Eigenvalue Estimates and Mutual Information for the Linear Time-Varying Channel

1 Eigen v alue Estimates and Mutual Information for the Linear T ime-V arying Ch annel Brendan Farrell, Member , IEEE, and Tho mas S trohmer Abstract —W e consid er li near time-varying ch annels wit h ad- ditive wh ite Gaussian n oise. For a large class of such channels we derive rigor ous estimates of the eigen values of th e corre lation matrix of the effective channel in terms of the sampled time- vary ing transfer function and, thus, p ro vide a theoretical justi- fication f or a relationship that h as been frequently observ ed in the literatur e. W e then use this eigenv alue estimate t o derive an estimate of the mutual inf ormation of th e c hannel. Our approach is constructive and is based on a careful balance of the trade-off between app roxima te operator di agonalization, signal dimension loss, and accuracy of eigen value estimates. Index T erms —Approxima te Diagonalization, Eigen value Es- timates, Mutual Info rmation, Time-V arying Channel, W eyl- Heisenberg S ystem I . I N T R O D U C T I O N A. M otivation The linear, time-inv ariant (L TI) channe l with impulse re- sponse h r ( t ) = Z h ( t − τ ) s ( τ ) dτ (1) and additive white Gaussian noise with variance σ 2 has normalized capacity 1 2 W Z W − W log 1 + | ˆ h ( ω ) | 2 σ 2 ! dω (2) for signals band-limited to [ − W , W ] . This classical result is, of course, d ue to Shannon [ 1], and is probab ly the most fundam ental result in infor mation theo ry . W e r efer to [2] for the mathematical steps and th e in formatio n-theor etic details for establishing (2). The linear, time-variant ( L TV) chan nel is g iv en by r ( t ) = Z h ( t, t − τ ) s ( τ ) dτ . (3) Motiv ated by Shann on’ s gro undbr eaking result, it has been a longstand ing desire o f engine ers and mathematician s to derive a characterizatio n of the cap acity o f time-varying cha nnels in terms o f the a ssociated time-varying tra nsfer functio n, analogo us to (2). While such a char acterization seems still B. Fa rrell was with the Department of Mathemati cs, Univ ersity of Califor - nia, Davis when the majority of this w ork w as compl eted. He is no w with the Lehrstuhl f ¨ ur Theoretisc he Information stechni k, T echnisc he Uni versi t ¨ at M ¨ unchen, Arcisstr . 21, 80333 M ¨ unch en, Germany . T . S trohmer is with th e Departmen t of Mathematic s, U ni versity of California, Davis, CA, 95616, USA. E-mail: farre ll@tu m.de, strohmer@mat h.ucda vis.edu B. Farrel l w as support ed by NSF VIGRE grant DMS-01353 45. B. F arrell and T . Strohmer were supported by NSF grant DMS-051146 1, and T . Strohmer was supported by AFOSR grant no. 5-36230.5710 and NSF grant DMS- 0811169. quite out of r each for the gen eral case, our aim in this paper is to g et one step closer to this amb itious goal. Th e math ematical found ation f or Sha nnon’ s famous resu lt is the fact that in the time- in variant case the (gener alized) eigenv alues of the channel m atrix a re dir ectly re lated to samp les o f the transfer function . T hus it is natural to ask to what extent such a relationship carrie s over to the time-varying case, which is what we plan to answer in this p aper . For inform ation-theo retic studies of so me spe cial cases of time-varying chan nels we refer the rea der to [3] and its vast list of references. In this paper we focus on the class of time-varying chan nels wh ose spreading function d ecays at an exponential rate both in time and frequen cy . This channel class is motivated by physical properties of chan nel propag ation and includes for instance und erspread ch annels [4], [5]. B. Contributions A precise f ormulation of th e results of this paper requires se veral steps of prepa ration. Theref ore we dela y the rigorous presentation of o ur results to later sections, and instead give an informal description of our contr ibutions. The m ain result o f o ur p aper shows that th e eig en values of the corre lation matrix of the effectiv e cha nnel can be well approx imated via samp ling values of the autoco rrelation o f the time-varying tran sfer functio n. W e derive rigor ous b ounds for the accuracy of th is a pproxim ation. Ou r a pproach is constructive and is ba sed on a careful balance of the trade-off between appr oximate matrix diagonalization, signal dimension loss, and accur acy of eigenv alu e estimates. While the p roof o f the eigenv alu e e stimate is q uite delicate, th is will come as no surprise to the expert in pseudo differential opera tor theor y , since charac terizing the spectrum of a pseudo differential op- erator (wh ich is essentially an op erator of the fo rm (3)) via its symbol has always been a difficult task. W e then show how this eigenv alu e e stimate can be used to der i ve a n estimate o f the mu tual inform ation of these channels. Recall that for th e tim e-in variant case the m utual informa tion (and thus in turn the capacity) is precisely captur ed by the sampled Fourier transform of the auto correlation of the impulse respo nse, as the time interval is extend ed to infin ity . Building on ou r eigenvalue estimates, we rigorou sly relate the mutual inform ation to samples of th e Fourier transfor m o f the “twisted auto-co n volved” spreading functio n. C. R emarks on the pr o of strate gy A few co mments on the proof strategy seem in or der . T wo different types of signal sets will play an importan t role: W eyl- Heisenberg signals an d p rolate spher oidal wa ve fun ctions. The 2 reader may wonder why we do n ot stick with just one of these two types. T he reaso n is that each o f the tw o has some major advantages, but also some significant limitations. Th us, by introdu cing bo th ty pes, W eyl-Heisenberg sig nals and pro late spheroida l wave f unctions (PSWFs), we can fully utilize the positive prop erties of ea ch set, while mitigating its negative proper ties with th e other set. For the eigenv alue est imate we rely on a set of well localized W eyl-Heisenberg signals whose span is close to th e span of the PSWFs in a sense th at will b e f ormalized in the pro of. While the PSWFs a re optima lly lo calized in an L 2 -sense, their lac k of sufficient temp oral decay (except f or the first few PSWFs) proh ibits us f rom linking the eigenv alues of A ∗ A , the correlation matrix of the effecti ve channel, to the associated time-varying tran sfer function . Th e off-diagon al entr ies of th e resulting ma trix would have at best linea r decay , wh ich is sim- ply insufficient for any reasonable estimate. On the other hand , the excellent localization properties of the W eyl-Heisenberg set yield an appro ximate diago nalization of the chan nel, so that the off-diag onal entries o f A ∗ A decay exponentially , which allows u s to obtain a rather tight eigenv alue estimate. The mutual informatio n will d epend on the type and num ber of tran smission signals. W e use a sign aling set consisting of abo ut 2 T W mutually or thonorm al W -bandlimited sig- nals which are “essentially localized” to a time in terval o f length T . Th e associated signal space, rigorou sly defined in Definition 2.1, will be denoted by L 2 ( T , W, ε ) . It is not difficult to co nstruct a linear indep endent, well-loca lized set of W eyl-Heisenberg sign als. Howe ver d ue to the infamo us Balian-Low theo rem (see Su bsection I II-A) such a set will be necessarily inco mplete in L 2 ( R ) , wh ich in turn implies that the num ber of W eyl-Heisenberg sign als inside L 2 ( T , W, ε ) is somewhat smaller than 2 T W + 1 , the approximate dimen sion of L 2 ( T , W, ε ) . T his dim ension loss m akes a dir ect estimate of th e mu tual inform ation somewhat cu mbersome . And that is where PSWFs come into play . W e (ap proxim ately) represent L 2 ( T , W, ε ) via the PSWFs, and then quantif y the (small) dimension loss between the W eyl-Heisenberg set a nd the PSWFs. Combining this estimate with o ur e igen value estimate enables us then to estimate the mutu al inform ation in term s of the time-varying transfer function. D. Connection s to p rior work Our work is related to previous research on two aspects of time-varying channels. Previous autho rs have discussed diagona lizing the channel and g iving the capacity in terms of singular values [6], [7], [8], and o ther a uthors have focused o n determinin g transmission signals with various useful properties [9], [1 0]. Our pap er is proba bly c losest in spirit to [11], wh ere the authors d erive estimates for the non-cohere nt cap acity for certain tim e-varying channels by c arefully co mbining sig nal design with approx imate diagonalization. Much of th e mathematical app roach to time-varying chan- nels from a time-frequ ency analysis perspective origina ted with K ozek [9], [12], [ 13]. While he add resses issues such as the com position and estimation of time-varying chann el operator s and the time-fr equency localization of transmission signals, his focus is a WSSUS mod el. Here we work with a deterministic channel. The r emainder o f the paper is o rganized as fo llows. At the e nd of this section we introd uce mathem atical tools and notation used throu ghout the paper . Section II describes our setup, the channel model and th e signal model. W e derive the eigenv alue estimate in Sectio n III and present th e estimate of the mutual inform ation in Section IV. E. Mathematica l too ls and notation Let f be a fu nction in L 2 ( R ) . The modulatio n operator M ω is defined by M ω f ( t ) = e 2 π iω · t f ( t ) (4) and the translation operator T x is defined by T x f ( t ) = f ( t − x ) (5) for all f ∈ L 2 ( R ) . The F ourier transform of a fu nction f ∈ L 2 ( R ) is given by ( F f )( ω ) = Z f ( t ) e − 2 π iωt dt. (6) W e also write ˆ f for F f . T he Fourier tr ansform of a function in two variables is d efined by extending (6) in the usual way to two dim ensions. Sometimes we n eed to take th e Fourier transform of a f unction f ( t 1 , t 2 ) with respect to the first or the second variable only . In this case we write F 1 f or F 2 f , respectively . When no interval is gi ven , integration is over all of R . For a co mplex-valued function f , we deno te its comp lex conjuga te b y ¯ f . Th e eigenvalues of a matr ix A are denoted by λ j ( A ) . The W e yl pseudod iffer ential operator L σ is defined as L σ f ( t ) = Z Z ˆ σ ( ω , x ) e − π ixω T − x M ω f ( t ) dω dx. (7) Here σ is called the symbol and its Fourier transfor m, ˆ σ , is called th e spr ea ding function . W e can exp ress th e com position of two pseud odifferential oper ators L σ , L τ in terms of the ir symbols. There ho lds L σ L τ = L σ♯τ , where σ ♯τ = F − 1 ( ˆ σ ♮ ˆ τ ) denotes the twisted pr odu ct of σ and τ , and ( ˆ σ ♮ ˆ τ )( ω , x ) = RR ˆ σ ( ω ′ , x ′ ) ˆ τ ( ω − ω ′ , x − x ′ ) e − πi ( xω ′ − ω x ′ ) dω ′ dx ′ is called the twisted convolution of ˆ σ and ˆ τ , see [14]. This can b e seen as a gen eralization of the composition rule of two time-inv arian t oper ators via o rdinary conv o lution. W e set S = σ ♯σ , wh ich is the Fourier transform of the “twisted autocor relation” of ˆ σ . Since S takes values in R , S + ( u, v ) is de fined by S + ( u, v ) = max( S ( u, v ) , 0) . I I . C H A N N E L M O D E L A N D S I G N A L M O D E L W e first derive an equivalent representation of the channel model (3). W e set σ ( t, ω ) = F 2 h ( t, · ) . Several m anipulation s and application s of the Fourier tra nsform yield [14] Z h ( t, t − τ ) s ( τ ) dτ = Z Z ˆ σ ( ω , x ) M ω T − x s ( t ) dω dx. (8) 3 This allows u s to equiv ale ntly expr ess th e linear time-varying channel as a pseudod ifferential o perator L σ s ( t ) = Z Z ˆ σ ( ω , x ) e − π ixω T − x M ω s ( t ) dω dx. (9) The integral in equ ation (9) has the interpretatio n that th e received signal is a weigh ted su m of shifted and m odulated copies o f th e o riginal sign al. Using the W eyl form allows us to expr ess the chan nel as a n o perator that h as fu rther useful relationships to oth er forms that will b e helpful in o ur pro of. See [14] for furthe r b ackgro und on such op erators. Our mo del is no w given b y the following steps an d is illustrated in equatio ns (10-14). First the rand om variable x ∈ C N , x ∼ N C (0 , I N ) is mapp ed to a set of o rthono rmal transmission sig nals φ i as co efficients (10). The signa l passes throug h the chann el g iv en by L σ (11) and is co rrupted b y A WGN (12). The r eceiv ed signal is ma pped to a sequence of r andom variables y by taking the inner pro duct with the detection signals ψ r k (13). N C (0 , I N ) ∼ x Φ → N X i =1 x i φ i (10) L σ → L σ N X i =1 x i φ i (11) ⊕ noise → L σ N X i =1 x i φ i + n (12) C → {h L σ N X i =1 x i φ i + n , ψ r k i} k ∈ Z (13) = y . (14) The re ader will hav e n oticed that we u se a different set of signals at the transmitter and th e receiver . The mu tual infor- mation between x an d y , I ( x, y ) depends on the transmission signals { φ i } N i =1 and the n umber of tran smission sign als, but as long as { ψ r k } k ∈ Z is a n o rthono rmal basis fo r L 2 ( R ) o r a tight f rame, then I ( x, y ) is indepen dent of the rec ei ve sign als. It is clear that the transmission signa ls { φ i } N i =1 should for m a linearly ind ependen t set. As already b riefly in dicated, later the Balian-L ow theo rem will force u s to select th e lin early indepen dent set of tr ansmission signals fr om a set of fu nctions that is a lso incomplete in L 2 ( R ) . Obviously , this implies a dimension loss of the signal space which manifests itself in an addition al err or term in o ur main e stimate of the mutual informa tion. An additional d imension loss would occur if we also used an inco mplete signalin g set at the receiver . Howe ver, at the receiver we are no t restricted to lin early in depend ent signaling sets (th us the Balian-Low theorem is no lo nger a n obstacle) and therefore we w ill use a different, and in fact overcomplete, signalin g set at the receiver . Now we introduce and discuss ou r requirem ent o n the transmission signals. W e requ ire th at th ey are L 2 -localized to a time-f requency r ectangle, which we fo rmalize with th e following d efinition. Definition 2.1: W e define the space L 2 ( T , W, ε ) by L 2 ( T , W, ε ) = n f ∈ L 2 ( R ) : Z T 0 | f ( t ) | 2 dt ≥ (1 − ε 2 ) k f k L 2 ( R ) and Z W − W | ˆ f ( ω ) | 2 dω ≥ (1 − ε 2 ) k f k L 2 ( R ) o . Giv en the intervals [0 , T ] , [ − W, W ] we deno te by { ϕ n } ∞ n =0 the associated PSWFs similar to [15], [25] 1 . Let P be the orthog onal p rojection onto the span of ϕ 0 , . . . , ϕ 2 T W . By Theorem 12 in [15] for every f ∈ L 2 ( T , W, ε ) , k f − P f k 2 ≤ 7 ε k f k 2 . (15) In other words, L 2 ( T , W, ε ) is well a pprox imated by the first 2 T W + 1 elemen ts of the PSWFs and L 2 ( T , W, ε ) is essentially (2 T W + 1) -d imensional. There are several reasons for restricting our tran smission signals to this space. Firstly , any r eal-world co mmunica tion signal has finite duratio n and (essentially) finite b andwidth. The a bove model is a stand ard way to describe this p roperty in a mathe matically m eaningfu l way [15]. Secon dly , for tim e- varying chan nels it is mo re insightful to h av e exp ressions for eigen value estimates or mutual information for finite time intervals (an d of course fin ite bandwidth) than for in finite time, as is also reflected in the pa pers [ 2], [6], [1 1]. Th us it is u seful to req uire some f orm of time-freq uency localization o f the transmission signa ls. W e no te that we could have chosen the signal space with somewhat different localization cond itions, such as for instanc e using exactly time-limited signals. How- ev er , o ur sym metric loc alization cond ition in Definition 2.1 lends itself to a somewhat shorter pro of (ad mittedly , in spite of the overall length of our proo f, th e reader might fin d that using the term “shorter” is not appr opriate here). I I I . E I G E N V A L U E E S T I M A T E S F O R T I M E - V A RY I N G C H A N N E L S A. W eyl-Heisenber g systems, time-frequency localization and mutual information W e assum e that th e reader is familiar with frame theory and refer to [14] for backgr ound. Definition 3.1: For a given function φ ∈ L 2 ( R ) ( the win- dow function) and gi ven parameters a, b > 0 , we den ote the associated Gabor system o r W eyl-Heisenberg system by ( φ, a, b ) := { M bl T ak φ } k,l ∈ Z , a, b ∈ R + . The r edu ndan cy of this system is 1 ab . ( Note that ab ≤ 1 is necessary for ( φ, a, b ) to be a frame for L 2 ( R ) [14] . ) Pr oposition 3.2: Let g s ( t ) = (2 s ) − 1 / 4 e − π s t 2 , and set ψ s = S − 1 / 2 g s , wh ere S is the frame op erator corr espondin g to ( g s , a ρ , b ρ ) . Th en ( ψ s , ρ b , ρ a ) = ( ψ s , ρa, ρb ) ( ab = 1 and ρ > 1 ) is an ortho normal system and there exist constan ts C > 0 and 0 < D < 1 such tha t | ψ s ( t ) | ≤ C e − D π s | t | ∀ t ∈ R | c ψ s ( ω ) | ≤ C e − Dπ s | ω | ∀ ω ∈ R . 1 The minor and trivia l dif ference to [15], [25] is that we consi der [0 , T ] and not [ − T , T ] . 4 Pr oof: A fundam ental theorem du e to L y ubarskii, Seip and W allsten states that ( g s , b ρ , a ρ ) is a frame for L 2 ( R ) if and only if ab ρ 2 < 1 [ 17], [18], [19]. By Theorem 5.1 .6 and Corollary 7 .3.2 in [14], ( S − 1 / 2 g s , b ρ , a ρ ) = ( ψ s , b ρ , a ρ ) is a tight frame for L 2 ( R ) with fram e constant ρ 2 . Now we use the W eyl-Heisenbe rg biorthogon ality relations [20], [21], [ 22], which state that if S g,γ = P k,l ∈ Z h · , M β l T αk g i M β l T αk γ = I on L 2 ( R ) , then h γ , M l/α T k/β g i = αβ δ k, 0 δ l, 0 . A read y consequen ce of this essential theorem is that ( ψ s , ρ a , ρ b ) = ( ψ s , ρb, ρa ) ( ab = 1 ) is an or thonorm al set [14]. Note that ( ψ s , ρb, ρa ) does no t span L 2 ( R ) ). By The orem 5 in [23], u p to a factor 0 < D < 1 , the exponential decay of g s and ˆ g s is preserved in ψ s and ˆ ψ s . Finally , Theo rem IV .2 in [24] imp lies that if ψ 1 is the win dow function for the orthono rmal set based on the initial window g 1 , then ψ s is the corresp onding window function for g s . Let ψ s , a and b be a s in the pr evious pro position. W e construct our signals b y setting a = β α , b = α β and s = ( α β ) 2 . The signals are then defined by : D1) ψ t = ψ ( α β ) 2 D2) ψ r = 1 ρ ψ ( α β ) 2 D3) ψ r k,l = M 1 ρ bl T 1 ρ β α k ψ r D4) ψ t k,l = M ρ α β l T ρ β α k ψ t Here t stands for “transm it” and r stan ds for “receiv e”. Definition 3.3: A function f ∈ L 2 ( R ) is exponentially localized to the region [0 , T ] × [ − W , W ] if there exist constants c 1 , C 1 , c 2 and C 2 such that | f ( t ) | ≤ C 1 e − c 1 | t | and | ˆ f ( ω ) | ≤ C 2 e − c 2 | ω | (16) for all t / ∈ [0 , T ] and all ω / ∈ [ − W , W ] . The Balian-Low theorem [14] pr ecludes the existence of an orthono rmal W eyl-Heisenberg basis ( φ, a, b ) for L 2 ( R ) with well-loca lized window function. In particular, φ and ˆ φ could never h av e expo nential decay . O n the other hand (as for instance Pro position (3. 2) shows) it is not difficult to con struct an orthon ormal system that is in complete in L 2 ( R ) or an overcomplete system ( φ, a, b ) with a φ that is exponentially well localized in time and freque ncy . T hus, the Balian-Low theorem is the reason why we use a sign aling set at the transmitter drawn fro m an inco mplete sy stem for L 2 ( R ) (implying ρ > 1 ) and an overcomplete signaling set at the receiver . While mutual inf ormation is not the m ain topic of th is section, we take the opportu nity to ad dress a non -trivial aspect associated with mutual inf ormation that ar ises f rom using a tig ht f rame in stead o f an or thonor mal basis as receive function s. If we used an o rthono rmal basis at the receiver , then the noise covariance matrix, C N C ∗ N in the p roof belo w , would b e a mu ltiple of the iden tity , and this p roposition would be simple and stand ard. Using a unit-n orm tig ht frame rathe r than an orthon ormal b asis does not change the eigenv alues, but it does make the property addressed in the proposition below more de licate. The exponential lo calization at the r eceiv er an d the L 2 ( T , W, ε ) -p roperty at the transmitter, howev er , deliver the necessary appro ximations f or this propo sition to h old. Pr oposition 3.4: Let { φ kl } ( k,l ) ∈J , |J | < ∞ , b e orthono r- mal transmission sig nals c ontained in L 2 ( T , W, ε ) , and let { ψ r kl } k,l ∈ Z be a tight f rame of exponentially localized receiver signals (with frame bound 1 ). Let x ∼ N C (0 , I |J | ) and y kl = h L σ X k ′ l ′ ∈J x k ′ l ′ φ k ′ l ′ + n , ψ r kl i , for k , l ∈ Z , where n ( t ) is A WGN of variance η 2 . Denote A klk ′ l ′ = h L σ φ k ′ l ′ , ψ r kl i . (17) Then I ( x ; y ) = |J | X i =1 log  1 + λ i ( AA ∗ ) η 2  . (18) Pr oof: Let Φ : L 2 ( R ) → L 2 ( R ) be the orthog onal projec- tion onto span { φ kl } ( k,l ) ∈J , and let C : L 2 ( R ) → l 2 ( Z 2 ) and C N : L 2 ( R ) → C (2 N +1) 2 be th e co efficient o perators giv en by C f = {h f , ψ r kl i} k,l ∈ Z and C N f = {h f , ψ r kl i} | k | , | l | ≤ N for N ∈ N . The mutu al information I ( x ; y ) is I ( x ; y ) = lim N →∞ { log det( C N L σ Φ L ∗ σ C ∗ N + η 2 C N C ∗ N ) − log det( η 2 C N C ∗ N ) } . Assume Φ L σ L ∗ σ Φ has ran k k , an d arrange all eigen values in non-incr easing o rder . W e must show that lim N →∞ λ i ( C N L σ Φ L ∗ σ C ∗ N + C N C ∗ N ) λ i ( CL σ Φ L σ C ∗ ) + 1 for i = 1 , ..., k . Note that CL σ Φ L ∗ σ C ∗ and Φ L σ L ∗ σ Φ h av e the same nonzer o eigenvalues. Since ˆ σ de cays exp onentially in b oth variables and each φ k ′ l ′ ∈ L 2 ( T , W, ε ) , using the Cauchy -Schwartz inequality shows that each L σ φ k,l is expon entially localized a time- frequen cy rectang le slightly larger than [0 , T ] × [ − W, W ] . Thus th e rang e of L σ Φ is exponen tially loca lized in time and frequen cy , and so any eigen vectors o f L σ Φ L ∗ σ correspo nding to non zero eigen values, since they belo ng to the rang e of L σ Φ , are similarly exponen tially localized, which ho lds as well for C N L σ Φ L ∗ σ C ∗ N for all N . In particular, for all f in the range of L σ Φ , there exist positive constants c, C such th at k C ∗ N C N f − f k L 2 ( R ) ≤ C e − cN . Let u ( N ) i be an eig en vecto r of C N L σ Φ L ∗ σ C ∗ N correspo nding to the non zero eigenv alue λ i . Th en u ( N ) i = C N f ( N ) i for some f ( N ) i in the range of L σ Φ . Now , lim N →∞ ( C N f ( N ) i ) ∗ C N C ∗ N ( C N f ( N ) i ) (19) = lim N →∞ h f ( N ) i , C N C ∗ N C N C ∗ N f ( N ) i i (20) = lim N →∞ h f ( N ) i , f ( N ) i i (21) = 1 . The conv ergence in lin es (20) and (21) is expon ential. While exponential convergence is not necessary , witho ut sufficient localization of all the fu nctions in volved, con vergence at all does not hold a prio ri f or (20) and (21). For i = 1 , ..., k , lim N →∞ λ i ( C N L σ Φ L ∗ σ C ∗ N + η 2 C N C ∗ N ) = lim N →∞ λ i ( C N L σ Φ L ∗ σ C ∗ N ) + η 2 5 The remain ing eig en vectors of C N C ∗ N are in the kernel of C N L σ Φ L ∗ σ C ∗ N . Thus, lim N →∞ n (2 N +1) 2 X i =1 log( λ i ( C N L σ Φ L ∗ σ C ∗ N + η 2 C N C ∗ N )) − (2 N +1) 2 X i =1 log( λ i ( η 2 C N C ∗ N )) o = lim N →∞ k X i =1 log( λ i ( C N L σ Φ L ∗ σ C ∗ N ) + η 2 ) − lim N →∞ k X i =1 log( λ i ( η 2 C N C ∗ N )) ( 22) = lim N →∞ k X i =1 log(1 + λ i ( C N L σ Φ L ∗ σ C ∗ N ) η 2 ) (23) = k X i =1 log(1 + λ i (Φ ∗ L ∗ σ L σ Φ ∗ ) η 2 ) = k X i =1 log(1 + λ i ( A ∗ A ) η 2 ) = |J | X i =1 log(1 + λ i ( A ∗ A ) η 2 ) , where lines (22) and (23) are con sequences of th e first h alf of the proo f. B. Eigen value Estima tes W e are ready to give a rigor ous form ulation o f our main result, which states that the eigen values of the correlation matrix A ∗ A c an be well appr oximated by samples o f S , the twisted autocor relation of the time -varying tran sfer function. Theor em 3.5 (Eigen va lue estimate): Assume the same setup as in Propo sition 3 .4. Furtherm ore, su ppose that | ˆ σ ( ω , x ) | ≤ C e − β | ω |− α | x | . (24) Let S = σ ♯σ . Th en for j = 1 , ..., |J | , there exists an index pair ( k , l ) such that     λ j ( A ∗ A ) − S ( ρ β α k, ρ α β l )     ≤ O  e − ρ 2 ( β + α ) + 1 ( αβ D ) 2  . (25) Remark: Our decay condition (24) on the spreading function comprises the standar d condition s of exponen tial decay of delay sprea d and comp act support of the Doppler spread [ 5]. Moreover , we could have impo sed an underspr ea d co ndition on th e sp reading fu nction, see [16 ] for various notions of undersp read c hannels. It is not h ard to see that condition (41) includes (or can be easily ad apted to) several forms of u n- derspread ch annels. This would result in somewhat different constants in the error estimate at the cost of a slightly longer proof , but the essence of th e theorem would remain the same. Furthermo re, one can replace th e expone ntial deca y condition by some for m o f (prac tically le ss justified) polyn omial decay and show that the err or term in (25) would then decrease at a correspo nding po lynomial rate. T o p rove T heorem 3.5 we cannot use PSWFs, but instead introdu ce expo nentially localized signa ls. Th e reason is th at the PSWFs dec ay linearly [25] and , thus, do not perm it the bound s obtain ed in th e main two lemmas of th is section. This is heuristically explained by the fact tha t the PSWFs are the approx imate eigenfun ctions of the o perator that restricts in time and frequen cy , which is a mu ch different op erator than a time-varying channel, for which the expon entially localized signals are approxim ate e igenfun ctions. This is seen formally in the off-diagonal de cay in the matrix A in Proposition 3.7 below . Howev er , since b oth sets of signals are lo calized, the spaces that they span are close, which is a p oint th at we formalize late r in the proo f of Theo rem 4.1. Thu s, the general idea is the standard linear algebr a app roach o f working with the same space, but switching to a b asis that allows for approx imate diag onalization . W e first n eed an auxiliary result. Lemma 3.6: For f , g ∈ L 2 ( R ) , let W ( f , g ) and A ( f , g ) de- note their cross-amb iguity and cross-W igner distributions [14]. If | ψ ( x ) | ≤ C e − c 1 | x | and | ˆ ψ ( ω ) | ≤ C e − c 2 | ω | for c 1 , c 2 > 0 , then |W ( ψ , ψ )( x, ω ) | ≤ C 2 e − 1 4 ( c 1 | x | + c 2 | ω | ) and |A ( ψ , ψ )( x, ω ) | ≤ C 2 e − 1 4 ( c 1 | x | + c 2 | ω | ) . Pr oof: The proof is contain ed in the proof of Theorem 2.4 in [26], when one v iews both distributions as short-time Fourier transforms, as explained in [ 1 4 ] . A key ingr edient in ou r proo f of Theo rem 3 .5 is the following lemma, which shows that the en tries of the matr ix A defined in (27) decay exponentially fast as we move aw ay from the main d iagonal. The appro ximate d iagonalizatio n of A via a pro perly designed W eyl-Heisenberg systems is well known in a qualitative sense [13], [27], [11]. What is new in the fo llowing lemma is that we g i ve a p recise qu antitative formu lation of this statement. Th is quantitative version is importan t in the sub sequent step s, whe re it will g i ve rise to explicit and rigorou s boun ds on th e approx imation of the eigenv alues o f A ∗ A by sam ples of the twisted autoc orrelation S of th e time-varying transfer fu nction. Lemma 3.7: Assume that | ˆ σ ( ω , x ) | ≤ C e − β | ω |− α | x | , (26) that the signals are given according to prop erties D 1 − D 4 above an d that A klk ′ l ′ = h L σ ψ t k ′ l ′ , ψ r kl i . (27) Then | A klk ′ l ′ | ≤ C ( e − αρ | 1 ρ 2 l − l ′ | + e − π 4 D ( α β ) 2 ρ | 1 ρ 2 l − l ′ | ) × ( e − β ρ | 1 ρ 2 k − k ′ | + e − π 4 D ( β α ) 2 ρ | 1 ρ 2 k − k ′ | ) . Pr oof: The following two essential identities hold for pseudod ifferential o perators, cf. [14]: h L σ f , g i = h σ, W ( g , f ) i (28) |h L σ T u M η f , T v M γ g i| = | ( ˆ σ ∗ A ( f , g ))( u − v , η − γ ) | . (29) 6 The system is given b y ψ ( α β ) 2 = S − 1 / 2 g ( α β ) 2 , where g ( α β ) 2 ( t ) = (2( α β ) 2 ) − 1 / 4 e − π ( β α ) 2 t 2 , and by Proposition (3.2) | ψ ( α β ) 2 ( t ) | ≤ C e − π ( β α ) 2 D | t | (30) | ˆ ψ ( α β ) 2 ( ω ) | ≤ C e − π ( α β ) 2 D | ω | . (31) Lemma 3.6 implies |A ( ψ ( α β ) 2 , ψ ( α β ) 2 )( x, ω ) | ≤ C e − π 4 D ( α β ) 2 | x |− π 4 D ( β α ) 2 | ω | . (32) | A k,l,k ′ ,l ′ | = |h L σ ψ t k ′ l ′ , ψ r kl i| = |h L σ M ρ α β l ′ T ρ β α k ′ ψ , M 1 ρ α β l T 1 ρ β α k ψ i| = | ( ˆ σ ∗ A ( ψ , ψ ))( β α ( 1 ρ k − ρk ′ ) , α β ( 1 ρ l − ρl ′ )) | = | Z Z ˆ σ ( ω , x ) A ( ψ , ψ ) ( β α ( 1 ρ k − ρk ′ ) − x, α β ( 1 ρ l − ρl ′ ) − ω ) dxdω | ≤ C Z Z e − β | ω |− α | x | e − π 4 ( β α ) 2 D | β α ( 1 ρ k − ρk ′ ) − x |− π 4 ( α β ) 2 D | α β ( 1 ρ l − ρl ′ ) − ω | dω dx = C Z e − β | ω |− π 4 ( β α ) 2 D | α β ( 1 ρ l − ρl ′ ) − ω | dω × Z e − α | x |− π 4 ( α β ) 2 D | β α ( 1 ρ k − ρk ′ ) − x | dx ≤ C ( e − αρ | 1 ρ 2 l − l ′ | + e − π 4 α β Dρ | 1 ρ 2 l − l ′ | ) × ( e − β ρ | 1 ρ 2 k − k ′ | + e − π 4 β α Dρ | 1 ρ 2 k − k ′ | ) , where we have u sed the bound: Z e − c 1 | y | e − c 2 | X − y | dy ≤ C ( e − c 1 | X | + e − c 2 | X | ) . The fo llowing lemm a shows th at the eigenv alues of A ∗ A are well approx imated b y its diagon al en tries. Lemma 3.8: Assume again the hypothe ses of Proposi- tion 3.7. T hen f or j = 1 , ..., |J | , th ere exists an index pair ( k , l ) such that | λ j ( A ∗ A ) − ( A ∗ A ) klkl ) | ≤ O ( e − ρ 2 ( β + α ) ) . (33) Pr oof: ( A ∗ A ) klk ′ l ′ = X j,j ′ ∈ Z A j j ′ kl A j j ′ k ′ l ′ = X j,j ′ ∈ Z h L σ ψ t kl , ψ r j j ′ ih L σ ψ t k ′ l ′ , ψ r j j ′ i = X j,j ′ ∈ Z h L σ ψ t kl , ψ r j j ′ ih ψ r j j ′ , L σ ψ t k ′ l ′ i = h L σ ψ t k ′ l ′ , L σ ψ t kl i = h L S ψ t k ′ l ′ , ψ t kl i , where S = σ ♯σ was d efined in Section II. Using the estimate from the proof of Lem ma 3.7, we have that | ˆ S ( ω , x ) | ≤ C αβ e − β | ω |− α | x | . Using the identity in equa tion (29), |h L S ψ t k ′ l ′ , ψ t kl i| = | ( ˆ S ∗ A ( ψ t , ψ t ))( ρ β α ( k ′ − k ) , ρ α β ( l ′ − l )) | ≤ C ( e − αρ | l − l ′ | + e − π 4 α β Dρ | l − l ′ | ) × ( e − β ρ | k − k ′ | + e − π 4 β α Dρ | k − k ′ | ) . Next X k = − K,...,K,k 6 = k ′ l = − L,..., L,l 6 = l ′ | ( A ∗ A ) klk ′ l ′ | ≤ C X k = − K,.., K,k 6 = k ′ ( e − β ρ | k − k ′ | + e − π 4 D β α ρ | k − k ′ | ) × X l = − L,..,L ,l 6 = l ′ ( e − αρ | l − l ′ | + e − π 4 D α β ρ | l − l ′ | ) = O e − β ρ 1 − e − β ρ + e − π 4 D β α ρ 1 − e − π 4 D β α ρ ! × e − αρ 1 − e − αρ + e − π 4 D α β ρ 1 − e − π 4 D α β ρ !! . (34) = O e − ρ ( β + α ) (1 − e − π 4 D β α ρ )(1 − e − π 4 D α β ρ ) ! = O  e − ρ 2 ( β + α )  . (35) W e n ow have an estimate o n the off-diago nal sums of the matrix A ∗ A and may apply the Ger shgorin disc theore m to obtain the claim. Having established that the spectrum of A ∗ A is very close to its diagonal en tries, w e n ext show th at in turn the diagon al of A ∗ A is well app roximated by the samples o f the associated twisted autocor relation S . Lemma 3.9: Assume again the hypothe ses of Proposi- tion 3.4 and that | ˆ σ ( ω , x ) | ≤ C e − β | ω |− α | x | . Let S = σ ♯σ . Th en | ( A ∗ A ) klkl − S ( ρ β α k , ρ α β l ) | = O  1 ( αβ D ) 2  . Pr oof: W e first look at ( A ∗ A ) klkl . Th e diagon al entries of A ∗ A are ( A ∗ A ) klkl = X k ′ l ′ ∈ Z 2 |h L σ ψ t kl , ψ r k ′ l ′ i| 2 (36) = k L σ ψ t kl k 2 2 , (37) since ( ψ , 1 ρ a, 1 ρ b ) is a tight W eyl-Heisenberg frame (Prop osi- tion (3.2)). k L σ ψ t k 2 2 (38) = h L σ ψ t kl , L σ ψ t kl i (39) = h σ ♯σ , W ( ψ t kl , ψ t kl ) i (40) = Z R 2 S ( x, ω ) W ( ψ t , ψ t )( x − ρ β α k , ω − ρ α β l ) dω dx 7 Setting S ′ = ∂ x ∂ ω S , by the Riemann -Lebesgue Le mma, kS ′ k ∞ ≤ Z Z | ˆ S ( ω , x ) | dω dx ≤ C Z Z 1 αβ e − β 2 | ω |− α 2 | x | dω dx = C ( αβ ) 2 . W e use Lemma 3.6 and the f act that R R W ( ψ , ψ )( ω , x ) dω dx = k ψ k 2 2 = 1 , cf. [1 4]. |k L σ ψ t k 2 2 − S ( ρ β α k , ρ α β l ) | = | Z R 2 S ( x, ω ) W ( ψ , ψ )( x − ρ β α k , ω − ρ α β l ) dω dx −S ( ρ α β l , ρ β α k ) | = | Z R 2 S ( x + ρ β α k , ω + ρ α β l ) W ( ψ , ψ )( x, ω ) dω dx −S ( ρ α β l , ρ β α k ) | = | Z R 2 [ S ( x + ρ β α k , ω + ρ α β l ) − S ( ρ α β l , ρ β α k )] W ( ψ , ψ )( x, ω ) dωdx ) | ≤ k S ′ k ∞ Z R 2 ( | x | + | ω | ) W ( ψ , ψ )( x, ω ) | dω dx ≤ C 1 ( αβ ) 2 Z R 2 ( | x | + | ω | ) e − πsD 4 | x |− π 4 s D | ω | dω dx = C 1 ( αβ D ) 2 These two bounds prove the lemm a. Pr oof of The or em 3.5: The estimate (2 5) follows n ow readily by app lying the triang le inequ ality to the left-hand - side of (25), and then using Lemm a 3.8 and Lemma 3.9. Remark: In the pro of o f this th eorem we rely on u sing W eyl-Heisenberg systems. Instead we cou ld hav e resorted to orthon ormal W ilson bases [14], which do not suffer fr om the Balian-Low T heorem. Ho wev er it would hav e resulted in a less elegant relationship betwe en eigenv alu es and samples of S . In particular, equations (28) and (29) w ould hav e to be r eplaced by more comp licated e xpressions. I V . F RO M E S T I M A T I N G E I G E N V A L U E S T O E S T I M A T I N G M U T U A L I N F O R M A T I O N For th e time-inv ariant case, the mutu al info rmation is pre- cisely captured by samp les of the Fourier transform of the autocorr elation of the impulse response when one allows T → ∞ . At the co re of this relationship is th e fact that th e (genera lized) eige n values o f th e channe l are directly linked to samples o f the transfer function . It tu rns o ut that f or our class of time-varying channels a similar conn ection is true in an appro ximate sense. Using th e eigen value estimate fr om th e previous section we w ill show that one can obtain an estimate of the m utual inform ation via samples of the Fourier transform of the “twisted auto-co n volved” spr eading function . This is the contents of the following th eorem. Theor em 4.1 (Mutual information estimate): Assume that the spreading function ˆ σ in the system model satisfies | ˆ σ ( ω , x ) | ≤ C e − β | ω |− α | x | , (41) and the A WGN n ( t ) h as v ariance η 2 . Let Φ T ,W = { φ k } N k =1 be a set of o rthonor mal fun ctions contained in L 2 ( T , W, ε ) , where N = (1 − δ )(2 T W + 1) for some 0 ≤ δ < 1 . Let I Φ T ,W ( x, y ) denote th e resulting mutual info rmation of the system g i ven in lines (10-14). Then there exist constan ts 0 < D , 1 < ρ and small constants 0 ≤ δ 1 , δ 2 such that       I Φ T ,W ( x, y ) − K,L X k =0 ,l = − L log 1 + S + ( ρ β α k, ρ α β l ) η 2 !       (42) ≤ (2 T W + 1) log  1 + O  e − ρ 2 ( β + α ) + 1 ( αβ D ) 2  (43) + log  1 +  14 ε η 2 + (14 ε + δ ) η 2  kS k L ∞ ( R )  (44) + log  1 +  14 ε η 2 + 1 − (1 − 49 ε 2 ) ρ 2 + δ 1 ρ β α + δ 2 ρ α β η 2  kS k L ∞ ( R )  ! (45) where K = T ρ α β − δ 1 and L = 2 W ρ β α − δ 2 . The p arameters D and ρ have the relatio nship that D → 0 as ρ → 1 and ρ → ∞ as D → 1 . The numbe rs δ 1 and δ 2 depend on the parameter s α, β an d ε , but r emain small as T a nd W inc rease. Before we pro ceed to the pr oof of this theo rem, it seems pruden t to comment on the statement of this th eorem a nd the various eleme nts that come into play here. Remark 1: In a nutshell our theor em sho w s that I Φ T ,W ( x, y ) ≈ K,L X k =0 ,l = − L log  1 + S + ( ρ β α k , ρ α β l ) η 2  , and quantifies rigorou sly in wh ich sense this app roximation is true. The err or due to estimating the mutual inf ormation from the samp les is given in (4 3) and is the co nceptually mo re importan t one f or this pa per . The error in (44) results from the transition from the system Φ T ,W in L 2 ( T , W, ε ) to the PSWFs, and the error (45) is d ue to the fact that the nu mber of the con structed W eyl-Heisen berg signa ls used is less than the number of PSWFs co rrespond ing to the time-frequ ency region. Remark 2 : The factor ρ is n ecessary fo r our co nstruction and is g reater than 1 , see Propo sition 3.2 and the subsequent discussion. While taking ρ very close to 1 would ma ke the error in equatio n (45) very small, it would increase the erro r in eq uation (43). W e can, howe ver , take ρ to b e fairly close to 1 , such as ρ = 5 / 4 . Th is issue of the trade- off be tween time-frequ ency loc alization and loss of dimensions in signal space has also been poin ted out in [ 11]. W e need the f ollowing lemma for the pr oof of Theor em 4.1. Lemma 4.2: Let S = σ ♯σ and S + ( x, ω ) = ( S ( x, ω )) + . Then     log(1 + λ k,l ( A ∗ A )) − log (1 + S + ( ρ β α k , ρ α β l ))     = log  1 + O  e − ρ 2 ( β + α ) + 1 ( αβ D ) 2  8 Pr oof: Using L emmas 3.8 and 3.9,     log(1 + λ k,l ( A ∗ A )) − log (1 + S + ( ρ β α k , ρ α β l ))     ≤ | log(1 + λ k,l ( A ∗ A )) − log (1 + ( A ∗ A ) klkl ) | +     log(1 + ( A ∗ A ) klkl ) − lo g(1 + S + ( ρ β α k , ρ α β l ))     = lo g  1 + O  e − ρ 2 ( β + α ) + 1 ( αβ D ) 2  Pr oof of Theo r em 4.1: Let P den ote the pro jection of L 2 ( R ) onto the span of the 2 T W + 1 PSWFs corr espondin g to [0 , T ] × [ − W , W ] . From (15) we obtain k P f k 2 L 2 ( R ) ≥ 1 − 4 9 ε 2 k f k 2 L 2 ( R ) (46) for all f ∈ L 2 ( T , W, ε ) . W e write P Φ for th e projectio n onto the set { φ 1 , . . . , φ N } and G for the Gram matrix of { P φ 1 , ..., P φ N } , i.e. G i,j = h P φ j , P φ i i i, j = 1 , ..., N . (47) Then ran k ( PP Φ ) = ran k ( G ) . No te th at the diagonal entries of G are p ositiv e and, since { φ 1 , ..., φ N } are o rthono rmal, that the eig en values of G have ab solute value at most 1 . By inequality (46) N X j =1 G j,j = N X j =1 k P φ j k 2 ≥ N (1 − 49 ε 2 ) , so that rank ( G ) ≥ N (1 − 49 ε 2 ) . Therefo re, rank ( P ⊥ P Φ ) ≤ r ank ( P ) − r ank ( PP Φ ) ≤ (2 T W + 1) − (1 − 49 ε 2 ) N , and k P Φ P L σ L ∗ σ PP Φ − P L σ L ∗ σ P k HS = k − P ⊥ Φ P L σ L ∗ σ PP Φ + P L σ L ∗ σ PP Φ − P L σ L ∗ σ P k HS = k − P ⊥ Φ P L σ L ∗ σ PP Φ + P L σ L ∗ σ P − P L σ L ∗ σ PP ⊥ Φ − P L σ L ∗ σ P k HS ≤ k P ⊥ Φ P L σ L ∗ σ PP Φ k HS + k P L σ L ∗ σ PP ⊥ Φ k H S ≤ 2 k P ⊥ Φ P k HS k L σ k 2 ≤ 2 rank ( P ⊥ Φ P ) k L σ k 2 ≤ 2((2 T W + 1) − (1 − 49 ε 2 ) N ) k L σ k 2 . If 2 T W + 1 > N , then set λ j ( P Φ L σ L ∗ σ P Φ ) = 0 for N < j ≤ 2 T W + 1 . Le t π be a per mutation of the integers 1 , ..., 2 T W + 1 . Then    2 T W +1 X j =1 log  1 + λ j ( P Φ L σ L ∗ σ P Φ ) η 2  − 2 T W +1 X j =1 log  1 + λ π ( j ) ( P L σ L ∗ σ P ) η 2     ≤ 2 T W +1 X j =1 log  1 + | λ j ( P Φ L σ L ∗ σ P Φ ) − λ π ( j ) ( P L σ L ∗ σ P ) | η 2  ≤ 2 T W +1 X j =1 log  1 + | λ j ( P Φ L σ L ∗ σ P Φ ) − λ j ( P Φ P L σ L ∗ σ PP Φ ) | η 2 + | λ j ( P Φ P L σ L ∗ σ PP Φ ) − λ π ( j ) ( P L σ L ∗ σ P ) | η 2  . W e co nsider the first eigenv alue difference in th e expression above. Apply ing Theorem A.46 in [28] we obtain | λ j ( P Φ P L σ L ∗ σ PP Φ ) − λ j ( P Φ P L σ L ∗ σ P ) | ≤ k P Φ L σ L ∗ σ P Φ − P Φ P L σ L ∗ σ PP Φ k . (48) Let P Φ f = u + v where u ∈ r ange P an d v ∈ range P ⊥ . Then k ( P Φ L σ L ∗ σ P Φ − P Φ P L σ L ∗ σ PP Φ ) f k = k P Φ L σ L ∗ σ ( u + v ) − P Φ P L σ L ∗ σ u k ≤ k ( P Φ L σ L ∗ σ − P Φ P L σ L ∗ σ ) u k + k P Φ L σ L ∗ σ v k = k P Φ P ⊥ L σ L ∗ σ u k + k P Φ L σ L ∗ σ v k ≤ k P Φ P ⊥ kk L σ k 2 k u k + k P Φ L σ L ∗ σ P ⊥ P Φ f k ≤ 7 ε k L σ k 2 k f k + 7 ε k L σ k 2 k f k ≤ 14 ε k L σ k 2 k f k , where we have used (1 5) in the penultimate step. Hence | λ j ( P Φ P L σ L ∗ σ PP Φ ) − λ j ( P Φ P L σ L ∗ σ P ) | ≤ 14 ε k L σ k 2 . ( 49) Concernin g the second dif ference of eigen values recall th at accordin g to Th eorem A.3 7 of [28] th ere exists a permutation π such that 2 T W +1 X j =1 | λ j ( P Φ P L σ L ∗ σ PP Φ ) − λ π ( j ) ( P L σ L ∗ σ P ) | 2 ≤ k P Φ P L σ L ∗ σ PP Φ − P L σ L ∗ σ P k 2 HS ≤ 4 ((2 T W + 1) − (1 − 49 ε 2 ) N ) 2 k L σ k 4 . (50 ) Using (49), (50) and the c oncavity of the log function we compute 2 T W +1 X j =1 log  1 + | λ j ( P Φ L σ L ∗ σ P Φ ) − λ j ( P Φ P L σ L ∗ σ PP Φ ) | η 2 + | λ j ( P Φ P L σ L ∗ σ PP Φ ) − λ π ( j ) ( P L σ L ∗ σ P ) | η 2  ≤ 2 T W +1 X j =1 log  1 + 14 ε k L σ k 2 η 2 + | λ j ( P Φ P L σ L ∗ σ PP Φ ) − λ π ( j ) ( P Φ P L σ L ∗ σ P ) | η 2  ≤ 2 T W +1 X j =1 log  1 + 14 ε k L σ k 2 η 2 + 2((2 T W + 1) − (1 − 49 ε 2 ) N ) k L σ k 2 η 2 (2 T W + 1)  9 W e will r eturn to (5 1) twice, taking N to be the cardin ality of Φ T ,W and of our constru cted set. W e lo ok at the system { ψ t k,l } from Section III-A. The signal ψ t k,l is exponen tially localized arou nd the p oint ( ρ α β l , ρ β α k ) . W e select those signals that are con tained in L 2 ( T , W, ε ) . F or some positive constants δ 1 and δ 2 , these are tho se signals with indices 0 ≤ k ≤ T ρ α β − δ 1 and 0 ≤ | l | ≤ W ρ β α − δ 2 . W e set K = T ρ α β − δ 1 and L = W ρ β α − δ 2 . W e den ote by P K,L the pro jection operator from L 2 ( R ) onto the span of { ψ t k,l } K,L k =0 ,l = − L . Now we u se (5 1) twice: once with N = (1 − δ )(2 T W + 1) fo r the cardinality of the set Φ T ,W , as assumed in the statement o f th e theorem, and once for Ψ K,L , where the cardina lity satisfies K (2 L + 1) ≥ 2 T W + 1 ρ 2 − δ 1 2 W ρ β α − δ 2 T ρ α β . The arguments above then yield    2 T W +1 X j =1 log  1 + λ j ( P Φ L σ L ∗ σ P Φ ) η 2  (51) − 2 T W +1 X j =1 log  1 + λ π ( j ) ( P K,L L σ L ∗ σ P K,L ) η 2     ≤ (2 T W + 1)  log  1 +  14 ε η 2 + 2(49 ε 2 + δ ) η 2  k L σ k 2  + log  1 +  14 ε η 2 + 2(1 − (1 − 49 ε 2 ) ρ 2 + δ 1 ρ β α + δ 2 ρ α β ) η 2  k L σ k 2   (52) The estimation of the eigen values P K,L L σ L ∗ σ P K,L is giv en b y the Lem mas 3.8 an d 4.2. Applying these two lemm as to gether with inequa lity ( 52) complete the proof of the theor em. 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