On the Boundedness of the Support of Optimal Input Measures for Rayleigh Fading Channels

On the Boundedness of the Suppo rt of Optimal Input Measures fo r Rayleigh F ading Ch annels Jochen Sommerfeld 1 , Igor Bjelakovi ´ c 1 , 2 , and Holger Boche 1 , 2 1 Heinrich-Hertz-Chair for Mobile Communications T ec hnische Unive rsit ¨ at Berlin W erner-v on-Siemens-Bau (HFT 6), Einsteinufer 25, 10587 Berli n, Germany & 2 Institut f ¨ ur Mathematik T echnische Unive rsit ¨ at Berlin Straße des 17. Juni 136, 10623 Berlin, Germany Email: { jochen.somme rfeld, igor .bjelako vic, holger .boche } @mk.tu-berlin.de Abstract — W e consider transmission o ver a wir eless multiple antenna communication system operating in a Rayleigh flat fading en vironment with no channel state inform ation at the recei ver and the transmitter with coherence time T = 1 . W e show that, subject to th e av erage p ower constraint, th e support of the capacity achie ving input distribution is bounded. Mor eov er , we show by a simple example concerning the identity theorem (or uniqueness theor em) fro m the complex a nalysis in sev eral variables t hat some of the existing results in the fi eld are not rigoro us. I . I N T RO D U C T I O N W e show in this pa per by elemen tary means that t he sup port of the capacity achieving input measure for multiple-inpu t multiple-ou tput (MIMO) R ayleigh fading ch annels su bject to av erage power co nstraint with coheren ce time T = 1 is bound ed. A generalization of the result to coh erence intervals of size T > 1 seems to be hig hly no n-trivial and will p robably require a substantial extension of the techniq ues u sed here supplemen ted by som e results and m ethods from the “hard analysis”. Previous fund amental achie vements, e.g. [1], [3], [5], [6], follow the same pro cedure which can b e tr aced back to the classic paper [8] by Smith. The basic to ols are the Karu sh- Kuhn-T ucker ( KKT) conditions fro m th e th eory o f co n vex optimization supported by an applicatio n of the identity the- orem (also kno wn as th e uniquen ess th eorem) from com plex analysis. Our appr oach is based on the KKT co nditions to o but av oids the u sage of the id entity theor em. In [1] Ab ou-Faycal, Trott, and Shamai proved, using these technique s, that for a o ne-dimen sional Rayleigh fading channel the optimal inp ut measure subjec ted to an a verage po wer constraint to be discrete with a fin ite nu mber of mass po ints. In [3] Chan, Hr anilovic, and Ksch ischang showed for a MIMO Rayleigh blo ck-fading chan nel with i.i.d. ch annel matrix co ef- ficients that th e o ptimum in put distribution subjected to peak and av erage power co nstraint contains a finite nu mber o f mass points with respect to a spec ific norm . In addition Fozunb al, Mclaughlin , and Schafer argued in [ 5] that a bound ed supp ort of the cap acity m aximizer implies its sing ularity with respect to the Borel-Leb esgue measure. Th e appr oach in [5], [3] is based o n the iden tity theo rem for holom orphic fun ctions in se veral comp lex variables and use the assumption that an open set in R n fulfills the h ypothesis of the identity theor em in C n . W e show in section IV by a simple example that the conclusion of the identity theorem fails in this setting. Consequently , these results are not rig orous. Sin ce, in contrast to th e com plex analysis in on e variable, it is still an o pen difficult proble m to characterize th e families of sets for which the identity the orem for holomo rphic functions in sev eral complex variables ho lds we cann ot hop e to under stand the proper ties of the cap acity max imizers in the present setting by an reduction to u niquen ess pro perties o f ho lomorp hic function s in higher dim ensions. Ther efore, it is likely that we will be forced to develop o r apply “rea l-analytic” tools for tackling this imp ortant commu nication-th eoretic problem. The paper is organize d as f ollows: Section II provides some basic d efinitions an d is followed by Sec tion I II wh ich co ntains the m ain resu lt of this pape r . As mentio ned above, in Section IV we give a n elementar y example that shows that th e a p- plication o f the identity theorem in h igher dimen sions is, in general, no t admissible if we want to un derstand the pro perties of capacity maximizer s of Rayleig h fading ch annels. Notation. Throug hout the paper we will denote the set of complex N -by- 1 matrices by M ( N × 1 , C ) and will freely identify this set with C N . ln stands fo r the log arithm to the base e . Capital letters X , Y , H are reserved for random variables. I I . R A Y L E I G H FA D I N G C H A N N E L W e co nsider a Rayleigh fading chann el with the coh erence time T = 1 which is d escribed by Y m = N X n =1 H mn X n + Z m (1) with coefficient matrices Y , Z ∈ M ( M × 1 , C ) , X ∈ M ( N × 1 , C ) and H ∈ M ( M × N , C ) , where th e th e channel H is assumed to be co mplex circularly symm etric Gaussian with zero m ean and with covariance ma trix Σ a nd the add iti ve no ise coefficients Z m are assumed to be i.i.d. complex circu larly sy mmetric Gaussian with C N (0 , σ 2 Z ) . Let P ( X ) b e the set of prob ability measure s o n ( M ( N × 1 , C ) , Σ B or el ( M ( N × 1 , C ))) . T hen the set µ g,a ( X ) = { µ ∈ P | Z ( g ( x ) − a ) dµ ( x ) ≤ 0 } (2) with the av erage power constraint of the tran smitted signal Z ( g ( x ) − a ) dµ ( x ) = Z 1 N N X n =1 | x n | 2 dµ ( x ) − a ≤ 0 (3) is weak* compact as it was shown in [5] an d [4 ]. If P ( Y ) is th e set of cond itional pro bability measures on ( M ( M × 1 , C ) , Σ B or el ( M ( M × 1 , C ))) we can determin e the channel b y a set { W ( ·| x ) ∈ P ( Y ) | x ∈ M ( N × 1 , C ) } , where W ( ·| x ) is absolutely c ontinuou s w ith respe ct to Borel- Lebesgue measur e. For the Rayleigh fadin g chann el the cond i- tional probab ility density o f th e received signals y con ditioned on the input symb ol x is given by p ( y | x ) = e − tr [ ( σ 2 Z 1 M +( 1 M ⊗ x H )Σ( 1 M ⊗ x )) − 1 y y H ] π M det ( σ 2 Z 1 M + ( 1 M ⊗ x H )Σ( 1 M ⊗ x )) (4) with covariance matrix Σ of H Σ = E ( H ⊗ H ∗ ) . (5) Let µ ∈ µ g,a ( X ) be a prob ability measur e and defin e f µ ( y ) := Z p ( y | x ) µ ( dx ) . (6) Then th e mutual information of th e channel with n o CSI a t the re ceiv er is given by I ( µ ; W ) = Z p ( y | x ) log p ( y | x ) f µ ( y ) dy dµ ( x ) . (7) The m utual Inform ation is a weak* con tinuous fu nctional o n the weak* comp act and conve x set µ g,a ( X ) (see [ 6]). T hus the f unctional I ( µ ; W ) achieves its m aximum on µ g,a ( X ) by the fo llowing Theor em 2.1 ( Cf. [1]): Let f be a weak * continu ous re al- valued function al on a weak* comp act subset S of X ∗ . Then f is bo unded on S and ach iev es its maxim um on S . The m utual information is strictly conc av e functio nal on µ g,a ( X ) up to equiva lence of measures. Hereby , tw o measur es µ, ν ∈ µ g,a ( X ) are called equiv alent if f µ ( y ) = f ν ( y ) . So its m aximum on µ g,a ( X ) is achieved by a unique input distribution up to eq uiv alence defined above [6]. He nce, with C ( a ) = sup µ ∈ µ g,a ( X ) I ( µ ; W ) (8) there exists a measure µ 0 ∈ µ g,a ( X ) that achiev es the capacity of th e c hannel and is unique u p to eq uiv alence of measures. The aim of this paper is to show that subjected to an average power con straint the capacity achieving distribution of the channel has an bou nded sup port. I I I . B O U N D E D S U P P O RT O F O P T I M A L I N P U T D I S T R I B U T I O N The p urpose o f this sectio n is to show that the sup port o f the capacity achieving input measur e for the channe l g iv en in (4), with c oherence time T = 1 , is bound ed. For r 1 , r 2 ∈ R with 0 ≤ r 1 < r 2 we set B ( r 1 , r 2 ) := { x ∈ M ( N × 1 , C ) : r 1 ≤ tr ( xx H ) ≤ r 2 } , (9) with h x, x i := tr ( xx H ) = k x k 2 . Lemma 3.1: Le t r 1 , r 2 ∈ R with 0 ≤ r 1 < r 2 and µ ( B ( r 1 , r 2 )) > 0 with µ ∈ µ g,a ( X ) be given. Then Z p ( y | x ) log f µ ( y ) dy ≥ log µ ( B ( r 1 , r 2 )) π M Π − M ( σ 2 Z + λ min x H x ) ( σ 2 Z + λ max r 1 ) (10) with Π := max x ∈ B ( r 1 ,r 2 ) det ( σ 2 Z 1 M + ( 1 M ⊗ x H )Σ( 1 M ⊗ x )) and λ min > 0 and λ max > 0 are the m inimum and m aximum eigenv alues of th e covariance matrix Σ . Pr oo f: By the defining relation (6) we have f µ ( y ) := Z p ( y | x ) µ ( dx ) ≥ Z B ( r 1 ,r 2 ) p ( y | x ) µ ( dx ) (11) Next we define Π := max x ∈ B ( r 1 ,r 2 ) det ( σ 2 Z 1 M + ( 1 M ⊗ x H )Σ( 1 M ⊗ x )) , (12 ) whereas the maximum of the function is achieved on B ( r 1 , r 2 ) because of the comp actness of B ( r 1 , r 2 ) . Hence , for x ∈ B ( r 1 , r 2 ) we obtain p ( y | x ) ≥ e − tr [ ( σ 2 Z 1 M +( 1 M ⊗ x H )Σ( 1 M ⊗ x )) − 1 y y H ] π M Π . (13) For every x ∈ M ( M × 1 , C ) we have ( σ 2 Z + λ min x H x ) 1 M ≤ ( σ 2 Z 1 M + ( 1 M ⊗ x H )Σ( 1 M ⊗ x )) ≤ ( σ 2 Z + λ max x H x ) 1 M (14) where λ min > 0 and λ max > 0 are th e m inimum and maximum eigenv alues of the herm itian and strictly positive covariance matrix Σ . By the d efinition of B ( r 1 , r 2 ) we have r 1 ≤ tr ( xx H ) = x H x = k x k 2 ( x ∈ B ( r 1 , r 2 )) . (15) Hence, it follows that σ 2 Z + λ min x H x ≥ σ 2 Z + λ min r 1 . (16) For t wo operators A, B ∈ M ( N , C ) with A ≤ B an d a positi ve operator R ∈ M ( N , C ) we have tr ( AR ) ≤ tr ( B R ) . (17) Due to the fact th at the operato rs in (14) are he rmitian an d positive and the same h olds for y y H and because th e f unction f ( A ) = − A − 1 is operator monoton e for all positiv e o perators [2], we h av e tr  (( σ 2 Z + λ min r 1 ) 1 M ) − 1 y y H  ≥ tr  (( σ 2 Z + λ min x H x ) 1 M ) − 1 y y H  ≥ tr  ( σ 2 Z 1 M + ( 1 M ⊗ x H )Σ( 1 M ⊗ x )) − 1 y y H  . (18) W ith (13) it follows tha t for x ∈ B ( r 1 , r 2 ) p ( y | x ) ≥ e − tr [ (( σ 2 Z + λ min r 1 ) 1 M ) − 1 y y H ] π M Π (19) Inserting this in to (11) yields f µ ( y ) ≥ µ ( B ( r 1 , r 2 )) π M Π e − tr [ (( σ 2 Z + λ min r 1 ) 1 M ) − 1 y y H ] . (20) Therewith we get Z p ( y | x ) log f µ ( y ) dy ≥ Z p ( y | x ) log  µ ( B ( r 1 , r 2 )) π M Π e − tr [ (( σ 2 Z + λ min r 1 ) 1 M ) − 1 y y H ]  dy = log A − Z tr  (( σ 2 Z + λ min r 1 ) 1 M ) − 1 y y H  p ( y | x ) dy = log A − Z k y k 2 ( σ 2 Z + λ min r 1 ) p ( y | x ) dy = log A − tr ( σ 2 Z 1 M + ( 1 M ⊗ x H )Σ( 1 M ⊗ x )) ( σ 2 Z + λ min r 1 ) ≥ log A − tr (( σ 2 Z + λ max x H x ) 1 M ) ( σ 2 Z + λ min r 1 ) = log A − M ( σ 2 Z + λ max k x k 2 ) σ 2 Z + λ min r 1 (21) with A := µ ( B ( r 1 , r 2 )) π M Π . Determining the cap acity ach ieving input distrib ution sub- jected to average power constraint is a conve x o ptimization problem . Necessary co nditions f or the op timal inp ut distri- bution can b e der i ved from th e local Karu sh-K uhn-T u ck er condition s. T ogether with th e fact th at the mutu al in formatio n is a concave fu nctional and the co n vexity of th e con straint function al we obtain (see [7] and [6 ]), that µ achieves capacity if and only if γ ( 1 N k x k 2 − a ) + C ( a ) − Z p ( y | x ) log p ( y | x ) f µ ( y ) dy ≥ 0 (22) with e quality if x ∈ supp ( µ ) , where γ = γ ( a ) ≥ 0 de notes the Lag range m ultiplier and Z 1 N X n | x n | 2 dµ ( x ) ≤ a is the constraint und er co nsideration. It is fairly standard fact that Z p ( y | x ) log p ( y | x ) dy = − log  ( π e ) M det( σ 2 Z 1 M + ( 1 M ⊗ x H )Σ( 1 M ⊗ x ))  and (22) c an be therefor e rewritten as γ ( 1 N k x k 2 − a ) + C ( a ) + lo g( π e ) M + + log det( σ 2 Z 1 M + ( 1 M ⊗ x H )Σ( 1 M ⊗ x ))+ + Z p ( y | x ) log f µ ( y ) dy ≥ 0 (23) with equality if x ∈ su pp ( µ ) . Let K K T ( x ) := γ ( 1 N k x k 2 − a ) + C ( a ) + log( π e ) M + + log det( σ 2 Z 1 M + ( 1 M ⊗ x H )Σ( 1 M ⊗ x ))+ + Z p ( y | x ) log f µ ( y ) dy (24) Then (22) ca n be r ephrased as K K T ( x ) ≥ 0 f or x ∈ M ( N × 1 , C ) and K K T ( x ) = 0 if x ∈ supp ( µ ) . The following the orem g i ves a sufficient condition for the bound edness o f the suppor t of the capacity achieving m easure in terms of the Lagran ge multiplier γ . Lemma 3.2: Le t a ∈ R + be given and let µ b e a cap acity achieving input measu re subject to the av erage p ower con - straint a fo r the chan nel (4). Then γ ( a ) > 0 imp lies that supp ( µ ) is boun ded. Pr oo f: Th e p roof is by contradictio n. Suppose that γ ( a ) = γ > 0 a nd that supp ( µ ) is not bo unded. By our assumptions we can find r 1 , r 2 ∈ R with the following proper ties: µ ( B ( r 1 , r 2 )) > 0 (25) γ − M N λ max σ 2 Z + λ min r 1 > 0 . (26 ) Applying Lemma 3.1 to the fun ction K K T ( x ) d efined in ( 24) we ob tain the following inequality . K K T ( x ) ≥ γ ( 1 N k x k 2 − a ) + C ( a ) + log( π e ) M + + log det( σ 2 Z 1 M + ( 1 M ⊗ x H )Σ( 1 M ⊗ x ))+ + log A − M ( σ 2 Z + λ max k x k 2 ) σ 2 Z + λ min r 1 = k x k 2 ( γ N − M λ max σ 2 Z + λ min r 1 ) − γ a + C ( a ) + log( π e ) M + + log det( σ 2 Z 1 M + ( 1 M ⊗ x H )Σ( 1 M ⊗ x ))+ + log A − M σ 2 Z σ 2 Z + λ min r 1 (27) Combining th e Kar ush-Kuhn-Tucker co nditions and (2 7) we obtain that for any x ∈ supp ( µ ) 0 = K K T ( x ) ≥ k x k 2 ( γ N − M λ max σ 2 Z + λ min r 1 ) − γ a + C ( a ) + log( π e ) M + + log det( σ 2 Z 1 M + ( 1 M ⊗ x H )Σ( 1 M ⊗ x ))+ + log A − M σ 2 Z σ 2 Z + λ min r 1 (28) But this last inequ ality with ou r assum ption th at supp ( µ ) is not bou nded, (26), and the fact that k x k 2 ( 1 N γ − M λ max σ 2 Z + λ min r 1 ) → ∞ as x → ∞ and log det( σ 2 Z 1 M + ( 1 M ⊗ x H )Σ( 1 M ⊗ x )) → ∞ as x → ∞ implies tha t 0 ≥ ∞ , which is the d esired con tradiction. In view of Lemma 3.2 o ur rem aining goal is to show that γ ( a ) > 0 f or e ach a ∈ R + . For example in [1] Abo u-Faycal, T rott, and Shamai showed this in th e scalar case. Our pr oof of th e c orrespon ding result in MI MO case b elow is strongly motiv ated b y their ap proach via Fano’ s ine quality . Lemma 3.3: For the channel given in (4) w e have γ ( a ) > 0 for each a ∈ R + . Pr oo f: As men tioned above th e proof is an extension of th e argum ent given in [1]. The c apacity fun ctional C ( · ) is a non -decreasing and co ncave fu nction of the argumen t a ∈ R + . It was observed in [ 1] using global Karu sh-Kuhn-T ucker condition s that γ ( a ) is the slope o f the tangent line to C ( · ) at a (cf. [1], Section III .B and Appendix II.A). Thu s, sin ce C ( a ) is non-d ecreasing and concave, it can be shown that γ ( a ) = 0 implies C ( a ′ ) = C ( a ) f or all a ′ ≥ a 1 . Conseq uently , we can rule out the p ossibility th at γ ( a ) = 0 by showing the existence of a seq uence of input measur es su ch that th e correspo nding sequence of mutua l info rmations appro aches ∞ . W e will be done if ther e is λ > 0 such that for each n ∈ N we can fin d distinct x 1 = x 1 ( n ) , . . . , x n = x n ( n ) ∈ C N and disjoint measu rable sets B 1 = B 1 ( n ) , . . . , B n = B n ( n ) ⊂ C M such that Z B i p ( y | x i ) dy ≥ λ for all i = 1 , . . . , n . Because a simp le applicatio n of Fano’ s inequality with block length 1 shows then that for the input measures µ n := 1 n P n i =1 δ x i ( δ x i is the po int measure co n- centrated on x i ) we have I ( µ n , W ) ≥ λ log n − 1 . Now we d efine λ := 1 2 ω 2 M λ min 2 π M λ max e − σ 2 Z + λ min λ min > 0 where ω 2 M denotes the sur face area of the unit sph ere in C M ≃ R 2 M and λ min , λ max are th e sm allest an d the largest eigenv alues of Σ and let n ∈ N be given. W e will now pr esent the constru ction of the vectors x 1 = x 1 ( n ) , . . . , x n = x n ( n ) ∈ C N and the decod ing sets B 1 = B 1 ( n ) , . . . , B n = B n ( n ) . Let x ∈ C N with || x || = 1 be fixed and consider a large p ositi ve real number K = K ( n ) ≥ 1 that will b e specified later . Set x i := K i x for i = 1 , . . . , n where K i := K 2 i . 1 This implicat ion is not obvious since C ( · ) need not be dif ferenti able. Ho wev er , C ( · ) is dif ferenti able a.e. due to the monotonicity and concavit y . The proof that C ( a ) = C ( a ′ ) for all a ′ ≥ a follo ws a standard line of reasoning from the real analysis and is skipped due to the space limitat ion. The full argument will be gi ven elsewhe re. Let λ min denote the smallest eigen value of Σ . F or i = 1 , . . . , n we set r i = r i ( K ) := q σ 2 Z + λ min K i (29) and B i := D ( r i , r i +1 ) where D ( r i , r i +1 ) = { y ∈ C M : r i ≤ tr ( y y H ) = h y , y i < r i +1 } . As shown in the proof of Lemma 3 .1 we have p ( y | x ) ≥ e − h y,y i σ 2 Z + λ min k x k 2 π M det( σ 2 Z + λ max k x k 2 1 M ) . (30) Using (30) an d transformin g to spherical coo rdinates in C M ≃ R 2 M we obtain Z B i p ( y | x i ) dy ≥ ω 2 M π M ( σ 2 Z + λ max K 2 i ) M × Z r i +1 r i e − a i r 2 r 2 M − 1 dr , (31) where ω 2 M denotes the surface ar ea of the unit sphere in C M and a i = a i ( K ) := 1 σ 2 Z + λ min K 2 i . After th e substitution t = a i r 2 in the integral on the RHS of the ineq uality (31) we arrive at Z B i p ( y | x i ) dy ≥ ω 2 M ( σ 2 Z + λ min K 2 i ) M 2 π M ( σ 2 Z + λ max K 2 i ) M × Z a i r 2 i +1 a i r 2 i e − t t M − 1 dt. (32) In what fo llows we u se the abbreviation F ( K i ) := ω 2 M ( σ 2 Z + λ min K 2 i ) M 2 π M ( σ 2 Z + λ max K 2 i ) M . (33) The defining relation (2 9) an d our assumption that K ≥ 1 ensure that a i r 2 i ≥ 1 . Using th is and ( 32) we ar e led to Z B i p ( y | x i ) dy ≥ F ( K i ) Z a i r 2 i +1 a i r 2 i e − t t M − 1 dt ≥ F ( K i ) Z a i r 2 i +1 a i r 2 i e − t dt = F ( K i )( e − a i r 2 i − e − a i r 2 i +1 ) , (34 ) for all i = 1 , . . . , n . Now , since K i = K 2 i , a i = a i ( K ) := 1 σ 2 Z + λ min K 2 i , an d r i = r i ( K ) := p σ 2 Z + λ min K i it is c lear that a i r 2 i +1 → ∞ as K → ∞ , a i r 2 i = σ 2 Z + λ min λ min , as K → ∞ and fro m (3 3) we have F ( K i ) → ω 2 M λ min 2 π M λ max as K → ∞ for all i = 1 , . . . , n . Thu s if we c hoose our K sufficiently large (34) and these limit re lations ensure th at Z B i p ( y | x i ) dy ≥ 1 2 ω 2 M λ min 2 π M λ max e − σ 2 Z + λ min λ min = λ > 0 , for all i = 1 , . . . , n . Mor eover it is c lear that the sequ ence of second moments of the measures µ n = 1 n P n i =1 δ x i can be made arbitra rily large for large K ( n ) . This con cludes ou r proof by the remar ks given at the b eginning of the argum ent. Now , we can sum marize ou r results obtained so far in th e following fashion: Theor em 3.4: W e con sider the chann el defin ed by (4). Then the support of the capacity achieving input measure is bound ed. Pr oo f: Simply apply Lem ma 3 .3 and L emma 3.2. I V . D I S C U S S I O N W ith the emb edding f unction ξ : C N → R 2 N ∈ C 2 N with z i = Re ( x i ) an d z i +1 = Im ( x i ) and the tran sformed chan nel we get an extension of the f unction K K T ( x ) : M ( N × 1 , C ) → R to K K T ( z ) : M (2 N × 1 , C ) → C where K K T ( z ) := γ ( 1 N z T z − a ) + C ( a ) − Z ˜ p ( ˜ y | z ) log ˜ p ( ˜ y | z ) f µ ( ˜ y ) d ˜ y . (35) ˜ p and ˜ y ∈ M (2 M × 1 , R ) are obta ined by changing the channel matrix and the chan nel ou tput accord ing the transfor mation of th e inp ut under ξ (in [3] p . 20 81, [5]). Moreover it is easily seen u sing Fu bini’ s theorem from measure th eory and Mor era’ s theor em from the co mplex a nalysis in several variables (cf. [9]) that this extension of th e fu nction K K T is holomo rphic. But, un fortuna tely , it is not true that th e iden tity theorem (also kn own a s th e un iqueness the orem) h olds for open sets in R 2 N as the fo llowing standard example shows: Example. W e con sider the simp lest n on-trivial case C 2 . Let { e 1 , e 2 } denote the stand ard basis of C 2 and let f : C 2 → C be defined as f ( z ) := z T e 2 = z 1 · 0 + z 2 · 1 = z 2 where T denotes the transpo se and z 1 , z 2 are th e c oordina tes of z ∈ C 2 with respect to the basis { e 1 , e 2 } . Clearly , f is holomo rphic and th e set o f zeros of f is N ( f ) = { C · e 1 } ≃ R 2 . In wh at fo llows we identify N ( f ) with R 2 . R 2 is, b y defini- tion, open in the natu ral topology on R 2 (but it is not open in the natu ral topolog y of C 2 , it is a clo sed linear subspace of C 2 ), an d the functio n f is, appa rently , not identically zer o on C 2 . Note that th is exam ple with the identical arguments shows also that the conclusion of the identity theorem is not valid for open balls, say , in R 2 ⊂ C 2 . If B ⊂ R 2 ⊂ C 2 is any op en ball in R 2 then f ( z ) = 0 f or all z ∈ B but, ag ain, f 6 = 0 on C 2 . The rea son is, as b efore, th at an open ball in R 2 (with th e natural top ology of R 2 ) is not o pen in th e topolog y of C 2 . This last example shows that the proof of Propo sition 4.3 in [5] is not corr ect, since it a ssumes the validity of the id entity theorem in exactly this setting . It is th is Proposition 4.3 in [5] which would a llow us to con clude that the suppor t of th e capacity achieving input measur e contains no op en sets (in C N ≃ R 2 N ) pr ovided we know that this sup port is b ound ed. Actually , th e authors of this pap er ar e con vinced that we need different mathematical techniq ues to tackle the problem of characterizatio n of the o ptimal inputs for m ultiple antenna Rayleigh fading systems no t r elying on the id entity th eorem. One r eason for this opinio n is the fact th at the characterization of sets for wh ich the ide ntity theor em holds (so called sets of uniquen ess) in the setting of se veral com plex variables is a long standing challen ging o pen problem in comp lex analysis. V . C O N C L U S I O N S A N D F U T U R E W O R K W e have shown that f or a Rayleigh fading chann el with coheren ce time T = 1 the sup port of the cap acity ach ieving input measure is bou nded. Our method of proof does not allow to extend th e resu lts to the case T > 1 . In fact the technique s we have used have to be substantially sharpene d and supplem ented by add itional n ew tools. Furtherm ore we have shown th at the ap proach based on the application of the identity theorem from the complex analysis in se vera l variables is not admissible. Ther efore, it seem s highly likely for us that the techniq ues nee ded shou ld be “real- analytic” in spirit. A C K N O W L E D G M E N T This w ork is supported by the Deu tsche Forschun gsge- meinschaft DFG via pro ject BO 173 4/16-1 ”Entwurf von geometrisch -algebraisch en und analytischen Method en zur Optimierun g von MIMO Kommunikationssystemen ”. R E F E R E N C E S [1] I.C. Abou-Fayca l, M.D. Trott and S. Shamai (Shitz), “The Capacity of Discrete -Time Memoryless Rayleigh-F ading Channels”, IEEE T rans. Inform. Theory , vol. 47(4), pp. 1290-1301, May 2001 [2] R.Bhatia, “Matrix Analysis”, Graduate T exts in Mathemati cs;169, Springer -V er lag, Berlin 1997 [3] T . H. Chan, S. Hranilovi c and F .R. 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