On the Symbol Asynchronous Gaussian Multiple-Access Channel

Transcription

1 On the Symbol Asynchronous Gaussian Multiple-Access Channel Hon-Fah Chong and Mehul Motani Department of Electrical and Computer Engineering ational University of Singapore, Singapore Abstract We consider the symbol asynchronous Gaussian multiple access channel in which each user is allowed to linearly modulate a set of orthogonal waveforms and the symbol periods for each user are not aligned at the receiver This models the case in which asynchronous users may employ quadrature signaling in a multiple access scenario The case in which each user is only allowed to linearly modulate a fixed waveform in each symbol period was considered by Verdu He explicitly evaluated the capacity region of this class of multiple access channels for the case the transmitters know the symbol period offset and also extended it to the case the transmitters have no knowledge of the offset In this paper, we characterize the capacity region for the scenario in which each user is allowed to modulate K orthogonal waveforms and the users know the symbol period offset We note that the orthogonal waveforms need not be identical for both users Similar to the case each user is allowed to modulate a fixed waveform, the result holds regardless of whether or not the transmitters are frameasynchronous I ITRODUCTIO In the information-theoretic study of multiple access channels MAC, there are two types of asynchronism that may occur, namely, frame asynchronism and symbol asynchronism In discrete MAC, frame asynchronism occurs when the codewords for the different users are not aligned at the receiver as shown in Fig The frame offset is an integer satisfying <, is the frame/block length For continuous-time waveform MAC, besides the possibility of frame asynchronism, symbol asynchronism may also occur when the symbol periods for the different users are not aligned at the receiver A two-user MAC is said to be frame synchronous and symbol asynchronous if the offset between the two frames is less than a symbol period see Fig The capacity region of the discrete memoryless MAC DM- MAC with frame synchronism was completely characterized by Ahlswede ] and Liao ] in the early 97s and is given by the following: Theorem : The capacity region of a DMMAC X X, p y x, x, Y is the closure of the convex hull of all R, R satisfying R < I X ; Y X R < I X ; Y X R R < I X X ; Y 3 for some product distribution p x p x on X X Fig Frame asynchronous discrete MAC roof: Refer to proof of 3, Theorem 53] Cover 4] and Wyner 5] gave an explicit expression for the capacity region of the memoryless Gaussian MAC with both frame and symbol synchronism The capacity region is given by the pentagon satisfying R < log 4 R < log 5 R R < log 6 is the power constraint on sender and is the power constraint on sender The restriction of frame synchronism was first removed for the DMMAC by Cover, McEliece and osner 6] They determined the capacity region of the two-user frame asynchronous DMMAC for the case the offset between the two frames as a fraction of the frame length goes to zero as the frame length increases, ie, as They showed that the capacity region remains unchanged in this case and is given by Theorem oltyrev 7], and Hui and Humblet 8] determined the capacity region of the totally frame asynchronous DMMAC They showed that the effect on the capacity region in this case is the removal of the convex hull operation from Theorem Verdu considered the continuous-time waveform memoryless Gaussian MAC with symbol asynchronism 9] each transmitter is allowed to linearly modulate a fixed signature waveform User j, j {, }, transmits a codeword b j, b j,,b j ] R by sending the signal b j ns j t nt 7 n=

2 Fig Frame synchronous symbol asynchronous continuous-time waveform MAC Fig 3 Quadrature Amplitude Modulation s j t is a fixed signature waveform in the interval, T The channel output is given by y t = b ns t τ nt n= b ns t τ nt n t 8 n= τ, τ, T and n t is white Gaussian noise with power spectral density equal to Verdu first considered the case the transmitters have knowledge of the symbol period offset before extending the result to the case the transmitters have no knowledge of the offset In this paper, we extend the result, the transmitters have knowledge of the symbol period offset at the receiver, to the case each user is allowed to modulate K orthogonal waveforms, instead of only one signature waveform This allows us to model a larger class of asynchronous Gaussian MAC For example, this encompasses the situation each Fig 4 Dual Channel Direct Sequence Spread Spectrum user is allowed to linearly modulate two phase-quadrature carriers in each symbol period and the carrier frequency for each user may be different see Fig 3 This may also encompass the dual channel direct-sequence spread spectrum system In this system, each user has two data streams that linearly modulate two phase-quadrature signals and each stream uses a different spreading code see Fig 4 Even though the two spreading codes for each user may not be orthogonal, from Lemma, the dual channel direct-sequence spread spectrum system is equivalent to the continuous-time waveform MAC each user linearly modulates two orthogonal waveforms Our extension closely follows the approach of Verdu In 9], an explicit expression for the capacity region relied on the evaluation of the eigenvalues of Toeplitz matrices In our case, the extension relies on the evaluation of the eigenvalues of block Toeplitz matrices rather than Toeplitz matrices Hence, we will also give some relevant background information on the spectrum of Hermitian block Toeplitz matrices The paper is organized as follows: In Section II, we first introduce some of the mathematical preinaries necessary to understand the paper In Section III, we review some of the results in 9] In Section IV, we describe the channel model and also the equivalent channel model with discrete time outputs In Section V, we describe the main result, Theorem 4, for the case transmitters have knowledge of the mutual offset In Section VI, we show that the capacity region given in Theorem 4 is obtained with stationary inputs Hence, the same capacity region also holds for the case the transmitters are frame-asynchronous In Section VII, we give the main details of the proof of our main result II MATHEMATICAL RELIMIARIES In this section, we give a brief review of some results on the spectrum of Hermitian block Toeplitz matrices ] A otation and preinary considerations We denote by R the set of real numbers, by C the set of complex numbers and by H K K the set of Hermitian matrices of size K K In the following, we will only consider Lebesgue-integrable Hermitian matrix-valued functions defined almost every over the interval Q =, π If A is a square matrix, we denote by A F its Frobenius norm In addition, if A H K K, we indicate by λ k A, k =,,, K, all the eigenvalues of A, counted with their multiplicities, numbered in non-decreasing order We also indicate by σ A the spectrum of A, ie, the set of eigenvalues of A It is again understood that each eigenvalue is counted according to its multiplicity For p, we denote by L p Q, H K K the Banach space of all K K Hermitian matrix-valued functions which are p-integrable on Q, that is, f L p Q, H K K f θ H K K and f p L = p π Q f θ p F dθ < We also denote

3 by L Q, H K K the Banach space of Hermitian matrixvalued functions which are essentially bounded over Q, ie, f is measurable and f L = inf {y R : f θ F < y for ae θ Q} < 9 B Asymptotic spectra of Hermitian block Toeplitz matrices A Hermitian matrix T has a block Toeplitz structure with K K blocks if A A A T = A A CK K A A A with A n C K K, A n = A n We consider the case the blocks A n are the Fourier coefficients of a K K Hermitian matrix-valued function f : Q H K K, which is integrable on Q, that is f L Q, H K K and A n = π f θe înθ dθ, n =, ±, ±, î is the imaginary unit The integration is understood to be carried out on each entry of the K K matrix For every natural number, we associate the block Toeplitz matrix with the Hermitian matrix-valued function f, and we say that {T } is the set of block Toeplitz matrices generated by f Each matrix T has a block Toeplitz structure and each block is a K K matrix with complex entries no structure is imposed upon these blocks If f : Q H K K is a Hermitian matrix-valued measurable function such that f θ has real eigenvalues, for Q Q, we set inf f = Q sup {y R : λ f θ > y for almost every θ Q } 3 sup f = Q { inf y R : λ K f θ < y for almost every θ Q } 4 Theorem : Suppose that f L Q, H K K, and let {T } be the set of Hermitian block Toeplitz matrices generated by f Then, for any natural number, if λ is an eigenvalue of T, it holds that inf f λ sup f 5 Q roof: Refer to proof of, Theorem 3] Q Theorem 3: Suppose that f L Q, H K K and that {T } is the set of block Toeplitz matrices generated by f; then for any function F, continuous on the interval inf f, supf], the following holds: K K l= F λ l T = π K K F λ k f θdθ roof: Refer to proof of, Theorem 33] III A REVIEW OF AST RESULTS Verdu first considered the frame synchronous and symbol asynchronous Gaussian MAC each user knows the symbol period offset In addition, each user is allowed to linearly modulate a fixed waveform s j t, j {, }, of unit energy defined on the interval, T Hence, the channel output is given by 8 subject to the power constraint n= b j n j, j {, } 6 n= ext, Verdu obtained a discrete channel model with an equivalent channel capacity by considering the projection of the channel output y t along the direction of the signals s j t, j {, }, and their T -shifts, ie, y j n = ntτj ntτ j y ts j t nt τ j dt 7 By defining the cross correlations between the assigned waveforms as follows assuming without loss of generality that τ τ : ρ = ρ = T T s ts t τ τ dt 8 s ts t T τ τ dt 9 the equivalent discrete MAC can be expressed as b n b n b n Y = R b n b n b n R is given by ρ ρ ρ R = ρ ρ ρ ρ ρ ρ

4 and the noise vector is Gaussian with zero-mean and covariance matrix R Verdu then made use of the iting characterization of the discrete MAC with memory, Theorem 3] to characterize the capacity of this channel The channel capacity of the frame synchronous and symbol asynchronous Gaussian MAC is given by C = R, R R R R R S R jθ, θ Q π π Sjθdθ=j j {,} log S θ σ log S θ σ log S θ dθ 3 dθ 4 S θ S θs θ ρ ρ ρ ρ cosθ σ 4 dθ 5 Finally, Verdu extended the result to the case each user does not know the symbol period offset Each user only knows that the cross correlation coefficients belong to an uncertainty set Ψ = {ρ, ρ }, ρ, ρ depends on the symbol period offset and on the fixed waveform chosen by each user In this case, the capacity region is given by C = R, R 6 S R jθ, θ Q π π Sjθdθ=j j {,} R log S θ dθ 7 R log S θ dθ 8 π R R inf log S θ ρ,ρ Ψ S θ S θs θ ρ ρ ρ ρ cosθ dθ σ 4 9 For both cases the transmitters know the symbol period offset and the transmitters only have knowledge of the uncertainty set Ψ, the channel capacities are achieved with stationary inputs Hence, the channel capacities for both cases also hold regardless of whether the transmitters are frame asynchronous or not In this paper, we will only consider the case the transmitters know the symbol period offset However, each user is allowed to modulate K orthogonal waveforms instead of only one fixed waveform and the set of K waveforms need Fig 5 Cross correlation coefficients between waveform k for first user and waveform k for the second user not be identical for both users Our proof follows along the lines of the proof of Verdu and we will give the relevant extensions necessary IV CHAEL MODEL For simplicity of notation, we denote n as an element taking values in the set {,,, }, j as an element taking values in the set {, }, and k and k as elements taking values in the set {,,, K} In this paper, we assume the two-user scenario in which user j linearly modulates each of its K orthogonal waveforms s jk t The waveforms occupy the symbol period, T and are assumed to be of unit energy The symbol periods for the K orthogonal waveforms are also assumed to be aligned for each user Hence, we can write the channel output as follows: y t = n= k=k n= n= k=k n= b k ns k t nt τ b k ns k t nt τ n t 3 τ, τ, T, n t is white Gaussian noise with power spectral density equal to and b k n, b k n R satisfying K n= n= k=k b jk n j 3 We obtain an equivalent model with discrete time outputs by passing the received waveform through a matched filter for each of the signals s jk t as follows: ntτj y jk n = y ts jk t nt τ j dt 3 ntτ j { } n= The discrete outputs obtained, {y k n} k=k and { } n= n= {y k n} k=k, are sufficient statistics for the transmitted messages See 9, g 735] or ] Since the K assigned n= waveforms for each user are assumed to be orthogonal, we only need to define the cross correlations between assigned waveforms for different users Referring to Fig 5, we may define the cross correlation coefficients as follows assume that τ τ : α kk = T s k ts k t τ τ dt 33

5 T ρ kk = s k ts k t T τ τ dt 34 For compactness of description, let us define the matrices, α = α kk ] and ρ = ρ kk ], in R K K We also denote X n, X n, n, n, Y n and Y n as column vectors in R K We define X n, X n, Y n and Y n as follows: X n = b n,b n,, b K n] T 35 X n = b n,b n,, b K n] T 36 Y n = y n,y n,, y K n] T 37 Y n = y n,y n,, y K n] T 38 It is easy to verify that the -block asynchronous Gaussian MAC, Y = Y T, Y T,, Y T, Y T] T, can be written as follows: X X X Y = M X 39 X X I K denotes the identity matrix of size K and the noise vector is Gaussian with zero-mean and covariance matrix M and M is given by I K α α T I K ρ ρ T I K α M = α T I K ρ I K α T 4 α We note that the noise sequence thus obtained is correlated However, to invoke, Theorem 3] for the iting characterization for capacity regions of discrete MAC with memory, we need the outputs to be conditionally independent given the inputs Following the Gram-Schmidt orthonormalization procedure, we can obtain an equivalent discrete MAC Refer to Appendix A, with the same capacity region and the noise process is independent Hence, we can directly make use of coding theorems the outputs are conditionally independent given the inputs V CAACITY REGIO Theorem 4: When the transmitters know the mutual offset, the capacity region of the energy-constrained asynchronous Gaussian MAC, each user has K orthogonal waveforms I K to transmit, is given by C = R R R R S jk θ, θ Q R π K π K S jk θdθ= j j {,} K K R, R 4 log S k θ dθ 4 log S k θ dθ 43 K log S k θ S k θ S k θ S k θ d k θ σ 4 dθ 44 d k θ, k = {,,, K} and θ Q are the eigenvalues of the following Hermitian matrix: Γθ = αα T ρρ T αρ T eîθ ρα T e îθ 45 Remark : We note that Γ is a Hermitian matrix-valued function Γ : Q H K K Hence, for every, there is an associated block Toeplitz matrix generated by Γ See Section II VI ACHIEVABILITY OF THE CAACITY REGIO BY STATIOARY IUTS We assume that the inputs to channel 39 are stationary Gaussian processes the power spectral density matrices S j θ, j {, }, are non-negative definite Hermitian matrices satisfying π π The mutual information rates are given by K trs j θ] dθ j 46 I X ; Y X = log det I K σ S θ ] dθ 47 I X ; Y X = log det I K σ S θ ] dθ 48 I X X ; Y = log det I K ] ] S θ O K B θ dθ O K S θ 49

6 B θ = I K α T ρ T eîθ ] α ρe îθ I K 5 and O K denotes the zero matrix of size K K We note that it is difficult to obtain a closed form expression directly as in 9, 39] However, we may perform singular value decomposition to α T ρ T eîθ as follows: α T ρ T eîθ = UθDθV θ 5 Dθ is the diagonal matrix of the square roots of the eigenvalues of Γθ Using the orthogonality of Uθ and V θ, we can express the determinant in the rate-sum constraint as det I K ] ]] Λ θ O K IK Dθ 5 Λ θ Dθ I K O K we have set Λ θ = V θs θvθ and Λ θ = U θs θuθ As singular value decomposition is a continuous and wellbehaved function of its matrix argument see 3, Section 3], we note that S j θ is a continuous function of θ as long as Λ j θ is a continuous function of θ In addition, S j θ for a fixed θ is a non-negative definite matrix if and only if Λ j θ is a non-negative definite matrix Finally, 46 is satisfied if and only if π π K trλ j θ] dθ j 53 is satisfied Hence, for a real and non-negative diagonal matrix Λ j θ satisfying 53, j {, } and θ Q, we have = = = I X ; Y X K log I X ; Y X K log I X X ; Y K log Λ k θ Λ k θ Λ k θ dθ 54 dθ 55 Λ k θ Λ k θ Λ k θ d k θ σ 4 d k θ are the eigenvalues of Γθ and π dθ 56 k=k Λ jk θdθ j 57 K Since the capacity region is achieved with stationary inputs, Theorem 4 holds also for the case the transmitters are frame asynchronous VII ROOF OF THE CAACITY REGIO The transmitters know the actual symbol period offset at the receiver and hence, the exact cross-correlation matrices α and ρ ext, let us define the following vectors: X = X T, X T,, X T] T 58 X = X T, X T,, X T] T 59 X = X T, X T,, X T, X T] T 6 As the capacity region of the asynchronous Gaussian MAC is equivalent to a discrete MAC with memory the outputs are conditionally independent given the inputs,, Theorem 3] is applicable in evaluating the capacity region of the asynchronous Gaussian MAC Hence, a iting characterization for the capacity region of the asynchronous Gaussian MAC is given by C = closure inf C 6 with C = R R I X ; Y X, R : R I X ; Y X X R,X R I X X ; Y 6 X and X are independent random variables satisfying E ] Xj T Xj j We can bound the first two terms in 6 as follows: 63 I X ; Y X I X ; Y X log det I K σ ] Σ log det I K σ ] Σ 64 Σ and Σ represents the covariance matrices of X and X, respectively We note from 39 that the output covariance matrix is given by ME X X T] M M Hence, we can also bound the third term as follows: I X X ; Y log det I K σ E X X T] M ] = log det I K ] ]] Σ O K IK S T O K Σ S I K 65 S is given by S = I α ρ 66 and denotes the Kronecker product The equality above follows along the lines of the proof of the identity in 9,

7 Lemma ] Refer to Lemma in Appendix B for the proof that the equality still holds true in our present case by replacing the scalars ρ and ρ in 9, Lemma ] with their equivalent K K matrices α and ρ, respectively ext, we can perform singular value decomposition to the matrix S = UDV, U and V are orthogonal matrices and D is a diagonal matrix of the singular values {d l } l=k l= of S T S, which is given by αα T αρ T ρα T αα T ρρ T αρ T αρ T 67 ρα T αα T ρρ T After obtaining 65, we can proceed exactly as follows in 9, g ] to obtain C = R, R 68 R R K φ jl, l={,,,k} l=k K l= φ jl j j {,} R K R l= log l= K l= log φ l log φ l φ l φ l φ lφ l σ 4 d l 69 We note that the imization over all matrices of size K is now reduced to a imization over all diagonal matrices of size K Since C is convex, each of the areto-optimal rate pairs rate pairs on the boundary of the region in C satisfy αr α R = R,R C φ jl l=k K l= φ jl j j {,}, l={,,,k} φ jl l=k K l= φ jl j j {,}, l={,,,k} l=k l= l=k l= f φ l,φ l,d l, if α f φ l,φ l,d l, if α 7 f z, z, d = α log z αlog z z z z d 7 f z, z, d = αlog z α log z z z z d 7 If we fix α, following the -dimensional optimization in 9, Appendix IV], we have φ jl, l={,,,k} l=k K l= φ jl j j {,} l= l=k f φ l, φ l, d l l= = l=k f γ d l, β, β,γ d l, β, β, d l = l=k g d l, β, β 73 l= β j, j {, }, are scalar positives Lagrange multipliers in the K-dimensional optimization problem and γ j d l, β, β are continuous functions of d l such that See proof of 9, Lemma 3] l=k γ j d l, β, β = j K 74 l= Following the same approach with the convex set C, given in Theorem 4, for every α, each of its areto-optimal pair attains R,R C αr αr = Φ jk θ R K π Q K Φ jk θdθ j j {,}, θ Q k=k Q f Φ k θ, Φ k θ,d k θdθ 75 and for every α, each of its areto-optimal pair attains R,R C αr αr = Φ jk θ R K π Q K Φ jk θdθ j j {,}, θ Q k=k Q f Φ k θ, Φ k θ,d k θdθ 76 d k θ, k = {,,, K}, are the eigenvalues of the Hermitian matrix Γθ By fixing α, and proceeding as in the Kdimensional optimization, we have αr αr R,R C = d σγθ g d, β, β dθ 77

8 β, and β are positive scalars Lagrange multipliers of the infinite-dimensional optimization problem such that π γ j d, β, β dθ = j, j = {, } π K σ d σγθ 78 ext, instead of considering the singular values of S T S, we can consider { } the singular values of the block Toeplitz matrix l=k T, ˆdl, obtained by substituting the first K K l= entries with αα T ρρ T It is indeed valid to carry this replacement because T and S T S differ only in the first K K entries Thus, they are equivalent asymptotically and since g, β, β and γ j, β, β are continuous functions, { } l=k their averages evaluated at {d l } l=k l= and ˆdl coincide l= as 4, Corollary ] We note that the sequence of symmetric, block Toeplitz matrices, {T }, can be generated by the K K Hermitian matrix-valued function Γ We can readily check that the functions Γ, γ j and g satisfy the conditions of Theorem 3 Hence, for every fixed positive pair β, β, we have the following: = = π g d l, β, β d l σt π g d, β, β dθ 79 d σγθ γ j d l, β, β K d l σt π γ j d, β, β dθ, j = {, } K d σγθ This completes the proof of the capacity region 8 AEDIX A EQUIVALET DISCRETE MAC WITH IDEEDET OISE ROCESS We first state the following theorem necessary to obtain the equivalent discrete MAC model with independent noise process: Lemma : Given K finite-energy functions {s k t} k=k, ie, T s k t dt <, defined on, T, there exists M K orthonormal functions {φ m t} m= defined on, T, ie, { T, k m φ k tφ m tdt = δ km = 8, k = m such that every function in the set {s k t} k=k can be represented in the form s k t = m= s km φ m t, k {,,, K} 8 for each k and m, s km = T s k tφ m tdt roof: Refer to 5, g 63] ext, we define the following waveforms: s L k t = { s k t, t, τ τ, t τ τ, T s R k t = {, t, τ τ s k t, t τ τ, T s L k t = { s k t T τ τ, t, τ τ, t τ τ, T s R k t = {, t, τ τ s k t τ τ, t τ τ, T k {,,, K} see also Fig 5 From Lemma, there exists M L orthonormal functions { φ L m t } L m= defined on, T such that every function in the set { { } } j= k=k s L jk t can be represented in the form j= L s L jk t = m= s L jkm φl m t, j {, }; k {,,, K} 87 for each j, k and m, s L jkm = T sl jk tφl m tdt Similarly, there exists M R orthonormal functions } m {φ Rm t =MR defined on, T such that every m = { { } } j= k=k function in the set s R jk t can be represented j= in the form s R jk t = m =M R m = s R jkm φr t, j {, }; k {,,, K} m 88 for each j, k and m, s R = T jkm sr jk tφr tdt m The channel output can then be written as y t = n= k=k n= n= k=k n= n= k=k n= n= k=k n= L b k n m= L b k n m= L b k n L b k n m= s L km φl m t nt τ s R km φr m t nt τ m= s L km φl m t nt τ s R kmφ R m t nt τ n t 89 We then obtain an equivalent model with discrete time outputs by passing the received waveform through a matched filter for

9 each of the signals φ L m t and φ R t as follows: m y L m n = ntτ ntτ y tφ L m t nt τ dt, m {,,, M L } 9 ntτ y R m n = y tφ R m t nt τ dt, ntτ m {,,, M R } 9 It is easy to verify that the noise process thus obtained is conditionally independent given the inputs Lemma : AEDIX B ROOF OF LEMMA det I K σ E X X T] M ] ] ]] = det I K σ Σ O K IK S T O K Σ S I K 9 roof: Let us assume that A, B, C, D are matrices with elements in R K K and is a matrix with elements in R K K If we assume that the Kronecker product is carried out entry by entry and that the multiplication of the elements is taken to be the matrix multiplication of the elements, it is then straightforward see also 9, Appendix II] to check that A ] IK O K B O K O K D ] OK I K O K O K ] OK O K = I K O K ] OK O C K O K I K ] A B C D T 93 is the permutation matrix whose only non-o K entries are n = I K, n =,, 94 j n = I K, n =,, 95 Therefore, we have E X X T] ] ] IK O = Σ K OK O Σ O K O K K O K I K ] Σ O = K O K Σ T ] ] IK O M = J K OK O J O K O K K O K I K ] ] S T OK O K OK I S K I K O K O K O K ] IK S = T S I T 96 K O K We note that is also unitary in our present case and we obtain det I K σ E X X T] M ] ] ] ] = det I K σ Σ O K IK S T O K Σ S I T K ] ]] = det I K σ T Σ O K IK S T O K Σ S I K ] ]] = det I K σ Σ O K IK S T O K Σ S I K REFERECES 97 ] R Ahlswede, The capacity region of a channel with two senders and two receivers, Annals robabil, vol, no 5, pp 85 84, 974 ] H Liao, Multiple access channels, hd dissertation 3] T M Cover and J A Thomas, Elements of Information Theory Wiley Interscience, 6 4] T M Cover, Some advances in broadcast channels, Advances in Communication Systems, vol 4, pp 9 6, 975 5] A D Wyner, Recent results in the Shannon Theory, IEEE Trans Inform Theory, vol, no, pp, Jan 974 6] T Cover, R McEliece and E osner, Asynchronous multiple-access channel capacity, IEEE Trans Inform Theory, vol 7, no 4, pp 49 43, July 98 7] G S oltyrev, Coding in an asynchronous multiple-access channel, roblems Inform Transmission, pp, July-Sept 983 8] J Hui and Humblet, The capacity region of the totally asynchronous multiple-access channel, IEEE Trans Inform Theory, vol 3, no, pp 7 6, Mar 985 9] S Verdu, The Capacity Region of the Symbol-Asynchronous Gaussian Multiple-Access Channel, IEEE Trans Inform Theory, vol 35, no 4, pp , July 989 ] M Miranda and Tilli, Asymptotic Spectra of Hermitian Block Toeplitz Matrices and reconditioning Results, SIAM J Matrix Anal Appl, vol, no 3, pp , Feb ] S Verdu, Multiple-Access Channels with Memory with and without Frame-Synchronism, IEEE Trans Inform Theory, vol 35, no 3, pp 65 69, May 989 ] E L Lehmann, Testing Statistical Hypothesis ew York: Wiley, 959 3] A A Maciejewski and C A Klein, The singular value decomposition: Computation and applications to robotics, The Int l Journ Robotics Res, vol 8, no 6, pp 63 79, Dec 989 4] R M Gray, On the Asymptotic Eigenvalue Distribution of Toeplitz Matrices, IEEE Trans Inform Theory, vol IT-8, no 6, pp 75 73, ov 97 5] J G roakis, Digital Communications McGraw-Hill International Edition, J is a matrix with entries in R K K all the diagonal entries are I K and the rest of the entries are