Construction of MIMO MAC Codes Achieving the Pigeon Hole Bound

Transcription

1 Construction of MIMO MAC Codes Achieving the Pigeon Hole Bound Toni Ernvall and Roope Vehkalahti Turku Center for Computer Science and Department of Mathematics, University of Turku, Finland s: {tmernv, arxiv:100675v1 [mathra] 3 Feb 01 Abstract This paper provides a general construction method for multiple-input multiple-output multiple access channel codes (MIMO MAC codes) that have so called generalized full rank property The achieved constructions give a positive answer to the question whether it is generally possible to reach the so called pigeon hole bound, that is an upper bound for the decay of determinants of MIMO-MAC channel codes I INTRODUCTION In MIMO MAC the DMT optimality criteria can be given by splitting the whole error probability space to separate error events and then giving a criteria for each event separately In the case where each user is in error, the criteria given in [] closely resembles that of the classical NVD condition used in single user case This criteria would give us a natural goal if we would like to build DMT achieving codes Unfortunately in [5] it was proved that this criteria is too tight in the case where the codes of the single users are lattice space-time codes The pigeon hole bound in [5] proves that, irrespective of the code design, the determinants of the overall code matrices will decay with a polynomial speed In [3] it was proved that the two user single antenna code (BB-code) given in [9] do achieve the pigeon hole bound In this paper we are giving a wast generalization of BB-code to general MAC-MIMO The codes we will build fulfill the so called generalized rank criteria and do reach the pigeon hole bound The single user codes we are using are based on the multiblock codes from division algebras [6] This approach has been taken in several recent papers on MIMO-MAC However, in these papers the full rank criteria has been achieved by using either transcendental elements [9] or algebraic elements with high degree [4] Both of these methods make it extremely difficult to measure the decay of the codes and likely lead to bad decay Instead of the usual algebraic independence strategy we will use valuation theory to achieve the full rank condition This technical tool allows us to use algebraic elements with low degree By applying Galois theoretic method of Lu et al [5] and methods from Diophantine approximation, originally introduced by Lahtonen et al in [3], we will prove that our codes achieve good decay and in particular reach the pigeon hole bound As a warm up we discuss, when it is possible to achieve the generalized full rank condition for two users, each having a single transmit antenna In particular these results clarify the construction of BB-code in [9] Then in Section IV we will give our general construction and state that such an construction always exists and that each code does fulfill the generalized rank criteria In Section V we analyze the decay of the constructed codes In particular Theorem 54 gives a general lower bound and Corollary 55 shows that our codes achieves the pigeon hole bound In the last section we give few explicit examples of our codes II DECAY FUNCTION, MULTIUSER CODE, AND OTHER DEFINITIONS In this paper we are considering MIMO MAC code design in the Rayleigh flat fading channel We suppose that we have U users, each having n t antennas and that the receiver has n r antennas and complete channels state information We also suppose that the fading for each user stays stable for Un t time units Let us refer to the channel matrix of the ith user with H i M nr n t and let us suppose that each of these have iid complex Gaussian random variables with zero mean and unit variance as coefficients In this scenario the base station receives U Y = H i X i +N, i=1 where, X i M nt Un t (C), is the transmitted codeword from the ith user, and N M nr n tu(c) presents the noise having iid complex Gaussian random variables as coefficients In this scenario multiuser MIMO signal is a Un t Un t matrix where the rows (j 1)n t +1,(j 1)n t +,,(j 1)n t +n t represent jth user s signal (j = 1,,U) We suppose that each user applies a lattice space-time code L j M nt Un t,j = 1,,U We also assume that each user s lattice is of full rank r = Un t, and denote the basis of the lattice L j by B j,1,,b j,r Now the code associated with the jth user is a restriction of lattice L j { r } L j (N j ) = b i B j,i b i Z, N j b i N j, i=1 where N j is a given positive number Using these definitions the U-user MIMO MAC code is (L 1 (N 1 ), L (N ),,L U (N U )) For a matrix M(X 1,,X U ) = (X T 1,X T,,X T U )T where X j L j (N j ) that jth user has sent, we define D(N 1,,N U ) = min detm(x 1,,X U ) X j L j(n j)\{0}

2 This is called the decay function In the special case N 1 = = N U = N we write D(N) = D(N 1 = N,,N U = N) If we have a U-user code with decay function D(N 1,,N n ), such that D(N 1,,N n ) 0 for all N 1,,N U Z +, we say that the code satisfies (generalized) rank criterion III -USER CODE In [9] Badr and Belfiore introduced a -user single antenna MAC code where the matrix coefficients were from the field Q(i, 5) In the following we give a complete characterization of when their construction method leads to codes with generalized full rank property Theorem 31: Let /Q and L/ be two field extensions of degree and a,b,c,d L Let also σ be the non-trivial element in the Galois group Gal(L/) Define C = {( ax bσ(x) ) } x,y OL There exists a matrix in C with zero determinant if and only if N(a) N(b) N(c) N(d) = 0, (1) where the function N = N L/ denotes the norm of extension L/ Proof: Assume first that we have a matrix ( ) ax bσ(x) with zero determinant This means that adxσ(y) = bcσ(x)y, which gives N(a)N(d)N(x)N(σ(y)) = N(b)N(c)N(σ(x))N(y) Continuing we get N(a)N(d) = N(b)N(c) ie N(a)N(d) N(b)N(c) = 0 Assume then that N(a)N(d) N(b)N(c) = 0 If b or c is zero then N(a)N(d) = 0 ie a or d is zero and then for all x,y O L we have ax bσ(x) = 0 Otherwise N( ad bc ) = 1 Then by Hilbert 90 we have some z L such that ad bc = σ(z) z Then write z = w n with w O L and n Z This gives ad This means that the determinant of ( aw bσ(w) c1 dσ(1) is zero bc = σ(w) w ie adw bcσ(w) = 0 ) IV MULTI ACCESS CODE WITH SEVERAL TRANSMISSION ANTENNAS A Construction of U-user code for n t transmission antennas From now on we concentrate on the scenario where we have U Z + users and each user has n t Z + transmission antennas Throughout this section we assume to be an imaginary quadratic extension of Q with class number 1, L is a cyclic Galois extension of of degree Un t, such that L = (α) with α R, σ a generating element in Gal(L/) and p O an inert prime in L/ We also define τ = σ U and F to be the fixed field of τ So we have [L : F] = n t, [F : ] = U, Gal(L/F) =< τ >, and Gal(F/) =< σ F > where σ F is a restriction of σ in F Let v = v p be the p- adic valuation of the field L In this section, when we say that L/, p, and σ are suitable we mean that they are as above Due to the space limitations, we skip the proof of the following proposition Proposition 41: For every complex quadratic field, having class number 1, and for any U and for any n t we have a suitable degree n t U extension L/, prime p O and automorphism σ Gal(L/) Now we are ready for our construction Let x j O L for all j = 1,,n t Define M = M(x 1,x,,x nt ) to be x 1 pτ(x nt ) pτ (x nt 1) pτ nt 1 (x ) x τ(x 1 ) pτ (x nt ) pτ nt 1 (x 3 ) x 3 τ(x ) τ (x 1 ) pτ n 1 (x 4 ) x nt 1 τ(x nt ) τ (x nt 3) pτ nt 1 (x nt ) x nt τ(x nt 1) τ (x nt ) τ nt 1 (x 1 ) ie a matrix representation of an element in the cyclic algebra (L/F,τ,p) The following lemma proofs that (L/F,τ,p) is a division algebra Lemma 4: Let x j O L for all j = 1,,n t such that x l 0 for some l and min(v(x 1 ),,v(x nt )) = 0 Then we have det(m(x 1,x,,x nt )) 0 and v(det(m(x 1,x,,x nt ))) n t 1 Proof: Write M = M(x 1,x,,x nt ) and N = N L/F Assume first that v(x 1 ) = 0 Then the determinant is N(x 1 ) + py for some y O L and hence we have v(det(m)) = min(v(n(x 1 )),v(py)) = 0 Assume then that v(x 1 ),v(x ),,v(x l 1 ) > 0 andv(x l ) = 0 with1 < l n t Notice that in this case all the other elements a of matrix M than those in the left lower corner block of side length n t l + 1 have v(a) > 0 Either they have coefficient p or they are automorphic images of elements x 1,x,,x l 1 Now det(m) = ±p l 1 N(x l ) + p l z for some z O L since all the other terms except ±p l 1 N(x l ) have at most n t l factors from this left lower corner and hence at least n t (n t l) = l terms have factor p This gives that v(det(m)) = min(v(p l 1 N(x l )),v(p l z)) = l 1 n t 1

3 Definition 41: Define M j = M(x j,1,x j,,,x j,nt ) for all j = 1,,U In our multi access system the code C j of jth user consists of n t Un t matrices B j = ( Mj,σ(M j ),σ (M j ),,p k σ j 1 (M j ),,σ (M j ) ) where k is any rational integer strictly greater than U(nt 1) and x j,l 0 for some l Here k is same for all users The the whole code C U,nt consists of matrices A = B 1 B B U, where B j C j for all i = 1,,U This means that the matrices A C U,nt have form p k M 1 σ(m 1 ) σ (M 1 ) M p k σ(m ) σ (M ) M U σ(m U ) p k σ (M U ) We will skip the proof of the following proposition, stating that each of the single user codes satisfies the NVD condition and are therefore DMT optimal Proposition 43: Let C U,nt C U,nt and C j be the jth users code in the system C U,nt for some j 1,,U Then the code C j is a Un t -dimensional lattice code with the NVD property The code depends on how we did choose L/, p, σ, and k, so to be precise, we can also refer to C U,nt with C U,nt (L/,p,σ,k) Let us call the family of all such codes C U,nt (L/,p,σ,k) (ie codes constructed with any suitable L/, p, σ, and k) by C U,nt That is C U,nt = {C U,nt (L/,p,σ,k)} L/,p,σ,k where L/, p, σ, and k are any suitable ones According to Proposition 41 we can always find suitable L/, p, σ, and k for any U Z + and n t Z + We therefore have the following theorem Theorem 44: For any choice of U Z + and n t Z + we have C U,n Note that the code C U,1 = C U,1 (L/,p,σ,1) C U,1, a code for U users each having one transmission antenna, consists of matrices of form p 1 x 1 σ(x 1 ) σ (x 1 ) σ (x 1 ) x p 1 σ(x ) σ (x ) σ (x ) x 3 σ(x 3 ) p 1 σ (x 3 ) σ (x 3 ) x U σ(x U ) σ (x U ) p 1 σ (x U ) Note also that the code C 1,nt = C 1,nt (L/,p,σ,k) C 1,nt is a usual single user code multiplied by p k Theorem 45: Let C U,nt C U,nt The code C U,nt is a full rate code and satisfies the generalized rank criterion Proof: Let A C U,nt = C U,nt (L/,p,σ,k) We may assume that min(v(x j,1 ),,v(x j,nt )) = 0, for all j = 1,,U, because otherwise we can divide extra p s off That does not have any impact on whether det(a) = 0 or not The determinant of A is U p kunt det(σ l 1 (M l ))+y where v(y) k(un t ) We know that v(σ l 1 (det(m l ))) = v(det(m l )) because p is from, ie from the fixed field of σ, and det(m l ) 0 for all l Therefore U U v(p kunt det(σ l 1 (M l ))) = kun+ v(det(m l )) that is less or equal than kun t +U(n t 1) = U(n t 1 kn t ) by 4 But if we would have det(a) = 0 then U v(y) = v(p kunt det(σ l 1 (M l ))) and hence v(y) U(n t 1 kn t ) implying k(un t ) U(n t 1 kn t ) This gives k U(n t 1) ie k U(nt 1) a contradiction Remark 41: Using multiblock codes from division algebras as single user codes in the MIMO MAC scenario has been used before for example in [9], [4] and [5] In [9] the full rank condition for codes with n t > 1 is achieved by using transcendental elements In [4] the same effect is achieved with algebraic elements of high degree V ON THE DECAY FUNCTION OF CODES IN C U,nt In this chapter we will prove an asymptotic lower bound for the decay function of codes from C U,nt In [3] the authors give a general asymptotic upper bound for a decay function in the case that only one user is properly using the code ie N 1 can be anything but N = = N U = 1 are restricted We will see that in this special case our codes have asymptotically the best possible decay Lemma 51: Let C U,nt = C U,nt (L/,p,σ,k) C U,nt, A C U,nt, and let F be the fixed field of τ = σ U Then det(a) F Proof: As F is the fixed field of τ it is enough to prove that τ(det(a)) = det(a) Write again M j = M(x j,1,x j,,,x j,nt ) and notice thatτ(m j ), after switching first and last column and first and last row, is x j,1 τ(x j,nt ) τ (x j,nt 1) τ nt 1 (x j, ) px j, τ(x j,1 ) pτ (x j,nt ) pτ nt 1 (x j,3 ) px j,3 τ(x j, ) τ (x j,1 ) pτ nt 1 (x j,4 ) px j,nt τ(x j,nt 1) τ (x j,nt ) τ nt 1 (x j,1 )

4 Here we did an even number of row and column changes so The ring O L has a Z-basis {γ 1,,γ Unt } Each of these when calling the above matrix M j we see that τ(det(a)) is basis elements can be presented as p k M 1 σ(m 1) σ (M 1) σ n 1 (M 1) γ M p k σ(m ) σ (M ) σn 1 (M ) l = s l,a a, M 3 σ(m 3 ) p k σ (M 3 ) σn 1 (M 3 ) where s l,a Q for all l = 1,,Un t and a S 1 S Let A C U,n be M U σ(m U ) σ (M U ) p k σ n 1 (M U ) p k M 1 σ(m 1 ) σ (M 1 ) M p k σ(m ) σ (M ) M U σ(m U ) p k σ (M U ) Let us call the above matrix A We notice that matrices A and A are exactly similar apart from the fact that in M j elements in places (1,),(1,3),,(1,n t ) have coefficient p k and elements in places (,1),(3,1),,(n t,1) have not coefficient p k and in M j these are vice versa But this does not change the value of the determinant because in the expansion of the determinant each summand include a product of same number of elements from columns that are congruent to 1 modulo n t and from rows that are congruent to 1 modulo n t Theorem 5: [7] For any full-rateu-user lattice code, each user transmitting with n t antennas there exists a constant k > 0 such that D(N 1 = N,N = N 3 = = N U = 1) k N ()nt For the next theorem we need few definitions Let p(x) = p 0 + p 1 x + + p m x m Z[x] be a polynomial Then we say that H(p(x)) = max{ p j } is the height of the polynomial p(x) and for an algebraic number α we define H(α) = H(φ α ) where φ α is the minimal polynomial of α The next generalization of Liouville s theorem can be found from [8, p 31] Theorem 53: Let α R be an algebraic number of degree κ, H(α) h, H(P) H and deg(p(x)) = m Z + Then either P(α) = 0 or P(α) cm H κ 1 with c = 1 3 κ 1 h κ Now we are ready to give a lower bound for the decay function of our codes The proof can be seen as an extension to the analysis given for BB-code in [3] Theorem 54: For a code C U,nt C U,nt there exists constant > 0 such that Especially D(N 1,N,,N U ) D(N) (N 1 N N U ) ()nt N U()nt Proof: Let C U,nt = C U,nt (L/,p,σ,k) Field extension L/Q has a basis S 1 S where S 1 = {1,δ,δ,,δ,β,βδ,βδ,,βδ } is a basis of F/Q with δ R, = Q(β) and β = w for some positive integer w Notice that if L = F then S = and M j = M(x j,1,x j,,,x j,nt ) be x j,1 pτ(x j,nt ) pτ (x j,nt 1) pτ nt 1 (x j, ) x j, τ(x j,1 ) pτ (x j,nt ) pτ nt 1 (x j,3 ) x j,3 τ(x j, ) τ (x j,1 ) pτ nt 1 (x j,4 ) x j,nt τ(x j,nt 1) τ (x j,nt ) τ nt 1 (x j,1 ) as usual Now for any j = 0,,Un t 1 we have σ j (x m,h ) = Un t u m,h,l σ j (γ l ) where u m,h,l Z and u m,h,l N m for all m, h and l Then the determinant det(a) is a sum consisting of Un t! elements of form Un t Un t Un t p f σ j (x mj,h j ) = p f ( u mj,h j,lσ j (γ l )) where f kun t and m j gets exactly n t times all the values 1,,U and h j gets values from {1,,n t } Now substituting γ l = s l,a a gives that the determinant is a sum consisting of elements of form p f Un t We also write Un t ( u mj,h j,l σ j (a) = s l,a σ j (a)) t j,a a where t j,a Q for all j,a and find that p f Un t ( Un t u m j,h j,l s l,a σ j (a)) can be written as a sum of elements of form 1 p f u a a where 1 Q is some constant, u a Z, and u a = O((N 1 N U ) nt ) Writing also p using basis S 1 S we see that the whole determinant det(a) can be written as a sum of elements of form u a a

5 multiplied by some constant and here we have u a Z, and u a = O((N 1N U ) nt ) On the other hand we know that det(a) F so by uniqueness of basis representation we know that det(a) is a sum consisting of elements of form and hence a S 1 u a a = m=0 u δ mδm +β m=0 u δ m β δm det(a) = H l δ l +β J l δ l, where H l,j l Z and H l, J l are of size O((N 1 N U ) nt ) for all l = 0,,U 1 Using the fact that δ is real we get det(a) ( H l δ l + J l δ l ) Now using 53 and noticing that deg(δ) = U we have det(a) where is some positive constant (N 1 N U ) ()nt, Corollary 55: For a code C U,nt C U,nt there exists constants k > 0 and > 0 such that k D(N 1 = N,N = = N U = 1) N ()nt N ()nt VI EXAMPLES Let us now give few examples of our general code constructions In table Table 1 we have collected some examples of suitable fields and L and inert primes p, fulfilling the conditions of Proposition 41 If = Q(i) then p i O refers to the inert prime and if = Q( 3) then p 3 O is inert The inert primes and fields L are found by looking at totally real subfields of Q(ζ m )/Q and then composing them with the field TABLE I [L : ] L p i p 3 3 (ζ 7 +ζ 1 7 ) +i 3 4 (ζ 17 +ζ17 4 +ζ ζ 1 17 ) +i 3 5 (ζ 11 +ζ 1 11 ) 1+i (ζ 13 +ζ 1 13 ) 1+i (ζ 9 +ζ9 1 +ζ 1 9 +ζ 1 9 ) 1+i 3 A Actual examples We get a code C 3,1 = C 3,1 (Q(i,ζ 7 +ζ7 1 ),+i,σ,1) ie 3-user code with each user having 1 antenna by setting L = (ζ 7 +ζ7 1 ), = Q(i), p = +i, and Gal(L/) =< σ > Now the actual code consists of matrices p 1 x σ(x) σ (x) y p 1 σ(y) σ (y) z σ(z) p 1 σ (z) where x,y,z OL We get a code C, = C, (Q( 3,ζ 17 + ζ ζ 4 ζ ), 3,σ,) ie -user code with each user having antennas by setting L = (ζ 17 + ζ ζ ζ 1 17 ), = Q( 3), p = 3, k = > U(nt 1), and Gal(L/) =< σ > Now the actual code consists of matrices p x 1 p 1 σ (x ) σ(x 1 ) pσ 3 (x ) p x p σ (x 1 ) σ(x ) σ 3 (x 1 ) y 1 pσ (y ) p σ(y 1 ) p 1 σ 3 (y ) y σ (y 1 ) p σ(y ) p σ 3 (y 1 ) where x 1,x,y 1,y O L and x 1 0 or x 0 and y 1 0 or y 0 VII CONCLUSION We gave a general construction for MIMO MAC codes satisfying the generalized rank criteria and gave a lower bound for their decay As a corollary we got that all our codes do achieve the pigeon hole bound ACNOWLEDGEMENT The research of T Ernvall is supported in part by the Academy of Finland grant The research of R Vehkalahti is supported by the Academy of Finland grants and 5457 The authors would like to thank Jyrki Lahtonen for suggesting this problem REFERENCES [1] D Tse, P Viswanath, and L Zheng, Diversity and multiplexing tradeoff in multiple-access channels, IEEE Trans Inf Theory, vol 50, no 9, pp , 004 [] P Coronel, M Gärtner, and H Bölcskei, Selective-fading multiple-access MIMO channels, Diversity-multiplexing tradeoff and dominant outage event regions, preprint available at [3] J Lahtonen, R Vehkalahti, H-F Lu, C Hollanti, and E Viterbo, On the Decay of the Determinants of Multiuser MIMO Lattice Codes, Proc 010 IEEE Inf Theory Workshop, Cairo, Egypt, Jan 010 [4] H-F Lu, R Vehkalahti, C Hollanti, J Lahtonen, Y Hong, and E Viterbo, New Space-Time Code Constructions for Two-User Multiple Access Channels, IEEE J Selected Topics in Signal Processing: Managing Complexity in Multiuser MIMO System, vol 3, no 6, pp , Dec 009 [5] H F Lu, C Hollanti, R Vehkalahti, J Lahtonen, DMT Optimal Code Constructions for Multiple-Access MIMO Channel, IEEE Trans Inform Theory, vol 57, no 6, pp , Jun 011 [6] H F Lu, Constructions of multi-block space-time coding schemes that achieve the diversity-multiplexing tradeoff, IEEE Trans Inform Theory, vol 54, no 8, pp , Aug 008 [7] H-F Lu, J Lahtonen, R Vehkalahti, and C Hollanti, Remarks on the criteria of constructing MAC-DMT optimal codes, Proc 010 IEEE Inf Theory Workshop, Cairo, Egypt, Jan 010 [8] Andrei B Shidlovskii, Transcendental numbers, Studies in mathematics, vol 1, Walter de Gruyter, Berlin, New York, 1989 [9] M Badr and JC Belfiore, Distributed space-time block codes for the non-cooperative multiple-access channel, Proc 008 International Zurich Seminar on Communication, Zurich, Germany, Mar 008, pp