Multipath Channels of Bounded Capacity

Multipath Channels of Bounded Capaci ty T obias K och Amos Lapidoth ETH Zurich, Switzerland Email: { tkoch, lapidoth } @isi.ee.eth z.ch Abstract — The capacity of discrete-time, non-coherent, multi- path fading channels is consi dered. It is shown that if th e delay spread is large i n the sense t hat the variances of t he path gains do not decay faster than geometrically , then capacity is boun ded in the signal-to-noise ratio. I . I N T R O D U C T I O N This paper studies no n-coher ent multipath (frequ ency- selecti ve) fading chann els. Such chan nels have been in ves- tigated extensively in the wideband r egime where th e sign al- to-noise ratio (SNR) is ty pically small, and it was sh own th at in the limit as the a v ailable bandwidth tends to infinity the capacity o f the fading channe l is the same as the capa city of the additive white Gau ssian no ise (A WGN) chan nel of equal received p ower , see [1]. 1 When the SNR is large we encou nter a different situatio n. Indeed , it has been shown in [5] fo r n on-coh erent fr equency- flat fading chan nels that if the fading p rocess is regular in the sense that the pr esent fading cann ot be p redicted perfectly from its past, then at high SNR capacity only increases double- logarithm ically in the SNR. Th is is in stark c ontrast to the logarithm ic growth of the A WGN capacity . See [6], [ 7], [8], and [9] for extensions to multi-a ntenna systems, and see [10] and [ 11] f or e xtensions to non-regular fading, i.e., when the present fading c an be pred icted perf ectly f rom its past. Thus, commun icating over non-co herent flat-fadin g channels at hig h SNR is power inefficient. In this pap er , we sh ow that co mmunica ting over non- coheren t m ultipath fading channels at high SNR is not me rely power inefficient, but even worse: if the delay spre ad is large in th e sense that the variances of the path g ains do n ot decay faster than geometrically , then capacity is bound ed in the SNR. For such chan nels, capacity does not tend to infinity as the SNR tends to infinity . T o state this result precisely we begin with a math ematical descrip tion of the channel model. A. Chan nel Mode l Let C and Z + denote the set of com plex nu mbers and th e set of positi ve inte gers, respectively . W e co nsider a discrete- time multipath fading chann el wh ose channel outpu t Y k ∈ C at time k ∈ Z + correspo nding to the channel inpu ts 1 Ho we ver , in con trast to the infinite bandwi dth capacity of the A WGN channe l where the condit ions on the capacity achie ving input di s trib ution are not so stringent , the infinite bandwidth capaci ty of non-coherent fading channe ls can only be ac hiev ed by signalin g schemes which are “pea ky”; see also [2], [3], [4] and reference s therei n. ( x 1 , x 2 , . . . , x k ) ∈ C k is given by Y k = k X ℓ =1 H ( k − ℓ ) k x ℓ + Z k . (1) Here, H ( ℓ ) k denotes the time- k gain of the ℓ -th path, and { Z k } is a sequ ence o f indepen dent and iden tically distributed (IID), zero -mean, variance- σ 2 , cir cularly-sym metric, comp lex Gaussian random variables. W e assume that fo r each path ℓ ∈ Z + 0 (with Z + 0 denoting the set of non- negativ e integers) the stochastic p rocess  H ( ℓ ) k , k ∈ Z +  is a zero-m ean stationary process. W e den ote its variance and its differential entropy rate by α ℓ , E h   H ( ℓ ) k   2 i , ℓ ∈ Z + 0 and h ℓ , lim n →∞ 1 n h  H ( ℓ ) 1 , H ( ℓ ) 2 , . . . , H ( ℓ ) n  , ℓ ∈ Z + 0 , respectively . W e fu rther assume that sup ℓ ∈ Z + 0 α ℓ < ∞ and inf ℓ ∈L h ℓ > − ∞ , (2) where th e set L is defin ed as L , { ℓ ∈ Z + 0 : α ℓ > 0 } . W e finally assume that the p rocesses  H (0) k , k ∈ Z +  ,  H (1) k , k ∈ Z +  , . . . are indepen dent ( “uncorr elated scattering ”), that they are jointly indepe ndent of { Z k } , and that the join t law of  { Z k } ,  H (0) k , k ∈ Z +  ,  H (1) k , k ∈ Z +  , . . .  does not depend on the inp ut sequence { x k } . W e consid er a non-co her en t channel mo del where n either the transmitter n or the r eceiv e r is cognizan t of the realization of  H ( ℓ ) k , k ∈ Z +  , ℓ ∈ Z + 0 , but both are aware of their statistic. W e d o not a ssume that th e path gains are Gaussian. B. Chan nel Capa city Let A n m denote the sequen ce A m , A m +1 , . . . , A n . W e defin e the c apacity as C ( SNR ) , lim n →∞ 1 n sup I  X n 1 ; Y n 1  , (3) where th e maximization is over all joint distributions on X 1 , X 2 , . . . , X n satisfying the power co nstraint 1 n n X k =1 E  | X k | 2  ≤ P , (4) and where SNR is d efined as SNR , P σ 2 . (5) By F a no’ s in equality , n o rate ab ove C ( SNR ) is achiev able. 2 (See [13] for a d efinition o f an achiev a ble rate.) Notice that the above channel (1) is generally not stationary 3 since the n umber of terms (p aths) in fluencing Y k depend s on k . It is the refore prima facie not clear wh ether the liminf on the RHS of (3) is a limit. C. Main R esult Theor e m 1: Consider the above cha nnel mo del. Then  lim ℓ →∞ α ℓ +1 α ℓ > 0  = ⇒  sup SNR > 0 C ( SNR ) < ∞  , (6) where we defin e, for any a > 0 , a/ 0 , ∞ and 0 / 0 , 0 . For example, when { α ℓ } is a geom etric sequence, i.e. , α ℓ = ρ ℓ , ℓ ∈ Z + 0 for some 0 < ρ < 1 , then capacity is b ounded . Theorem 1 is proved in Sectio n I I where it is ev e n shown that (6) would co ntinue to hold if we repla ced the limin f in (3) by a limsup. Section III addresses briefly multipath chann els of unbo unded capacity . I I . P RO O F O F T H E O R E M 1 The p roof fo llows along the same lines as the proo f of [14, Thm. 1i)]. W e first note th at it follows from the left-han d side (LHS) of (6) that we can fin d an ℓ 0 ∈ Z + 0 and a 0 < ρ < 1 so tha t α ℓ 0 > 0 an d α ℓ +1 α ℓ ≥ ρ, ℓ = ℓ 0 , ℓ 0 + 1 , . . . . (7) W e contin ue with the ch ain r ule f or mutual in formatio n 1 n I ( X n 1 ; Y n 1 ) = 1 n ℓ 0 X k =1 I  X n 1 ; Y k   Y k − 1 1  + 1 n n X k = ℓ 0 +1 I  X n 1 ; Y k   Y k − 1 1  . (8) Each term in the first sum on the right- hand side ( RHS) of (8) is upper bo unded by 4 I  X n 1 ; Y k   Y k − 1 1  ≤ h ( Y k ) − h  Y k    Y k − 1 1 , X n 1 , H (0) k , H (1) k , . . . , H ( k − 1) k  ≤ log π e σ 2 + k X ℓ =1 α k − ℓ E  | X ℓ | 2  !! − log  π eσ 2  ≤ log 1 + sup ℓ ∈ Z + 0 α ℓ · n · SNR ! , (9) where the first inequ ality fo llows be cause conditio ning can not increase entropy; the secon d inequ ality follows from the 2 See [12] for conditi ons that guarante e that C ( SNR ) is achie va ble. 3 By a stat ionary channel we mean a channel where for any stationary input { X k } the pair { ( X k , Y k ) } is jointly s tatio nary . 4 Throughout this paper , l og( · ) denote s the natural logarithm function. entropy maxim izing prop erty of Ga ussian r andom variables [13, Th m. 9. 6.5]; and the last in equality fo llows b y up per bound ing α ℓ ≤ sup ℓ ∈ Z + 0 α ℓ , ℓ = 0 , 1 , . . . , k − 1 and from the p ower constrain t (4). For k = ℓ 0 + 1 , ℓ 0 + 2 , . . . , n , we upper bound I  X n 1 ; Y k   Y k − 1 1  using th e gener al upper bo und for mutu al informa tion [5, Thm. 5.1] I ( X ; Y ) ≤ Z D  W ( ·| x )   R ( · )  Q . ( x ) , (10) where D ( ·k· ) denotes relativ e entropy , W ( ·|· ) is the channel law , Q ( · ) denotes the distribution on the chan nel inpu t X , and R ( · ) is any d istribution on the ou tput alph abet. 5 Thus, any ch oice of output distribution R ( · ) yields an upper bo und on the mutual info rmation. For any gi ven Y k − 1 1 = y k − 1 1 , we cho ose the output distribution R ( · ) to be of den sity √ β π 2 | y k | 1 1 + β | y k | 2 , y k ∈ C , (11) with β = 1 / ( ˜ β | y k − ℓ 0 | 2 ) and 6 ˜ β = min  ρ ℓ 0 − 1 α ℓ 0 max 0 ≤ ℓ ′ ≤ ℓ 0 α ℓ ′ , α ℓ 0 , ρ ℓ 0  . (12) W ith th is ch oice 0 < ˜ β < 1 and ˜ β α ℓ ≤ α ℓ + ℓ 0 , ℓ ∈ Z + 0 . (13) Using (11) in (10), and averaging over Y k − 1 1 , we ob tain I  X n 1 ; Y k   Y k − 1 1  ≤ 1 2 E  log | Y k | 2  + 1 2 E h log  ˜ β | Y k − ℓ 0 | 2  i + E h log  ˜ β | Y k − ℓ 0 | 2 + | Y k | 2  i − h  Y k   X n 1 , Y k − 1 1  − E  log | Y k − ℓ 0 | 2  + log π 2 ˜ β . (14) W e bound the term s in ( 14) sep arately . W e b egin with E  log | Y k | 2  = E  E  log | Y k | 2   X k 1  ≤ E  log  E  | Y k | 2   X k 1  = E " log σ 2 + k X ℓ =1 α k − ℓ | X ℓ | 2 !# , (15) where the inequality follows from Jensen’ s inequality . L ike- wise, we use Jensen’ s ineq uality and ( 13) to upper bou nd E h log  ˜ β | Y k − ℓ 0 | 2  i ≤ E " log ˜ β σ 2 + ˜ β k − ℓ 0 X ℓ =1 α k − ℓ 0 − ℓ | X ℓ | 2 ! # ≤ E " log σ 2 + k − ℓ 0 X ℓ =1 α k − ℓ | X ℓ | 2 !# (16) 5 For channels with finite input and output alphabets this inequ ality follo ws by T opsøe’ s identi ty [15]; see also [16, Thm. 3.4]. 6 When y k − ℓ 0 = 0 , then the density of the Cauchy distribut ion (11) is undefined. Howe ver , this e vent is of zero probability and has there fore no impact on the mutual information I ` X n 1 ; Y k ˛ ˛ Y k − 1 1 ´ . and E h log  ˜ β | Y k − ℓ 0 | 2 + | Y k | 2  i ≤ E " log 2 σ 2 + 2 k − ℓ 0 X ℓ =1 α k − ℓ | X ℓ | 2 + k X ℓ = k − ℓ 0 +1 α k − ℓ | X ℓ | 2 !# ≤ lo g 2 + E " log σ 2 + k X ℓ =1 α k − ℓ | X ℓ | 2 !# , (17) where the second inequality follows because P k ℓ = k − ℓ 0 +1 α k − ℓ | X ℓ | 2 ≤ 2 P k ℓ = k − ℓ 0 +1 α k − ℓ | X ℓ | 2 . Next, we derive a lower bound on h  Y k   X n 1 , Y k − 1 1  . Let H k ′ =  H (0) k ′ , H (1) k ′ , . . . , H ( k − 1) k ′  , k ′ = 1 , 2 , . . . , k − 1 . W e have h  Y k   X n 1 , Y k − 1 1  ≥ h  Y k   X n 1 , Y k − 1 1 , H k − 1 1  = h  Y k    X n 1 , H k − 1 1  , (18) where the inequality follows b ecause conditioning cannot increase entropy , and w here th e equality follows because, condition al on  X n 1 , H k − 1 1  , Y k is indep endent of Y k − 1 1 . Let S k be define d as S k , { ℓ = 1 , 2 , . . . , k : min {| x ℓ | 2 , α k − ℓ } > 0 } . (19) Using the en tropy power ineq uality [13, Thm. 16.6. 3], an d using that the pr ocesses  H (0) k , k ∈ Z +  ,  H (1) k , k ∈ Z +  , . . . are indep endent an d jointly ind ependen t of X n 1 , it can be shown that for any given X n 1 = x n 1 h k X ℓ =1 H ( k − ℓ ) k X ℓ + Z k      X n 1 = x n 1 , H k − 1 1 ! ≥ lo g X ℓ ∈S k e h  H ( k − ℓ ) k X ℓ   X ℓ = x ℓ , { H ( k − ℓ ) k ′ } k − 1 k ′ =1  + e h ( Z k ) ! . (20 ) W e lower boun d the differential entrop ies on the RHS of (2 0) as follows. The differential e ntropies in the sum are lower bound ed by h  H ( k − ℓ ) k X ℓ    X ℓ = x ℓ ,  H ( k − ℓ ) k ′  k − 1 k ′ =1  = lo g  α k − ℓ | x ℓ | 2  + h  H ( k − ℓ ) k     H ( k − ℓ ) k ′  k − 1 k ′ =1  − log α k − ℓ ≥ lo g  α k − ℓ | x ℓ | 2  + inf ℓ ∈L ( h ℓ − log α ℓ ) , ℓ ∈ S k , (21) where the equ ality follows fr om the behavior of dif f erential entropy under scaling; an d where the in equality follows by the stationarity of the pr ocess  H ( k − ℓ ) k , k ∈ Z +  which implies that the differential entro py h  H ( k − ℓ ) k    H ( k − ℓ ) k ′  k − 1 k ′ =1  cannot be smaller tha n the dif f erential entropy rate h k − ℓ [13, Thms. 4.2.1 & 4.2.2], and by lower boundin g ( h k − ℓ − log α k − ℓ ) by inf ℓ ∈L ( h ℓ − log α ℓ ) (which holds for each ℓ ∈ S k because S k ⊆ L ). Th e last differential entro py on the RHS of (20) is lower bo unded b y h ( Z k ) ≥ inf ℓ ∈L ( h ℓ − log α ℓ ) + log σ 2 , (22) which follows by no ting that inf ℓ ∈L ( h ℓ − log α ℓ ) ≤ log ( π e ) . (23) Applying ( 21) & (2 2) to (20), and averaging over X n 1 , yields then h  Y k   X n 1 , Y k − 1 1  ≥ E " log σ 2 + k X ℓ =1 α k − ℓ | X ℓ | 2 !# + inf ℓ ∈L ( h ℓ − log α ℓ ) . (24) W e contin ue with the analysis o f (14) by lower bou nding E  log | Y k − ℓ 0 | 2  . T o this e nd, we write the expectation a s E   E   log      k − ℓ 0 X ℓ =1 H ( k − ℓ 0 − ℓ ) k − ℓ 0 X ℓ + Z k − ℓ 0      2       X k − ℓ 0 1     and lower bou nd the co nditional exp ectation by E   log      k − ℓ 0 X ℓ =1 H ( k − ℓ 0 − ℓ ) k − ℓ 0 X ℓ + Z k − ℓ 0      2       X k − ℓ 0 1 = x k − ℓ 0 1   = lo g σ 2 + k − ℓ 0 X ℓ =1 α k − ℓ 0 − ℓ | x ℓ | 2 ! − 2 · E    log       P k − ℓ 0 ℓ =1 H ( k − ℓ 0 − ℓ ) k − ℓ 0 x ℓ + Z k − ℓ 0 q σ 2 + P k − ℓ 0 ℓ =1 α k − ℓ 0 − ℓ | x ℓ | 2       − 1    ≥ lo g σ 2 + k − ℓ 0 X ℓ =1 α k − ℓ 0 − ℓ | x ℓ | 2 ! + log δ 2 − 2 ǫ ( δ, η ) − 2 η h −   P k − ℓ 0 ℓ =1 H ( k − ℓ 0 − ℓ ) k − ℓ 0 x ℓ + Z k − ℓ 0 q σ 2 + P k − ℓ 0 ℓ =1 α k − ℓ 0 − ℓ | x ℓ | 2   (25) for some 0 < δ ≤ 1 an d 0 < η < 1 , where h − ( X ) , Z { x ∈ C : f X ( x ) > 1 } f X ( x ) log f X ( x ) x . , (26) and where ǫ ( δ, η ) > 0 tends to zero as δ ↓ 0 . (W e w rite x ℓ in lo wer case to indicate that expectation and entropy are cond itional on X k − ℓ 0 1 = x k − ℓ 0 1 .) Here, the inequality follows by writing the expectation in the form E  log | A | − 1 · I {| A | > δ }  + E  log | A | − 1 · I {| A | ≤ δ }  (where I {·} d enotes the indicator functio n), an d by upper bound ing then the first expectation by − log δ and the secon d expectation u sing [5, Lem ma 6.7] . W e contin ue b y uppe r bound ing h −   P k − ℓ 0 ℓ =1 H ( k − ℓ 0 − ℓ ) k − ℓ 0 x ℓ + Z k − ℓ 0 q σ 2 + P k − ℓ 0 ℓ =1 α k − ℓ 0 − ℓ | x ℓ | 2   = h +   P k − ℓ 0 ℓ =1 H ( k − ℓ 0 − ℓ ) k − ℓ 0 x ℓ + Z k − ℓ 0 q σ 2 + P k − ℓ 0 ℓ =1 α k − ℓ 0 − ℓ | x ℓ | 2   − h   P k − ℓ 0 ℓ =1 H ( k − ℓ 0 − ℓ ) k − ℓ 0 x ℓ + Z k − ℓ 0 q σ 2 + P k − ℓ 0 ℓ =1 α k − ℓ 0 − ℓ | x ℓ | 2   ≤ 2 e + log( π e ) + log σ 2 + k − ℓ 0 X ℓ =1 α k − ℓ 0 − ℓ | x ℓ | 2 ! − h k − ℓ 0 X ℓ =1 H ( k − ℓ 0 − ℓ ) k − ℓ 0 x ℓ + Z k − ℓ 0 ! , (27) where h + ( X ) is d efined as h + ( X ) , h ( X ) + h − ( X ) . Here, we applied [5, Lemm a 6. 4] to u pper bound h +   P k − ℓ 0 ℓ =1 H ( k − ℓ 0 − ℓ ) k − ℓ 0 x ℓ + Z k − ℓ 0 q σ 2 + P k − ℓ 0 ℓ =1 α k − ℓ 0 − ℓ | x ℓ | 2   ≤ 2 e + log ( π e ) . (28) A veraging ( 27) over X k − ℓ 0 1 yields h −   P k − ℓ 0 ℓ =1 H ( k − ℓ 0 − ℓ ) k − ℓ 0 X ℓ + Z k − ℓ 0 q σ 2 + P k − ℓ 0 ℓ =1 α k − ℓ 0 − ℓ | X ℓ | 2       X k − ℓ 0 1   ≤ 2 e + log( π e ) + E " log σ 2 + k − ℓ 0 X ℓ =1 α k − ℓ 0 − ℓ | X ℓ | 2 !# − h k − ℓ 0 X ℓ =1 H ( k − ℓ 0 − ℓ ) k − ℓ 0 X ℓ + Z k − ℓ 0      X k − ℓ 0 1 ! ≤ 2 e + log( π e ) − inf ℓ ∈L ( h ℓ − log α ℓ ) , (29) where the secon d in equality follows by conditioning the dif- ferential entro py ad ditionally on Y k − ℓ 0 − 1 1 , an d b y u sing then lower bou nd ( 24). A lower bou nd o n E  log | Y k − ℓ 0 | 2  follows now by averaging (25) over X k − ℓ 0 1 , and by ap plying ( 29) E  log | Y k − ℓ 0 | 2  ≥ E " log σ 2 + k − ℓ 0 X ℓ =1 α k − ℓ 0 − ℓ | X ℓ | 2 !# + log δ 2 − 2 ǫ ( δ, η ) − 2 η  2 e + log( π e )  + inf ℓ ∈L 2 η ( h ℓ − log α ℓ ) . (30) Returning to the analysis o f (1 4), we obtain f rom (3 0), (2 4), (17), (16), and (15) I  X n 1 ; Y k   Y k − 1 1  ≤ 1 2 E " log σ 2 + k X ℓ =1 α k − ℓ | X ℓ | 2 !# + 1 2 E " log σ 2 + k − ℓ 0 X ℓ =1 α k − ℓ | X ℓ | 2 !# + log 2 + E " log σ 2 + k X ℓ =1 α k − ℓ | X ℓ | 2 !# − E " log σ 2 + k X ℓ =1 α k − ℓ | X ℓ | 2 !# − inf ℓ ∈L ( h ℓ − log α ℓ ) − E " log σ 2 + k − ℓ 0 X ℓ =1 α k − ℓ 0 − ℓ | X ℓ | 2 !# − log δ 2 + 2 ǫ ( δ, η ) + 2 η  2 e + log( π e )  − inf ℓ ∈L 2 η ( h ℓ − log α ℓ ) + log π 2 ˜ β ≤ K + E " log σ 2 + k X ℓ =1 α k − ℓ | X ℓ | 2 !# − E " log σ 2 + k − ℓ 0 X ℓ =1 α k − ℓ 0 − ℓ | X ℓ | 2 !# , (31) with K , − inf ℓ ∈L  1 + 2 η  ( h ℓ − log α ℓ ) + log 2 π 2 ˜ β δ 2 + 2 ǫ ( δ, η ) + 2 η  2 e + log( π e )  . (32) The second inequality in (31) follows because P k − ℓ 0 ℓ =1 α k − ℓ | X ℓ | 2 ≤ P k ℓ =1 α k − ℓ | X ℓ | 2 . In order to sho w that the capacity i s bound ed in the SNR, we apply ( 31) and (9) to (8) and use then that f or any seq uences { a k } and { b k } n X k = ℓ 0 +1 ( a k − b k ) = n X k = n − ℓ 0 +1 ( a k − b k − n +2 ℓ 0 ) + n − ℓ 0 X k = ℓ 0 +1 ( a k − b k + ℓ 0 ) . (33) Defining a k , E " log σ 2 + k X ℓ =1 α k − ℓ | X ℓ | 2 !# (34) and b k , E " log σ 2 + k − ℓ 0 X ℓ =1 α k − ℓ 0 − ℓ | X ℓ | 2 !# (35) we have for th e first sum o n th e RHS of (33) n X k = n − ℓ 0 +1 ( a k − b k − n +2 ℓ 0 ) = n X k = n − ℓ 0 +1 E " log σ 2 + P k ℓ =1 α k − ℓ | X ℓ | 2 σ 2 + P k − n + ℓ 0 ℓ =1 α k − n + ℓ 0 − ℓ | X ℓ | 2 !# ≤ ℓ 0 log 1 + sup ℓ ∈ Z + 0 α ℓ · n · SNR ! , (36) which follows by lower b oundin g th e deno minator by σ 2 , and by using the n Jensen’ s in equality along with th e last ineq uality in (9). For th e second sum on the RHS o f ( 33) we have n − ℓ 0 X k = ℓ 0 +1 ( a k − b k + ℓ 0 ) = n − ℓ 0 X k = ℓ 0 +1 E " log σ 2 + P k ℓ =1 α k − ℓ | X ℓ | 2 σ 2 + P k ℓ =1 α k − ℓ | X ℓ | 2 !# = 0 . (37) Thus, apply ing (31)–(3 7) and (9) to ( 8), we obta in 1 n I ( X n 1 ; Y n 1 ) ≤ ℓ 0 n log σ 2 + sup ℓ ∈ Z + 0 α ℓ · n · SNR ! + ℓ 0 n log σ 2 + sup ℓ ∈ Z + 0 α ℓ · n · SNR ! + n − 2 ℓ 0 n K , (38) which tends to K < ∞ a s n te nds to infinity . This proves Theorem 1. I I I . M U LT I P AT H C H A N N E L S O F U N B O U N D E D C A PAC I T Y W e have seen in Th eorem 1 that if th e variances of the p ath gains { α ℓ } do n ot decay faster than geo metrically , then ca- pacity is bound ed in the SNR. In this section, we d emonstrate that this need not b e the case when the variances of th e path gains decay faster th an geometrica lly . Th e f ollowing theorem presents a sufficient conditio n for the cap acity C ( SNR ) to be unbou nded in the SNR. Theor e m 2: Consider the above cha nnel mo del. Then  lim ℓ →∞ 1 ℓ log log 1 α ℓ = ∞  = ⇒  sup SNR > 0 C ( SNR ) = ∞  . (3 9) Pr oof: Omitted. Note: W e d o not cla im that C ( SNR ) is ac hiev able. Ho w- ev e r , it can be sh own that when, for example, th e p rocesses  H ( ℓ ) k , k ∈ Z +  , ℓ ∈ Z + 0 are IID Gaussian, th en the maxim um achiev ab le rate is unbou nded in the SNR, i.e., any r ate is achiev ab le for sufficiently large SNR. Certainly , the cond ition on the LHS of (39) is satisfied when the ch annel ha s fin ite m emory in the sense that for so me finite L ∈ Z + 0 α ℓ = 0 , ℓ = L + 1 , L + 2 , . . . . In this case, (1) b ecomes Y k =            k − 1 X ℓ =0 H ( ℓ ) k x k − ℓ + Z k , k = 1 , 2 , . . . , L L X ℓ =0 H ( ℓ ) k x k − ℓ + Z k , k = L + 1 , L + 2 , . . . . (40) This ch annel (40) was studied for g eneral (but finite) L in [17] where it was shown that its cap acity satisfies lim SNR →∞ C ( SNR ) log log SNR = 1 . 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