CMA 2007 Channel capacity estimation using free probability theory �yvind Ryan and Merouane Debbah January 2008 �yvind Ryan and Merouane Debbah Channel capacity estimation using free probability theory
CMA 2007 Channel capacity estimation using free probability theory Problem at hand The capacity per receiving antenna of a channel with n × m 1 channel matrix H and signal to noise ratio ρ = σ 2 is given by n C = 1 1 = 1 log 2 ( 1 + 1 � � m σ 2 HH H I n + n log 2 det � (1) σ 2 λ l ) n l = 1 where λ l are the eigenvalues of 1 m HH H . We would like to estimate C . To estimate C , we will use free probability tools to estimate the m HH H based on some observations ˆ eigenvalues of 1 H i �yvind Ryan and Merouane Debbah Channel capacity estimation using free probability theory
CMA 2007 Channel capacity estimation using free probability theory Observation model 1 The following is a much used observation model: H i = H + σ X i ˆ (2) where ◮ The matrices are n × m ( n is the number of receiving antennas, m is the number of transmitting antennas) ◮ ˆ H i is the measured MIMO matrix, ◮ X i is the noise matrix with i.i.d standard complex Gaussian entries. �yvind Ryan and Merouane Debbah Channel capacity estimation using free probability theory
CMA 2007 Channel capacity estimation using free probability theory Existing ways to estimate the channel capacity Several channel capacity estimators have been used in the literature: � L 1 � 1 H H � C 1 i = 1 log 2 det I n + m σ 2 ˆ H i ˆ = i nL 1 � 1 � L � I n + H i ˆ H H C 2 n log 2 det i = 1 ˆ (3) = L σ 2 m i � L � L 1 � σ 2 m ( 1 1 H i )( 1 H i ) H ) � C 3 n log 2 det I n + i = 1 ˆ i = 1 ˆ = L L Why not try to formulate an estimator based on free probability instead? �yvind Ryan and Merouane Debbah Channel capacity estimation using free probability theory
CMA 2007 Channel capacity estimation using free probability theory 2.6 2.4 2.2 2 1.8 Capacity 1.6 1.4 1.2 True capacity C 1 1 C 2 C 3 0.8 0 5 10 15 20 25 30 Number of observations Comparison of the classical capacity estimators for various number of observations. σ 2 = 0 . 1, n = 10 receive antennas, m = 10 transmit antennas. The rank of H was 3. �yvind Ryan and Merouane Debbah Channel capacity estimation using free probability theory
CMA 2007 Channel capacity estimation using free probability theory The Mar� chenko Pastur law The Mar� chenko Pastur law µ c : f µ c ( x ) = ( 1 − 1 ( x − a ) + ( b − x ) + � c ) + δ 0 ( x ) + (4) , 2 π cx where ( z ) + = max ( 0 , z ) , a = ( 1 − √ c ) 2 , b = ( 1 + √ c ) 2 , and δ 0 ( x ) is dirac measure (point mass) at 0. ◮ free cumulants: 1 , c , c 2 , c 3 , ... . ◮ µ c is the limit eigenvalue distribution of 1 N XX H , with X an n × N with independent standard complex Gaussian entries as N → ∞ , and n N → c . �yvind Ryan and Merouane Debbah Channel capacity estimation using free probability theory
CMA 2007 Channel capacity estimation using free probability theory Main free probability result we will use De�ne 1 N R n R H Γ n = n 1 N ( R n + σ X n )( R n + σ X n ) H , W n = where R n and X n are independent n × N random matrices, X n is complex, standard, Gaussian. Theorem If e . e . d . (Γ n ) → ν Γ , then e . e . d . ( W n ) → ν W where ν W is uniquely identi�ed by ν W � µ c = ( ν Γ � µ c ) ⊞ δ σ 2 ( � = "the opposite of ⊠ "). �yvind Ryan and Merouane Debbah Channel capacity estimation using free probability theory
CMA 2007 Channel capacity estimation using free probability theory Realization of the theorem for the problem at hand Form the compound observation matrix H 1 ... L + σ H 1 ... L X 1 ... L , where ˆ = √ L 1 � � H 1 ... L ˆ H 1 , ˆ ˆ H 2 , ..., ˆ H L = √ , L 1 H 1 ... L [ H , H , ..., H ] , √ = L X 1 ... L [ X 1 , X 2 , ..., X L ] . = For the problem at hand, the theorem takes the form � � (5) mL ≈ ν 1 1 ... L � µ n ν 1 1 ... L � µ n ⊞ δ σ 2 H H m H 1 ... L H H m ˆ H 1 ... L ˆ mL Since 1 1 ... L = 1 m H 1 ... L H H m HH H , we can now estimate the moments m HH H from the moments of the observation matrix of 1 1 H H H 1 ... L ˆ m ˆ 1 ... L , and thereby estimate the eigenvalues, and hence the channel capacity. �yvind Ryan and Merouane Debbah Channel capacity estimation using free probability theory
CMA 2007 Channel capacity estimation using free probability theory Free probability based estimator for the moments of the channel matrix Can also be written in the following way for the �rst four moments: h 1 + σ 2 ˆ h 1 = ˆ h 2 + 2 σ 2 ( 1 + c ) h 1 + σ 4 ( 1 + c ) h 2 = h 3 + 3 σ 2 ( 1 + c ) h 2 + 3 σ 2 ch 2 ˆ h 3 = 1 + 3 σ 4 � c 2 + 3 c + 1 h 1 � + σ 6 � c 2 + 3 c + 1 � (6) h 4 + 4 σ 2 ( 1 + c ) h 3 + 8 σ 2 ch 2 h 1 ˆ h 4 = + σ 4 ( 6 c 2 + 16 c + 6 ) h 2 + 14 σ 4 c ( 1 + c ) h 2 1 + 4 σ 6 ( c 3 + 6 c 2 + 6 c + 1 ) h 1 + σ 8 � c 3 + 6 c 2 + 6 c + 1 � , h i are the moments of the observation matrix 1 H H where ˆ m ˆ H 1 ... L ˆ 1 ... L , h i are the moments of 1 m HH H . �yvind Ryan and Merouane Debbah Channel capacity estimation using free probability theory
CMA 2007 Channel capacity estimation using free probability theory 2.6 2.4 2.2 2 1.8 Capacity 1.6 1.4 1.2 True capacity 1 C f C u 0.8 0 5 10 15 20 25 30 Number of observations Comparison of C f and C u for various number of observations. σ 2 = 0 . 1, n = 10 receive antennas, m = 10 transmit antennas. The rank of H was 3. �yvind Ryan and Merouane Debbah Channel capacity estimation using free probability theory
CMA 2007 Channel capacity estimation using free probability theory 4.5 4 3.5 True capacity, rank 3 C f , rank 3 Capacity True capacity, rank 5 3 C f , rank 5 True capacity, rank 6 C f , rank 6 2.5 2 1.5 0 10 20 30 40 50 60 70 80 90 100 Number of observations C f for various number of observations. No phase o�-set/phase drift. σ 2 = 0 . 1, n = 10 receive antennas, m = 10 transmit antennas. The rank of H was 3, 5 and 6. �yvind Ryan and Merouane Debbah Channel capacity estimation using free probability theory
CMA 2007 Channel capacity estimation using free probability theory How would an algorithm for free convolution look? De�nition A family of unital ∗ -subalgebras ( A i ) i ∈ I is called a free family if a j ∈ A i j i 1 � = i 2 , i 2 � = i 3 , · · · , i n − 1 � = i n ⇒ φ ( a 1 · · · a n ) = 0 . (7) φ ( a 1 ) = φ ( a 2 ) = · · · = φ ( a n ) = 0 A family of random variables a i is called a free family if the algebras they generate form a free family. ◮ How do we implement this in terms of moments? ◮ From the previous result, we are basically interested in computing the moments of ab , when ab are free, and b is free Poisson. �yvind Ryan and Merouane Debbah Channel capacity estimation using free probability theory
CMA 2007 Channel capacity estimation using free probability theory Implementation of main result The following formula can be used for incremental calculation of the moments of the measure µ ⊠ µ c , from the moments of the measure µ : m [ coef k ]( cM µ )[ coef m − k ]( 1 + cM µ ⊠ µ c ) k . (8) [ coef m ]( cM µ ⊠ µ c ) = � k = 1 Here, ◮ M µ ( z ) = µ 1 z + µ 2 z 2 + ... , where µ i are the moments of µ . ◮ coef k means the coe�cient of z k in the polynomial. ◮ The power series coe�cient can be computed through k -fold discrete (classical) convolution. ◮ (8) is proved by �rst proving that cM µ ⊠ µ c = ( cM µ ) ⋆ Zeta . �yvind Ryan and Merouane Debbah Channel capacity estimation using free probability theory
CMA 2007 Channel capacity estimation using free probability theory Observation model 2 A more general observation model is: H i = D r i HD t ˆ i + σ X i , (9) where D r i and D t i are n × n and m × m diagonal matrices which represent phase o�-sets and phase drifts (impairments due to the antennas and not the channel) at the receiver and transmitter given respectively by diag [ e j φ i 1 , ..., e j φ i D r n ] , and = i diag [ e j θ i 1 , ..., e j θ t D t m ] = i where the phases φ i j and θ i j are random. We assume all phases independent and uniformly distributed. �yvind Ryan and Merouane Debbah Channel capacity estimation using free probability theory
CMA 2007 Channel capacity estimation using free probability theory Problem when extending to phase o�-set and phase drift ◮ In the compund observation matrix we now put 1 D r i HD t i , D r i HD t i , ..., D r i HD t H 1 ... L = � � √ , i L The moments of 1 m H 1 ... L H H 1 ... L are now in general di�erent from the moments of 1 m HH H ! ◮ In other words stacking the observations and using the free convolution framework does not give us what we want A way to resolve this: ◮ Don't stack the observations at all. ◮ Perform convolution through exact formulas for the mixed moments of matrices and Gaussian matrices of lower order. ◮ Unbiased capacity estimator. �yvind Ryan and Merouane Debbah Channel capacity estimation using free probability theory
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