Spectral Efficiency of CDMA Cellular Networks

Spectral Efficiency of CDMA Cellular Networks N. Bonneau, M. Debbah, E. Altman and G. Caire INRIA Eurecom Institute nicolas.bonneau@sophia.inria.fr

Outline 2 Outline Uplink CDMA Model: Single cell case Multiple cells case Useful tools. Performances with various filters. Simulations results. Extensions of the work.

One Dimensional (1D) Single Cell Network 3 One Dimensional (1D) Single Cell Network a K users in the cell. Code Division Multiple Access (CDMA): Simultaneous communication of all the users to the base station using different codes.

One Dimensional (1D) Single Cell Network Model 4 One Dimensional (1D) Single Cell Network Model The N 1 received signal y at the base station has the form: y = w 1... w K s + n s = (s 1,..., s K ) is the emitted symbol vector. w i is the N 1 ith user code. N is the spreading length. n is a N 1 white complex gaussian noise vector of variance σ 2. The single cell case has been studied thoroughly by the information theory community.

One Dimensional (1D) Cellular Network 5 Uplink cellular model a Infinite base station deployment (L ). Each base station is supposed to cover a region of length a (inter-cell distance). N is the spreading length (fixed). d is the density of the network (number of users per meter) (fixed). We let α = d N. The number of users per cell is: K = da.

Uplink CDMA Cellular Model 6 Uplink CDMA Cellular Model The N 1 received signal y at the base station has the form: y = Ws + W + s + + n W and W + are the spreading code matrices, of size N K and N 2LK.

Uplink CDMA Cellular Model 7 Uplink CDMA Cellular Model The N 1 received signal y at the base station has the form: y = WPs + W + P + s + + n P and P + are the power attenuation diagonal matrices, of size K K and 2LK 2LK. [ ] P = diag h 1 P (x1 ),..., h K P (xk ) ] P + = diag [h K+1 P (xk+1 ),..., h (2L+1)K P (x (2L+1)K ) {h k } k=1...(2l+1)k and P (x) represent respectively flat fading and path loss.

Uplink CDMA model 8 Assumptions Code structure model: Entries of W and W + are modeled as random i.i.d. elements, for example: { 1 N ; + 1 N }. Scaled Gaussian. Channel model: Flat fading parameters {h k } k=1...(2l+1)k are i.i.d. Gaussian random variables. Path loss: The attenuation is of the polynomial form: (β is the attenuation factor). P (x) = P (1 + x ) β Shadowing, albeit important, is not considered.

Uplink CDMA model 9 Estimation of the SINR for a linear filter Let us denote: W = ] [w 1 W, P = [ h 1 P (x1 ) P ], s = [ ] s 1 s Let G be a 1 N vector linear filter, the received signal at the output of the filter is given by: Gy = h 1 P (x1 )Gw 1 s 1 + GW P s + GW + P + s + + Gn The SINR of user 1 in cell p is given by: SINR(x 1, p) = h 1 2 P (x 1 )Gw 1 w H 1 G H σ 2 GG H + GW P P H W H G H + GW + P + P H +W H + G H

Uplink CDMA model 1 Performance Metric The mean spectral efficiency of cell p is given by: [ γ p = 1 K ] NT E x log 2 (1 + SINR(x i, p)) The mean spectral efficiency of cell p can be rewritten: i=1 γ p = K NT E x [log 2 (1 + SINR(x, p))] The measure of performance is the number of bits per second per hertz per meter (b/s/hz/m) the system is able to deliver: T is the chip time, which is set to 1. γ = 1 a γ p = d N E x [log 2 (1 + SINR(x))]

Uplink CDMA model 11 Spectral Efficiency: asymptotic analysis The spectral efficiency is the number of bits per second per hertz per meter (b/s/hz/m) the system is able to deliver: γ = d N E x [log 2 (1 + SINR(x))] (1) For a fixed d (or K = da) and N, it is extremely difficult to get some insight on expression (1). We analyze (1) in the asymptotic dense regime (N, d but d N show that SINR(x) converges almost surely to a deterministic value. α) and The results are based on tools of random matrix theory.

Random matrix theory 12 Random matrices: Some definitions Let M be a N N Hermitian random matrix. We denote F M the empirical distribution of its eigenvalues: F M (λ) = 1 N {eigenvalues λ} The Stieltjes transform m of a distribution F is defined as: 1 m(z) = df (λ) λ z

Random matrix theory 13 Theorem (J. Silverstein) Given a sequence of random Hermitian matrices of the form XTX H such that X is N K with i.i.d. entries scaled by 1/ N. T is K K diagonal with real entries. F T converges almost surely in distribution to a pdf H as N. K/N c > as N. The empirical eigenvalue distribution of XTX H converges to a limiting distribution F, as N. The Stieltjes transform of F is the solution of the equation: m(z) = 1 z c λdh(λ) 1+λm(z)

Random matrix theory 14 Application Example: N = 496 K = 124 Consider XX H such as X is N K with i.i.d. entries scaled by 1/ N. Result: When N and K N c, the limiting distribution F is given by: F (λ) = 1 (λ ) ( (1 c) 2 (1 + c) 2πcλ 2 λ ), (1 c) 2 < λ (1 + c) 2.9.8 distributiob of the eigenvalues for alpha=.25 and SNR=1db simulations with matrix of size 496*124 theory.7.6.5.4.3.2.1.5 1 1.5 2 2.5 3 3.5 4 eigenvalues

Random matrix theory 15 Random matrices: Trace Given a sequence of random matrices of the form w H Mw such that w is a N 1 vector with i.i.d. entries scaled by 1/ N. M is a N N matrix independant of w. The following result holds: w H Mw N 1 N Trace (M) a.s.

Spectral efficiency: Matched Filter 16 Matched Filter The Matched filter to user 1 is G = w H 1. α, the mean spectral efficiency with i.i.d. ran- Proposition: When N and d N dom spreading and Matched filter is: E h,p (x) [γ] = 2α a ( ) where p(t) = P h 2 = t and I = 2α + a/2 + + log 2 ( 1 + t p(t )P (x )dx dt ) tp (x) p(t)dxdt σ 2 + I The asymptotic spectral efficiency depends only on a few meaningful parameters: α, σ 2, p(t), P (x) and a!

Spectral efficiency: Full knowledge Wiener filter 17 Full knowledge Wiener filter The Wiener filter to user 1 is G = w H 1 (R + R + + σ 2 I) 1. Let R = W P P H W H, R = WPP H W H, R + = W + P + P H +W H +, and R tot = R + R +. Proposition: When N and d N α, the mean spectral efficiency with random spreading and full knowledge Wiener filter is: E h,p (x) [γ] = 2α + a/2 ) log a 2 (1 + tp (x)m Rtot ( σ 2 ) p(t)dxdt where m Rtot (z) is the Stieltjes transform of the empirical distribution function of the eigenvalues of R tot given by: m Rtot (z) = 2α + + 1 tp(t)p (x)dxdt 1+tP (x)m Rtot (z) z

Spectral efficiency: Partial knowledge Wiener filter 18 Partial knowledge Wiener filter The Wiener filter to user 1 is G = w H 1 (R + σ 2 I) 1. Proposition: When N and d N α, the mean spectral efficiency with i.i.d. Gaussian random spreading and partial knowledge Wiener filter is: E h,p (x) [γ] = 2α + ( ) a/2 log a 2 1 + tp (x)mr ( σ 2 ) 2 p(t)dxdt m R ( σ 2 ) + I where m R (z) is the Stieltjes transform of the empirical distribution function of the eigenvalues of R given by: and I = 2α m R (z) = + + a/2 2α + a/2 1 tp(t)p (x)dxdt. tp(t)p (x)dxdt 1+tP (x)m R (z) z [ mr (z) z ] σ 2

Spectral efficiency: Optimum filter 19 Optimum filter The mutual information between y and s at the output of the optimum receiver is given by: I(s, y) = H(y) H(y/s) = log 2 det(2πe(r tot + σ 2 I)) log 2 det(2πe(r + + σ 2 I)) = log 2 det(r tot + σ 2 I) log 2 det(r + + σ 2 I) Proposition: When N and d N spreading and optimum filter is: γ = 1 a ln(2) σ 2 + α, the spectral efficiency with i.i.d. random ( ) m Rtot (z) m R + (z) dz where m R + (z) is the Stieltjes transform of the empirical distribution functions of the eigenvalues of R + given by: m R + 1 (z) = 2α + + tp(t)p (x)dxdt a/2 1+tP (x)m R +(z) z

Spectral efficiency: Other bounds 2 Others bounds Proposition: When N and d N α, the spectral efficiency with i.i.d. random spreading, optimum filter and no inter-cell interference is: γ = 1 σ 2 ( m R (z) 1 ) dz a ln(2) z + Proposition: When N and d N α, the spectral efficiency with i.i.d. random spreading, optimum filter and joint multi-cell processing is: γ = 1 σ 2 ( m Rtot (z) 1 ) dz a ln(2) z +

Simulation results 21 Flat fading, β = 2, P = 1, σ 2 = 1 7.12 Spectral efficiency versus the inter cell distance with polynomial path loss.1 spectral efficiency (bits/s/hz/m).8.6.4 Matched filter.2 5 1 15 2 25 3 35 4 45 5 cell size a (m)

Simulation results 22 Flat fading, β = 2, P = 1, σ 2 = 1 7.12 Spectral efficiency versus the inter cell distance with polynomial path loss.1 spectral efficiency (bits/s/hz/m).8.6.4 Matched filter.2 5 1 15 2 25 3 35 4 45 5 cell size a (m)

Simulation results 23 Flat fading, β = 2, P = 1, σ 2 = 1 7.12 Spectral efficiency versus the inter cell distance with polynomial path loss.1 spectral efficiency (bits/s/hz/m).8.6.4 Partial Knowledge Wiener filter Matched filter.2 5 1 15 2 25 3 35 4 45 5 cell size a (m)

Simulation results 24 Flat fading, β = 2, P = 1, σ 2 = 1 7.12 Spectral efficiency versus the inter cell distance with polynomial path loss.1 spectral efficiency (bits/s/hz/m).8.6.4 Partial Knowledge Wiener filter Matched filter.2 5 1 15 2 25 3 35 4 45 5 cell size a (m)

Simulation results 25 Flat fading, β = 2, P = 1, σ 2 = 1 7.12 Spectral efficiency versus the inter cell distance with polynomial path loss.1 Full Knowledge Wiener filter spectral efficiency (bits/s/hz/m).8.6.4 Partial Knowledge Wiener filter Matched filter.2 5 1 15 2 25 3 35 4 45 5 cell size a (m)

Simulation results 26 Flat fading, β = 2, P = 1, σ 2 = 1 7.12 Spectral efficiency versus the inter cell distance with polynomial path loss.1 Full Knowledge Wiener filter spectral efficiency (bits/s/hz/m).8.6.4 Partial Knowledge Wiener filter Matched filter.2 5 1 15 2 25 3 35 4 45 5 cell size a (m)

Simulation results 27 Flat fading, β = 2, P = 1, σ 2 = 1 7.12 Spectral efficiency versus the inter cell distance with polynomial path loss Optimum filter.1 Full Knowledge Wiener filter spectral efficiency (bits/s/hz/m).8.6.4 Partial Knowledge Wiener filter Matched filter.2 5 1 15 2 25 3 35 4 45 5 cell size a (m)

Simulation results 28 Flat fading, β = 2, P = 1, σ 2 = 1 7.12 Spectral efficiency versus the inter cell distance with polynomial path loss Optimum filter.1 Full Knowledge Wiener filter spectral efficiency (bits/s/hz/m).8.6.4 Partial Knowledge Wiener filter Matched filter.2 5 1 15 2 25 3 35 4 45 5 cell size a (m)

Simulation results 29 Flat fading, β = 2, P = 1, σ 2 = 1 7.3 Spectral efficiency versus the inter cell distance with polynomial path loss Optimum filter, no interference spectral efficiency (bits/s/hz/m).2.1 5 1 15 2 25 3 35 4 45 5 cell size a (m)

Simulation results 3 Flat fading, β = 2, P = 1, σ 2 = 1 7 35 Spectral efficiency versus the inter cell distance with polynomial path loss 3 spectral efficiency (bits/s/hz/m) 25 2 15 1 Optimum joint multi cell processing 5 5 1 15 2 25 3 35 4 45 5 cell size a (m)

Results 31 Conclusion: Uplink CDMA Random matrix theory is an efficient tool to analyze the behavior of cellular networks. Spectral efficiency increases substantially: With the use of non-linear detectors based only on the knowledge of intra-cell codes. With the use of Wiener filtering based on the knowledge of the inter-cell interference. Problems are currently being worked at: Frequency selective fading. Orthogonal codes.

Extensions 32 Frequency selective fading The N 1 received signal y at the base station has the form: where is the Hadamard product. y = ( HP W ) s + ( H + P + W + ) s+ + n Frequency selective fading: H and H + respectively of size N K and N 2LK, where L tends to infinity: h (1) 1 h (2) 1... h (K) 1 H =... H + = h (1) N h(2) N h (K+1) 1 h (K+2) h (K+1) N... h(k) N 1... h ((2L+1)K) 1... h (K+2) N... h ((2L+1)K) N

Extensions 33 Orthogonal codes (Single Cell Network) a The N 1 received signal y at the base station has the form: y = H 1 w 1 s 1 + H 2 w 2 s 2 + + H K w K s K + n = ( H W ) s + n Code structure model: User codes w k are columns extracted from a Haar unitary (random) matrix W.

Extensions 34 Orthogonal codes (Single Cell Network) a Haar unitary matrix theory is an efficient tool to analyze the behavior of orthogonal spreading. SINR always increases with the use of orthogonal codes.