Zero-Forcing Precoding for Frequency Selective MIMO Channels with $H^infty$ Criterion and Causality Constraint
We consider zero-forcing equalization of frequency selective MIMO channels by causal and linear time-invariant precoders in the presence of intersymbol interference. Our motivation is twofold. First, we are concerned with the optimal performance of c…
Authors: S, er Wahls, Holger Boche
Zero-F oring Preo ding for F requeny Seletiv e MIMO Channels with H ∞ Criterion and Causalit y Constrain t ⋆ Sander W ahls ∗ Holger Bo he T e hnishe Universität Berlin, Heinrih-Hertz-L ehrstuhl für Mobilkommunikation, W erner-von-Siemens-Bau (HFT6), Einsteinufer 25, 10587 Berlin, Germany V olk er P ohl T e hnion - Isr ael Institute of T e hnolo gy, Dep artment of Ele tri al Engine ering, Haifa 32000, Isr ael Abstrat W e onsider zero-foring equalization of frequeny seletiv e MIMO hannels b y ausal and linear time-in v arian t preo ders in the presene of in tersym b ol in terfer- ene. Our motiv ation is t w ofold. First, w e are onerned with the optimal p erfor- mane of ausal preo ders from a w orst ase p oin t of view. Therefore w e onstrut an optimal ausal preo der, whereas on trary to other w orks our onstrution is not limited to nite or rational impulse resp onses. Moreo v er w e deriv e a no v el n umerial approa h to omputation of the optimal p erfomane index a hiev able b y ausal pre- o ders for giv en hannels. This quan tit y is imp ortan t in the n umerial determination of optimal preo ders. Key wor ds: MIMO, In tersym b ol In terferene, Filterbank, Preo der, Equalizer, Causalit y, Bezout Iden tit y, Matrix Corona Problem, Minim um Norm ⋆ This w ork w as supp orted b y the German Resear h F oundation (DF G) under gran ts BO 1734/11-1 and BO 1734/11-2 ∗ Corresp onding author. F on: +49 30 314 28559. F ax: +49 30 314 28320. Email addr esses: sander.wahlsmk.tu-berlin.de (Sander W ahls), holger.bohemk.tu-berlin.de (Holger Bo he), pohlee.tehnion.a.il (V olk er P ohl). Preprin t submitted to Elsevier April 26, 2022 1 In tro dution Man y of to da ys state-of-the-art wireless systems adopt m ultiple-input m ultiple-output (MIMO) transmission to inrease sp etral eieny together with m ulti-arrier metho ds to op e with in tersym b ol in terferene (ISI). Multi- arrier metho ds simplify hannel equalization b eause they deomp ose fre- queny seletiv e hannels in to m ultiple at fading hannels (the so alled ar- riers), whi h an b e easily equalized. While m ulti-arrier transmission oers man y adv an tages inluding eetiv e hannel equalization, it also exhibits some dra wba ks regarding the p eak-to-a v erage p o w er ratio (P APR). Often single- arrier transmission, where the frequeny seletiv e hannel is approa hed di- retly , is onsidered as an alternativ e to m ulti-arrier transmission [1,2,3℄. While therefore single-arrier transmission is in teresting on its o wn, it has b een further sho wn in [4,5℄, that in fat man y ommon m ulti-arrier, o de-m ultiplex and spae-time blo k-o de systems an b e mo deled as single-arrier systems b y virtual enhanemen t of the MIMO system. V arious authors used this ap- proa h to deriv e new equalization metho ds based on single-arrier equalization in order to exploit join t equalization of spatial, time and o de or frequeny domains [4,5,6,7 ℄. There, and generally for linear time-in v arian t (L TI) equal- ization of single-arrier systems with zero-foring and ausalit y onstrain t, one usually solv es the so-alled Bezout Identity H ( e iθ ) G ( e iθ ) = I (0 ≤ θ < 2 π ) , where the matrix-v alued transfer funtion H of a stable and ausal L TI system (the frequeny seletiv e MIMO hannel) is giv en, and a transfer funtion G of a stable and ausal L TI preo der, whi h equalizes H , has to b e omputed [8℄. T ransmitters ma y use su h G to pre-equalize the hannel. Alternativ ely , reeiv ers an also solv e the Bezout Iden tit y for the transp osed hannel H T (i.e. H T G = I ) and equalize the hannel H with the transp osed solution G T . The main diult y in solving the Bezout Iden tit y is the ausalit y of G , b eause the naiv e approa h G ( e iθ ) = H ( e iθ ) ∗ [ H ( e iθ ) H ( e iθ ) ∗ ] − 1 (0 ≤ θ < 2 π ) of a pseudoinverse generally results in a non-ausal preo der [9℄. If the n um b er of hannel inputs equals the n um b er of hannel outputs, the pseudoin v erse is the unique solution to the Bezout Iden tit y . The situation hanges if the n um- b er of inputs of H is larger than the n um b er of outputs. No w preo ders for H no longer ha v e to b e unique. Usually this non-uniqueness then is exploited to ho ose a ausal G that is optimal in some sense. The t w o ommon optimal- it y onditions are minimality of the e qualizers ener gy and minimality of the e qualizers p e ak value , resp etiv ely . The minimal energy ondition orresp onds to the lassial approa h of signal-to-noise ratio (SNR) maximization [5,7,10℄. Ho w ev er, this approa h is only feasible if the statistial prop erties of the noise 2 are kno wn. F or unkno wn noise statistis, it annot b e applied. Pi king up an idea from robust on trol (see e.g. [11℄), where one is onerned with unpre- ditable errors that arise e.g. due to unertain mo deling, instead minimization of the equalizers p eak v alue has b een prop osed [6℄. As w e shall see later, this minimizes the w orst ase error instead of the a v erage error, whi h annot b e determined due to the unkno wn noise statistis. In this pap er w e are in terested in the optimal p erformane that ausal pre- o ders and equalizers an ar hiv e regarding the w orst ase error. Therefore w e sho w ho w a solution to the Bezout Iden tit y with minimal p eak v alue an b e onstruted. W e disuss wh y this giv es the b est upp er b ounds on v arious p erturbations in the system. Con trary to other w a ys to solv e the Bezout Iden- tit y , our onstrution holds for the most general ase of systems with innite impulse resp onses (whi h are not required to b e rational) and ev en innite input and output v etors, i.e. w e allo w systems to ha v e innite temp oral as w ell as innite spatial dimension. W e further giv e a new result on the n umer- ial omputation of the minim um p eak v alue a hiev able b y ausal solutions to the Bezout Iden tit y if the n um b ers of inputs and outputs are nite. This is imp ortan t b eause for all metho ds kno wn to the authors that solv e the Be- zout Iden tit y with minimal p eak v alue in a n umerially exploitable w a y , the minimal p eak v alue has to b e kno wn in adv ane [ 6 ,12℄. Therefore eien t om- putation of the minimal p eak v alue is imp ortan t for n umerial solution of the Bezout Iden tit y . W e p oin t out that the optimization approa h in [13℄ requires no prior kno wledge of the minimal p eak v alue. Ho w ev er, it only omputes nite impulse resp onse solutions to the Bezout Iden tit y , whi h are generally sub optimal. W e pro eed as follo ws. In Setion 2 w e giv e our problem statemen t after w e ha v e in tro dued some notation and neessary basi mathematial onepts. W e further disuss the pratial in terpretation of our problem statemen t. In Setion 3 w e deriv e our results on the n umerial omputation of the minimal p eak v alue a hiev able b y ausal solutions to the Bezout Iden tit y . Then a op- timal ausal preo der is onstruted in Setion 4. W e nally dra w onlusions in Setion 5. 2 Preliminaries 2.1 Notation W e denote the omplex n um b ers b y C , the omplex matries with m ro ws and n olumns b y C m × n and the omplex olumn v etors b y C m := C m × 1 . The omplex unit dis is giv en as D := { z ∈ C : | z | < 1 } , its b order is the unit irle 3 T := { z ∈ C : | z | = 1 } . Complex onjugation is denoted b y ¯ ( · ) , taking adjoin ts in a Hilb ert spae b y ( · ) ∗ . F urthermore H , E and E ∗ denote separable Hilb ert spaes with salar pro duts h· , ·i H , h· , ·i E and h· , ·i E ∗ , resp etiv ely . By H ⊕ E w e mean the diret Hilb ert sum, i.e. the spae H × E equipp ed with salar pro dut h h ⊕ e, g ⊕ f i H⊕E := h h, g i H + h e, f i E . The spae of b ounded linear op erators b et w een E and E ∗ is denoted b y L ( E , E ∗ ) . It is equipp ed with the op erator norm k T k op := sup e ∈E , k e k E =1 k T e k E ∗ . On an y spae the iden tit y op erator is written as I . F or matries A ∈ C m × n the smallest and largest singular v alue will b e denoted b y σ min ( A ) and σ max ( A ) , resp etiv ely . The losure of a set M is denoted b y closure M , the spae spanned b y all linear om binations of its elemen ts b y span M . As usual, L p T ( X ) denotes the spae of (equiv alene lasses of ) p -in tegrable funtions on T with v alues in a Bana h spae X . The norm in L p T ( X ) is k f k p p := R 2 π θ = 0 k f ( e iθ ) k p X dθ 2 π for 1 ≤ p < ∞ and k f k ∞ := esssup ζ ∈ T k f ( ζ ) k X = inf { m > 0 : µ ( { ζ ∈ T : k f ( ζ ) k X > m } ) = 0 } for p = ∞ , where µ denotes the Leb esgue measure. W e refer to [14, Setion 3.11℄ and the referenes therein for details on in tegration of v etor- and op erator-v alued funtions. If p = 2 , L 2 T ( E ) equipp ed with the salar pro dut h f , g i 2 := R 2 π θ = 0 h f ( e iθ ) , g ( e iθ ) i E dθ 2 π is a Hilb ert spae. F or F ∈ L ∞ T ( L ( E , E ∗ )) w e denote the p oin t-wise adjoin t b y F ∗ , i.e. F ∗ ( ζ ) = ( F ( ζ )) ∗ almost ev erywhere on the unit irle. 2.2 Basi R esults and Con epts 2.2.1 Har dy Sp a es and T o eplitz Op er ators W e in tro due the usual Har dy sp a es on the dis b y H 2 D ( E ) := ( u : D → E : u analyti , k u k 2 2 := sup 0 0 . Then some G ∈ H ∞ ( E ∗ , E ) with k G k ∞ ≤ δ − 1 and F ( z ) G ( z ) = I for al l z ∈ D exists if and only if k T F ∗ u k 2 ≥ δ k u k 2 for al l u ∈ H 2 ( E ∗ ) . 2.2.2 Shur Class F untions in the unit ball of H ∞ ( E , E ∗ ) , the so-alled Shur lass S ( E , E ∗ ) := { F ∈ H ∞ ( E , E ∗ ) : k F k ∞ ≤ 1 } , 5 ha v e some sp eial prop erties, whi h will turn out to b e useful in the onstru- tion of a minim um norm righ t in v erse. Ev ery S h ur funtion an b e fatorized as follo ws. Theorem 2 ([17, Th. 2.1℄) L et F : D → L ( E , E ∗ ) . Then F ∈ S ( E , E ∗ ) if and only if ther e exists a holomorphi funtion W : D → L ( H , E ∗ ) suh that I − F ( z ) F ( w ) ∗ = (1 − z ¯ w ) W ( z ) W ( w ) ∗ ( z , w ∈ D ) . Note that W an b e giv en expliitly , see [17, Se. 3.3℄. W e nish with the observ ation that also ertain blo k op erators dene S h ur funtions. Lemma 3 ([12, Lem. 2℄) L et T ∈ L ( H ⊕ E , H ⊕ E ∗ ) with k T k op ≤ 1 . Then T has a unique blo k matrix r epr esentation T = A B C D : H E → H E ∗ and the funtion F : D → L ( E , E ∗ ) , F ( z ) := D + C z ( I − z A ) − 1 B is Shur, i.e. F ∈ S ( E , E ∗ ) . F untions dened as F in the Lemma ab o v e are kno wn in op erator theory as harateristi funtions, while unitary op erators lik e T are kno wn as unitary olligations. Those onepts resem ble m u h the onept of a transfer funtion and a state-spae realization in on trol theory . W e refer to [17,18 ℄ for details. 2.3 Pr oblem F ormulation Before w e giv e an exat problem form ulation w e in tro due and disuss the target ob jetiv e γ opt ( H ) := inf ( {k G k ∞ : G ∈ H ∞ ( E ∗ , E ) , H ( z ) G ( z ) = I for all z ∈ D } ∪ {∞ } ) , whi h is, as w e will see, a tigh t lo w er b ound on the w orst-ase transmit energy enhanemen t of ausal preo ders for the hannel H , and a measure for the a hiev able robustness against imp erfetly kno wn hannel transfer funtions. Note that in partiular γ opt ( H ) = ∞ if and only if H has no righ t in v erse in H ∞ . W e alw a ys assume H ∈ H ∞ ( E , E ∗ ) unless w e expliitly men tion otherwise. It w as sho wn in [19℄ that if dim E ∗ < ∞ , existene of a righ t in v erse in H ∞ is 6 further equiv alen t to H ( z ) H ( z ) ∗ ≥ δ 2 I for some δ > 0 and all z ∈ D . It is somewhat surprising that although b y the result from [ 19 ℄ γ opt < ∞ if and only if δ c := sup n δ ≥ 0 | H ( z ) H ( z ) ∗ ≥ δ 2 I for all z ∈ D o > 0 , δ c has no diret onnetion to γ opt , i.e. γ opt annot b e omputed from δ c [9℄. Ho w ev er, as w e will see, it is imp ortan t to kno w γ opt in adv ane of the onstrution of an optimal preo der. Therefore w e deriv e a new metho d for n umerial omputation of γ opt and then solv e the follo wing problem. Problem 4 L et γ opt ( H ) < ∞ . How an G ∈ H ∞ ( E ∗ , E ) with H ( z ) G ( z ) = I for al l z ∈ D and k G k ∞ = γ opt ( H ) b e onstrute d? W e lose this setion with a short disussion in whi h sense minimization of the innit y norm in Problem 4 giv es optimal lters. The input-output relation of a frequeny seletiv e MIMO hannel is giv en b y y ( ζ ) = H ( ζ ) x ( ζ ) + n ( ζ ) ( ζ ∈ T ) , where H denotes the hannel, x the transmitted signals and y and n the reeiv ed signals and additiv e noise, resp etiv ely . If a preo der G for H is used to pre-distort the transmitted signals, this input-output relation hanges to y ( ζ ) = H ( ζ ) G ( ζ ) x ( ζ ) + n ( ζ ) = x ( ζ ) + n ( ζ ) ( ζ ∈ T ) . There are t w o adv an tages in minimizing the innit y norm of the preo der. The rst adv an tage is minimization of the transmit signals energy . The energy neessary to transmit a signal x using the preo der G is giv en b y k Gx k 2 2 . Without loss of generalit y , let us assume that k x k 2 2 = 1 . Then, it an b e sho wn that the transmit energy neessary in the w orst ase is exatly k G k 2 ∞ , i.e. sup x ∈ H 2 ( E ∗ ) , k x k 2 =1 k Gx k 2 2 = k G k 2 ∞ . Th us, minimizing k G k ∞ guaran tees the lo w est amoun t of neessary transmit energy . If equalizers instead of preo ders are onsidered, i.e. y ( ζ ) = G ( ζ )[ H ( ζ ) x ( ζ ) + n ( ζ )] = x ( ζ ) + G ( ζ ) n ( ζ ) ( ζ ∈ T ) , this is equiv alen t to minimal w orst ase noise enhanemen t. The seond adv an tage of minimization of the innit y norm is robustness. As- sume an imp erfetly kno wn hannel transfer funtion H ∆ = H + ∆ with righ t in v erse G ∆ , where H is the orret hannel and ∆ is a p erturbation. Using 7 the same argumen t as b efore, w e see that the energy of the w orst error that an result from the p erturbation equals sup x ∈ H 2 ( E ∗ ) , k x k 2 =1 k x − H G ∆ x k 2 2 = sup x ∈ H 2 ( E ∗ ) , k x k 2 =1 k ∆ G ∆ x k 2 2 = k ∆ G ∆ k 2 ∞ . Sine it holds k ∆ G ∆ k 2 ∞ ≤ k ∆ k 2 ∞ k G ∆ k 2 ∞ , and this inequalit y an b eome sharp e.g. for ∆ = δ I , w e see that minimizing k G ∆ k ∞ also minimizes the w orst ase error that results from an imp erfetly kno wn hannel transfer funtion. This argumen t applies to equalizers in the same w a y it applies to preo ders. 3 Computation of the Optimal Norm This setion deals with the omputation of the optimal norm γ opt a hiev able b y solutions to the Bezout Iden tit y . Sine man y algorithms whi h diretly solv e Problem 4 only ompute sub optimal solutions, i.e. giv en γ > γ opt they ompute a righ t in v erse G γ with norm k G γ k ∞ < γ , it is imp ortan t to kno w the optimal v alue for γ in adv ane [6,12℄. W e p oin t out that omputation of γ opt also arises in other on texts, see e.g. Remark 1 in [20℄ (with the next orollary in mind). W e start with an exat (but inomputable) form ula for γ opt . The next t w o orollaries are diret onsequenes of Theorem 1. Corollary 5 F or ρ ( H ) := inf u ∈ H 2 ( E ∗ ) , k u k 2 =1 k T H ∗ u k 2 , it holds γ opt ( H ) = ρ ( H ) − 1 . Cor ol lary 6 If γ opt ( H ) < ∞ , a right inverse G ∈ H ∞ ( E ∗ , E ) with k G k ∞ = γ opt ( H ) exists. The in teresting thing ab out Corollary 5 is that it sho ws us wh y the optimal ausal equalizer annot p erform b etter than the optimal non-ausal one. Note that the optimal norm for non-ausal equalizers is giv en b y inf u ∈ H 2 ( E ∗ ) , k u k 2 =1 k H ∗ u k 2 ! − 1 (see [9℄), whi h is the same form ula as Corollary 5 , exept for the additional Riesz pro jetion P + : γ opt ( H ) = inf u ∈ H 2 ( E ∗ ) , k u k 2 =1 k P + ( H ∗ u ) k 2 ! − 1 . It is no w lear that ausal equalizers p erform w orse b eause the signal energy of u whi h is mapp ed in to the non-ausal part of H ∗ u is ut o. Ho w m u h 8 energy is shifted in to the non-ausal part thereb y dep ends on the F ourier o eien ts of H ∗ , whi h are related to H b y d H ∗ k = ˆ H ∗ − k for k ∈ Z . W e no w deriv e a omputable appro ximation of γ opt . The main idea will b e to appro ximate the relation γ opt = ρ − 1 from Corollary 5. In order to ompute γ opt , w e try to appro ximate ρ with ρ N ( H ) := inf u ∈ P N H 2 ( E ∗ ) , k u k 2 =1 k P N T H ∗ u k 2 , i.e. w e restrit domain and image of T H ∗ to p olynomials of degree N and tak e the inm um for this restrition. Beause P N T H ∗ P N is linear and nite dimensional, it an b e represen ted b y a matrix. The main result of this setion is the follo wing. Theorem 7 The se quen e { ρ N ( H ) } N is monotoni al ly de r e asing and on- ver ges with limit lim N →∞ ρ N ( H ) = ρ ( H ) = γ opt ( H ) − 1 . If H ∈ H ∞ ( C m × n ) with m ≤ n , 1 and Γ H,N := ˆ H ∗ 0 ˆ H ∗ 1 . . . ˆ H ∗ N 0 ˆ H ∗ 0 . . . ˆ H ∗ N − 1 . . . . . . . . . . . . 0 . . . 0 ˆ H ∗ 0 ∈ C n ( N +1) × m ( N +1) , ρ N an b e ompute d as ρ N ( H ) = σ min (Γ H,N ) . PR OOF. W e only sk et h the pro of here, the full pro of is giv en in the ap- p endix. It onsists of three main steps. The rst step is to sho w that the sequene { ρ N ( H ) } N is monotonially dereasing and lo w er b ounded b y ρ ( H ) . The main idea is that the relation ρ N ( H ) = inf u ∈ P N H 2 ( E ∗ ) , k u k 2 =1 k P N T H ∗ u k 2 = inf u ∈ P N H 2 ( E ∗ ) , k u k 2 =1 k T H ∗ u k 2 holds for ev ery N ∈ N and th us the inm um is alw a ys tak en o v er the same target ob jetiv e, but o v er a spae whi h inreases with N . This is done in the app endix in Prop osition 14 . In a seond step it is sho wn that the lo w er b ound ρ ( H ) for { ρ N ( H ) } N is sharp. Therefore for arbitrary ǫ > 0 a sequene { u N } N su h that u N ∈ P N H 2 ( E ∗ ) , k u N k 2 = 1 and lim N →∞ k P N T H ∗ u N k 2 ≤ ρ ( H ) + ǫ 1 Note that trivially γ opt ( H ) = ∞ for m > n . 9 is onstruted in the app endix in Prop osition 15. Th us ρ N ( H ) on v erges to ρ ( H ) , whi h is equal to γ opt ( H ) − 1 b y Corollary 5. Finally Prop osition 16 in the app endix giv es the form ula for omputation of ρ N ( H ) via singular v alue deomp osition if H is matrix-v alued. Sine the argumen ts used to pro v e Theorem 7 hold analogously if w e appro x- imate sup u ∈ H 2 ( C m ) , k u k 2 =1 k T H ∗ u k 2 = k T H ∗ k op = k T ∗ H k op = k T H k op = k H k ∞ instead of ρ N ( H ) = inf u ∈ H 2 ( C m ) , k u k 2 =1 k T H ∗ u k 2 , w e also see that for H ∈ H ∞ ( C m × n ) the sequene { σ max (Γ H,N ) } N is monotonially inreasing and on- v erges with limit lim N →∞ σ max (Γ H,N ) = k H k ∞ . W e note that the w ell-kno wn fat that the limit k H k ∞ of σ max (Γ H,N ) an b e found b y p erforming a grid sear h o v er all frequenies, i.e. lim N →∞ σ max (Γ H,N ) = k H k ∞ = esssup ζ ∈ T σ max ( H ( ζ )) , do es not arry o v er to omputation of γ opt ( H ) . Here, in general w e ha v e lim N →∞ σ min (Γ H,N ) = γ opt ( H ) − 1 6 = essinf ζ ∈ T σ min ( H ( ζ )) . This di hotom y results from the fat that while indeed sup u ∈ H 2 ( E ∗ ) , k u k 2 =1 k H ∗ u k 2 = sup u ∈ H 2 ( E ∗ ) , k u k 2 =1 k P + ( H ∗ u ) k 2 , in general w e ha v e inf u ∈ H 2 ( E ∗ ) , k u k 2 =1 k H ∗ u k 2 6 = inf u ∈ H 2 ( E ∗ ) , k u k 2 =1 k P + ( H ∗ u ) k 2 . This an b e easily seen in the next example. Example 8 Set H ( ζ ) = ζ for ζ ∈ T . Then by Parseval's R elation inf u ∈ H 2 ( C ) , k u k 2 =1 k H ∗ u k 2 = inf u ∈ H 2 ( C ) , k u k 2 =1 k u k 2 = 1 , however for u ( z ) = 1 we have ( H ∗ u )( ζ ) = ¯ ζ and ther efor e k P + ( H ∗ u ) k 2 = k 0 k 2 = 0 . It is also imp ortan t to note that Theorem 7 do es not generalize to the ase H ∈ L ∞ T . W e giv e an example where ρ ( H ) = 1 , a in v erse in H ∞ exists, but the smallest singular v alues of the nite setions do not on v erge to ρ ( H ) . 10 Example 9 Set H ( ζ ) := ¯ ζ for ζ ∈ T . Then by Parseval's R elation ρ ( H ) = inf u ∈ H 2 ( C ) , k u k 2 =1 k T H ∗ u k 2 = inf u ∈ H 2 ( C ) , k u k 2 =1 k T ζ u k 2 = inf u ∈ H 2 ( C ) , k u k 2 =1 k u k 2 = 1 . F urther, H has a inverse in H ∞ , i.e. G ( ζ ) = ζ . However, σ min ˆ H ∗ 0 ˆ H ∗ 1 . . . ˆ H ∗ N ˆ H ∗ − 1 ˆ H ∗ 0 . . . . . . . . . . . . . . . ˆ H ∗ 1 ˆ H ∗ − N . . . ˆ H ∗ − 1 ˆ H ∗ 0 = σ min 0 . . . . . . 0 1 . . . . . . . . . . . . . . . 1 0 = 0 for al l N ∈ N . 4 Constrution of the Optimal Causal Preo der In this setion w e onstrut a minim um norm solution to the Bezout Iden tit y , i.e. w e solv e Problem 4 . The ma jor idea of the pro of is the follo wing. W e rst sho w ho w to onstrut righ t in v erses with norm at most one. Then giv en an y H ∈ H ∞ ( E , E ∗ ) , w e apply this te hnique to the saled funtion γ opt H . Appropriate resaling of the obtained in v erse will result in a minim um norm righ t in v erse. 4.1 Shur R ight Inverse The rst step is onstrution of a S h ur righ t in v erse. Therefore w e fatorize the funtion to b e in v erted similar to Theorem 2 and use this fatorization to onstrut a on tration of the form of T in Lemma 3. The harateristi funtion of this on tration then is the w an ted righ t in v erse. This is a v arian t of the te hnique kno wn as lurking isometry metho d, whi h has b een in tro dued b y Ball and T ren t [17, Th. 5.2℄ and indep enden tly Agler and MCarth y [ 21℄ to solv e the Bezout Iden tit y . W e start with the fatorization. Lemma 10 L et H have a right inverse G ∈ S ( E ∗ , E ) . Then ther e exits a holomorphi funtion W : D → L ( H , E ∗ ) suh that H ( z ) H ( w ) ∗ − I = ( 1 − z ¯ w ) W ( z ) W ( w ) ∗ ( z , w ∈ D ) . (1) 11 PR OOF. By Theorem 2 there exists a holomorphi funtion ˜ W : D → L ( ˜ H , E ) su h that I − G ( z ) G ( w ) ∗ = (1 − z ¯ w ) ˜ W ( z ) ˜ W ( w ) ∗ . Th us H ( z ) H ( w ) ∗ − H ( z ) G ( z ) G ( w ) ∗ H ( w ) ∗ = (1 − z ¯ w ) H ( z ) ˜ W ( z ) ˜ W ( w ) ∗ H ( w ) ∗ . Sine H G = I w e obtain with W ( z ) := H ( z ) ˜ W ( z ) that H ( z ) H ( w ) ∗ − I = ( 1 − z ¯ w ) W ( z ) W ( w ) ∗ . W e an no w in tro due the appropriate blo k op erator. Denition 11 L et H have a de omp osition like ( 1 ) in L emma 10 . W e dene the sets D 0 := closure span ¯ w W ( w ) ∗ H ( w ) ∗ e ∗ : w ∈ D , e ∗ ∈ E ∗ ⊂ H ⊕ E , R 0 := closure span W ( w ) ∗ I e ∗ : w ∈ D , e ∗ ∈ E ∗ ⊂ H ⊕ E ∗ , and a funtion V 0 : D 0 → R 0 by ∞ X k =0 c k ¯ w W ( w ) ∗ H ( w ) ∗ e ∗ k 7→ ∞ X k =0 c k W ( w ) ∗ I e ∗ k . Note that it an b e easily sho wn with (1 ) that V 0 is a isometry , i.e. * V 0 h e , V 0 h e + H⊕E ∗ = * h e , h e + H⊕E for all h e ∈ H ⊕ E . Later w e will use this fat when w e apply Lemma 3 to an extension of V 0 . The w an ted righ t in v erse an no w b e giv en expliitly . Theorem 12 L et H have a de omp osition like (1) in L emma 10 and onstrut V 0 as in Denition 11 . Denote by V 00 = A B C D : H E → H E ∗ 12 the ontinuation of V 0 with zer o, i.e. V 00 d = V 0 d , d ∈ D 0 0 , d / ∈ D 0 . Then the funtion G ( z ) := D ∗ + B ∗ ( I − z A ∗ ) − 1 z C ∗ ( z ∈ D ) is a Shur right inverse of H , i.e. G ∈ S ( E ∗ , E ) and H G = I . PR OOF. Let w ∈ D . By onstrution of V 00 it holds A B C D ¯ w W ( w ) ∗ H ( w ) ∗ e ∗ = W ( w ) ∗ I e ∗ , for all e ∗ ∈ E ∗ , whi h is equiv alen t to A ¯ w W ( w ) ∗ + B H ( w ) ∗ = W ( w ) ∗ (2) and C ¯ w W ( w ) ∗ + D H ( w ) ∗ = I . (3) Sine k V 00 k op ≤ k V 0 k op = 1 b eause V 0 is an isometry , w e ha v e k A k op ≤ 1 and th us k A ¯ w k op < 1 . Th us I − A ¯ w is in v ertible, and ( 2) yields W ( w ) ∗ = ( I − A ¯ w ) − 1 B H ( w ) ∗ . Plugging this represen tation of W ( w ) ∗ in to (3 ) results in C ¯ w ( I − A ¯ w ) − 1 B H ( w ) ∗ + D H ( w ) ∗ = I . T aking adjoin ts and replaing w b y z sho ws that H ( z ) h D ∗ + B ∗ ( I − z A ∗ ) − 1 z C ∗ i = I . This righ t in v erse is S h ur b y Lemma 3. 4.2 Minimum Norm R ight Inverse The extension of Theorem 12 from an upp er b ound one on righ t in v erses to arbitrary b ounds is a simple saling argumen t. Note that in partiular the upp er b ound γ = γ opt ( H ) is v alid due to Corollary 6, and results in a minim um norm righ t in v erse of H . 13 Corollary 13 L et γ opt ( H ) ≤ γ < ∞ . Denote by ˜ G ∈ S ( E ∗ , E ) the right in- verse to ˜ H := γ H as given by The or em 12. Then G := γ ˜ G is a right inverse of H with k G k ∞ ≤ γ . PR OOF. Sine γ opt ( H ) ≤ γ < ∞ , a righ t in v erse ˇ G ∈ H ∞ ( E ∗ , E ) of H with k ˇ G k ∞ ≤ γ exists b y Corollary 6. Th us γ H γ − 1 ˇ G = I , k γ − 1 ˇ G k ∞ ≤ 1 , whi h sho ws that ˜ H = γ H has a righ t in v erse in S ( E ∗ , E ) . Let ˜ G ∈ S ( E ∗ , E ) denote the righ t in v erse of ˜ H giv en b y Theorem 12. Then G = γ ˜ G holds k G k ∞ = γ k ˜ G k ∞ ≤ γ as w ell as H G = γ − 1 ˜ H γ ˜ G = I . 5 Conlusions In this pap er w e onsidered the problem of the onstrution of a ausal pre- o der with optimal robustness for a stable and ausal L TI system with m ulti- ple inputs and outputs. This problem is equiv alen t to nding a solution to the Bezout Iden tit y with minimized p eak v alue, for whi h w e ga v e an expliit on- strution. W e deriv ed a no v el metho d for n umerial omputation of the lo w est p eak v alue a hiev able in this problem, b eause it has to b e kno wn prior to the onstrution of the optimal preo der. This metho d is based on omputation of a singular v alue deomp osition of the nite setion of a ertain innite blo k T o eplitz matrix, whi h is diretly onstruted from the F ourier o eien ts of the systems transfer funtion. App endix The omplete pro of of Theorem 7 follo ws splitted in three prop ositions. The rst prop osition sho ws that { ρ N ( H ) } N is monotonially dereasing and on v erges with a limit not lo w er than ρ ( H ) . Prop osition 14 It holds ρ N ( H ) ≥ ρ N +1 ( H ) ≥ ρ ( H ) for al l N ∈ N . 14 PR OOF. Let u ∈ H 2 ( E ∗ ) . W e set v := P N u and w := T H ∗ v . A simple omputation sho ws that the F ourier o eien ts of w = P + ( H ∗ v ) are giv en b y ˆ w k = P ∞ j =0 ˆ H ∗ j ˆ v k + j , k ≥ 0 0 , k < 0 . Sine b y onstrution ˆ v k = 0 for k > N , w e see that ˆ w k = 0 for k > N . Th us k T H ∗ P N u k 2 2 = k w k 2 2 = ∞ X k =0 k ˆ w k k 2 2 = N X k =0 k ˆ w k k 2 2 = k P N w k 2 2 = k P N T H ∗ P N u k 2 2 (4) holds b y P arsev al's Relation for ev ery u ∈ H 2 ( E ∗ ) . Beause trivially P N H 2 ( E ∗ ) ⊂ P N +1 H 2 ( E ∗ ) ⊂ H 2 ( E ∗ ) , w e obtain with (4 ), that ρ N ( H ) = inf u ∈ P N H 2 ( E ∗ ) , k u k 2 =1 k P N T H ∗ u k 2 = inf u ∈ P N H 2 ( E ∗ ) , k u k 2 =1 k T H ∗ u k 2 ≥ inf u ∈ P N +1 H 2 ( E ∗ ) , k u k 2 =1 k T H ∗ u k 2 (= ρ N +1 ( H )) ≥ inf u ∈ H 2 ( E ∗ ) , k u k 2 =1 k T H ∗ u k 2 = ρ ( H ) . W e no w ensure that the limit of { ρ N ( H ) } N also is not greater than ρ ( H ) . Prop osition 15 F or every ǫ > 0 ther e exists K ∈ N suh that ρ N ( H ) ≤ ρ ( H ) + ǫ for al l N > K . PR OOF. W e assume H 6 = 0 sine the ase H = 0 is trivially true. Let ǫ > 0 and ho ose ˇ u ∈ H 2 ( E ∗ ) with k ˇ u k 2 = 1 su h that |k T H ∗ ˇ u k 2 − ρ ( H ) | = k T H ∗ ˇ u k 2 − inf u ∈ H 2 ( E ∗ ) , k u k 2 =1 k T H ∗ u k 2 ≤ ǫ 6 . (5) Sine ˇ u ∈ H 2 ( E ∗ ) , T H ∗ ˇ u ∈ H 2 ( E ) and k ˇ u k 2 = 1 , P arsev al's Relation sho ws that lim N →∞ k P N ˇ u − ˇ u k 2 = lim N →∞ k P N T H ∗ ˇ u − T H ∗ ˇ u k 2 = 0 , lim N →∞ k P N ˇ u k 2 = 1 . 15 Th us K ∈ N exists su h that k P N ˇ u − ˇ u k 2 ≤ ǫ 6 k T H ∗ k − 1 op , (6) k P N T H ∗ ˇ u − T H ∗ ˇ u k 2 ≤ ǫ 6 and (7) k P N ˇ u k 2 ≥ ρ ( H ) + ǫ 2 ρ ( H ) + ǫ (8) for all N > K . Then for N > K it follo ws that k P N T H ∗ P N ˇ u − T H ∗ ˇ u k 2 ≤ k P N T H ∗ ( P N ˇ u − ˇ u ) k 2 + k T H ∗ ˇ u − P N T H ∗ ˇ u k 2 ≤ k P N T H ∗ k op | {z } ≤k T H ∗ k op k P N ˇ u − ˇ u k 2 | {z } ≤ ǫ/ (6 k T H ∗ k op ) b y ( 6 ) + k T H ∗ ˇ u − P N T H ∗ ˇ u k 2 | {z } ≤ ǫ/ 6 b y ( 7 ) ≤ ǫ 3 (9) and therefore k P N T H ∗ P N ˇ u k − inf u ∈ H 2 ( E ∗ ) , k u k 2 =1 k T H ∗ u k 2 ≤ |k P N T H ∗ P N ˇ u k 2 − k T H ∗ ˇ u k 2 | | {z } ≤ ǫ/ 3 b y (9 ) + k T H ∗ ˇ u k 2 − inf u ∈ H 2 ( E ∗ ) , k u k 2 =1 k T H ∗ u k 2 | {z } ≤ ǫ/ 6 b y ( 5 ) ≤ ǫ 2 . W e see that k P N T H ∗ P N ˇ u k 2 ≤ inf u ∈ H 2 ( E ∗ ) , k u k 2 =1 k T H ∗ u k 2 + ǫ 2 = ρ ( H ) + ǫ 2 . (10) Sine k P N ˇ u k 2 > 0 for N > K b y (8), the sequene { ˇ u N } N >K giv en b y ˇ u N := P N ˇ u k P N ˇ u k 2 ∈ P N H 2 ( E ∗ ) is w ell-dened. W e obtain the in tended result 16 ρ N ( H ) = inf u ∈ P N H 2 ( E ∗ ) , k u k 2 =1 k P N T H ∗ u k 2 ≤ k P N T H ∗ ˇ u N k 2 = k P N T H ∗ P N ˇ u k 2 k P N ˇ u k 2 (b y (10 )) ≤ ρ ( H ) + ǫ 2 k P N ˇ u k 2 (b y (8)) ≤ ρ ( H ) + ǫ for all N > K . W e kno w no w b y the Prop ositions 14 and 15 that the sequene ρ N on v erges to ρ for N → ∞ . Ho w ev er it is still unlear, ho w ρ N an b e omputed expliitly . The next prop osition giv es a simple form ula for the n umerial omputation of ρ N . Prop osition 16 L et H ∈ H ∞ ( C m × n ) with m ≤ n and set Γ H,N := ˆ H ∗ 0 ˆ H ∗ 1 . . . ˆ H ∗ N 0 ˆ H ∗ 0 . . . ˆ H ∗ N − 1 . . . . . . . . . . . . 0 . . . 0 ˆ H ∗ 0 ∈ C n ( N +1) × m ( N +1) . Then ρ N ( H ) = σ min (Γ H,N ) . PR OOF. Let U S V ∗ = Γ H,N denote a singular v alue deomp osition of Γ H,N with singular v alues σ 1 ≥ · · · ≥ σ m ( N +1) ≥ 0 . Then U ∈ C n ( N +1) × n ( N +1) and V ∈ C m ( N +1) × m ( N +1) are unitary matries and S ∈ C n ( N +1) × m ( N +1) is of the form S = σ 1 . . . σ m ( N +1) . Let u ∈ P N H 2 ( C n ) and set v := P N T H ∗ u . W e sa w already in the pro of of 17 Prop osition 14 , that the non-zero F ourier o eien ts of v are uniquely deter- mined b y the relation ˆ v 0 ˆ v 1 . . . ˆ v N = ˆ H ∗ 0 ˆ H ∗ 1 . . . ˆ H ∗ N 0 ˆ H ∗ 0 . . . ˆ H ∗ N − 1 . . . . . . . . . . . . 0 . . . 0 ˆ H ∗ 0 ˆ u 0 ˆ u 1 . . . ˆ u N = Γ H,N ˆ u 0 ˆ u 1 . . . ˆ u N . Th us b y P arsev al's Relation ρ N ( H ) = inf u ∈ P N H 2 ( C m ) , k u k 2 =1 k P N T H ∗ u k 2 = inf u ∈ C m ( N +1) , k u k 2 =1 k Γ H,N u k 2 = inf u ∈ C m ( N +1) , k u k 2 =1 k S u k 2 = σ m ( N +1) = σ min (Γ H,N ) . Referenes [1℄ D. F aloner, S. 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