Channel Dependent Mutual Information in Index Modulations

CHANNEL DEPENDENT MUTU AL INFORMA TION IN INDEX MODULA TIONS P ol Henar ejos ⋆ , Ana P ´ er ez-Neira ⋆ , Anxo T ato † , Carlos Mosquera † ⋆ Centre T ecnol ` ogic de T elecom unicacions de Catalunya (CTTC), Castelldefels, Spain † atlanTT ic Research Center , Univ ersity of V igo, Spain Email: { pol.henarejos,ana.perez } @cttc.es ⋆ , { anxotato,m osquera } @gts.uvi g o.es † ABSTRA CT Mutual Information is the m etric that is used to perfo rm link adaptation , which allows to ach iev e rates near c a pacity . The computatio n of adaptive tran smission mod es is achieved by employing th e mapping between th e Signal to No ise Ratio and th e Mutual I nform a tion. Due to the high c omplexity of the compu tation o f the Mutual Info rmation, this pr o cess is perfor med off-line v ia Monte Carlo simulations, who se r e- sults are store d in look -up tables. Ho wev er , in Ind ex Modu - lations, such as Spatial Modulation or Polarized Mod ulation, this is not f easible since the constellation an d the Mutual In - formation are channel depend ent and it would require to com- pute this metric at each time instant if the ch annel is time varying. In this pa p er , we pro p ose different app r oximation s in ord er to obtain a simple closed- form expression that allows to c o mpute the Mutual Inf o rmation at each time instant and thus, making feasib le the link ad aptation. Index T erms — Mutual I nform ation, Spatial Modu lation, Polarized Modu lation, In dex Modu lations, Link Adaptatio n 1. INTR ODUCTION Link Adap tation in modern commun ications is perfo rmed by comp u ting th e Effecti ve Sig n al to No ise (SNR) Mapping (ESM) based o n Mutual Inf ormation (MI-ESM) [1]–[3]. For instance, the work described in [4] describes the pro c edure of computing M I-ESM in Single-Input Single-Output sys- tems for IEEE 802.16e standard . Analogou sly , authors o f [5] describe the MI-E SM algo rithm fo r L ong T erm Evolution (L TE) networks. All of these works ha ve in common the co m- putation of the Mu tual Infor mation (MI) , wh ich in volves an expectation o f a func tion of a Rando m V ariable ( R V) withou t closed-for m solution. In the literature, th e co mputation o f th e expectatio n of MI is p erform ed off-line v ia Monte Carlo simulatio ns and the re - sults are stored in to a look-u p table (LUT). After this step, the received SNR o f ea ch sym bol within a cod e b lock or frame is mapped to th e LUT to obtain the MI correspon ding to the SNR. This work is funded by projects MYRAD A (TEC2016-75103-C2-2-R), ELISA (TEC2014-5925 5-C3-1-R) and TERESA (TEC2017-9009 3-C3-1-R). In I ndex M odulation s (IM ), such as Sp atial Mod ulation [6] o r Polarized Modulatio n [7], the inform ation is transmit- ted not only with a fixed constellation , such as Qu adratur e Amplitude Modu lation (QAM), but also with the channel hops. Due to the d ependen ce o n the chan nel, th e MI com- putation cann o t b e perfor med off-line sin ce the expr essions contain th e ch annel rea lization [8]. The solution is to compute the MI cur ve in each time in stant, dep ending o n the channe l realization. Du e to the h igh computatio n al comp lexity of MI computatio n, this ap proach is no t feasible. This paper presen ts closed- f orm expr essions ba sed on d if- ferent order app roximatio ns o f th e MI of IM . Based on the works [9]–[11], which c o mpute the capacity of I M, we aim at solving the difficulty of finding a closed-for m expression of MI. Thanks to th is expression, we are a b le to co m pute the MI at each time instan t with mu c h less comp utational co mplexity and ma king the p roblem of adaptive IM affordab le. Hence, the M I estimated is u sed to select the Modu la tio n and Coding Scheme in the link ad aptation alg orithm pro cess. 2. SYSTEM MODEL AND MUTU AL INFORMA TION Giv en a discrete time instant, the IM over an arb itrary Multiple-In put Multiple-Output (MI MO) chan nel realizatio n , with t in puts and r outpu ts, is defined as y = √ γ H x + w , (1) where y ∈ C r is the received vector, γ is the average SNR, x = l s , l is the all-zero vecto r excep t at po sition l that is 1, H = [ h 1 . . . h t ] ∈ C r × t is the channe l ma trix, l ∈ [1 , t ] is the hopp ing ind ex, s ∈ C is the complex symb ol from the constellation S . T h e A WGN noise is modeled as vector w ∈ C r ∼ C N ( 0 , I r ) . In other words, x has only o ne com p onent different fr om zero ( l th compo nent) an d its value is s ; that is, the transmitted sym b ol hop s amo ng the different chann els. Differently from previous works, in this pap er we do n ot analyze the statistics of H , as we are only interested in the MI giv en a chan nel realization. H models the effects and specific impairmen ts of the employed do main (spatial, po la r ization, frequen cy , etc.). h ( s, l | y ) = − X s ∈S t X l =1 Z Y f S , L , Y ( s, l, y ) log 2  f S , L | Y ( s, l, y )  d y = X s ∈S t X l =1 Z Y f S , L , Y ( s, l, y ) log 2  f Y ( y ) f S , L , Y ( s, l, y )  d y = X s ∈S t X l =1 Z Y f Y | S , L ( y , s, l ) p S ( s ) p L ( l ) × log 2 P s ′ ∈S P t l ′ =1 f Y | S , L ( y , s ′ , l ′ ) p S ( s = s ′ ) p L ( l = l ′ ) f Y | S , L ( y , s, l ) p S ( s ) p L ( l ) ! d y = 1 tS X s ∈S t X l =1 I E Y | S , L ( log 2 P s ′ ∈S P t l ′ =1 f Y | S , L ( y , s ′ , l ′ ) f Y | S , L ( y , s, l ) !) (3) Since the transmitted vector is deter mined by ( s, l ) , it is possible to r ewrite (1) as y = √ γ h l s + w . (2) Thus, th e MI be twe e n the r eceiv ed signal and ( s, l ) is ex- pressed as I ( y ; s, l ) = I ( y ; s | l ) + I ( y ; l ) = H ( s | l ) − h ( s | l , y ) + H ( l ) − h ( l | y ) = H ( s ) + H ( l ) − h ( s, l | y ) (4) where the third equality assumes that s and l are indep endent R V , H ( X ) = − P x ∈ X p X ( x ) log 2 ( p X ( x )) is the entr opy of X and h ( X ) = − R ∞ −∞ f X ( x ) log 2 ( f X ( x )) d x is the differential entropy of X . No te that, in contrast to [11], where the ca p acity is ob ta in ed, in our case the sy mbol s is not maximize d a n d belongs to a particu la r constellation . The en tropy of s and l is expressed as H ( s ) = lo g 2 S and H ( l ) = lo g 2 t , where S is the number of symb o ls defined in the constellation. Th e expression of the differential en tropy h ( s, l | y ) is deno ted in (3), wh ere Y is the do m ain of y , I E X {·} is the expectation of X , f S , L , Y ( s, l, y ) is the joint pr obability density fu nction (pdf ) of s , l and y , f Y | S , L ( y , s, l ) is the con- ditional pdf of y condition ed to s an d l , f Y ( y ) is the pd f of y , p S ( s ) = 1 /S and p L ( l ) = 1 /t are th e pro babilities o f symbo l s a n d index l , respectively , R Y d y . = R Y 1 · · · R Y r d y 1 . . . d y r , and Y i is the d omain of th e i th co m ponen t of y . The pdf o f y c ondition e d to s an d l is o btained by as- suming s and l to be d eterministic in (2). In th is case, it is clear that y is a multivariate complex Ga u ssian R V , with mean eq ual to √ γ h l s and id entity covariance. T h us, the con- ditioned pdf is expressed as f Y | S , L ( y , s, l ) = 1 π r e −k y − √ γ h l s k 2 . (5) Note that we assume that s and l are eq uiprob able. By substi- tuting (5) in (3), th e expectation can be d escribed as I E Y | S , L ( log 2 P s ′ ∈S P t l ′ =1 f Y | S ′ , L ′ ( y , s ′ , l ′ ) f Y | S , L ( s, l, y ) !) = I E W ( log 2 X s ′ ∈S t X l ′ =1 e − γ    h l s + w √ γ − h l ′ s ′    2 + γ    w √ γ    2 !) = I E W ′ ( log 2 X s ′ ∈S t X l ′ =1 e − γ  k h l s − h l ′ s ′ + w ′ k 2 − k w ′ k 2  !) , (6) where W ′ ∼ C N  0 , 1 γ I  and, thus, the condition ed R V Y | S , L ≡ W ′ . Computing (6) is achieved nu merically by generating a very large nu mber of realizations of W ′ and averaging the re- sults via Monte Carlo simulations. Howe ver , this c an only be feasible in scenario s where fixed constellations are employed. In the ca se of IM, the constellation depend s on the chan nel realization. Hence, the expectatio n has to b e calculated at each tim e in stant, requiring high compu tational complexity and making the pr oblem o f link adap ta tio n u naffordable. Our approa c h overcomes this problem , since it do es not requ ire off-line compu tations and presents clo sed-form expressions. Once f W ′ is d efined, we ap ply the same p rocedu re as d e - scribed in [11], which uses th e T aylor Series Expa n sion (TSE) to appro ximate the exp ectation of a fun ction by its mo ments. The centr al mo ments of W ′ are defined by µ W ′ i, ℜ = µ W ′ i, ℑ = 0 ϑ n W ′ i, ℜ = ϑ n W ′ i, ℑ = ( ( n − 1)! ! 1 (2 γ ) n 2 = if n is even 0 if n is od d , (7) where W ′ i, ℜ and W ′ i, ℑ are th e real and imaginar y parts of the i th com ponent of the R V W ′ . By assuming that g sl ( w ′ ) = lo g 2 X s ′ ∈S t X l ′ =1 e − γ  k x sl − x s ′ l ′ + w ′ k 2 − k w ′ k 2  ! , (8) I ( y ; s, l ) = log 2 ( tS ) − 1 tS X s ∈S t X l =1 log 2 ( D sl ) − 1 tS X s ∈S t X l =1 ∞ X n =1 1 (2 γ ) n (2 n )! ! r X m =1 ∂ 2 n g sl ∂ w ′ 2 n m, ℜ ( µ W ′ ) + ∂ 2 n g sl ∂ w ′ 2 n m, ℑ ( µ W ′ ) ! (10) r X m =1 ∂ 2 g sl ∂ w ′ 2 m, ℜ ( µ W ′ ) + ∂ 2 g sl ∂ w ′ 2 m, ℑ ( µ W ′ ) ! = 4 γ P s ′ ∈S P t l ′ =1 D sl,s ′ l ′ log 2  D − 1 sl,s ′ l ′  D sl − (2 γ ) 2 log(2) r X m =1   P s ′ ∈S P t l ′ =1 ( x m,s ′ l ′ , ℜ − x m,sl, ℜ ) D sl,s ′ l ′ D sl ! 2 + P s ′ ∈S P t l ′ =1 ( x m,s ′ l ′ , ℑ − x m,sl, ℑ ) D sl,s ′ l ′ D sl ! 2   = − 4 γ log 2  G sl  D D sl,s ′ l ′ sl,s ′ l ′  A sl ( D sl,s ′ l ′ ) − (2 γ ) 2 log(2) D 2 sl r X m =1  D 2 m,sl, ℜ + D 2 m,sl, ℑ  , (11) I (2) ( y ; s, l ) ≃ log 2  tS G ( D sl )  + A   log 2  G sl  D D sl,s ′ l ′ sl,s ′ l ′  A sl ( D sl,s ′ l ′ ) + γ log(2) D 2 sl r X m =1  D 2 m,sl, ℜ + D 2 m,sl, ℑ    (12) we defin e the TSE of function g sl ( w ′ ) in th e vicin ity of µ W ′ as g sl ( w ′ ) = T ( g sl , w ′ , µ W ′ ) = P N ( g sl , w ′ , µ W ′ ) + R N ( g sl , w ′ , ξ ) , wh e r e P N is th e T aylor po lynomia l of degree N and R N is the remaind e r term of d egree N . Thus, the expectation o f (6) is e q ual to I E W ′ { T ( g sl , w ′ , µ W ′ ) } = g sl ( µ W ′ ) + ∞ X n =1 1 (2 γ ) n (2 n )! ! r X m =1 ∂ 2 n g sl ∂ w ′ 2 n m, ℜ ( µ W ′ ) + ∂ 2 n g sl ∂ w ′ 2 n m, ℑ ( µ W ′ ) ! . = P N ( g sl , w ′ , µ W ′ ) + R N ( g sl , w ′ , ξ ) , (9) where ξ ∈ [ µ W ′ , w ′ ] and P N ( g sl , w ′ , µ W ′ ) = I E W ′ { P N ( g sl , w ′ , µ W ′ ) } = g sl ( µ W ′ ) + ⌊ N/ 2 ⌋ X n =1 1 (2 γ ) n (2 n )! ! r X m =1 ∂ 2 n g sl ∂ w ′ 2 n m, ℜ ( µ W ′ ) + ∂ 2 n g sl ∂ w ′ 2 n m, ℑ ( µ W ′ ) ! R N ( g sl , w ′ , ξ ) = I E W ′ { R N ( g sl , w ′ , ξ ) } . (13) Hereinafter, for the sake of clarity , we introdu ce the following definitions: x sl . = h l s D sl,s ′ l ′ . = e − γ k x sl − x s ′ l ′ k 2 D sl . = X s ′ ∈S t X l ′ =1 D sl,s ′ l ′ = X s ′ ∈S t X l ′ =1 e − γ k x sl − x s ′ l ′ k 2 . (14) The first term of (9) is d escribed as g sl ( µ W ′ ) = lo g 2 X s ′ ∈S t X l ′ =1 e −k x sl − x s ′ l ′ k 2 ! = log 2 ( D sl ) . (15) Thus, b y u sin g (9), (15) an d sub stituting them into (4), then the MI can be expr essed in a closed-fo rm as in (10) an d it can be ap p roximate d by considerin g addition al terms. The simplest expression is the first order appro ximation, which is obtained by o mitting the thir d term in ( 10). Consequen tly , the first ord er approx imation is d enoted b y I (1) ( y ; s, l ) ≃ log 2 ( tS ) − 1 tS X s ∈S t X l =1 log 2 ( D sl ) = log 2  tS G ( D sl )  , (16) where G ( D sl ) an d A ( D sl ) are th e g eometric an d arithm etic mean, respectively , i.e., G ( D sl ) =  Q s ∈S Q t l =1 D sl  1 tS and A ( D sl ) = 1 tS P s ∈S P t l =1 D sl . The second order appro x imation in volves the secon d deriv ativ e o f g sl at x sl . Thus, after some ma thematical ma - nipulation s, the secon d term is expressed as (11), wh e r e D m,sl, ℜ = X s ′ ∈S t X l ′ =1 ( x m,s ′ l ′ , ℜ − x m,sl, ℜ ) D sl,s ′ l ′ D m,sl, ℑ = X s ′ ∈S t X l ′ =1 ( x m,s ′ l ′ , ℑ − x m,sl, ℑ ) D sl,s ′ l ′ (17) and A sl ( D sl,s ′ l ′ ) = 1 tS X s ′ ∈S t X l ′ =1 D sl,s ′ l ′ G sl  D D sl,s ′ l ′ sl,s ′ l ′  = Y s ′ ∈S t Y l ′ =1 D D sl,s ′ l ′ sl,s ′ l ′ ! 1 tS (18) are th e ar ithmetic and geom etric means over s ′ and l ′ by keep- ing s and l fixed. Hence, by plugg in g (11) in ( 1 0), the second order approx imation of MI is described by (1 2). 2.1. Bounds of approximated Mutual Information TSE applied to the expectation of a fu n ction of a R V allows to express it as a fu nction of its momen ts instead of the R V ; thus, making mo re efficient the com putation by successiv e a pprox - imations. An impor tant remar k is th at the expectation of TSE is lower or u pper bo u nded by the first o rder appr oximation , depend ing on its conv exity or concavity , respectively . In our case, this can be proven by examining the co n vexity of (8) an d ap plying the Jensen’ s inequ ality , wh ich r esults that the expectation of TSE is lower b ound ed by (15). This can be proven by using the Jensen’ s inequality a s follows P 1 ( f , x , µ X ) = f ( µ X ) = f (I E X { x } ) ≤ I E X { f ( x ) } . (19) Note that, du e to the minus sign in (4), the lower bound of Jensen’ s inequ a lity b ecomes an upper boun d, which is in- creased by the factor log 2 ( tS ) an d averaged by t and S . 3. RESUL TS In this section, we illu strate th e results derived fro m the p re- vious section s. W e com pare the perform ance of first and sec- ond o rder ap proxim ations, i.e., (1 6) an d ( 12), r espectively , by simulating th e curves of MI with the integral- based expres- sion (3), (4). In this simulation , we g e nerate 1 0 3 indepen d ent ch annel realizations following a Ray leig h distribution and av erage th e -20 -15 -10 -5 0 5 10 15 20 [dB] 0 0.5 1 1.5 2 2.5 3 Capacity [b/s/Hz] 1st Order 2nd Order Integral-based Expression (a) 2 × 2 , S = 4 -20 -15 -10 -5 0 5 10 15 20 [dB] 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Capacity [b/s/Hz] 1st Order 2nd Order Integral-based Expression (b) 2 × 2 , S = 16 -20 -15 -10 -5 0 5 10 15 20 [dB] 0 0.5 1 1.5 2 2.5 3 3.5 4 Capacity [b/s/Hz] 1st Order 2nd Order Integral-based Expression (c) 4 × 4 , S = 4 -20 -15 -10 -5 0 5 10 15 20 [dB] 0 1 2 3 4 5 6 Capacity [b/s/Hz] 1st Order 2nd Order Integral-based Expression (d) 4 × 4 , S = 16 Fig. 1 . Comparison of th e MI fo r different or der appr oxima- tions and the integral-based expre ssion, i.e., (16), ( 12) and (4), respectively . results to obtain a sin g le smo oth curve. Note th at we d o not average over n oise r ealizations since we obtain e d m ath- ematical expressions th at are not f unctions o f a noise R V . W e also d epict different inpu t/o utputs config u rations an d differ- ent constellation s. Particularly , we c o nsider QPSK and 16 - QAM constellation s. Fig. 1 illustrates the M I of first and second orde r ap- proxim a tions, ( 16) and (12), r e sp ectiv ely , compa r ed w ith the integral-based expression, (3), (4). First, as we deno ted in Section 2. 1, the fir st o rder app r oximation is, at the same time, the up p er boun d of the integral-ba sed expression. Addition- ally , we can observe that, a s expected , th e second order ap- proxim a tion produ c e s tighter curve com pared with th e first order app roximatio n . 4. CONCLUSIONS In this paper we in troduce the problem of implemen ting link adaptation in I ndex Mod u lations, such as Spatial Mo dula- tion or Polariz e d Modulatio n, wher e th e info rmation is mod- ulated with fixed constellation s and dynamic ch annel hops. If the channe l is time varying, it is u naffordable to com pute the Mutual Inf ormation at each time instant. W ith our ap- proach it is possible to o btain a smooth curve by usin g c lo sed- form expressions, decre a sing the co mputation al co mplexity and allowing to p erform the link ad a ptation. Finally , we de- pict th e first and second orde r appr oximation s comp ared with integral-based exp r ession f or several configura tions and co n- stellation size. 5. REFERENCES [1] T . T ao and A. Czylwik, “Perf o rmance analysis of link adaptation in L TE systems, ” in Pr oc. Int. I TG W o rk- shop Smart Antenn as , Feb. 201 1, pp . 1–5. [2] S. A. Cheema, M . Gro ssman n, M. Landman n, G. D. Galdo, and G. D. 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