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Aris Moustakas, University of Athens

WARNING THIS IS NOT A TALK ABOUT FREE PROBABILTY THEORY

Aris Moustakas, University of Athens

WARNING RIGOR LEVEL: ∞

∞-

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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)

Introduction

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Aris Moustakas, University of Athens

• Why random matrices in communications?

– Multi-antenna channels – 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

–

• Information Capacity

–

• SINR linear MMSE

I = Tr log2 h I + ρHH†i SINR = ρw†H† h I + ρHUU†H†i−1 Hw y = Hx + z

Introduction

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Aris Moustakas, University of Athens

• Important Statistics of random quantities

– Mean – Variance – Higher cumulant moments (vanish for large matrix sizes = CLT)

• Methods

– Free probability

• Asymptotic freeness

– Canonical RMT

• Stieljes transforms

– Replicas

• Gaussian matrices (?)

– Diagrammatics (= Free probability??)

• Gaussian matrices
• Unitary matrices

– Other methods

Cons:

• Non rigorous
• Non-general

(Gaussian – Unitary) Pros:

• Back-of-the

envelope

• Easy

Methods:

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Aris Moustakas, University of Athens

• 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

Application: MMSE SINR for synchronous transmission

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Aris Moustakas, University of Athens

• Channel Model:

– N time-slots, 2 bases with K & K’ users – Each user gets a code to transmit – Synchronous transmission (downlink):

• 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:

• Elements of U are i.i.d.
• Approximate this by Gaussian i.i.d. matrix

– Assume U, U’ independent

y = Hx + H0x0 + z x = PK

k=1 wkdk

U = [w1 w2 . . . wK . . . wN] U†U = UU† =

±1 √ N

IN

Application: MMSE SINR for synchronous transmission

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Aris Moustakas, University of Athens

• Channel Model:

– H, H’: Channel matrices

• Diagonal with independent coefficients:

– Fast fading (time-variability) – Independent frequency channels

• Toeplitz form

– Delayed paths

– z: receiver thermal noise (white)

y = Hx + H0x0 + z

Application: MMSE SINR for synchronous transmission

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Aris Moustakas, University of Athens

• Optimal linear receiver for user 1(several caveats):

– Multiply y with optimal vector – J, J’ are input power covariance matrices – Resulting SINR

• Aim: Calculate asymptotic properties of β (i.e. evaluate its mean)
• Compare with effective interference

– Averaged over codes – Averaged over time & codes

g = w†

1H† h

σ2IN + HUJU†H† + H0U0J0U0†H0†i−1 η = w†

1H† h

σ2IN + HUJU†H† + H0U0J0U0†H0†i−1 Hw1 β =

η 1−η

H0U0J0U0†H0† = E h H0H0†i TrJ0/N H0U0J0U0†H0† = H0H0†TrJ0/N

Diagrammatic Approach

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Aris Moustakas, University of Athens

• Start with simple problem:

– tr[.] = Tr[.]/N

i j i j

g = E · tr h I − AUBU†i−1 A ¸ = P∞

n=0 trE

h³ AUBU†´n A i

– Matrices A, B as lines

• Trace corresponds connecting solid lines
• Averaging over U: Connect dashed lines in all possible ways

– Gives 1/N for each U, U* pair

• Represent each matrix in the expansion:

– Uij as two dashed lines with two external lines

EXACT APPROACH

(Brouwer –Beenakker) (Argaman – Zee)

Diagrammatic Approach

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Aris Moustakas, University of Athens

• 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) – where red blob represents all other terms

• Self-energy = “R-transform”
• Leading terms in Weingarten function of each power of U’s
• Resum terms to get final result

– Functional form of m encodes statistics of U

• mg=1 for Gaussian U

g = tr

A I−Afmu f = tr B I−Bgmu

mu = √

1+4fg−1 2fg Differences between Gaussian & NonGaussian Generating function

• f (2k)!/(k!2(2k-1))

Results

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Aris Moustakas, University of Athens

• Generalize to current problem with one interferer

gi = tr · HiHi

† ³

I + H1H1

†f1 ¯

m1u + H2H2

†f2 ¯

m2u ´−1¸ fi = tr

J†

i

I+J†

i gi ¯

miu

i = 1, 2

¯ miu =

1−√ 1−4figi 2figi β β+1 = η = Nf1g1 ¯ m1u K

• Note: m1 does not have to be the same as m2 (e.g. = 1)
• “In principle”, above result cannot be obtained using free probability (?)

– Depends on relative eigenvector space of H1, H2 , not only on their eigenvalues

r(z) = trE ·³ z − H1U1J1U†

1H† 1 − H2U2J2U† 2H† 2

´−1¸ = tr ·³ z − H1H1

†f1m1u − H2H2 †f2m2u

´−1¸

Results

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Aris Moustakas, University of Athens

• α=Κ1/Ν=Κ2/Ν

ρ=1/σ2

• Asymptotic theory: introduce fast fading on each channel

10 20 30 40 50 60 70 80 90 100 0.5 1 1.5 2 2.5 3 3.5 4 4.5

ρ (=ρ1=ρ2)

SINR SINR=η/(1-η) vs ρ; N=32 α = 0.625 unitary Hadamard theor asympt theor

Results

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Aris Moustakas, University of Athens 10 20 30 40 50 60 70 80 90 100 0.5 1 1.5 2 2.5 3 3.5 4 ρ (=ρ1=ρ2) SINR SINR=η/(1-η) vs ρ; N=64 α = 0.625 unitary Hadamard theor asympt theor

• α=Κ1/Ν=Κ2/Ν

ρ=1/σ2

• Asymptotic theory: introduce fast fading on each channel

Results

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Aris Moustakas, University of Athens 10 20 30 40 50 60 70 80 90 100 0.5 1 1.5 2 2.5 3 3.5 ρ (=ρ1=ρ2) SINR SINR=η/(1-η) vs ρ; N=128 α=0.625 unitary Hadamard theor asympt theor

• α=Κ1/Ν=Κ2/Ν

ρ=1/σ2

• Asymptotic theory converges

Hadamard=good approximation

Results

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Aris Moustakas, University of Athens 10 20 30 40 50 60 70 80 90 100 0.5 1 1.5 2 2.5 3 3.5 4 4.5 ρ (=ρ1=ρ2) SINR SINR=η/(1-η) vs ρ; N=32 α=0.625; interference=gaussian unitary Hadamard theor

• α=Κ1/Ν=Κ2/Ν

ρ=1/σ2

• Gaussian interference

Results

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Aris Moustakas, University of Athens 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 4 6 8 10 12 14 16 18 MMSE SINR vs loading; ρ1=ρ2=20dB; interference loading α2=0.6 α SINR (dB) Unitary Interference Gaussian Interference Unspread Interference Effective Noise Blue line “interpolates” between green and red with α2 (α2=1 Interference cancels unitary matrices – α2<<1 Unitary looks Gaussian

Results

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Aris Moustakas, University of Athens 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 4 6 8 10 12 14 16 18 20 MMSE SINR vs loading; ρ1=ρ2=20dB; interference loading α2=0.3 α1 SINR (dB) Unitary Interference Gaussian Interference Unspread Interference Effective Noise

Results

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Aris Moustakas, University of Athens 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

• 1

1 2 3 4 5 6 7 8 MMSE SINR vs loading; ρ1=ρ2=10dB; interference loading α2=0.6

α1

SINR (dB) Unitary Interference Gaussian Interference Unspread Interference Effective Noise

Second Order Moments using Diagrammatics

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Aris Moustakas, University of Athens

• Calculate second order statistics of eigenvalues (=differentiate below)
• Two traces – two lines (closed)
• Apply same principles (more diagrams)

– Intra-circle – Cross-circle – 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)

1 2

Diagrammatic Approach

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Aris Moustakas, University of Athens

• The x-circle connections characterized by

their neighbors (gf, fg, ff, gg)

• Thus we are left to just sum over

“effective” quantities Γ, F, G

• By symmetry Γfg = Γgf
• Γ’s involve same terms as mu but are

broken into disjoint terms

– Some go to Γff, some go to Γgf

• If H is Gaussian Γgg = Γff = 0

F G G Γfg Γgg Γgf

Result

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Aris Moustakas, University of Athens

• After resumming we finally get

1

Results

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Aris Moustakas, University of Athens

Example: variance of MMSE SINR for synchronous downlink (no interference for simplicity)

• Variance for N= 128,

SNR = 10

• Cell Loading

α=Κ/Ν=0.5

• Channel matrix H with

iid Gaussian elements of size xaxis*N

• Good agreement with

theory

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Variance MMSE SINR; Ntot=128 N1= 64; SNR = 10 unitary Hadamard

• rthogonal

theor

Results

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Aris Moustakas, University of Athens

• Approach may

be invalid due to lack of “randomization”

• f Haar

eigenvalue matrix for channel

• Second order

statistics no longer follow unitary asymptotic results !!!

For channel matrix of Toeplitz form Hadamard behavior quite different

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.005 0.01 0.015 0.02 0.025 0.03 load α Variance(SINR) Variance MMSE SINR; Ntot=64; SNR = 1; Toeplitz channel 32 i.i.d. tap delays unitary Hadamard

• rthogonal

theor

Methods:

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Aris Moustakas, University of Athens

• Replicas

– Originally applied to dirty magnetic systems (1970’s) – Calculate moment generating function – Replica trick: Calculate MGF for integer values of ν – Analytically continue for real values of ν – Technical Assumption: Replica Symmetry

• Not always valid
• Valid for random matrices with continuous symmetries (H complex Gaussian w/

SU(N) rotational symmetry)

– Applications: Calculation of mean, variance, higher order moments of trace, variance, higher moments of trlog (and hence of MMSE SINR)

g(ν) = E h³ I + ρHH†´νi

Results

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Aris Moustakas, University of Athens

• Outage

probability well behaved down to small errors.

• For increasing N

behavior better

• Better when

N>M Mutual Information distribution of MMSE SINR for MIMO Gaussian channels

• Need to calculate E[βiβj] and then calculate MI
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5 10 15 20 25 10

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10

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10

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10

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10

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10

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10 Outage probability for M=10, N=20 SNR(dB) Outage Probability

Conclusions – Open Questions

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Aris Moustakas, University of Athens

• Applied diagrammatic approach to calculate asymptotic mean and

variance of MMSE SINR with/without interference

• Applications

– MMSE SIR for synchronous channels with (a)synchronous interference – MMSE SIR for synchronous downlink channels

• Works well for orthogonal, unitary matrices
• Hadamard matrices do not fare well WHY?

– MMSE SIR capacity for MIMO channels

• Reasonable waterfall curves, even for large SNRs.
• Crossover to bad behavior is function of SNR, N
• Open Questions

– Spectrum of AUBU’ + CUDU’?

• Known only for Gaussian U (using replicas)