Optimal Radix-2 FFT Compatible Filters for GFDM

1 Optimal Radix-2 FFT Compatible Fil ters for GFDM Ahmad Nimr , Maximilian Matth ´ e, Dan Zhang, Gerhard Fettweis V odafone Chair Mobile Communication Systems, T echnische Univ ersit ¨ at Dresden , Ge rmany { first name.last name } @ifn.et.tu-dres den.de Abstract —Fo r a linear wa vef orm, a fin ite condition n umber of the corr esponding modulation matrix is necessary f or the wa vef orm to con vey the message with out ambiguity . Based on the Zak transform, thi s letter presents an analytical approach to compu t e th e condition number of the modulation matrix fo r th e mu lti-carrier wav efo rm g eneralized f requency division multiplexing (GFDM). On to p, we further propose a filter design that yields non-singu l ar m odulation matrices for an ev en number of subcarriers and subsymbols, which is not achievable fo r any previous work. Su ch new design h as significant impact on implementation complexity , as th e radi x-2 FF T operations for con ventional multicarrier wav eforms can readily be em ployed fo r GFDM. Additionally , we analytically derive the optimal filter that minimizes the condition number . W e further numerically e valuate the signal-to-interference ratio (SIR) and noise-enhancement factor (NEF) for matched filter (MF) and zero-f orcing (ZF) GFDM r eceivers f or such design, respectiv ely . Index T erms —GFDM, Z ak transform , pul se shape, conditional number , ev en number of subsymbols. I . I N T RO D U C T I O N Among se veral wa vefo rm altern ati ves to orthog onal fre- quency di visio n multiplexing (OFDM) [1], considerable research on detection algorithms, perfo r mance and low- complexity implemen tations has been cond u cted for GFDM [2]. GFDM, as a n on-or thogon al filtered multicarrier system with K subcarrier s employs circular filtering of M subsym bols within each bloc k to keep the signal con fin ed within the blo c k duration of K M samples. Naturally , the c hoice of the pulse shaping filter strongly influences the system per f ormanc e , as it controls system orthogo nality and inter ference structure. In [3], it is p roved b y means o f th e discrete Zak tr ansform (DZT) [4] of the tran smit filter that the tr ansmit signal beco mes ambiguo us and the GFDM modulatio n matrix A is singular when a re a l-valued symmetric filter with ev en M an d K is employed. The authors of [5] extended this result by showing that A has exactly o ne zer o eigen value, sugg esting that one GFDM bloc k can mostly conve y ( K M − 1) d ata symb ols. Hence, odd M is co mmonly adopted for d ata transmission in the literature. Even th ough several works on comp lexity reduction for GFDM modulation and demodu lation h ave been published [6], [7], od d M forbid s the N - p oint FFT to be im- plemented solely b y energy-efficient radix -2 based pro cessing. This also n a r rows the desig n space of GFDM as a flexible wa vefo rm gen erator [8]. In th is paper w e prop ose a filter d esign for GFDM to supports even values fo r bo th M a nd K , particular ly when they are power-of-two. T o this end, we introd uce a fraction al shift in the sampling of the contin uous frequency response of conv entional basis filters, such as raised- cosine (RC) filter , to allow bo th e ven-valued and odd- valued M , K to be der iv ed from the same filter response. As a function of this shift, a closed-fo rm expression f or the con dition nu mber o f A is provided and the optimal shif t fo r both ev en and od d M , K in ter ms o f the minimal co ndition nu mber is derived. T o verify the desig n we evaluate th e SIR for the MF rec e i ver and th e NEF of th e ZF receiver . sectionGFDM Modulation Matr ix Deco mposition One GFDM block conve ys the data sym bols { d k,m } via K sub- carriers an d M subsy m bols, yieldin g N = K M samples. The n th o ne a s the en try n o f x ∈ C N × 1 equals [2] [ x ] n = K − 1 X k =0 M − 1 X m =0 d k,m g [ h n − mK i N ] e j 2 π k K n , (1) where g [ n ] denotes the pulse shaping filter and correspo nds to the entr y n of g ∈ C N × 1 . With the N × N modula tio n matrix A constructed as [ A ] ( n,k + mK ) = g [ h n − mK i N ] e j 2 π k K n , Eq. (1) can b e fo rmed as x = Ad , where [ d ] k + mK = d k,m . A. Decomposition of A For a L × Q matr ix X , let x = vec L,Q ( X ) and u n vec L,Q ( x ) denote the vectorization oper ation an d its inverse. Let F N be the N -p oint d iscrete Fourier tr ansform (DFT) matrix with elements [ F N ] ( i,j ) = e − j 2 π ij N . Let th e unitar y m atrix U L,Q be U L,Q = 1 √ Q I L ⊗ F Q , (2) where ⊗ is the Kronec ker produ ct. Le t Π L,Q ∈ ℜ LQ × LQ be the p ermutation matrix that fu lfills fo r any L × Q matrix X vec ( X T ) = Π L,Q vec ( X ) . (3) The ( Q, L ) DZT [4] Z x = ( F Q ⊗ I L ) x , x ∈ C QL × 1 can be written as a matr ix Z ( x ) Q,L ∈ C Q × L with Z ( x ) Q,L = F Q V ( x ) Q,L = ˜ V ( x ) Q,L , (4) where V ( x ) Q,L = ( unv ec L × Q { x } ) T . (5) Resorting to the DZT of g and ˜ g = F N g , we can factorize A into two for ms A = Π T K,M U H K,M | {z } U ( g ) Λ ( g ) U K,M Π K,M U H M ,K | {z } V ( g ) H , (6) = F H N √ N Π T M ,K U H M ,K | {z } U ( ˜ g ) Λ (˜ g ) U M ,K Π M ,K U K,M Π M ,K | {z } V ( ˜ g ) H , (7) 2 where U (˜ g ) and V (˜ g ) are unitary matrices an d Λ ( g ) = diag n vec M ,K n √ K Z ( g ) M ,K oo , (8) Λ (˜ g ) = diag  vec M ,K  1 √ K Z ( ˜ g ) K,M  , (9) where Λ ( g ) and Λ (˜ g ) contain the DZT of g and ˜ g , respe c ti vely . As a result, p roperties of A are dictated by Λ (˜ g ) or Λ ( g ) . In this paper we focu s on p ulse sha p es that are sparse in frequen cy , h e n ce we emp loy Λ (˜ g ) for the subsequent analysis. B. P erformance ind icators Define the short- h and n otation z k,m = h Z (˜ g ) K,M i ( k,m ) . Then σ 2 k,m = | z k,m | 2 correspo n d to the squared singular values of A scaled by K . The condition al num ber o f A is given by [9] cond ( A ) = max k,m { σ k,m } min k,m { σ k,m } = σ max σ min . (10) Considering the r e ceiv ed signal in A WGN ch annel [10] the NEF of ZF and the SI R of the MF receiver can be wr itten as NEF = 1 N 2 k A k 2 F   A − 1   2 F = 1 N 2    Λ (˜ g )    2 F    Λ (˜ g ) − 1    2 F = 1 N 2   X k,m σ 2 k,m     X k,m 1 σ 2 k,m   , (11) SIR = 1 N     A H A k g k 2 − I N     2 F = 1 N     Λ (˜ g ) Λ (˜ g ) H k g k 2 − I N     2 F = 1 N X k,m    σ 2 k,m 1 N P k,m σ 2 k,m − 1    2 . (12) I I . G F D M P U L S E S H A P I N G FI LT E R D E S I G N The con ventional pulse shaping filter design f or the GFDM is to let g = F H N ˜ g with [ ˜ g ] n = H ( n N ) , where H ( ν ) stands fo r the discrete-time Fourier transform ( DTFT) of a pre- selected basis filter h [ n ] that is o f practical in te r ests, e.g ., RC or Root- Raised Cosine (RRC). Here, ν is the n ormalized freq uency and th u s the p eriod of H ( ν ) is eq ual to 1 . With su ch design of g , it has been shown in [3] that A becomes sing ular fo r ev en M , K and a real symm etric filter h [ n ] . This is caused by h Z ( ˜ g ) K,M i ( k,m ) = ( Z H ( ν ))( k K , m M ) , where ( Z H ( ν ))( f , t ) denotes the d iscrete-time Zak transfo rm of H ( ν ) an d f or any real symmetric filter we k now ( Z H ( ν ))( 1 2 , 1 2 ) = 0 . The re- quiremen t of o dd M or K impedes an efficient imp lementation in te r ms o f low-comp lexity r adix- 2 FFT o peration s. I n th e sequel, we propose a n ovel design appro ach that overcomes this restriction for any basis filter h [ n ] fulfilling the following condition s 1) h [ n ] is r eal-valued, i.e. H ( ν ) = H ∗ (1 − ν ) = H ∗ ( − ν ) . 2) H ( ν ) spans two sub carriers with in each pe riod, i.e. H ( ν ) = 0 , ∀ ν ∈ [ 1 K , 1 2 ] . 3) | H ( ν ) | is decreasing from 1 to 0 fo r ν ∈ [0 , 1 K ] . W e start f r om noting | ( Z H ( ν − η ))( f , t ) | = | ( Z H ( ν ))( f − η , t ) | , (13) namely , shifting the fr equency response of a filter also shifts the f r equency coor dinate of its Zak transfor m [4]. Hence, shifting H ( ν ) can h e lp us av oid to sample th e zero in ( Z H ( ν )) for ev en M , K . Acc ording ly , the samples of ˜ g are defined as [ ˜ g ] n ( λ )=    H  n + λ N  , 0 ≤ n < M − λ H ∗  N − n − λ N  , N − M − λ < n ≤ N − 1 0 , otherwise    , (14) for λ ∈ [0 , 1[ . ˜ g can be reshaped as in ( 5) to h V ( ˜ g ) K,M ( λ ) i ( k,m ) =    H  m + λ N  , k = 0 H ∗  M − m − λ N  , k = K − 1 0 , else where    . (15) Applying DFT acco rding to (4), we get z k,m ( λ ) = H  m + λ N  + H ∗  M − m − λ N  e j 2 π k K . (16) Due to th e sym m etry of H ( ν ) , we h av e z k,m (1 − λ ) = z ∗ k,M − 1 − m ( λ ) e j 2 π k K , σ 2 k,m (1 − λ ) = σ 2 k,M − 1 − m ( λ ) . (17) Hence, all re su lts regarding co nditional numb er , NEF an d SIR are symm etric aro und λ = 0 . 5 . Mo reover , Eq. (16) shows that z k,m (1 + λ ) = z k,m +1 ( λ ) . Hence, it suffices to stud y the c a se 0 ≤ λ ≤ 0 . 5 . Add itionally , we focus on K = 2 x for x > 1 . T o obtain clo sed-form s of the condition nu mber o f A , we subsequen tly focus on two particular families o f H ( ν ) , namely well-localized filters that fulfill the inter-symbol-in terference (ISI)-free cr iterion without or with matche d filterin g. A. ISI fr ee without matched filter In this case H ( ν ) additionally satisfies K − 1 X k =0 H  ν − k K  = 1 . (18) From the sym m etry and limited ban d of H ( ν ) it follows H ( ν ) + H ∗  1 K − ν  = 1 , ∀ ν ∈ [0 , 1 K ] (19) and H  m + λ N  + H ∗  M − m − λ N  = 1 . A lso , ther e exists a function f ( ν ) = r ( ν ) e j φ ( ν ) with f ( ν ) = − f ∗  1 K − ν  and H ( ν ) = 1 2 (1 + f ( ν )) , ∀ ν ∈ [0 , 1 K ] . (20) Let u s assume a re a l-valued f ( ν ) , i.e. φ ( ν ) = 0 and f ( ν ) = r ( ν ) 1 . Due to the constraint o f decreasing amplitud e H ( ν ) , r ( ν ) must be decre asing fro m 1 to − 1 for ν ∈ [0 , 1 K ] . Based on (19) an d (2 0) we get σ 2 A k,m ( λ ) = (1 + f 2 m ( λ ) 2 + (1 − f 2 m ( λ )) 2 cos  2 π k K  , ( 21) 1 Comple x f ( ν ) as in X ia-filters [11] is treated in the follo wing section. 3 where f m ( λ ) = f ( m + λ N ) = 2 H  m + λ N  − 1 . The singu lar values are sy mmetric with respect to k , an d decreasing with k = 0 , · · · , K 2 . Th erefore, σ 2 A 0 ,m ( λ ) = 1 and σ 2 A K 2 ,m ( λ ) = f 2 m ( λ ) are the max imum and minimum singular value w ith respect to k , respectively . Therefo re, σ 2 A max ( λ ) = 1 , b e c ause f 2 m ( λ ) ≤ 1 , an d σ 2 A min ( λ ) is obtained f rom min m { f m ( λ ) 2 } . Since f ( ν ) is d ecreasing and antisymm etric aroun d 1 2 K , f ( ν ) 2 is decreasing ∀ ν ∈ [0 , 1 2 K ] and increasing ∀ ν ∈ [ 1 2 K , 1 K ] . As a result, when M is even, 0 ≤ λ ≤ 0 . 5 , σ 2 A min is obtained at m = M / 2 , and when M is od d, it is ob tained at m = ( M − 1) / 2 . Th erefore, σ 2 A min ( λ ) = f 2  1 2 K + S ( λ ) 2 N  . (22) where S ( λ ) =  2 λ, M is e ven 1 − 2 λ, M is odd  . (2 3) From the increasing/decr easing intervals of f 2 ( ν ) , σ 2 A min ( λ ) increases with 0 ≤ λ ≤ 0 . 5 for even M and decr eases when M is odd . Hence, the condition number can b e expre ssed as cond( A A )( λ ) = 1    f  1 2 K + S ( λ ) 2 N     . (24) Similarly , co nd( A A )( λ ) is decr easing fo r even M an d incre a s- ing for odd M . Henc e, the b est c o ndition of A is attained at λ = 0 . 5 f or even M a nd λ = 0 f or odd M . B. ISI fr ee after matched filtering A filter H ( ν ) is ISI-free after ma tched filtering if K − 1 X k =0     H  ν − k K      2 = 1 . (25) By exploiting the symmetry and band limit, we get | H ( ν ) | 2 +     H ∗  1 K − ν      2 = 1 , ∀ ν ∈ [0 , 1 K ] , (26) and henc e   H  m + λ N    2 +   H ∗  M − m − λ N    2 = 1 . Furtherm ore, there exists a real-valued function f ( ν ) = − f  1 K − ν  , which is d ecreasing from 1 to − 1 in the interval ν ∈ [0 , 1 K ] with | H ( ν ) | 2 = 1 2 (1 + f ( ν )) , ∀ ν ∈ [0 , 1 K ] . (27) Adding an (arb itrary) phase φ ( ν ) yield s the original H ( ν ) by H ( ν ) = e j φ ( ν ) r 1 2 (1 + f ( ν )) , ∀ ν ∈ [0 , 1 K ] . (28) Using ( 26) and (28), H  m + λ N  = e j φ a,m ( λ ) r 1 2 (1 + f m ( λ )) , H ∗  M − m − λ N  = e j φ b,m ( λ ) r 1 2 (1 − f m ( λ )) . (29) where φ a m ( λ ) = φ  m + λ N  , φ b m ( λ ) = − φ  M − m − λ N  , an d f m ( λ ) = f  m + λ N  . As special cases, we stud y the p hase in the form φ ( ν ) = − φ  1 K − ν  + β π 2 , β = 0 , 1 , 2 , 3 . Th en e j φ a,m ( λ ) = j β e j φ b,m ( λ ) . No ISI with and with out MF, as the Xia filters [11] p rovide, is obtained with f ( ν ) = cos (2 φ ( ν )) and β = 2 or , eq ually , φ ( ν ) = 1 2 acos( f ( ν ) . From (16), we get σ 2 B k,m ( λ ) = 1 + p 1 − f 2 m ( λ ) cos 2 π k − β K 4 K ! . (30) The m aximum singular value with r espect to k is located at k max = β K 4 and the m inimum one at k min = ( β + 2 mo d 4) K 4 . This re quires that K is a mu ltiple of 4 for β = 1 , 3 . σ 2 B k max ,m ( λ ) = 1 + p 1 − f 2 m ( λ ) , σ 2 B k min ,m ( λ ) = 1 − p 1 − f 2 m ( λ ) . (31) Follo wing the same argu ment as previously , based on the proper ties of f ( ν ) , both σ 2 B min ( λ ) and σ 2 B max ( λ ) are ob tained at m = M / 2 for even M and m = M − 1 2 and f or o dd M . Thus, σ 2 B max ( λ ) = 1 + s 1 − f 2  1 2 K + S ( λ ) 2 N  , σ 2 B min ( λ ) = 1 − s 1 − f 2  1 2 K + S ( λ ) 2 N  , (32) and the co nditional n umber can th e n b e written as cond( A B )( λ ) =    f  1 2 K + S ( λ ) 2 N     1 − r 1 − f 2  1 2 K + S ( λ ) 2 N  . (33) cond( A B )( λ ) is decreasing for even M and increasing for odd M with λ ∈ [0 , 0 . 5] . When using the same fu nction f ( ν ) in cases A and B, we notice that σ 2 B max ( λ ) ≥ 1 = σ 2 A max and σ 2 B min ≤ f 2 m max ( λ ) = σ 2 A min , and hence cond( A A )( λ ) ≤ cond( A B )( λ ) , (34) proving that th e con d ition num ber is smaller when u sing an ISI-free filter, com pared to using its square ro ot, which has been numerically shown in [12]. I I I . N U M E R I C A L E X A M P L E In this section, we stud y the family o f pro totype filters with roll-off factor α , b eing ob tained with the gener ator fu n ction f ( ν ) =    1 , 0 ≤ ν ≤ 1 − α 2 K f a  2 K α [ ν − 1 2 K ]  , 1 − α 2 K < ν ≤ 1+ α 2 K − 1 , 1+ α 2 K < ν ≤ 1 K    . (35) f a is rea l-valued anti-sy m metric ( f a ( x ) = f a ( − x ) ), and decreasing from 1 to − 1 f or x ∈ [ − 1 , 1] . T herefor e, f ( ν ) = − f ( 1 K − ν ) . Hence , f ( ν ) can con struct pulse shap es acco rding to (20 ) or (28). From ( 33), (24) and u sing (35), we find that for M α ≤ S ( λ ) , cond( A A ) = cond( A B ) = 1 . For S ( λ ) ≤ M α , cond( A A )( λ ) = 1    f a  S ( λ ) αM     , cond( A B )( λ ) =    f a  S ( λ ) αM     1 − q 1 − f a 2 ( S ( λ ) αM ) . (36) The condition n umber is inde penden t of K and, based on the proper ties of f a , increa ses with αM . As a particular example, 4 0 0.2 0.4 0.6 0.8 1 Sampling offset 2 4 6 8 10 12 14 Conditional number = 0.5, K = 64 M=8 M=8 M=9 M=9 M=9 M=9 RC-Sim RC-Closed-form RRC-Sim RRC-Closed-form Fig. 1: Conditiona l number . 0 0.2 0.4 0.6 0.8 1 Sampling offset 1 2 3 4 5 6 7 8 Noise enhancment [dB] = 0.5, K = 64 M=8 M=8 M=9 M=9 M=9 M=9 RC RRC Fig. 2: Noise en hancmen t Factor . 5 10 15 20 25 30 35 40 M -15 -10 -5 0 NEF/SIR [dB] = 0.5, K = 64 RC RRC NEF SIR Fig. 3: NEF an d SIR for o ptimal λ . RC and RRC use the fun ction f a ( x ) = − sin( π 2 x ) . Replacing in (36) we g et, cond( A RC )( λ ) =  sin  π 2 S ( λ ) αM  − 1 , cond( A RRC )( λ ) =  tan  π 4 S ( λ ) αM  − 1 . (37) Fig. 1 shows the co n dition number of A f or different sam- pling shift λ , and validates the closed-f o rm expressions ( 37) numerically . As sho wn, λ = 0 is o ptimal fo r o dd M and λ = 1 2 for even M when K is also e ven. In addition, as pr oved in ( 37), using RC yields a better cond itioned A than RRC. Further more, num e r ically o btained values for NEF as shown in Fig. 2 behave similarly as th e cond ition number . This can be explained by the influence of the smaller singular value on the noise en h ancemen t. In both cases, th e condition number as well as the smallest singular value depen d on    f a  S ( λ ) αM     = sin  π 2 S ( λ ) αM  . Considerin g th e op timum λ , Fig. 3 illu strates N E F and SIR with different M . The proper choice of λ with respect to M preserves the trend of NEF which increases with M . On the other ha n d, SIR is indepen dent of M when M is big enou gh. In fact, SIR approa c h es the interferen ce value that can be directly obtain e d from SIR = 2 R 1 2 1 2 K | H ( ν ) | 2 dν , which is in d epende n t of λ and K but depe n ds on α . I V . C O N C L U S I O N For the wa vefo rm GFDM, the con dition number of its modulatio n matrix is fully characte r ized by the ado pted pulse shaping filter . In this letter , we obser ved th at a frequency- domain shift o f the f requency response of the pulse sha ping filter can y ield a ch ange in th e condition number . By de riving a closed-form expression of the conditio n number, we can find the o p timal shift that min im izes th e cond ition n umber for GFDM modulatio n. This yields a filter desig n that per- mits GFDM to h av e an arb itrary n umbers o f subcarriers an d subsymbo ls per subcarrier, in pa rticular power-of-two values become possible. W e num e rically verified the obtained closed- form expression an d computed the NEF and SIR with respect to ZF and MF receivers in an additive white Gaussian noise (A WGN) chan nel, in dicating that an optimal con dition num b er yields a lso o ptimal N E F v alues. A C K N OW L E D G M E N T The work pr e sen ted in th is paper has bee n perfor med in the framework of the SA TURN projec t with contr act no. 10023 5995 f unded by the Euro paischer Fonds f ¨ ur regionale Entwicklun g (EFRE) and by the Federal Ministry of Educatio n and Research within the p rogram me ”T wenty20 - Partnership for Innovation” u nder co ntract 03ZZ05 05B - ”fast wireless”. R E F E R E N C E S [1] P . Banelli et al. , “Modulation Formats and W ave forms for 5G Network s: Who W ill Be the Heir of OFDM?: An overvie w of alternat i ve modula- tion schemes for improve d spectr al ef ficienc y , ” IEEE Signal Proc essing Mag . , vol. 31, no. 6, pp. 80–93, Nov . 2014. [2] N. 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Matth´ e et al. , “Influence of Pulse Shaping on Bit Error Rate Performance and Out of Band Radiation of Generalized F requency Di vision Multiple xing, ” in ICC’14 - W orkshop on 5G T echnolo gies (ICC’14 WS - 5G) , Sydney , Australia, 2014, pp. 43—-48. [13] R. M. Gray et al. , “T oeplitz and circula nt matrices: A re view , ” F ounda- tions and T r ends R  in Communicatio ns and Information Theory , vol. 2, no. 3, pp. 155–239, 2006. 5 A P P E N D I X Let S ∈ C QL × QL be a block circu lant ma trix g enerated from d iagonal m atrices, such that S =    S 0 S Q − 1 · · · S 1 . . . . . . . . . S Q − 1 · · · S 0    , (38) where S q = diag { v q } ∈ C L × L , and v q is the q - th co lumn of a matrix V ∈ C L × Q , i.e. v q = [ V ] (: ,q ) , then [13] S = Π T L,Q U H L,Q Λ U L,Q Π L,Q . (39) where, Λ = dia g  vec  F Q V T  . (40 ) Using the notations x ( i ) = x [ < n − i > N ] an d definin g the rep letion matrix R L,Q ∈ ℜ LQ × L , R L,Q = 1 L ⊗ I Q , the transmitted GFDM block in ( 1) ca n b e expr essed in the following vector fo r m, x = K − 1 X k =0 M − 1 X m =0 d k,m diag n g ( mK ) o R M ,K  F H K  (: ,k ) , = M − 1 X m =0 √ K diag n g ( mK ) o R M ,K 1 √ K F H K d m = S ( M ) U H M ,K vec { D } = A · vec { D } . (41) Here, A = S ( M ) U H M ,K and S ( M ) √ K = h diag n g (0 K ) o R M ,K , · · · , diag n g (( M − 1) K ) o R M ,K i (42) is block circular matrix as in (38), with S ( M ) m = diag  √ K h V ( g ) K,M i (: ,m )  . From (39), we g e t S ( M ) = Π T K,M U H K,M Λ ( g ) U K,M Π K,M , (43) Λ ( g ) = √ K diag  vec  F M  V ( g ) K,M  T  . (44) As a r esult we get A defined in (6). The N -FFT of (1) results in ˜ x [ n ] = K − 1 X k =0 M − 1 X m =0 d k,m ˜ g [ < n − k M > N ] e − j 2 π m M n . (45) Follo wing similar steps we get ˜ x = F N A · vec { D } . (46) Here F N A = S ( K ) U K,M Π M ,K and S ( K ) √ M =  diag  ˜ g (0 M )  R K,M , · · · , diag  ˜ g (( K − 1) M )  R K,M  . (47) By replacing S ( K ) as in (44), th e n 1 √ N F N A = Π T M ,K U H M ,K Λ (˜ g ) U M ,K Π M ,K U K,M Π M ,K . (48) Λ (˜ g ) = 1 √ K diag  vec  F K  V ( ˜ g ) M ,K  T  . (49) And finally b y m u ltiplying whit 1 √ N F N , we get (7 ).