A Novel Power Allocation Scheme for Two-User GMAC with Finite Input Constellations

1 A No v el Po we r Allocation Scheme for T wo-User GMA C with Finite Inpu t Constellations J. Harshan, Member , I EEE , an d B. Sundar Rajan, Senior Member , IEEE Abstract — Constellation Constrained (CC) capacity regions of two-user Gaussian Multiple Access Channels (GMA C) have been recently r eported, wherein an appropriate angle of rotation between the constellations of the two u sers is sh own to enlarge the CC capacity r egion. W e refer to such a scheme as th e Constellation Rotation (CR) scheme. In this paper , we p ropose a nov el scheme called the Constellation Power Allocation (CP A) scheme, wherein the inst antaneous transmit power of the two users are varied by maintain ing their av erage power constraints. W e show that the CP A scheme offers CC sum capacities equal (at low SNR values) or close (at high S NR values) to those offered by the CR scheme with reduced decoding complexity for QAM constellations. W e stud y the robustness of the CP A scheme for random ph ase offsets in the chan nel and unequ al avera ge power constraints for the two u sers. With random phase offsets i n th e channel, we sh ow that the CC sum capacity offer ed by the CP A scheme is more than the CR scheme at hi gh SNR values. With unequal a verage power constraints, we sho w that the CP A scheme prov ides maximum gain wh en the power lev els are close, and the advantage diminish es with the increase in the power dif ference. Index T erms — Constellation constrained capacity regions, mul- tiple access channels, power allocation, fin ite constellations. I . I N T R O D U C T I O N A N D P R E L I M I N A R I E S In traditional networks, infor mation exchang e between the subscriber de vices a nd the base station is realized by schedul- ing the transmissions in disjoint set o f point-po int chan nels, i.e., b y separating the subscribers through TDM A, FDMA, or CDMA. Over the last few d ecades, eno rmous research has taken place to loo k beyond channel sepa ration , an d d ev elop advanced p hysical layer technique s that can pr ovide capacity gains at the cost of join t-processing of the signals fro m different subscriber s [1]-[7]. Along that direction, determin- ing the c apacity regions of multi-term inal networks [8], and designing appro priate low-comp lexity signalling schemes [9 ] has received a lo t of atten tion. T ill date, ca pacity region s are known only for a certain class of network con figuration s s uch as the multiple a ccess chan nels (MA C) [ 3], [4], and Gau ssian broad cast chann els (BC) [ 10], to name a few . Thoug h, contributions o n such configu rations started for channels with fe w users (such as tw o-user MA C/BC or three-user MAC/B C), the resu lts hav e now been generalized to arbitrary nu mber of users [6 ], [7], [11]. Further , for Gaussian MA C (GMA C), the capa city achieving input distribution is known to be Gaussian, which is a continuo us distribution [8]. Part of this work is in the proceeding s of IEEE Internationa l Symposium on Personal, Indoor and Mobile Radio Communicati ons (PIMRC 2011) held at T oronto, Canada, 11-14 September , 2011. J. Harshan is with the Dept. of ECSE, Monash Uni versity , Australia. B. Sundar Raja n is with the Dept. of ECE, Indian Institute of Scienc e, India. Email: harshan.jagadee sh@monash.edu, bsrajan@ec e.iisc.ernet .in. Howe ver, in prac tice, the input constellations are finite in size and are unifo rmly distributed. A s a result, the kn own results for continuo us input does not shed light on the actual perform ance of fin ite con stellations. This difference in the nature of the input alphabet h as motiv ated research ers to revisit the GMAC (and similar ch annels), and study them fr om the view point o f finite input constellations. As a first step, some results h av e been reported fo r ch annels with few users [13]-[27]. This p aper is alo ng that direction , and we deal with two-u ser GMA C with finite input co nstellations. Influenced by some prelimin ary works on MA C with finite inputs constellations [1 3], [14], a deta iled study on the CC capacity [28] r egions of two-user GMA C has be en reported in [15]. It is shown in [15] that introdu cing an appro priate an gle of r otation between the constellations p rovides enlargeme nt in CC capac ity regio n. Furth er , an gles of rotation which maximizes the CC sum capacity have b een provided fo r some known c onstellations. W e refer to such a method of enlarging the CC capac ity region as the Con stellation Rotatio n (CR) scheme. Note that th e CR scheme is a No n-Ortho gonal Multiple Access (NO-MA) scheme, wher ein th e two users transmit d uring the same time and in the same bandwid th. It is also shown in [15] that th e CC cap acity region of the CR schem e strictly enclo ses the CC cap acity r egion o f the FDMA and th e TDMA. I t is highligh ted in [15] that th e above behaviour is no t observed with Gaussian inputs, which in turn shows the importanc e of studying these channels with finite inp ut constellations. Oth er than two-u ser GMA C, similar research on finite input constellation s have also bee n r eported for MAC with quantization an d fading [ 16]-[18], Gaussian broadc ast channels [19], interfer ence channe ls [20], [21], relay channels [2 2], [24], po int-poin t MIMO channels [ 26], and secrecy chann els [27]. In the schemes p roposed in [15]-[27], either the chan nels are assumed to be fixed, o r some form o f chan nel state informa tion is av ailab le at the transmitter s. For instance, th e CR scheme [15] assumes fixed ch annels, and the tech nique is sen siti ve to th e cho ice o f the relati ve angle of rotation. If th e ch annels intro duce ran dom phase offsets (say , due to clock syn chroniza tion p roblems), the n the resultant relative angle need no t pr ovide maxim um gains. In such a case, even if th e ran dom pha se offsets are mad e av ailable to the re ceiv er , the CR scheme becomes ineffecti ve as the transmitters do not have the knowledge of the ph ase values. As a r esult, ther e is a ne ed for signa l design wh ich is r obust to random phase offsets. In this paper, we propose a NO-MA schem e called th e Constellation Po wer Allocation (CP A) scheme [16], wherein instead of introducin g rotatio n between the constellations, we 2 vary the transmit powers of the two users. Unlike the CR scheme, the pr oposed CP A scheme is ro bust to r andom phase offsets in the cha nnel. The contributions o f this paper are summarized as b elow: • W e propose a novel transmitter side techniq ue called the CP A sche me to ob tain enlargemen t in th e CC capacity region of two-user GMAC. I n the pr oposed scheme, the transmit power of the two users are varied while re taining the average power constraint for each u ser . • For the CP A sch eme, the transmit power of the users are varied through a scale factor α ∈ [0 , 1] . For such a model, we prop ose the prob lem of finding an approp riate α that maximizes the CC sum cap acity . For simpler compu ta- tions, we pro pose a deterministic metric to com pute the scale factor such that the CC sum capacity is maximized at high Signa l-to-Noise Ratio (SNR) values. W e com pute the CC sum cap acities for some known constellation s, and show th at the CC su m capacities offered b y the CP A scheme ar e equal (at low SNR values) or close (at high SNR values) to those offered by th e CR sch eme in [15]. (Section III) • For r egular QAM constellation s, we first identify th at the CP A scheme provides a sum co nstellation S sum whose in-phase an d the quadratu re comp onents are sep arable, and subseque ntly show th at the CP A scheme provides lower d ecoding co mplexity th an the CR sche me. T his advantage is shown to come with n o sign ificant reductio n in the CC sum capacity . T o exploit the redu ced dec oding complexity offered by the CP A scheme, we propo se indepen dent coding along the in-phase and the q uadratur e compon ents f or each u ser . W e also p ropose TCM based code pair s to appr oach the CC sum capa city fo r 16- QAM c onstellation. W e po int out that the low de coding complexity advantages of the CP A are applicab le only for GMA C with no rando m pha se offsets. (Sectio n IV ) • W e study th e robustness of the CP A scheme f or two-user GMA C with rando m p hase offsets in the channel. For such chan nels, it is clear that the CR scheme does no t improve the CC sum cap acity due to th e ra ndom phase offsets u nknown to th e transmitter s. W e sho w that the CC sum capacity offered b y the CP A scheme is mo re than the CR sch eme at high SNR values. W e also study the robustness of the CP A sch eme for un equal average power c onstraints for th e two users, where we show that the gains o f th e CP A scheme decreases as the power difference between the users in crease. (Section V) Notations : Throug hout the pap er , boldface letters and capi- tal boldface letter s are used to represent vectors and matrices, respectively . For a rand om variable X which takes value fr om the set S , w e assume some order ing of its elem ents an d use X ( i ) to represent the i -th eleme nt o f S , i.e., X ( i ) represents a realization of the r andom variable X . W e use the symbo l ı to r epresent √ − 1 . Cardinality of a set S is deno ted by |S | . Absolute value of a complex n umber x is denoted by | x | , and E [ x ] denotes the expe ctation of the ran dom variable x . A circu larly symm etric comp lex Gau ssian ran dom vector, x with mean µ and covariance matr ix Γ is denoted by x ∼ C S C G ( µ , Γ ) . I I . T W O - U S E R G M AC : S I G N A L M O D E L A N D C C S U M C A PAC I T Y The model of two-user GMAC consists of two users that need to convey in depend ent informatio n to a single destination. It is assumed that User- 1 and User- 2 commu nicate to the destination at the same time and in th e same freq uency ban d (the two users e mploy a NO-MA sch eme). Symbo l level synchro nization is assum ed at the destination. In this section, we assume no rando m p hase offsets introdu ced by the channel. The two users are equip ped with finite complex constellations S 1 and S 2 of size N 1 and N 2 , respectively such tha t for x i ∈ S i , we have E [ | x i | 2 ] = 1 . Let P i denote th e av erage power constrain t f or User- i . When User- 1 and User- 2 tr ansmit symbols √ P 1 x 1 and √ P 2 x 2 simultaneou sly , the destination receives the co mplex symbo l y given by y = p P 1 x 1 + p P 2 x 2 + z , where z ∼ C S C G  0 , σ 2  , (1 ) and σ 2 2 is the variance of the A WGN in each dim en- sion. The C C sum capacity of two-user GMA C [15] is I  √ P 1 x 1 + √ P 2 x 2 : y  , wh ich is given in (2) at the top of the n ext pag e. In [15], an appropr iate angle of ro tation between the con- stellations is sh own to increase the CC sum capacity . In this paper, we intro duce th e CP A schem e to increase the CC sum capacity . Before intro ducing the CP A scheme, we recall the definition of the sum constellation an d u niquely de codable (UD) c onstellation p airs [ 15]. Given two constellations S 1 and S 2 , the sum constellation of S 1 and S 2 is g iv en by S sum , n p P 1 x 1 + p P 2 x 2 | ∀ x 1 ∈ S 1 , x 2 ∈ S 2 o . The adder channe l in the two-user GMA C can be viewed as a mapping φ g iv en by φ  p P 1 x 1 , p P 2 x 2  = p P 1 x 1 + p P 2 x 2 . A constellation pair ( S 1 , S 2 ) is said to be uniq uely d ecodable (UD) if the map ping φ is on e-one. I I I . C O N S T E L L A T I O N P O W E R A L L O C A T I O N S C H E M E F O R T W O - U S E R G M AC In Section II, we have assumed u nit av erage p ower on S 1 and S 2 . Th erefore, to meet the av erage power constrain t P i , User- i can transmit symbols of the for m √ P i x i . W e now propo se an alternate method to tran smit the symb ols o f S i by main taining the average power constraint P i . T o explain the n ew sch eme, we let T = { 1 , 2 , 3 , · · · , T − 1 , T } de note the set o f the ind ices of complex cha nnel u se, wher e T is the total num ber of ch annel u ses. Assuming T to b e an ev en n umber, let T odd = { 1 , 3 , 5 · · · , T − 3 , T − 1 } and T e ven = { 2 , 4 , 6 · · · , T − 2 , T } deno te the set of odd and ev en indices of chann el use, respectively . W e use the variable t to deno te the in stantaneous channe l use ind ex. W e also use α ∈ [0 , 1 ] to den ote a real valued variable which is used to vary the tr ansmit power of eac h user . Using the pseud o co de representatio n, we explain the CP A scheme below . 3 I ( p P 1 x 1 + p P 2 x 2 : y ) = log 2 ( N 1 N 2 ) − 1 N 1 N 2 N 1 − 1 X k 1 =0 N 2 − 1 X k 2 =0 E " log 2 " P N 1 − 1 i 1 =0 P N 2 − 1 i 2 =0 exp  −| √ P 1 x 1 ( k 1 ) + √ P 2 x 2 ( k 2 ) − √ P 1 x 1 ( i 1 ) − √ P 2 x 2 ( i 2 ) + z | 2 /σ 2  exp ( −| z | 2 /σ 2 ) ## (2) Q ( ¯ α ) = N 1 − 1 X k 1 =0 N 2 − 1 X k 2 =0 log 2   N 1 − 1 X i 1 =0 N 2 − 1 X i 2 =0 exp  −| p (2 − ¯ α ) P 1 ( x 1 ( k 1 ) − x 1 ( i 1 )) + p ¯ αP 2 ( x 2 ( k 2 ) − x 2 ( i 2 )) | 2 / 2 σ 2    (3) I (1) ( p P L x 1 + p P S x 2 : y ) = N 1 − 1 X k 1 =0 N 2 − 1 X k 2 =0 E         log 2   N 1 − 1 X i 1 =0 N 2 − 1 X i 2 =0 exp  −| p P L x 1 ( k 1 ) + p P S x 2 ( k 2 ) − p P L x 1 ( i 1 ) − p P S x 2 ( i 2 ) + z | 2 /σ 2    | {z } β ( k 1 ,k 2 ,z )         | {z } λ ( k 1 ,k 2 ) (4) IF t ∈ T odd • User- 1 transmits p (2 − α ) P 1 x 1 for x 1 ∈ S 1 • User- 2 transmits √ αP 2 x 2 for x 2 ∈ S 2 ELSE • User- 1 transmits √ αP 1 x 1 for x 1 ∈ S 1 • User- 2 transmits p (2 − α ) P 2 x 2 for x 2 ∈ S 2 END During o dd indice s, the de stination receives a symbol y t of the f orm y t = p (2 − α ) P 1 x 1 + √ αP 2 x 2 + z . Similarly , during e ven ind ices, the destination receiv es a symbol y t of the f orm y t = √ αP 1 x 1 + p (2 − α ) P 2 x 2 + z . W ith the above mentioned power allo cation, th e average power for User- 1 is (2 − α ) P 1 and αP 1 in o dd and even channel uses, respectively . Similarly , the average p ower f or User- 2 is αP 2 and (2 − α ) P 2 in odd and even channel uses, respectiv ely . W ith this, the av erage power P i is maintained for User - i . When the tw o users employ identical constellations and equal average power , an approp riate value of α can provide the UD pr operty at th e receiver for every chann el use. W e now proceed to find th e o ptimal α that m aximizes th e CC sum ca pacity of th e CP A scheme. A. Op timal α for the CP A Scheme W e consider the identical co nstellation case, i.e., S 1 = S 2 , and focus on find ing α such th at the CC sum capacity is maximized . Throu gh the CP A scheme, the two users switch the scale factors on altern ate chan- nel uses. Du ring odd ch annel uses, the destination views the sum con stellation S sum,odd giv en b y S sum,odd , n p (2 − α ) P 1 x 1 + √ αP 2 x 2 | ∀ x 1 , x 2 o . During even chan- nel uses, the destination vie ws S sum,even giv en by S sum,even , n √ αP 1 x 1 + p (2 − α ) P 2 x 2 | ∀ x 1 , x 2 o . If the two u sers have equal average power con straint, th e destination views the same sum constellation on e very ch annel use. On the other h and, if th e two users have u nequal average power constraint, then the sum constellation seen b y the de stination is not th e same du ring the even and odd indices of th e c hannel use. Con sidering the g eneral case o f u nequal a verage p ower constraints, the CC sum capacity of the CP A schem e is given by 1 2 X ¯ α ∈ Ω I  p (2 − ¯ α ) P 1 x 1 + p ¯ αP 2 x 2 : y  , where Ω = { α, 2 − α } . Note that the CC sum capac ity can be in creased by choosing an ap propriate α ∈ [0 , 1] . Howe ver , we know th at α = 0 corr esponds to sing le-user transmission. As a r esult, hencefor th, we con sider selecting an ap propr iate α in the interval (0 , 1] . T hus, the CC su m capa city can be maximized by c hoosing α opt giv en by , α opt = arg max α ∈ (0 , 1] 1 2 X ¯ α ∈ Ω I  p (2 − ¯ α ) P 1 x 1 + p ¯ αP 2 x 2 : y  . (5) Note th at th e above ob jectiv e function in volves expectation of a non- linear fu nction of the rand om v ariable z , and henc e, its closed form expression is not av ailable . Th erefore, co mputing α opt is n ot straigh tforward. For h igh values of P 1 σ 2 and P 2 σ 2 , the following theo rem provides a deterministic metric (which is indepen dent of the variable z ) to cho ose α such that the CC capacity is max imized. Henc eforth, high SNR values imply high values of P 1 σ 2 and P 2 σ 2 . Theor em 1 : At h igh SNR values, the optimum scale factor α required to maximize th e CC sum capacity is app roximated closely by α ∗ where α ∗ = arg min α ∈ (0 , 1] X ¯ α ∈ Ω Q ( ¯ α ) , where Q ( ¯ α ) is given in (3) at the top of this page. Pr oof: Since N 1 and N 2 are c onstants, we have the following equality , arg max α ∈ (0 , 1] X ¯ α ∈ Ω I ( p (2 − ¯ α ) P 1 x 1 + p ¯ αP 2 x 2 : y ) = arg min α ∈ (0 , 1] X ¯ α ∈ Ω I (1) ( p (2 − ¯ α ) P 1 x 1 + p ¯ αP 2 x 2 : y ) , where I (1) ( p (2 − ¯ α ) P 1 x 1 + √ ¯ αP 2 x 2 : y ) is given in (4), and we use P L to deno te (2 − ¯ α ) P 1 , and P S to denote ¯ αP 2 . Note 4 that th e individual te rms λ ( k 1 , k 2 ) of I (1) ( √ P L x 1 + √ P S x 2 : y ) are o f the form E [ log 2 ( β ( k 1 , k 2 , z ))] for a ran dom variable β ( k 1 , k 2 , z ) . Apply ing Jensen’ s inequ ality: E [ log 2 ( β ( k 1 , k 2 , z ))] ≤ log 2 [ E ( β ( k 1 , k 2 , z ))] on λ ( k 1 , k 2 ) , we hav e I (1) ( p P L x 1 + p P S x 2 : y ) ≤ Q ( ¯ α ) , (6) where Q ( ¯ α ) is gi ven by (3). I n th e rest of the proof , we sho w that at high SNR values, the following a pprox imation holds good: arg min α ∈ (0 , 1] X ¯ α ∈ Ω Q ( ¯ α ) ≈ a rg min α ∈ (0 , 1] X ¯ α ∈ Ω I (1) ( p P L x 1 + p P S x 2 : y ) . Note th at the ter m I (1) ( √ P L x 1 + √ P S x 2 : y ) can be wr itten as in (7) and ( 8) where µ ( k 1 , k 2 , i 1 , i 2 ) = p P L x 1 ( k 1 ) + p P S x 2 ( k 2 ) − p P L x 1 ( i 1 ) − p P S x 2 ( i 2 ) , and M ( k 1 , k 2 ) = |M ( k 1 , k 2 ) | such that M ( k 1 , k 2 ) is g iv en by M ( k 1 , k 2 ) = { ( i 1 , i 2 ) 6 = ( k 1 , k 2 ) | µ ( i 1 , i 2 , k 1 , k 2 ) = 0 } . Removing independe nt ter ms of the form exp  − | z | 2 σ 2  in ( 8), we have arg min α ∈ (0 , 1] X ¯ α ∈ Ω I (1) ( p P L x 1 + p P S x 2 : y ) = arg min α ∈ (0 , 1] X ¯ α ∈ Ω I (2) ( p P L x 1 + p P S x 2 : y ) , where I (2) ( √ P L x 1 + √ P S x 2 : y ) is g iv en in (10). Further, at high SNR values, we hav e th e app roximatio n, arg min α ∈ (0 , 1] X ¯ α ∈ Ω I (2) ( p P L x 1 + p P S x 2 : y ) ≈ arg min α ∈ (0 , 1] X ¯ α ∈ Ω I (3) ( p P L x 1 + p P S x 2 : y ) , where I (3) ( √ P L x 1 + √ P S x 2 : y ) is given b y ( 11). At high SNR values, each term γ ( k 1 , k 2 , z ) in (11) is small, and hence we use the appro ximation log 2 (1 + γ ( k 1 , k 2 , z )) ≈ log 2 ( e )( γ ( k 1 , k 2 , z )) to obtain (12). Evaluating the expectation in ( 12), we g et (1 3). Once again, applying the a pprox imation log 2 (1 + δ ( k 1 , k 2 )) ≈ log 2 ( e )( δ ( k 1 , k 2 )) in (13), we g et (14), which is d enoted b y I (4) ( √ P L x 1 + √ P S x 2 : y ) . Now , we con sider the term I (5) ( √ P L x 1 + √ P S x 2 : y ) given in (15) and sh ow the following equality: arg min α ∈ (0 , 1] X ¯ α ∈ Ω I (5) ( p P L x 1 + p P S x 2 : y ) = arg min α ∈ (0 , 1] X ¯ α ∈ Ω I (4) ( p P L x 1 + p P S x 2 : y ) . (9) Once the above equ ality is proved, the statemen t of this theorem also gets proved since I (5) ( √ P L x 1 + √ P S x 2 : y ) is a scaled version of Q ( ¯ α ) (as shown in (16)) . T owards proving the eq uality in (9), note that at h igh SNR values, δ ( k 1 , k 2 ) is small fo r all v alues of α . For th ose values of α wh ich provid e the UD p roperty , we have log 2 (1 + M ( k 1 , k 2 )) = 0 ∀ k 1 , k 2 . Howe ver, for tho se values of α wh ich d o not provid e the UD prop erty , we have log 2 (1 + M ( k 1 , k 2 )) ≥ 1 for some k 1 , k 2 . Fur ther , at high SNR, f or a ll α , log 2 (1 + δ ( k 1 , k 2 )) << 1 , ∀ k 1 , k 2 . Due to these re asons, the values of α which do not pr ovide the UD property do not min imize I (5) ( √ P L x 1 + √ P S x 2 : y ) as well as I (4) ( √ P L x 1 + √ P S x 2 : y ) . As a result, the optimal value of α must b elong to the set which provides the UD property . For such values of α , we ha ve I (5) ( √ P L x 1 + √ P S x 2 : y ) = I (4) ( √ P L x 1 + √ P S x 2 : y ) , and hence the equ ality in (9) holds. This completes the pr oof. Using the r esults of Theorem 1, for high SNR v alues, we propo se to find α ∗ which minimizes P ¯ α ∈ Ω Q ( ¯ α ) , a tight upper bo und on P ¯ α ∈ Ω I (1) ( √ P L x 1 + √ P S x 2 : y ) . Howe ver , note that for small to mode rate values of SNR , the v alues of α ∗ obtained by solv ing (3) need not max imize P ¯ α ∈ Ω I ( √ P L x 1 + √ P S x 2 : y ) since the bo und in (6) is n ot kn own to be tight. It is also clear that solving (3) is easier than solving (5) since P ¯ α ∈ Ω Q ( ¯ α ) is deterministic a nd independ ent of th e term z . B. Num erical Resu lts In this section, we first compu te α ∗ for the case of eq ual av erage p ower constraint for the two user s (i.e., P 1 = P 2 ). For the simulation results, we use σ 2 = 1 an d SNR = P 1 . The v alues o f α ∗ are obtained by varying α fr om 0 to 1 in steps of 0.01 . In T able I , the values of α ∗ are presen ted for some known co nstellations. Th e CC sum capacities o f QPSK and 8 -PSK are also pr ovided in Fig. 1 and Fig. 2 , respectively for th e f ollowing schemes: ( i ) CP A scheme, ( ii ) CR scheme, ( iii ) neither CP A nor CR, and ( iv ) with both CP A and CR. For the scheme “with both CP A and CR”, th e pairs ( α ∗ , θ ∗ ) are computed using a metric wh ich is ob tained on th e similar lin es of Theo rem 1. From the figures, no te that the CP A scheme provides CC sum capacities clo se ( or eq ual) to the CR scheme for all SNR values. Note that the scheme “with both CP A and CR” does not provide sign ificant CC capa city gain over th e CP A scheme. 0 5 10 15 20 25 30 1.5 2 2.5 3 3.5 4 QPSK Constellation SNR in dB CC sum capacity CPA Neither CPA nor CR CR With both CPA and CR Fig. 1. CC sum capa city of QPSK constellati on with equal a verag e po wer constrai nt and no random phase off sets. T o verify the results o f Th eorem 1, we p lot th e CC sum capacity of QPSK and 8 -PSK with α opt and α ∗ in Fig. 5 I (1) ( p P L x 1 + p P S x 2 : y ) = N 1 − 1 X k 1 =0 N 2 − 1 X k 2 =0 E         log 2         exp  −| z | 2 /σ 2  + N 1 − 1 X i 1 =0 N 2 − 1 X i 2 =0 | {z } ( i 1 ,i 2 ) 6 =( k 1 ,k 2 ) exp  −| µ ( k 1 , k 2 , i 1 , i 2 ) + z | 2 /σ 2                  (7) = N 1 − 1 X k 1 =0 N 2 − 1 X k 2 =0 E         log 2         exp  −| z | 2 /σ 2  (1 + M ( k 1 , k 2 )) + N 1 − 1 X i 1 =0 N 2 − 1 X i 2 =0 | {z } µ ( k 1 ,k 2 ,i 1 ,i 2 ) 6 =0 exp  −| µ ( k 1 , k 2 , i 1 , i 2 ) + z | 2 /σ 2                  (8) I (2) ( p P L x 1 + p P S x 2 : y ) = N 1 − 1 X k 1 =0 N 2 − 1 X k 2 =0 E         log 2 (1 + M ( k 1 , k 2 )) + log 2         1 + 1 (1 + M ( k 1 , k 2 )) N 1 − 1 X i 1 =0 N 2 − 1 X i 2 =0 | {z } µ ( k 1 ,k 2 ,i 1 ,i 2 ) 6 =0 exp  −| µ ( k 1 , k 2 , i 1 , i 2 ) + z | 2 + | z | 2 σ 2                  (10) I (3) ( p P L x 1 + p P S x 2 : y ) = N 1 − 1 X k 1 =0 N 2 − 1 X k 2 =0 E               log 2 (1 + M ( k 1 , k 2 )) + log 2               1 + 1 (1 + M ( k 1 , k 2 )) N 1 − 1 X i 1 =0 N 2 − 1 X i 2 =0 | {z } µ ( k 1 ,k 2 ,i 1 ,i 2 ) 6 =0 exp  −| µ ( k 1 , k 2 , i 1 , i 2 ) + z | 2 σ 2  | {z } γ ( k 1 ,k 2 ,z )                             (11) 0 5 10 15 20 25 30 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 8 PSK Constellation SNR in dB CC sum capacity CPA Neither CPA nor CR CR With both CPA and CR Fig. 2. CC sum capacit y of 8-PSK constel lation with equal av erage power constrai nt and no random phase offsets. 3. Fro m th e fig ure, we see that α ∗ provides app roximately same CC sum capac ity values not on ly at high SNR but also at moderate SNR values. I n addition, we a lso present the corr espondin g values of α opt and α ∗ in T ab le. II, which shows that α opt and α ∗ are d ifferent f or lo w SNR values, and are appr oximately same f or high SNR values. These re sults demonstra te that the metric P ¯ α ∈ Ω Q ( ¯ α ) is a tight upper bound on P ¯ α ∈ Ω I (1) ( √ P L x 1 + √ P S x 2 : y ) at high SNR values. For the uneq ual average power ca se, we co mpute α ∗ for the QPSK con stellation. W e u se SNR = P 1 (with σ 2 = 1 ) . For the 0 5 10 15 20 25 30 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 CC sum capacity SNR in dB with α opt with α * Fig. 3. CC sum capacity of QPSK and 8-PSK constellati ons with α opt and α ∗ for GMA C with equal avera ge po wer constraint and no random phase of fsets. simulation results, we ha ve considered th e following relations between P 1 and P 2 : ( i ) P 2 = 0 . 3 P 1 , ( ii ) P 2 = 0 . 5 P 1 and ( iii ) P 2 = 0 . 75 P 1 , and ( iv ) P 2 = 0 . 9 P 1 . In Fig. 4, the C C sum capacities of QPSK are provided for the CP A scheme, the CR scheme, and the “ neither CP A n or CR” sche me. When comp ared to the “neithe r CP A nor CR” scheme, the CP A scheme provides increased CC sum capacities at moderate to high SNR values, when P 1 and P 2 are clo se. When compared to the CR scheme, the CP A schem e provid es marginal improvement in the CC 6 I (3) ( p P L x 1 + p P S x 2 : y ) ≈ N 1 − 1 X k 1 =0 N 2 − 1 X k 2 =0         log 2 (1 + M ( k 1 , k 2 )) + E         log 2 ( e ) (1 + M ( k 1 , k 2 )) N 1 − 1 X i 1 =0 N 2 − 1 X i 2 =0 | {z } µ ( k 1 ,k 2 ,i 1 ,i 2 ) 6 =0 exp  −| µ ( k 1 , k 2 , i 1 , i 2 ) + z | 2 σ 2                  (12) = N 1 − 1 X k 1 =0 N 2 − 1 X k 2 =0               log 2 (1 + M ( k 1 , k 2 )) + log 2 ( e ) 2               1 (1 + M ( k 1 , k 2 )) N 1 − 1 X i 1 =0 N 2 − 1 X i 2 =0 | {z } µ ( k 1 ,k 2 ,i 1 ,i 2 ) 6 =0 exp  −| µ ( k 1 , k 2 , i 1 , i 2 ) | 2 2 σ 2  | {z } δ ( k 1 ,k 2 )                             (13) I (4) ( p P L x 1 + p P S x 2 : y ) = N 1 − 1 X k 1 =0 N 2 − 1 X k 2 =0         log 2 (1 + M ( k 1 , k 2 )) + 1 2 log 2         1 + 1 (1 + M ( k 1 , k 2 )) N 1 − 1 X i 1 =0 N 2 − 1 X i 2 =0 | {z } µ ( k 1 ,k 2 ,i 1 ,i 2 ) 6 =0 exp  −| µ ( k 1 , k 2 , i 1 , i 2 ) | 2 2 σ 2                  (14) I (5) ( p P L x 1 + p P S x 2 : y ) = N 1 − 1 X k 1 =0 N 2 − 1 X k 2 =0 1 2         log 2 (1 + M ( k 1 , k 2 )) + log 2         1 + 1 (1 + M ( k 1 , k 2 )) N 1 − 1 X i 1 =0 N 2 − 1 X i 2 =0 | {z } µ ( k 1 ,k 2 ,i 1 ,i 2 ) 6 =0 exp  −| µ ( k 1 , k 2 , i 1 , i 2 ) | 2 2 σ 2                  (15) = 1 2 N 1 − 1 X k 1 =0 N 2 − 1 X k 2 =0 log 2   N 1 − 1 X i 1 =0 N 2 − 1 X i 2 =0 exp  −| p P L x 1 ( k 1 ) − p P L x 1 ( i 1 ) + ( p P S x 2 ( k 2 ) − p P S x 2 ( i 2 )) | 2 / 2 σ 2    = 1 2 Q ( ¯ α ) (16) sum capacities only at hig h SNR values, when P 1 and P 2 are close. The figur e shows that the gain s of the CP A scheme diminishes wh en the p ower difference is more tha n 3 dB (as shown for the case P 2 = 0 . 5 P 1 and P 2 = 0 . 3 P 1 ). C. Reduced ML Decodin g Complexity for QAM Constellations with CP A In this subsection , we highligh t the ad vantage of using CP A over CR for the class of regular QAM co nstellations. For unco ded transmission, when CR is em ployed for QAM constellations, the in-phase and the quadratur e c ompon ents of the sym bols o f S sum are entangled . Howe ver, with CP A, since the scale factor α is re al valued, the in-phase and qua drature compon ents of the sym bols in S sum do not g et entang led. As a result, S sum is separable, a nd can be wr itten as th e cross p roduct o f in -phase an d quad rature comp onents of its symbols. In particu lar , if the two users employ square regular M -QAM constellation, th en there a re M points in S sum along the in-phase and the quadratu re c ompon ent, respecti vely . Since S sum is sep arable, the destination can decode the in-phase and the quadrature com ponen ts independen tly . Therefo re, the worst case ML decodin g comp lexity is O ( M ) . Howe ver , with CR scheme, the worst case ML decod ing co mplexity is O ( M 2 ) . Therefo re, for th e class of QAM constellations, CP A provides lower decodin g complexity with n egligible loss in the CC sum capacity wh en c ompared with CR. I V . C H A N N E L C O D I N G F O R Q A M C O N S T E L L A T I O N S W I T H C P A In this section, we design co de pairs based on TCM (Trellis Coded Modulation ) [12] to achiev e sum-r ates close to the CC sum-capacity of QAM co nstellations using the CP A scheme . Throu ghout this section, w e assume equal average power constraints fo r the two users with P 1 = P 2 = P . If S 1 and S 2 represent regular M -Q AM constellations, then the in-p hase and the quad rature comp onents of S sum are r espectiv ely of the f orm p (2 − α ) P x 1 I + √ αP x 2 I and p (2 − α ) P x 1 Q + √ αP x 2 Q , where x iI ∈ S iI , x iQ ∈ S iQ and S iI and S iQ denote th e cor respond ing √ M -P AM co nstellations for the i -th user along the in -phase an d the qua drature dimension , respec- ti vely . The set of in-p hase and quadrature symbols of S sum are respectively den oted by S sum,I and S sum,Q , an d a re given by S sum,I = n ( p (2 − α ) P x 1 I + √ αP x 2 I ) | x iI ∈ S iI o , and S sum,Q = n ( p (2 − α ) P x 1 Q + √ αP x 2 Q ) | x iQ ∈ S iQ o . Since th e CP A schem e makes the in- phase and quad rature compon ents of S sum separable, the symb ols of S sum,I can be deco ded indepen dent of the symbols of S sum,Q , ther eby reducing the deco ding com plexity . T o facilitate this, User- i 7 T ABLE I N U M E R I C A L LY C O M P U T E D α ∗ W I T H E Q UA L A V E R A G E P O W E R C O N S T R A I N T A N D N O R A N D O M P H A S E O F F S E T S . SNR in dB QPSK 8-PSK 1 6-PSK 16-QAM 0 0.74 0.65 0.65 0.48 2 0.65 0.85 0.85 1.00 4 0.52 0.74 0.74 1.00 6 0.46 0.67 0.67 1.00 8 0.43 0.65 0.66 0.84 10 0.41 0.64 0.67 0.74 12 0.41 0.59 0.70 0.68 14 0.40 0.53 0.74 0.46 16 0.40 0.50 0.79 0.62 18 0.40 0.49 0.78 0.13 20 0.37 0.49 0.59 0.12 22 0.24 0.49 0.58 0.12 24 0.15 0.49 0.56 0.12 26 0.10 0.49 0.55 0.12 28 0.06 0.48 0.55 0.12 30 0.04 0.13 0.55 0.12 T ABLE II N U M E R I C A L LY C O M P U T E D VAL U E S O F α W I T H E Q U A L AVE R AG E P OW E R C O N S T R A I N T A N D N O R A N D O M P H A S E O F F S E T S . SNR in dB α opt for QPSK α ∗ for QPSK α opt for 8-PSK α ∗ for 8-PSK 0 0.88 0.74 0 .92 0.65 4 0.55 0.52 0 .83 0.74 8 0.44 0.43 0 .66 0.65 12 0.41 0.41 0 .60 0.59 16 0.41 0.40 0 .52 0.50 20 0.38 0.37 0 .49 0.49 24 0.19 0.15 0 .49 0.49 0 10 20 30 1 1.5 2 2.5 3 3.5 4 P 2 = 0.3P 1 CC sum capacity SNR in dB CPA scheme CR scheme neither CPA nor CR 0 10 20 30 1 1.5 2 2.5 3 3.5 4 P 2 = 0.5P 1 CC sum capacity SNR in dB CPA scheme CR scheme neither CPA nor CR 0 10 20 30 1 1.5 2 2.5 3 3.5 4 P 2 = 0.75P 1 CC sum capacity SNR in dB CPA scheme CR scheme neither CPA nor CR 0 10 20 30 1.5 2 2.5 3 3.5 4 P 2 = 0.9P 1 CC sum capacity SNR in dB CPA scheme CR scheme neither CPA nor CR Fig. 4. CC sum capacit ies of QPSK with unequal ave rage power constraint and no random phase offset s. should make su re that the sym bols o f S iQ and S iI are not coded jointly . Therefo re, each user can h ave two e ncoders one along ea ch dimen sion. Let the sub script X ∈ { I , Q } denote either th e in-ph ase d imension o r quad rature dimension . For each i ∈ { 1 , 2 } , let User- i be eq uipped with a con volutional encoder C iX with m iX input bits and m iX + 1 ou tput bits. Throug hout the section, we co nsider co n volutional codes which add only 1 -bit r edunda ncy . Let the m iX + 1 outp ut bits of C iX take values from √ M -P AM constellation S iX such that |S iX | = 2 m iX +1 . He nceforth , the set o f cod ew ords generated fr om C 1 X and C 2 X are r epresented b y tr ellises T 1 X and T 2 X respectively . W e assume that the d estination perform s joint de coding of the in-phase symbols of User - 1 and User - 2 by decoding for a sequence over S sum,I on the sum trellis, T sum,I (see [1 5] for the de finition of the sum trellis). Similarly , joint decod ing of the quadratur e sym bols of User - 1 and User- 2 is perfo rmed by dec oding for a sequen ce over S sum,Q on the sum tr ellis, T sum,Q . Due to the existence o f an equivalent A WGN ch annel in the GMAC set-up, the sum trellis, T sum,X has to be labeled with the elements of S sum,X satisfying the design rules in [12]. Howe ver, such a labeling rule can be obtain ed o n T sum,X only through the pairs ( T 1 X , T 2 X ) and ( S 1 X , S 2 X ) . Hence, we propo se labeling r ules o n T 1 X and T 2 X using S 1 X and S 2 X respectively such that T sum,X is labeled with the elements o f S sum,X as per Ungerb oeck rules. Since the numb er o f inp ut bits to C iX is m iX , ther e are 2 m iX edges diverging from (or conver ging to ; hencefo rth, we only r efer to diverging edges) each state of T iX . Also, as there is only one bit redund ancy adde d b y the encoder, an d as |S iX | = 2 m iX +1 , the ed ges diverging from each state hav e to be labeled with the elem ents of a su bset of S iX of size 2 m iX . Th erefore, fo r each i , S iX has to b e partition ed into two sets S 1 iX and S 2 iX , an d the diverging ed ges from each state of T iX have to be labeled with the elements of either S 1 iX or S 2 iX . From th e definition of sum trellis, ther e are 2 m 1 X + m 2 X edges diver ging from each state o f T sum,X and these edges get labeled with the elemen ts of o ne of th e following sets, A = n S i 1 X + S j 2 X | i, j ∈ { 1 , 2 } o . T o satisfy Ungerboec k 8 design rules, the tr ansitions o riginating from the sam e state of T sum,X must be assigned symb ols tha t a re separated by largest minim um distance. Therefor e, th e p roblem a ddressed is to find a partitioning of S iX ( √ M -P AM co nstellations) into two sets S 1 iX and S 2 iX of eq ual cardinality such that the minimum Eu clidean distance (denote d by d min ) o f each on e of the sets in A is maxim ized. Howev er , since d min values of the sets in A can poten tially be different, we find a partitioning such that th e minimu m o f the d min values of the sets in A is maximized. A. De signing TCM S chemes with 16-QA M Constellation The set partitioning pr o blem described above d epends on the structur e of the sum constellation . As a result, th e solution to the set partitionin g pr o blem depend s on the cho ice of α . For arbitrary v alues o f M and α , we are una ble to solve the set partition ing problem due to lack of structure o n S sum of two QAM con stellations. Howe ver, through compute r simulations, we have fo und a solution for the above problem for 16-QAM con stellation. For 16-QAM constellation, S iI and S iQ are of th e form S iI = 1 √ 10 {− 3 , − 1 , 1 , 3 } and S iQ = 1 √ 10 {− 3 ı, − ı, ı, 3 ı } . For this set-up, we obtain a two way par- tition of S iI and S iQ such that the minimum of the d min values of the sets in A is maximized. In particu lar , the o ptimal parti- tion is obtained for dif ferent P values. For each value of P , the optimal partition is obtain ed b y using the co rrespond ing α ∗ as giv en in the last column of T able I. The optimal p artitions fo r all values of P are f ound to be S 1 iI = 1 √ 10 {− 3 , 1 } , S 2 iI = 1 √ 10 {− ı, 3 ı } , S 1 iQ = 1 √ 10 {− 3 , 1 } and S 2 iQ = 1 √ 10 {− ı, 3 ı } , which is the Ungerb oeck partition ing. W ith this set partition- ing, trellis code pa irs based on T CM can be de signed in order to transmit 2 bits for each user (for each user m iX = 1 bit is transmitted along each dimen sion) using th e 16-QAM constellation. In the precedin g sections, we hav e studied the advantage of the CP A scheme for two-user GMAC with n o pha se offsets in the channel. In the next section , we study the r obustness o f the CP A sch eme for r andom phase offsets in the channel. V . C P A S C H E M E F O R T W O - U S E R G M AC W I T H R A N D O M P H A S E - O FF S E T S In this section, we consider a two-user G MA C with random phase offsets introd uced by th e chann el for bo th the users [14]. Similar to th e signal mo del in Section I I, the two u sers are eq uipped with constellatio ns S 1 and S 2 of size N 1 and N 2 , respectiv ely . When User - 1 and User - 2 transmit sym bols √ P 1 x 1 and √ P 2 x 2 simultaneou sly , the destination receives the symbol y given b y y = e ıθ 1 p P 1 x 1 + e ıθ 2 p P 2 x 2 + z , where z ∼ C S C G  0 , σ 2  , (17) and θ 1 , θ 2 are i.i.d . ran dom variables distributed unif ormly over (0 , 2 π ) . W e assume that on ly the destination has the knowledge of θ 1 and θ 2 . If User- i h as the knowledge of θ i , then the phase offset c an be compensated by each user which in turn results in the signal mod el discu ssed in Sectio n II. The CC capacity r egion fo r the above m odel can be co mputed along the similar lines of th e model in Section II. In addition to the steps needed to compu te the mu tual in formatio n values in Section I I, in this ca se, we have to take expectation of the mu- tual infor mation values over θ 1 and θ 2 . Hen ce, the CC sum c a- pacity is given by E θ 1 ,θ 2  I  √ P 1 x 1 + √ P 2 x 2 : y | θ 1 , θ 2  , where I  √ P 1 x 1 + √ P 2 x 2 : y | θ 1 , θ 2  is given in ( 18). Note th at th e CC sum capacity is a fu nction o f th e distan ce distribution (DD) of the sum co nstellation S sum giv en by S sum , n p P 1 e ıθ 1 x 1 + p P 2 e ıθ 2 x 2 | ∀ x 1 ∈ S 1 , x 2 ∈ S 2 o . Despite having r andom ph ase offsets in the chan nel, the DD of S sum can b e change d b y scaling the input sym bols of on e user relativ e to the othe r . W ith such a ch ange in the DD o f S sum , the CC sum capacity ca n be incr eased by choosing a n ap propriate α . T ow ards that dire ction, we apply the CP A scheme (given in Section III) for this channel. W ith the CP A scheme, the CC sum cap acity is given by E θ 1 ,θ 2 h P ¯ α ∈ Ω I  p (2 − ¯ α ) P 1 x 1 + √ ¯ αP 2 x 2 : y | θ 1 , θ 2 i , where Ω = { α, 2 − α } . Since the CC sum capacity is a function of α , w e hav e to comp ute α such that E θ 1 ,θ 2 " X ¯ α ∈ Ω I  p P L x 1 + p P S x 2 : y | θ 1 , θ 2  # is maximized , where P L = (2 − ¯ α ) P 1 and P S = ¯ αP 2 . In o ther words, we have to solve the fo llowing optimization pr oblem, α opt = arg max α ∈ (0 , 1] E θ 1 ,θ 2 " X ¯ α ∈ Ω I ( √ P L x 1 + √ P S x 2 : y | θ 1 , θ 2 ) # . (21) Note th at th e closed form expression of E θ 1 ,θ 2 [ I ( √ P L x 1 + √ P S x 2 : y | θ 1 , θ 2 )] is n ot av ailable. Therefo re, computing α opt is not straigh tforward. On the similar lines of Th eorem 1, we use α ∗ = arg min α ∈ (0 , 1] P ¯ α ∈ Ω Q p ( ¯ α ) , to o btain the approp riate values of α , where Q p ( ¯ α ) is g iv en in (19), an d µ 1 ( k 1 , k 2 , i 1 , i 2 ) is giv en in (2 0). No te that Q p ( ¯ α ) is free from the ran dom variable z . Hence, it is comp utationally easier to obtain α ∗ . A. Num erical Resu lts For th e equal average power case (i.e., SNR = P 1 = P 2 with σ 2 = 1 ) , we co mpute th e values of α ∗ for QPSK, 8- PSK and 8-QAM con stellations. The correspon ding values of α ∗ are listed in T able III. The CC su m capacities for QPSK and 8-PSK are also pr ovided in Fig. 5 and Fig. 6 , respectively for the following two cases: ( i ) the CP A scheme with α = α ∗ , and ( ii ) the CR scheme. For the CR sch eme, th ough the two users employ ro tated con stellations, the cha nnel induc ed rotations (due to θ 1 and θ 2 ) make the effectiv e angle of rotation a ran dom v ariable. There fore, the CR scheme f or this chann el correspo nds to the CP A schem e with α = 1 . The figur es show that the CP A schem e pr ovides larger CC sum capac ities th an the CR sch eme a t high SNR values. For the une qual average power case, we have co mputed α ∗ for the QPSK co nstellation and for sev eral values o f SNR = P 1 σ 2 (where σ 2 = 1 ). W e use the following th ree 9 I  p P 1 x 1 + p P 2 x 2 : y | θ 1 , θ 2  = log 2 ( N 1 N 2 ) − 1 N 1 N 2 N 1 − 1 X k 1 =0 N 2 − 1 X k 2 =0 E " log 2 " P N 1 − 1 i 1 =0 P N 2 − 1 i 2 =0 exp  −| e ıθ 1 √ P 1 ( x 1 ( k 1 ) − x 1 ( i 1 )) + e ıθ 2 √ P 2 ( x 2 ( k 2 ) − x 2 ( i 2 )) + z | 2 /σ 2  exp ( −| z | 2 /σ 2 ) ## (18) Q p ( ¯ α ) = E θ 1 ,θ 2 " N 1 − 1 X k 1 =0 N 2 − 1 X k 2 =0 log 2 " N 1 − 1 X i 1 =0 N 2 − 1 X i 2 =0 exp  −| µ 1 ( k 1 , k 2 , i 1 , i 2 ) | 2 / 2 σ 2  ## (19) µ 1 ( k 1 , k 2 , i 1 , i 2 ) = e ıθ 1  p P L x 1 ( k 1 ) − p P L x 1 ( i 1 )  + e ıθ 2  p P S x 2 ( k 2 ) − p P S x 2 ( i 2 )  (20) relations be tween P 1 and P 2 : ( i ) P 2 = 0 . 5 P 1 , ( ii ) P 2 = 0 . 75 P 1 and ( iii ) P 2 = 0 . 9 P 1 . The corre sponding CC sum capacities are presented in Fig. 7 for the CP A and the CR scheme. Fr om the figures, n ote that the CP A scheme provid es in creased CC sum capacities at modera te to h igh SNR values, especially when P 1 and P 2 are close. Further, the advantage of CP A is n oticed to diminish wh en th e power difference is more than 3 d B. 0 5 10 15 20 25 30 1.5 2 2.5 3 3.5 4 SNR in dB CC sum capacity with CPA ( α = α * ) with CR Fig. 5. CC sum capacity of QPSK with equal average powe r constraint and random phase of fsets. 0 5 10 15 20 25 30 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 SNR in dB CC sum capacity with CPA ( α = α * ) with CR Fig. 6. CC sum capaci ty of 8-PSK with equal av erage power constraint and random phase of fsets. 0 10 20 30 1 1.5 2 2.5 3 3.5 4 P 2 = 0.5P 1 CC sum capacity SNR in dB CR scheme CPA scheme 0 10 20 30 1 1.5 2 2.5 3 3.5 4 P 2 = 0.75P 1 CC sum capacity SNR in dB CR scheme CPA scheme 0 10 20 30 1.5 2 2.5 3 3.5 4 P 2 = 0.9P 1 CC sum capacity SNR in dB CR scheme CPA scheme Fig. 7. CC sum capacit ies of QPSK with unequal ave rage power constraint and random phase offsets. V I . C O N C L U S I O N A N D D I R E C T I O N S F O R F U T U R E W O R K The pr oposed CP A scheme can be useful f or two-user fading M A C e specially when the chann el state inf ormation is available at the transmitters. For such a case, depen ding on the instantaneo us chann el states of the two users, we obtain one of th e channe l models discussed in th is paper . Therefo re, when ev er the fading amplitudes o f the tw o channels are app roximately close, the CP A sch eme will be beneficial. Hence, studyin g the impact of the CP A scheme fo r fading MA C is an interesting direction for future work. On the other hand, the CP A scheme can also be studied for GMAC with arbitrary numb er of users. Howe ver , large number of users in a NO-MA scheme is known to increase the decoding complexity at the d estination [9]. 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