M U2 U3 . . . Uk belongs to L+ 2 (T) for each k = 2, 3, . . . m. Taking the last product, k = m, we come to relation (9).
eiθ + z iθ log S(e ) dθ , eiθ − z
- 21 -
In order to compute χ+ = M U2 U3 . . . Um approximately we should be able to approximate a unitary matrix-function (13) for each F of form (11), (12). For this reason, we approximate F in L2 cutting the tails of negative Fourier coefficients of functions ϕj , j = 1, 2, . . . , k − 1, and compute a corresponding unitary matrix-function in the explicit form. Namely, let ∞ (N ) ϕj (z) = n=−N cn (ϕj )z n , j = 1, 2, . . . , k −1, where ∞ ϕ ∼ n=−∞ cn (ϕ)z n , z ∈ T, is the Fourier series expansion of ϕ ∈ L2 (T), and let, for a matrix-function F of form (11), (12), FN be the matrix-function where the (N ) (N ) (N ) last row in F is replaced by (ϕ1 , ϕ2 , . . . , ϕk−1 , f + ). Denote by UN a corresponding unitary matrix-function of form (13) which existence is claimed in Lemma 2, i.e. det UN = 1 and FN UN ∈ L+ 2 (T). Lemma 3: The functions uij , 1 ≤ i, j ≤ k, in the representation of the unitary matrix-function UN by the form (13) are analytic polynomials of order N , i.e.
If we factorize the left upper k×k submatrix of Mk−1 as
(k) (k) (k) µ12 ··· µ1,k−1 0 µ11 (k) (k) (k) µ22 ··· µ2,k−1 0 µ21 .. .. .. .. = .. . . . . . (k) (k) (k) µ µ · · · µ 0 k−1,1 k−1,2 k−1,k−1 ϕ1 ϕ2 ··· ϕk−1 f+ (k) (k) (k) µ12 ··· µ1,k−1 0 µ11 (k) (k) (k) µ22 ··· µ2,k−1 0 µ21 .. .. .. ·F, .. ... . . . . (k) (k) (k) µ µk−1,2 · · · µk−1,k−1 0 k−1,1 0 0 ··· 0 1 (k)
where F is the matrix-function (11) below, µij ∈ L+ 2 (T), i, j = 1, 2, . . . , k − 1, by assumption, ϕj := (k) µkj ∈ L2 (T), j = 1, 2, . . . , k − 1, and f + := fk+ ∈ H2O (see (7)), then it becomes clear that to accomplish our purpose it is important the following Lemma 2: For each k×k matrix-function F of form 1 0 0 ··· 0 0 0 1 0 ··· 0 0 0 0 1 ··· 0 0 F = . (11) .. , .. .. .. .. .. . . . . . 0 0 0 ··· 1 0 ϕ1 ϕ2 ϕ3 · · · ϕk−1 f +
uij (z) =
there exists a unitary matrix-function U of form u12 ··· u1k u11 u21 u22 ··· u2k .. . . .. , .. .. U = . . uk−1,1 uk−1,2 · · · uk−1,k uk1 uk2 ··· ukk uij ∈ L+ ∞ (T), i, j = 1, 2, . . . , k,
n a(ij) n z .
(15)
n=0
Proof: By virtue of Lemma 2 we know that uij ∈ k−1 (N ) + L∞ (T). Since i=1 ϕi uij + f + ukj ∈ L+ 2 (T) and k−1 (N ) N z N i=1 ϕi uij ∈ L+ (T), we have z u = Φj /f + kj 2 + N for some Φj ∈ L2 (T) = H2 . Since z ukj ∈ L∞ (T), by virtue of Theorem A, we can conclude that z N ukj ∈ L+ ∞ (T), j = 1, 2, . . . , k. Thus the representation (15)
is valid for the last row, i = k, in (13). We can then claim that the cofactor of every entry of UN has at most N negative Fourier coefficients, and since co(uij ) = uij , the representation (15) is valid for the entries of the upper rows in (13) as well. Obviously, FN − F L2 → 0 and we can prove that UN → U in measure, which guarantees that FN UN − F U H2 → 0 as N → ∞. Thus χ+ F = F U can be computed approximately. A certain system of algebraic linear equations of or(ij) der N leads to finding the coefficients an in (15). This system is never ill conditioned and always enjoys some nice structure which gives an opportunity to accelerate its solution. An explicit form of the system is a core of the calculation procedure of the proposed matrix spectral factorization algorithm. We do not disclose these equations in the present paper since the algorithm is currently under the intellectual property management in order to be commercially used in the industry. Instead, we completely demonstrate the main idea in the two dimensional case and offer for evaluation a software implementation of the algorithm to the interested reader.
where ϕj ∈ L2 (T), j = 1, 2, . . . , k − 1, and f + ∈ H2O ,
N
(12)
(13)
(14)
with determinant 1 such that F U ∈ L+ 2 (T). Proof: The existence of a unitary matrix-function U ∗ for which χ+ F = F U is a spectral factor of F F follows from the existence of the spectral factorization of F F ∗ . Since det F (z) = f + (z), z ∈ T, and f + is an outer + function, we have det χ+ F (z) = cf (z), z ∈ D, where |c| = 1. Hence det U (z) = c for a.a. z ∈ T and we can assume without lose of generality that c = 1. Since F U does not alter the first k − 1 rows of U and the entries of the last row of U are equal to the conjugates of their cofactors, (14) holds as well. It is clear that for each k = 2, 3, . . . , m the unitary matrix function Uk in (10) is the one whose existence is claimed in Lemma 2 for a matrix-function F , where ϕ1 , ϕ2 , ϕ3 , · · · , ϕk−1 , f + in (11) are the first k nonzero terms in the k-th row of the product M U2 U3 . . . Uk−1 .
5. ILLUSTRATION OF THE MAIN IDEA ON TWO-DIMENSIONAL MATRICES The main idea of the method is contained in following Theorem 1: Suppose 1 0 F (z) = , (16) ϕ(z) f + (z)
- 22 -
z ∈ T, where ϕ ∈ L2 (T), and f + ∈ H2O . If a matrixfunction
α(z) β(z) U (z) = , (17) −β(z) α(z)
Determining B from the first equation of (22), B = G−1 ΓA, and then substituting it into the second equation, we get the following system of algebraic linear equations
z ∈ T, where α, β ∈ L+ ∞ (T), is such that
where R = G−1 Γ · G−1 Γ + IN +1 . It is easy to check that G−1 Γ is symmetric. Thus R is positive definite with all eigenvalues more than or equal to 1, so that the system (23) is solvable. Furthermore, R always has a displacement structure (see [3], p. 808). Namely, R − ZRZ ∗ has rank 2, where Z is the upper triangular (N + 1)×(N + 1) matrix with ones on the first subdiagonal and zeroes elsewhere. This further accelerates a solution process of (23). Whenever we find the coefficients a0 , a1 , . . . , aN from (23), and then b0 , b1 , . . . , bN , we can normalize them so that to get the matrix-function (17), where α(z) and β(z) are defined from (21), unitary and with determinant 1. The proof of a convergence of the algorithm as N → ∞ see in [2].
F (z)U (z) ∈ L+ 2 (T),
R · A = l0−1 · E,
(18)
then U is unitary with determinant 1 times some positive constant. Proof: We have to show that |α(z)|2 + |β(z)|2 = C a.e. on T.
(19)
It follows from (16)-(18) that ϕ(z)α(z) − f + (z)β(z) =: Ψ1 (z) ∈ L+ 2 (T), + ϕ(z)β(z) + f (z)α(z) =: Ψ2 (z) ∈ L+ 2 (T).
(20)
Hence f + (|α|2 + |β|2 ) = Ψ2 α − Ψ1 β ∈ L+ 2 (T). Therefore |α(z)|2 + |β(z)|2 = Φ(z)/f + (z) for some Φ ∈ 2 2 L+ 2 (T) = H2 . Since |α(z)| + |β(z)| ∈ L∞ (T), we can use Theorem A to conclude that |α(z)|2 + |β(z)|2 ∈ L+ ∞ (T) and consequently (19) follows since the function is positive. Theorem 1 suggests that we need only to care about the condition (20) to be fulfilled. The matrix (17) will be automatically unitary (after normalization). This simplifies the process of explicit ∞ construction of U (z) whenever ϕ(z) = ϕ(N ) (z) = n=−N cn (ϕ)z n in (16). Namely, α(z) =
N n=0
an z n and β(z) =
N n=0
bn z n
6. CONCLUSIONS We present an absolutely new algorithm of matrix spectral factorization which actually reduces the problem of m×m matrix factorization to a LU triangular factorization of the matrix, to m times scalar spectral factorization, and to the solution of m − 1 linear systems of equations which may be of high orders, in order to achieve a good accuracy, but always have a positive definite coefficient matrices with displacement structure. All these three consistent components are extremely well developed to compare with original problem.
(21)
7. ACKNOWLEDGEMENT
are polynomials of order N in this case according to Lemma 3, and equating N negative coefficients of functions Ψ1 and Ψ2 in (20) to 0 (to avoid the trivial solution we take c0 (Ψ2 ) = 1), we lead to a system of linear algebraic equations with respect to coefficients an and bn . For simplicity, we use the matrix notation Γ · A − G · B = O, (22) Γ · B + G · A = E,
We are grateful to IT specialist G. Modebadze for creating a software implementation of our algorithm.
REFERENCES [1] L. Ephremidze, G. Janashia and E. Lagvilava, “A new computational algorithm of spectral factorization for polynomial matrix-functions”, Proc. A. Razmadze Math. Inst., Vol. 136, pp. 41-46, 2004. [2] G. Janashia and E. Lagvilava, “A method of approximate factorization of positive definite matrix functions”, Studia Mathematica, Vol. 137, No. 1, pp. 93100, 1999. [3] T. Kailath, A. H. Sayed and B. Hassibi, Linear Estimation, Prentice Hall, New Jersey, 2000. [4] P. Koosis, Introduction to Hp spaces, Cambridge University Press, 1980. [5] A. H. Sayed and T. Kailath, “A survey of Spectral Factorization Methods”, Numer. Linear Algebra Appl., Vol. 8, pp. 467-496, 2001. [6] N. Wiener and P. Masani, “The prediction theory of multivariate stochastic processes”, I, Acta Math. Vol. 98, pp. 111-150, 1957, II, Acta Math. Vol. 99, pp. 93-137, 1958.
where (γn = c−n (ϕN ) and ln = cn (f + ) below) γ0 γ1 γ2 · · · γN −1 γN γ1 γ2 γ3 · · · γN 0 γ2 γ3 γ4 · · · 0 0 Γ= , .. .. .. .. .. .. . . . . . . 0 0 γN 0 0 · · · lN l0 l1 l2 · · · lN −1 0 l0 l1 · · · lN −2 lN −1 G = 0 0 l0 · · · lN −3 lN −2 , .. .. .. .. .. .. . . . . . . 0 0 0 ··· 0 l0 A = (a0 , a1 , a2 , . . . , aN )T ,
(23)
B = (b0 , b1 , b2 , . . . , bN )T ,
O = (0, 0, 0, . . . , 0)T and E = (1, 0, 0, . . . , 0)T .
- 23 -
