ON THE BOUNDEDNESS BEHAVIOR OF THE SPECTRAL FACTORIZATION IN THE WIENER ALGEBRA FOR FIR DATA

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1 ON THE BOUNDEDNESS BEHAVIOR OF THE SPECTRAL FACTORIZATION IN THE WIENER ALGEBRA FOR FIR DATA Holger Boche and Volker Pohl Technische Universität Berlin, Heinrich Hertz Chair for Mobile Communications Werner-von-Siemens Bau, Einsteinufer 25, 0587 Berlin, Germany phone: , fax: , {holgerboche, ABSTRACT It is known that the spectral factorization mapping is unbounded in the Wiener algebra, in general However in applications, the given data are often polynomials For such finite dimensional spectra, the spectral factorization mapping is bounded, of course, but the boundedness constant depends on the degree of the given polynomial spectra This paper presents lower and upper bounds for this boundedness constant depending on the degree N of the given data INTRODUCTION Spectral factorization is the process by which a real valued function f given on the unit circle T = {z C : z = } is written as a product f = f + f in which the spectral factor f + (z) is an analytic function for all z inside the unit disk D = {z C : z < } and without any zero in D The second spectral factor f is analytic and without any zero for all z > This spectral factorization operation arises in many different areas of communications, signal processing, control or information theory, eg in interpolation and extrapolation of stochastic processes [], channel equalization [2, 3, 4], in spectral estimation [5], or in H and quadratic optimal control [6, 7] The boundedness behavior of the spectral factorization mapping S : f f + is of fundamental importance in practical applications since it gives an upper bound on the norm of the spectral factor f + depending on the norm of the given spectrum f If the spectral factor is related to the transfer function of a certain filter, like in many signal processing or communication applications, the norm of f + is linked to the stability behavior of this filter with respect to disturbances in its input data In [8], Jacob and Partington studied the continuity and boundedness of the spectral factorization for so called decomposing Banach algebras They found that the spectral factorization mapping is continuous but unbounded on these algebras (at least for all the practical important algebras) The most prominent example of these decomposing Banach algebra is the Wiener algebra, ie the space of all absolutely convergent Fourier series This algebra is also of considerable significance in system theory, since it can be identified with all bounded-input bounded-output (BIBO) stable transfer functions of linear systems In applications, it is often assumed that the given spectra are polynomials, or the nonpolynomial spectra are approximated by a polynomial since for such spectra there exist several efficient algebraic methods for the spectral factorization [9] Therefore, this paper investigates the boundedness behavior of the spectral factorization mapping S for all trigonometric polynomials of a certain degree N in the Wiener algebra Since the set of all polynomials with a degree of at most N is finite dimensional, S will be bounded on the space of this polynomials However, the boundedness behavior will depend strongly on the degree N of the polynomials This paper derives a lower and an upper bound on boundedness of the spectral factor depending on the degree N They will show that the BIBO norm of the spectral factor grows at least proportional to N /4 and at most proportional with N For N this shows, in accordance with [8] that the spectral factorization mapping is unbounded on the Wiener algebra, in general 2 PRELIMINARY RESULTS This section recalls shortly the notations of the spectral factorization and the Wiener algebra as far as it will be needed in this paper Moreover, it states some known results which are needed subsequently 2 Notations & Wiener algebra As usual, the imaginary unit is denoted by j = and z is the conjugate complex of z The set C (T) of all continuous functions on T is a Banach space under the usual supremum norm f = sup ζ T f (ζ ) The Banach space of p- integrable functions ( < p < ) on T with the usual norm f p := ( T f (ζ ) p dζ ) /p is denoted by L p (T) Moreover, L (T) denotes the set of all measurable and essentially bounded functions on T For any function f L (T), the Fourier coefficients are defined by ˆf (k) = f (e jτ )e jkτ dτ, k = 0,±,±2, () wherein the integration goes from π to π (If not stated otherwise, all integrals in this paper are taken over this interval) Every real function f L p with p > can be written as f (θ) = a N k= a k cos(kθ) + b k sin(kθ) (2) with the real Fourier coefficients a k = π f (θ)cos(kθ)dθ, bk = π f (θ)sin(kθ)dθ (3) and where the degree N may become infinity Of particular interest will be the set P N C (T) of all trigonometric polynomials (2) with a degree of at most N By W, we denote the set of all functions of the form f (e jω ) = ˆf (k)e jkω with f W := ˆf (k) < (4) and for ω [ π,π) Moreover, W + denotes the set of all f W for which ˆf (k) = 0 for all k < 0 Both, W and W EURASIP 76

2 are commutative Banach algebras with unity under pointwise multiplication W is known as the Wiener algebra Note that every function f W + of the form (4) can be identified with a function f (z) = ˆf (k)z k which is analytic for all z D The next property of W is easily verified (see eg [0]): PROPOSITION : The Wiener algebra W is continuously embedded in C (T) with f f W for all f W The Wiener algebra plays an important role is system theory: Assume that L is a linear discrete time system with impulse response { ˆf (k)} k= and with the transfer function f (e jω ) = ˆf k= (k)e jkω Let {x(k)} k= be an input sequence of L, then the output sequence of L will be given by y(k) = n= ˆf (n)x(k n) The sequence x = {x(k)} is called bounded if x l := sup k x(k) <, and the linear system L is said to be bounded-input bounded-output (BIBO) stable if every bounded input sequence x yields a bounded output sequence y = {y(k)}, ie if L x l L BIBO := sup < x l x l It is well known, that this BIBO-norm of L is equal to the Wiener norm (4) of its transfer function, ie it holds that L BIBO = f W Thus, the Wiener norm of the transfer function f of a linear system L characterizes the maximal amplification of the input signal (and in particular of additional disturbances of the input signal) in the l -norm Thus, it characterizes the BIBO stability of L 22 Spectral factorization in the Wiener algebra DEFINITION : An element f W possesses a spectral factorization in W if there exist functions f + W + with f (z) 0 for all z D and f (z) = f + (/z) such that f (e jω ) = f + (e jω ) f (e jω ) for all ω [ π,π) The functions f +, f are called the spectral factors of f and every f W which possesses a spectral factorization is called a spectral density Since f + W +, it is the Fourier transform of a causal sequence { ˆf + (k)} whereas f corresponds to an anti-causal sequence { ˆf (k)} 0 k= Which elements of W are spectral densities and how can the spectral factor be determined? The answer is well known and given by the following proposition PROPOSITION 2: An element f W is a spectral density if and only if it satisfies the Paley-Wiener condition log f (e jω )dω > (5) Moreover, one spectral factor is given by f + (z) = (S f )(z) = exp( ) 4π log f (e jτ ) e jτ +z e jτ z dτ (6) This spectral factor is unique up to a constant of modulus In the following, we denote by W PW the set of all spectral densities in W, ie the set of all f W which satisfy the Paley-Wiener condition (5) 3 BOUNDEDNESS BEHAVIOR OF THE SPECTRAL FACTORIZATION MAPPING This section investigates the boundedness of the spectral factorization mapping (6) on the Wiener algebra W DEFINITION 2: The spectral factorization S : W W + is said to be bounded if there exists a constant C 0 < such that for all f W PW with f W always holds f + W = S f W C 0 (7) It is easily verified (see eg [8]) that condition (7) for the boundedness of the spectral factorization is equivalent to S f 2 W C f W for all f W PW (8) with the boundedness constant C = C0 2 It was shown in [8] that the spectral factorization is unbounded on W Thus, there exists no constant C such that (8) holds However in applications, the given spectra f are often polynomials of finite degree N and since P N is a finite dimensional space, the spectral factorization will be bounded for all f P N Thus there exists a constant C such that (8) holds for all f P N However, the constant C = C(N) will depend strongly on the degree N of the given spectra In this paper, we derive a lower and an upper bound on the constant C(N) in (8) depending on N These bounds will show that C(N) grows at least proportional to N and at most proportional to the degree N of the spectra Thus, we study the boundedness constant C(N) in (8) which is obviously defined by C(N) := 3 Upper bound sup f W PW P N f W = S f 2 W (9) First, we derive an upper bound on the boundedness constant C = C(N) in (8) Such an upper bound is easily obtained, as the following first lemma shows LEMMA 3: Consider the spectral factorization on W for all trigonometric polynomials P N Then, for the boundedness constant C(N) in (8) holds that C(N) < N + Proof: Let f P N W PW be an arbitrary spectral density of degree N and let f + = S f the corresponding spectral factor By Cauchy-Schwarz inequality holds that f + W = N ˆ f + (k) ( N ) /2 ( N ) /2 f ˆ + (k) 2 (0) = N + f + 2 N + f + Since f = f + f +, it is clear that f (ζ ) = f + (ζ ) 2 for almost all ζ T and consequently that f + = f /2 Using also Proposition, one obtains that f + 2 W (N + ) f W which shows that C(N) N + It remains to show that equality can not hold In (0) equality holds only for constant spectral factors f +, ie for all constant spectra f However, for every constant spectrum f holds that f + 2 W = f W and therefore f + 2 W < (N + ) f W for all N EURASIP 77

3 Next we derive a lower bound on C(N) which will be much more demanding than the upper bound 32 Lower bound This section derives a lower bound on the boundedness constant C(N) in (8) for all spectra f P N To this end, we consider a special polynomial g N W + of degree N given by g N (e jω ) = N+ N e j k2 π N+ e jkω, ω [ π,π) () With the function g N, we define the trigonometric polynomial f N P N by f N (e jω ) := g N (e jω )g N (e jω ) = N k= N ˆf N (k)e jkω (2) Since ĝ N (k) = N+ exp( j k2 π N+ ) with k = 0,,,N are the Fourier coefficients of g N, a straight forward calculation shows that the Fourier coefficients of f N are given by ˆf N (k) = = N k ĝ N (l)ĝ N (l + k) l=0 πk2 exp( j N+ ) exp( j k2 N+ )exp( jk) (N + ) 2 exp( j k N+ ) and by ˆf N ( k) = ˆf N (k) for all k = 0,,2,,N The next lemma summarizes some properties of the functions g N and f N, which will be needed later LEMMA 4: For the functions g N and f N as defined by () and (2), respectively, holds that g N W = and f N W < 3+2log([N+]/2) N+ Proof: The first statement is obvious The norm of f N is given by f N W = N+ +2 N k= ˆf N (k) and for the modulus the Fourier coefficients holds ( ) sin π k(n+) k2 N+ ˆf N (k) = (N + ) 2 sin( N+ πk ) (N + ) 2 sin( N+ πk ) Now, let L N be the largest integer for which L N < (N + )/2 and note that sin(πx) 2x for all x [0,/2] Therewith, one obtains the upper bound f N W N (N + ) 2 ( ) L N k N+ k=2 L N k= N + 2k 3+2logL N N+ for the norm of f N, which is equivalent to the statement of the lemma since L N < (N + )/2 Assume that g N is a spectral factor of f N, then both bounds of Lemma 4 together with (8) would give the desired lower bound for constant C = C(N) in (8): C(N) N+ 3+2log([N+]/2) Figure : The modulus of the function g N for different degrees N However, even though f N = g N g N, the function g N is not a spectral factor, because g N has zeros inside the unit disk D Nevertheless, we use g N to define the two functions ϕ N := g N + g N and ψ N := ϕ N ϕ N (3) By this definition, it is clear that ϕ N W + has no zeros inside D and therefore ϕ N is a spectral factor of the trigonometric polynomial ψ N P N W PW Similar, as Lemma 4, Lemma 7 below will present bounds on the Wiener norm of ϕ N and ψ N Based on these bounds, we will be able to derive the desired lower bound on the constant C(N) in (8) However, since the definition of ϕ N is based on the infinity norm of g N, we will need an upper bound for g N This bound is given by the following lemma LEMMA 5: Let g N W + be the function defined by (), then there exists a constant C such that for all N N g N C N+ Note, that the polynomials g N have a very special property: By Lemma 4, the functions g N have a constant Wiener norm g N W = independent on the degree N of the polynomial By Lemma 5, on the other hand, the peak value of these polynomials is upper bounded by C / N + and decreases with increasing degree N of the polynomial (cf Fig ) Thus, for N, the peak value of g N tends to zero whereas the BIBO norm remains fixed equal to one To prove Lemma 5, one has to find an upper bound on the modulus of an exponential sum of the form S(q ω ;a,b) = b k=a exp( j q ω(k)) wherein in our case, the particular function q ω (k) is given by q ω (k) = k2 2(N+) + kω and the bounds of the summation index are a = 0 and b = N Here, ω [ π,π] is just a fixed parameter and one has to find an upper bound on S(q ω ;a,b) for all parameters in this 2007 EURASIP 78

4 interval It turns out that it is sometimes simpler to get an upper bound on the corresponding integral I(q ω ;a,b) = b a exp( j q ω(k))dk and by a famous result of van der Corput, it is possible to control the difference D(q ω ;a,b) := I(q ω ;a,b) S(q ω ;a,b) between the sum and the integral, under some conditions on the function q ω : LEMMA 6: (van der Corput) If q ω (k) is monotone and if q ω(k) ε for some ε > 0 and for all k [a,b], then there exist constants C 2 = 4/π and C 3 = + 4/π such that D(q ω ;a,b) C 2 ε +C 3 (4) This lemma due to van der Corput is taken from [, Chapter V, Lemma 44] were also a proof can be found By means of this van der Corput lemma, we shortly sketch the proof of Lemma 5 Sketch of proof (Lemma 5): We consider the functions q ω, S(q ω,a,b), and I(q ω,a,b) as defined above Therewith, the function g N given by () can be written as g N (e jω ) = N+ S(q ω,0,n) (5) To apply Lemma 6, one has to check whether q ω satisfies the conditions of Lemma 6 Since q ω(k) = N+ k + ω, one sees that q ω(k) may become larger than for certain parameters ω [ π,π) and certain k [0,N] However, since the polynomials g N are -periodic, one can consider the sum S(q ω ;0,N) and the integral I(q ω ;0,N) equivalently on other intervals for the parameter ω Therefore, we consider ω on the interval T = [,0) and split up this interval into three parts: T = [, + /(N + ) /4 ), T 2 = [ + /(N + ) /4, /(N + ) /4 ), and T 3 = [ /(N + ) /4,0) As an example, we consider here only the interval T 2 If ω T 2, one obtains for the modulus of the first derivative of q ω (k) with k [0,N] that q ω(k) (N+) /4 Therefore, one can apply Lemma 6 with ε = (N + ) /4 which gives S(q ω ;0,N) I(q ω ;0,N) +C 2 (N + ) /4 +C 3 It remains to find an upper bound for the integral I(q ω ;0,N) An elaborate but not complicated calculation shows that I(q ω ;0,N) N + +O((N +) /4 ) Together with (5), one obtains therefore that g N (e jω ) N+ + O((N + ) 3/4 ), ω T 2 (6) A similar analysis has to be done for ω T and ω T 3 but because of the limited space this is not done here However, also for these cases, one obtains a result similar to (6) but with other constants Therewith, the statement of Lemma 5 is proved Now, we have all ingredients to derive upper bounds on the W -norms of the trigonometric polynomial ψ N and its spectral factor ϕ N defined in (3) LEMMA 7: Let ϕ N W + and ψ N P N be the two functions defined in (3), then there exist two constants C 4 and such that holds C 4 N+ ϕ N W + C 4 N+ and ψ N W N+ Proof: The first statement is a simple consequence of Lemma 5 By the definition (3) of ϕ N one has ϕ N W g N W + g N + C N+ and ϕ N W g N W g N C N+ To prove the second statement, note that by the definition of ψ N one has ψ N = g N g N +(g N +g N ) g N + g N 2 Applying Lemma 4 and 5 one obtains therefore the upper bound ψ N W 3+2log(N/2) N+ + 2C N+ + C2 N+ With these preparations, we are able now to state and to prove our main result on a lower and an upper bound of the boundedness constant C(N) of the spectral factorization THEOREM 8: Consider the spectral factorization on W for all trigonometric polynomials P N with a degree of at most N Then, there exists a constant C 6 such that for the boundedness constant C(N) in (8) holds that C 6 N + C(N) < N + (7) Proof: The upper bound was already derived in Lemma 3 To prove the lower bound, we consider the trigonometric polynomial ψ N W P N and its spectral factor ϕ N = Sψ N W +, both defined in (3) Then, by (8) and by the boundedness of the spectral factorization on P N there exists a constant C(N) such that ϕ N 2 W C(N) ψ N W If one applies the lower and and upper bound of ϕ N and ψ N as given by Lemma 7, respectively, one obtains the lower bound C(N) N+ 2 C 4 + C2 4 N+ for the boundedness constant which is equivalent to (7) with an appropriately chosen constant C 6 Note the following remarkable property of the function ϕ N which is essential for all consideration in this paper: For sufficiently large degree N, the norm ϕ N W of ϕ N is almost equal to whereas the norm of the product ψ N = ϕ N ϕ N becomes arbitrarily small as N The function g N showed already the same behavior, but g N was not minimum phase Therefore, we constructed ϕ N (3) by adding g N to g N The proof of Lemma 7 shows that the convergence of ϕ N ϕ N W as N is mainly determined by the convergence behavior of g N (cf Lemma 5) which is slower than the convergence behavior of g N g N W In the subsequent concluding section, we discuss some consequences and implications of Theorem EURASIP 79

5 4 CONSEQUENCES AND DISCUSSIONS Theorem 8 gives lower and upper bounds on the boundedness constant of the spectral factorization mapping for trigonometric polynomials with a finite degree N in the Wiener algebra W Given an f P N with f W =, the upper bound of Theorem 8 shows that the BIBO norm of the spectral factor f + becomes never larger than S f W N + On the other hand, the lower bound of Theorem 8 shows that for every degree N, there exist trigonometric polynomials f P N for which the norm of the spectral factor becomes larger than C6 (N + ) /4 Both, upper and lower bound tend to infinity as N, which shows in particular that the spectral factorization is unbounded on W : COROLLARY 9: The spectral factorization is not bounded on the Wiener algebra W Proof: By the definition (9) of the boundedness constant C(N) holds that to every ε > 0 there exists an f W PW P N with f W = and such that S f 2 W C(N) ε C 6 N + ε using the lower bound of Theorem 8 Therefore, to every constant C 0 < there exists an N < and an f W PW with f W = such that S f 2 W C 0 This shows that S is unbounded on W Thus, the spectral factorization mapping S is unbounded on the Wiener algebra W This means that for certain functions f W PW the norm of the corresponding spectral factor may become arbitrarily large Next, we want to discuss the influence of the minimum value of f (e jω ), ω [ π,π] on the norm f + W of the spectral factor To this end, we consider for a constant c 0 > 0 the set M (c 0 ) := { f W : f W = ; f (e jω ) > c 0, ω [ π,π)} For the determination of the spectral factor by (6), the logarithm of the given spectrum f has to be determined Therefore, the next lemma investigates first the Wiener norm of log f for f M (c 0 ) depending on the minimum c 0 LEMMA 0: ) Assume that c 0 > /2 then it holds that log f W log( 2c 0 ) for all f M (c 0) 2) To every c 0 < /4 and every M > 0 there exists an f M (c 0 ) such that log f W > M Thus, if the minimum of the given spectra is sufficiently large (> /2), the norm of the logarithm is always bounded as given by the lemma If on the other hand, the minimum is smaller than /4, the norm of the logarithm may become arbitrarily large, ie the operation of taking the logarithm is unbounded on M (c 0 ) with c 0 < /4 For the range /4 c 0 /2 no definite statement can be made on the norm of the logarithm Of course, this behavior of the logarithm has consequences for the spectral factorization on W as given in the next theorem THEOREM : Let c 0 > /2 and f M (c 0 ) arbitrary then it holds that f + W = S f W 2c0 Sketch of proof: By the definition of the spectral factorization holds f = f + f Taking the logarithm gives log f = log f + + log f Since c 0 > 0 [2] log f W and one can write log f as a Fourier series: (log f )(e jω ) = k= a k e jkω Define g + (e jω ) = a 0 /2 + k= a k e jω and g (e jω ) = a 0 /2 + k= a k e jω Since log f is real valued, a k = a k and consequently logg + W = 2 log f W Moreover, it is clear that f + = expg + such that, using Lemma 0, one has f + W = expg + W exp g + W This is what the theorem claims 2c0 Even though the spectral factorization is unbounded on W, one can find subsets of functions in W such that the spectral factorization is bounded for all functions in these subsets Apart form the subsets W P N which were investigated in this paper, Theorem gives a characterization of such subsets of W in terms of the minimum value of the functions REFERENCES [] N Wiener, Extrapolation, Interpolation, and Smoothing of Stationary Time Series with Engineering Applications The MIT Press and John Wiley & Sons, Inc, 949 [2] N Al-Dhahir and J Cioffi, MMSE Decision-Feedback Equalizers: Finite-Length Results, IEEE Trans Inform Theory, vol 4, no 4, pp , July 995 [3] A T Erdogan, B Hassibi, and T Kailath, MIMO Decision Feedback Equalization from an H Perspective, IEEE Trans Signal Processing, vol 52, no 3, pp , 2004 [4] H Boche and V Pohl, Structural Properties of the Wiener Filter Stability, Smoothness Properties, and FIR Approximation Behavior, IEEE Trans Inform Theory, vol 52, no 9, pp , Sept 2006 [5] C I Byrnes, T T Georgiou, and A Lindquist, A New Approach to Spectral Estimation: A Tunable High- Resolution Spectral Estimator, IEEE Trans Signal Processing, vol 48, no, pp , Nov 2000 [6] B Francis, A Course in H Control Theory New York: Springer Verlag, 987 [7] O J Staffans, Quadratic optimal control of stable systems through spectral factorization, Math Control Signals Systems, vol 8, pp 67 97, 995 [8] B Jacob and J R Partington, On the Boundedness and Continuity of the Spectral Factorization Mapping, SIAM J Control Optim, vol 40, no, pp 88 06, 200 [9] A H Sayed and T Kailath, A Survey of Spectral Factorization Methods, Numer Linear Algebra Appl, vol 8, pp , 200 [0] V V Peller and S V Khrushchev, Hankel operators, best approximations, and stationary Gaussian processes, Russian Math Surveys, vol 37, no, pp 6 44, 982 [] A Zygmund, Trigonometric Series Cambridge: University Press, 968 [2] N Wiener, Tauberian Theorems, Ann of Math, vol 33, pp 00, EURASIP 720