On the Optimum Asymptotic Multiuser Efficiency of Randomly Spread CDMA

On the Optimum Asymptotic Multiuser Efficiency of Randomly Spread CDMA Ralf R. Müller Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU) Lehrstuhl für Digitale Übertragung 13 December 2014

1. Introduction Linear Vector Channel Consider a linear vector channel y = 1 N Hb + n with b {±1} K n N(0, σ 2 I) and H being an N K random matrix with iid entries of unit variance and vanishing odd moments. 1 / 21

1. Introduction Multiuser Efficiency Bit error probability without interference with interference η Multiuser efficiency measures bitwise error probability. η [0; 1] Signal to noise ratio 2 / 21

1. Introduction More on Multiuser Efficiency For unit variance Gaussian data, the multiuser efficiency is trivially related to the Stieltjes transform G(s) in the large matrix limit: η(σ 2 ) = lim s σ 2 sg( s). 3 / 21

1. Introduction More on Multiuser Efficiency For unit variance Gaussian data, the multiuser efficiency is trivially related to the Stieltjes transform G(s) in the large matrix limit: η(σ 2 ) = lim s σ 2 sg( s). In general, the multiuser efficiency depends on both the eigenvalue spectrum of the random matrix HH and the distribution of the data. 3 / 21

2. Results from Literature Large System Result Let N, K, but β = K/N fixed. Then, the multiuser efficiency with optimal detection is a solution to the fixed point equation 1 η = 1+ β σ 2 1 η 2πσ 2 R ( η ) tanh σ 2 x exp ( ) η(x 1)2 2σ 2 dx. In case the fixed point equation has multiple solutions, the correct one is that solution for which the term η σ 2 + η log η [ η ( η ) ] ) 2β 2πσ 2 log cosh σ 2 x η(x 1)2 exp ( 2σ 2 dx is smallest (Tanaka 2002). R 4 / 21

2. Results from Literature A Phase Transition in CDMA 10 0 10 1 optimal detection K = 2N 1 10 2 without interference bit error probability 10 3 10 4 10 5 10 6 0 2 4 6 8 10 12 signal to noise ratio [db] For large SNR, i.e. σ 2 0, we have η 1. 5 / 21

2. Results from Literature Asymptotic Multiuser Efficiency The asymptotic multiuser efficiency is given by η = lim η = min x H Hx σ 2 0 x {±1,0} K \{0} N (Verdú 1986). 6 / 21

2. Results from Literature Asymptotic Multiuser Efficiency The asymptotic multiuser efficiency is given by η = lim η = min x H Hx σ 2 0 x {±1,0} K \{0} N (Verdú 1986). Theorem: The optimum asymptotic multiuser efficiency converges to 1 almost surely as N, K, but fixed (Tse & Verdú 2000). β = K N < 6 / 21

2. Results from Literature Asymptotic Multiuser Efficiency The asymptotic multiuser efficiency is given by η = lim η = min x H Hx σ 2 0 x {±1,0} K \{0} N (Verdú 1986). Theorem: The optimum asymptotic multiuser efficiency converges to 1 almost surely as N, K, but fixed (Tse & Verdú 2000). β = K N < What about β? 6 / 21

2. Results from Literature Uniquely Detecting Matrices Definition An N K matrix H is called uniquely detecting for an alphabet B, iff for all b 1 B K b 2. Hb 1 = Hb 2 b 1 = b 2 7 / 21

2. Results from Literature Logarithmically Infinite Overload Theorem: Let N, K, but ζ = K N log 3 K fixed. Then, there does not exist any H {±1} N K such that it is uniquely detecting for binary data, iff multiuser efficiency is greater, ζ > log 3 4 0.79 (Lindström 1966). 8 / 21

2. Results from Literature Logarithmically Infinite Overload Theorem: Let N, K, but ζ = K N log 3 K fixed. Then, there does not exist any H {±1} N K such that the optimum asymptotic multiuser efficiency is greater than zero, if ζ > log 3 4 0.79 (Lindström 1966). 8 / 21

2. Results from Literature An Existence Result Theorem: Let N, K, but ζ = K N log 3 K fixed. Let H {±1} N K. Then, the probability that H is uniquely detecting for binary data is 1, if (Erdős & Rényi 1963). ζ < 1 2 9 / 21

2. Results from Literature An Existence Result Theorem: Let N, K, but ζ = K N log 3 K fixed. Let H {±1} N K. Then, the optimum asymptotic multiuser efficiency converges to a nonzero value almost surely, if (Erdős & Rényi 1963). ζ < 1 2 9 / 21

2. Results from Literature Summary of Known Results 1 η Tse-Verdú Lindström 0 Erdös-Rényi 0 ζ 0.5 0.79 10 / 21

3. New Results A New Binary Existence Result Theorem: Let N, K, but ζ = K N log 3 K fixed. Let H {±1} N K. Then, the optimum asymptotic multiuser efficiency converges to 1 almost surely if ζ < 3 8 (Sedaghat, RRM, Marvasti 2013). 11 / 21

3. New Results Generalization of the Binary Existence Result Theorem: Let N, K, but ζ = K N log 3 K fixed. Let H {±1} N K. Then, the optimum asymptotic multiuser efficiency converges to value which is greater or equal to min{1, 4 8ζ} (Sedaghat, RRM, Marvasti 2013). 12 / 21

3. New Results Summary of Results 1 η Tse-Verdú Lindström 0 Erdös-Rényi 0 ζ 0.5 0.79 13 / 21

3. New Results Summary of Results 1 Sedaghat-RRM-Marvasti η Lindström 0 0 ζ 0.5 0.79 13 / 21

3. New Results Minimum Distance By Erdős and Rényi, the minimum distance is non-zero almost surely as long as ζ < 1 2 (despite logarithmically infinite overload). By the new result, the minimum distance is almost surely at least as big as in an hypercube downscaled by a factor of β as long as ζ < 3 8 (despite logarithmically infinite overload). 14 / 21

3. New Results Minimum Distance By Erdős and Rényi, the minimum distance is non-zero almost surely as long as ζ < 1 2 (despite logarithmically infinite overload). By the new result, the minimum distance is almost surely at least as big as in an hypercube downscaled by a factor of β as long as ζ < 3 8 (despite logarithmically infinite overload). The projection reduces the noise variance from K to N. 14 / 21

3. New Results Minimum Distance By Erdős and Rényi, the minimum distance is non-zero almost surely as long as ζ < 1 2 (despite logarithmically infinite overload). By the new result, the minimum distance is almost surely at least as big as in an hypercube downscaled by a factor of β as long as ζ < 3 8 (despite logarithmically infinite overload). The projection reduces the noise variance from K to N. In a hypercube the kissing number (number of nearest neighbors is linear in K. After the projection it could eventually be exponential in K degrading the asymptotic multiuser efficiency. 14 / 21

3. New Results A Gaussian Existence Result Theorem: Let N, K, but ζ = K N log 3 K fixed. Let H have Gaussian entries. Then, the optimum asymptotic multiuser efficiency converges to 1 almost surely if ζ < 1 2 (Sedaghat, RRM, Marvasti 2013). 15 / 21

4. Proofs Sketch of Proof We define R = H H/N and start from Verdú 1986 ( ) Pr(η < 1) = Pr min x {±1,0} K \{0} x Rx < 1. The union bound gives Pr(η < 1) x {±1,0} K \{0} ( ) Pr x Rx < 1 We split the summation into shells of equal Hamming weight (zero norm): x {±1,0} K \{0} ( ) = K w=1 x {±1,0} K : x 0 =w ( ) 16 / 21

4. Proofs Lemma on Odd Weights Lemma 1: For any vector x {±1, 0} K with odd weight, Proof: x Rx = 1 N x Rx 1. N K H ij x j i=1 j=1 2 } {{ } 1 The squared bracket is a positive integer, since the sum contains an odd number of binary antipodal (+1, 1) terms. 17 / 21

4. Proofs Lemma on Odd Weights Lemma 1: For any vector x {±1, 0} K with odd weight, Proof: x Rx = 1 N x Rx 1. N K H ij x j i=1 j=1 2 } {{ } 1 The squared bracket is a positive integer, since the sum contains an odd number of binary antipodal (+1, 1) terms. W. l. o. g., we can consider vectors with even weights. 17 / 21

4. Proofs Lemma on Even Weights Lemma 2: For any vector x {±1, 0} K with even weight and we have x Rx < 1, Hx 0 < N 4 Proof: x Rx = 1 N N K H ij x j i=1 j=1 2 } {{ } / {1,2,3} 18 / 21

4. Proofs Lemma on Even Weights Lemma 2: For any vector x {±1, 0} K with even weight and we have Proof: x Rx = 1 N x Rx < 1, Hx 0 < N 4 N K H ij x j i=1 j=1 2 } {{ } / {1,2,3} If the empirical average is less then 1, the number of terms which are 4 or more must be less than N/4. 18 / 21

4. Proofs Subshell Uniformity Lemma 3: For any vectors x, y {±1, 0} K with we have Proof: x 0 = y 0, ( ) ( ) Pr x Rx < 1 = Pr y Ry < 1 x Rx = 1 N ( N K H ij x j i=1 j=1 } {{ } =:u i ) 2 19 / 21

4. Proofs Subshell Uniformity Lemma 3: For any vectors x, y {±1, 0} K with we have Proof: x 0 = y 0, ( ) ( ) Pr x Rx < 1 = Pr y Ry < 1 ( x Rx = 1 N K ) 2 H ij x j N i=1 j=1 }{{} =:u i Characteristic function: K φ ui (t) = φ Hij x j (t) = [φ H11 (t)] x 0 j=1 19 / 21

4. Proofs Back to the Main Proof After splitting into subshells, we had: K Pr(η < 1) Pr ( x Rx < 1 ) w=1 x {±1,0} K : x 0=w 20 / 21

4. Proofs Back to the Main Proof After splitting into subshells, we had: Pr(η < 1) K w=1 x {±1,0} K : x 0=w With the three lemmas, we get Pr(η < 1) K/2 w=1 ( K 2w Pr ( x Rx < 1 ) ) ( 2 2w x 0 Pr Hx 0 < N ) =2w 4 20 / 21

4. Proofs Back to the Main Proof After splitting into subshells, we had: Pr(η < 1) K w=1 x {±1,0} K : x 0=w With the three lemmas, we get with Pr(η < 1) = K/2 w=1 K/2 w=1 ( K 2w ( K 2w p w = Pr 2w j=1 Pr ( x Rx < 1 ) ) ( 2 2w x 0 Pr Hx 0 < N ) =2w 4 ) 2 2w N/4 1 i=0 H 1j = 0 = ( N i ( 2w w ) p N i w (1 p w ) i ) 2 2w. A few more bounds lead to the desired result. 20 / 21

5. References References Erdős & Rényi On two problems of information theory, Magyar Tud. Akad. Mat. Kutató Int. Közl, vol. 8, pp. 229 243, 1963. Lindström On a combinatorial detection problem II, Studia Scientiarum Mathematicarum Hungarica, vol. 1, pp. 353 361, 1966. Tse & Verdú Optimum asymptotic multiuser efficiency of randomly spread CDMA, IEEE Trans. Inform. Theory, vol. 46, no. 7, 2000. Tanaka A statistical mechanics approach to large-system analysis of CDMA multiuser detectors, IEEE Trans. Inform. Theory, vol. 48, no. 11, 2002. Sedaghat, RRM, Marvasti On Optimum Asymptotic Multiuser Efficiency of Randomly Spread CDMA, arxiv:1410.8366v1. 21 / 21

On the Optimum Asymptotic Multiuser Efficiency of Randomly Spread CDMA Ralf R. Müller Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU) Lehrstuhl für Digitale Übertragung 13 December 2014