WARNING THIS IS NOT A TALK ABOUT FREE PROBABILTY THEORY Aris - PowerPoint PPT Presentation

WARNING THIS IS NOT A TALK ABOUT FREE PROBABILTY THEORY Aris Moustakas, University of Athens

WARNING ∞ - -1 1 RIGOR LEVEL: ∞ Aris Moustakas, University of Athens

Synchronous MMSE SIR with interference: Diagrams & Replicas Aris Moustakas (Univ. Athens, Greece) Collaborators: M. Debbah (Supelec, France) R. Kumar, G. Caire (USC, USA) Aris Moustakas, University of Athens

Introduction • Why random matrices in communications? – Multi-antenna channels y = Hx + z – Code matrices • Random i.i.d. – Scrambling codes (+-+-….) – Approximated by random Gaussian matrices • Orthogonal – Hadamard-Walsh codes [++++], [++- -], [+ - + -], [+ - - +] – Fourier transform matrices – Approximated by unitary Haar matrices • Two standard functions of matrices h I + ρ HH † i – I = Tr log 2 • Information Capacity SINR = ρ w † H † h I + ρ HUU † H † i − 1 – Hw • SINR linear MMSE 4 Aris Moustakas, University of Athens

Introduction • Important Statistics of random quantities – Mean – Variance – Higher cumulant moments (vanish for large matrix sizes = CLT) • Methods Cons: – Free probability • Non rigorous • Asymptotic freeness • Non-general – Canonical RMT (Gaussian – Unitary) • Stieljes transforms – Replicas • Gaussian matrices (?) Pros: – Diagrammatics (= Free probability??) • Back-of-the envelope • Gaussian matrices • Easy • Unitary matrices – Other methods 5 Aris Moustakas, University of Athens

Methods: • Diagrammatic Method – Important method in high-energy physics since 1930’s • Applied to mesoscopic systems (1980’s) – Applies mostly to Gaussian & Unitary matrices • Also matrices “ close ” to these • Non-hermitian matrices – Expand resolvent (Stieljes transform) in powers of random matrix and calculate average and then resum (!) – For large N, only a certain type of diagrams survive (planar approximation) – Applications: Calculation of mean and variance of resolvent – Similar to free probability methods 6 Aris Moustakas, University of Athens

Application: MMSE SINR for synchronous transmission • Channel Model: y = Hx + H 0 x 0 + z – N time-slots, 2 bases with K & K’ users – Each user gets a code to transmit x = P K k =1 w k d k – Synchronous transmission (downlink): U † U = UU † = I N U = [ w 1 w 2 . . . w K . . . w N ] • Matrix is unitary • In reality: U is a Hadamard-Walsh matrix • For OFDMA systems U is the Fourier transform basis matrix • Approximate this with Haar-distributed unitary matrices – Alternative (uplink): Asynchronous transmission: ± 1 √ • Elements of U are i.i.d. N • Approximate this by Gaussian i.i.d. matrix – Assume U , U’ independent 7 Aris Moustakas, University of Athens

Application: MMSE SINR for synchronous transmission • Channel Model: y = Hx + H 0 x 0 + z – H , H’ : Channel matrices • Diagonal with independent coefficients: – Fast fading (time-variability) – Independent frequency channels • Toeplitz form – Delayed paths – z : receiver thermal noise (white) 8 Aris Moustakas, University of Athens

Application: MMSE SINR for synchronous transmission • Optimal linear receiver for user 1(several caveats): – Multiply y with optimal vector 1 H † h σ 2 I N + HUJU † H † + H 0 U 0 J 0 U 0† H 0† i − 1 g = w † – J , J’ are input power covariance matrices – Resulting SINR η β = 1 − η 1 H † h σ 2 I N + HUJU † H † + H 0 U 0 J 0 U 0† H 0† i − 1 η = w † Hw 1 • Aim: Calculate asymptotic properties of β (i.e. evaluate its mean) • Compare with effective interference H 0 U 0 J 0 U 0† H 0† = H 0 H 0† Tr J 0 /N – Averaged over codes h H 0 H 0† i H 0 U 0 J 0 U 0† H 0† = E Tr J 0 /N – Averaged over time & codes 9 Aris Moustakas, University of Athens

Diagrammatic Approach • Start with simple problem: · ¸ h I − AUBU † i − 1 h³ AUBU † ´ n i = P ∞ g = E tr A n =0 trE A – tr[.] = Tr[.]/N • Represent each matrix in the expansion: – U ij as two dashed lines with two external lines – Matrices A, B as lines i j i j • Trace corresponds connecting solid lines • Averaging over U : Connect dashed lines in all possible ways EXACT – Gives 1/N for each U, U* pair APPROACH (Brouwer –Beenakker) (Argaman – Zee) 10 Aris Moustakas, University of Athens

Diagrammatic Approach • In large N limit only planar diagrams survive – All crossed (non-planar) diagrams are subleading in N • This allows us to write the trace in disconnected parts – no dashed is allowed to escape (not even the other U’s) Differences – where red blob represents all other terms between • Self-energy = “R-transform” Gaussian & NonGaussian • Leading terms in Weingarten function of each power of U ’s B A • Resum terms to get final result I − A fm u f = tr g = tr I − B gm u √ – Functional form of m encodes statistics of U 1+4 fg − 1 m u = • m g =1 for Gaussian U 2 fg Generating function of (2k)!/(k! 2 (2k-1)) 11 Aris Moustakas, University of Athens

Results • Generalize to current problem with one interferer · ´ − 1 ¸ † ³ † f 1 ¯ † f 2 ¯ g i = tr H i H i I + H 1 H 1 m 1 u + H 2 H 2 m 2 u 1 − √ J † 1 − 4 f i g i i = 1 , 2 f i = tr m iu = ¯ i I + J † 2 f i g i i g i ¯ m iu β +1 = η = Nf 1 g 1 ¯ β m 1 u K • Note: m 1 does not have to be the same as m 2 (e.g. = 1) • “In principle” , above result cannot be obtained using free probability (?) – Depends on relative eigenvector space of H 1 , H 2 , not only on their eigenvalues ·³ ´ − 1 ¸ z − H 1 U 1 J 1 U † 1 H † 1 − H 2 U 2 J 2 U † 2 H † r ( z ) = trE 2 ·³ ´ − 1 ¸ † f 1 m 1 u − H 2 H 2 † f 2 m 2 u z − H 1 H 1 = tr 12 Aris Moustakas, University of Athens

Results • α = Κ 1 / Ν = Κ 2 / Ν ρ =1/ σ 2 • Asymptotic theory: introduce fast fading on each channel SINR= η /(1- η ) vs ρ ; N=32 α = 0.625 4.5 unitary Hadamard 4 theor asympt theor 3.5 3 SINR 2.5 2 1.5 1 0.5 0 10 20 30 40 50 60 70 80 90 100 ρ (= ρ 1 = ρ 2 ) 13 Aris Moustakas, University of Athens

Results • α = Κ 1 / Ν = Κ 2 / Ν ρ =1/ σ 2 • Asymptotic theory: introduce fast fading on each channel SINR= η /(1- η ) vs ρ ; N=64 α = 0.625 4 unitary Hadamard 3.5 theor asympt theor 3 2.5 SINR 2 1.5 1 0.5 0 10 20 30 40 50 60 70 80 90 100 ρ (= ρ 1 = ρ 2 ) 14 Aris Moustakas, University of Athens

Results • α = Κ 1 / Ν = Κ 2 / Ν ρ =1/ σ 2 • Asymptotic theory converges Hadamard=good approximation SINR= η /(1- η ) vs ρ ; N=128 α =0.625 3.5 unitary Hadamard theor 3 asympt theor 2.5 SINR 2 1.5 1 0.5 0 10 20 30 40 50 60 70 80 90 100 ρ (= ρ 1 = ρ 2 ) 15 Aris Moustakas, University of Athens

Results • α = Κ 1 / Ν = Κ 2 / Ν ρ =1/ σ 2 • Gaussian interference SINR= η /(1- η ) vs ρ ; N=32 α =0.625; interference=gaussian 4.5 unitary Hadamard 4 theor 3.5 3 2.5 SINR 2 1.5 1 0.5 0 0 10 20 30 40 50 60 70 80 90 100 ρ (= ρ 1 = ρ 2 ) 16 Aris Moustakas, University of Athens

Results MMSE SINR vs loading; ρ 1 = ρ 2 =20dB; interference loading α 2 =0.6 18 Unitary Interference Gaussian Interference 16 Unspread Interference Blue line Effective Noise 14 “interpolates” between green 12 and red with α 2 ( α 2=1 Interference SINR (dB) 10 cancels unitary matrices – α 2<<1 8 Unitary looks Gaussian 6 4 2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 α 17 Aris Moustakas, University of Athens

Results MMSE SINR vs loading; ρ 1 = ρ 2 =20dB; interference loading α 2 =0.3 20 Unitary Interference Gaussian Interference 18 Unspread Interference Effective Noise 16 14 SINR (dB) 12 10 8 6 4 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 α 1 18 Aris Moustakas, University of Athens

Results MMSE SINR vs loading; ρ 1 = ρ 2 =10dB; interference loading α 2 =0.6 8 Unitary Interference Gaussian Interference 7 Unspread Interference Effective Noise 6 5 SINR (dB) 4 3 2 1 0 -1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 α 1 19 Aris Moustakas, University of Athens

Second Order Moments using Diagrammatics • Calculate second order statistics of eigenvalues (=differentiate below) • Two traces – two lines (closed) • Apply same principles (more diagrams) 2 – Intra-circle – Cross-circle 1 – Given x-circle connections • Can sum over all possible intra-circle ones • Then can sum over all cross-circle positions • Thus get from Go -> G and Fo->F • Variance is O(1) 20 Aris Moustakas, University of Athens

Diagrammatic Approach • The x-circle connections characterized by their neighbors (gf, fg, ff, gg) Γ fg F Γ gf • Thus we are left to just sum over G “effective” quantities Γ , F, G • By symmetry Γ fg = Γ gf Γ gg • Γ ’s involve same terms as mu but are broken into disjoint terms G – Some go to Γ ff , some go to Γ gf • If H is Gaussian Γ gg = Γ ff = 0 21 Aris Moustakas, University of Athens

Result • After resumming we finally get 1 22 Aris Moustakas, University of Athens