MIMO Capacities : Eigenvalue Computation through Representation Theory

Transcription

1 MIMO Capacities : Eigenvalue Computation through Representation Theory Jayanta Kumar Pal, Donald Richards SAMSI Multivariate distributions working group

2 Outline 1 Introduction 2 MIMO working model 3 Eigenvalue computations 4 Representation theory of unitary groups 5 Computation of the m.g.f.

3 Introduction Outline 1 Introduction 2 MIMO working model 3 Eigenvalue computations 4 Representation theory of unitary groups 5 Computation of the m.g.f.

4 Introduction What is MIMO? MIMO : Multiple-Input-Multiple-Output. Multiple antennas used to transmit and receive signals in wireless communications. Use of multiple antennas increase information throughput substantially.

5 Introduction What is MIMO? MIMO : Multiple-Input-Multiple-Output. Multiple antennas used to transmit and receive signals in wireless communications. Use of multiple antennas increase information throughput substantially.

6 Introduction What is MIMO? MIMO : Multiple-Input-Multiple-Output. Multiple antennas used to transmit and receive signals in wireless communications. Use of multiple antennas increase information throughput substantially.

7 Introduction Asymptotic ergodic capacity of the channels Mutual information I averaged over channel realizations for large number of antennas. We are interested in the moments of the capacity and the probability of outage. Here we discuss a method for computing the moment generating function of I. Use of representation theory to calculate the joint probability distribution of eigenvalues.

8 Introduction Asymptotic ergodic capacity of the channels Mutual information I averaged over channel realizations for large number of antennas. We are interested in the moments of the capacity and the probability of outage. Here we discuss a method for computing the moment generating function of I. Use of representation theory to calculate the joint probability distribution of eigenvalues.

9 Introduction Asymptotic ergodic capacity of the channels Mutual information I averaged over channel realizations for large number of antennas. We are interested in the moments of the capacity and the probability of outage. Here we discuss a method for computing the moment generating function of I. Use of representation theory to calculate the joint probability distribution of eigenvalues.

10 Introduction Asymptotic ergodic capacity of the channels Mutual information I averaged over channel realizations for large number of antennas. We are interested in the moments of the capacity and the probability of outage. Here we discuss a method for computing the moment generating function of I. Use of representation theory to calculate the joint probability distribution of eigenvalues.

11 MIMO working model Outline 1 Introduction 2 MIMO working model 3 Eigenvalue computations 4 Representation theory of unitary groups 5 Computation of the m.g.f.

12 MIMO working model System model Signals are complex realizations of the form re iθ. n t : # transmitter antennas. n r : # receiver antennas. x : transmitted signals. y : received signal. y = Gx + z G : Complex n r n t matrix of Channel coefficients. i.e. G ij = channel coefficient : transmitter j to receiver i. z : additive noise.

13 MIMO working model System model Signals are complex realizations of the form re iθ. n t : # transmitter antennas. n r : # receiver antennas. x : transmitted signals. y : received signal. y = Gx + z G : Complex n r n t matrix of Channel coefficients. i.e. G ij = channel coefficient : transmitter j to receiver i. z : additive noise.

14 MIMO working model System model Signals are complex realizations of the form re iθ. n t : # transmitter antennas. n r : # receiver antennas. x : transmitted signals. y : received signal. y = Gx + z G : Complex n r n t matrix of Channel coefficients. i.e. G ij = channel coefficient : transmitter j to receiver i. z : additive noise.

15 MIMO working model Model assumptions x, z independent. Their elements are i.i.d. We assume mean 0, variance 1, Gaussian structure. Arbitrary covariances : easy extensions. G is known to receiver, not transmitter. G has complex Gaussian entries. Covariance assumptions later.

16 MIMO working model Model assumptions x, z independent. Their elements are i.i.d. We assume mean 0, variance 1, Gaussian structure. Arbitrary covariances : easy extensions. G is known to receiver, not transmitter. G has complex Gaussian entries. Covariance assumptions later.

17 MIMO working model Mutual information Capacity I(y; x G) = log det(i + G G) I expressed in nats. Moment generating function : g(z) = E[e zi ] = E G [{det(i + G G)} z ] Probability of outage : P out = E G [Θ(I I out )] = where Θ is the Heaviside function. g(iz) 2πi e iziout z i0 + dz

18 MIMO working model Mutual information Capacity I(y; x G) = log det(i + G G) I expressed in nats. Moment generating function : g(z) = E[e zi ] = E G [{det(i + G G)} z ] Probability of outage : P out = E G [Θ(I I out )] = where Θ is the Heaviside function. g(iz) 2πi e iziout z i0 + dz

19 MIMO working model Mutual information Capacity I(y; x G) = log det(i + G G) I expressed in nats. Moment generating function : g(z) = E[e zi ] = E G [{det(i + G G)} z ] Probability of outage : P out = E G [Θ(I I out )] = where Θ is the Heaviside function. g(iz) 2πi e iziout z i0 + dz

20 Eigenvalue computations Outline 1 Introduction 2 MIMO working model 3 Eigenvalue computations 4 Representation theory of unitary groups 5 Computation of the m.g.f.

21 Eigenvalue computations Eigenvalues of G G Clearly, g(z) = E [ N (1 + λ i ) z] = i= (1 + λi ) z p(λ)dλ where λ = (λ 1,..., λ N ) are the positive eigenvalues of G G. N = min(n t, n r ). Also, let M = max(n t, n r ) Simple case : G i.i.d. with zero mean, i.e. p(g) e tr(g G) then, p(λ) = C M,N (λ) 2 [λ M N i e λ i ]

22 Eigenvalue computations Eigenvalues of G G Clearly, g(z) = E [ N (1 + λ i ) z] = i= (1 + λi ) z p(λ)dλ where λ = (λ 1,..., λ N ) are the positive eigenvalues of G G. N = min(n t, n r ). Also, let M = max(n t, n r ) Simple case : G i.i.d. with zero mean, i.e. p(g) e tr(g G) then, p(λ) = C M,N (λ) 2 [λ M N i e λ i ]

23 Eigenvalue computations Channels with non-trivial distributions Semi-correlated channels : non-zero correlation in the transmitters (alternately receivers). p(g) = c(t)e trt 1 G G for some T > 0. Non-zero means : G has mean G 0 p(g) e tr { (G G 0 ) (G G 0 )} Fully correlated channels : non-zero correlation in the transmitters and the receivers. p(g) = c(t, R)e tr{t 1 GR 1 G }

24 Eigenvalue computations Channels with non-trivial distributions Semi-correlated channels : non-zero correlation in the transmitters (alternately receivers). p(g) = c(t)e trt 1 G G for some T > 0. Non-zero means : G has mean G 0 p(g) e tr { (G G 0 ) (G G 0 )} Fully correlated channels : non-zero correlation in the transmitters and the receivers. p(g) = c(t, R)e tr{t 1 GR 1 G }

25 Eigenvalue computations Channels with non-trivial distributions Semi-correlated channels : non-zero correlation in the transmitters (alternately receivers). p(g) = c(t)e trt 1 G G for some T > 0. Non-zero means : G has mean G 0 p(g) e tr { (G G 0 ) (G G 0 )} Fully correlated channels : non-zero correlation in the transmitters and the receivers. p(g) = c(t, R)e tr{t 1 GR 1 G }

26 Eigenvalue computations Singular Value Decomposition of G The singular values of G Let Ω = diag(µ 1,..., µ N ) U, V are unitary matrices, µ i = λ i G = UΩV Using normalized Haar measures, p(λ) = C M,N (λ) 2 N i=1 λ M N i p(uωv )dudv, integrating over the unitary groups of order n t and n r.

27 Eigenvalue computations Singular Value Decomposition of G The singular values of G Let Ω = diag(µ 1,..., µ N ) U, V are unitary matrices, µ i = λ i G = UΩV Using normalized Haar measures, p(λ) = C M,N (λ) 2 N i=1 λ M N i p(uωv )dudv, integrating over the unitary groups of order n t and n r.

28 Representation theory of unitary groups Outline 1 Introduction 2 MIMO working model 3 Eigenvalue computations 4 Representation theory of unitary groups 5 Computation of the m.g.f.

29 Representation theory of unitary groups Representation theory: recap ρ : G V, a homomorphism from a group G to a group of invertible matrices V. Example : GL(M)- complex M M invertible matrices, U(M)- its subgroup of unitary matrices. ρ is irreducible if it has no non-trivial decomposition. The irreducible polynomial representations of U(M) are parametrized by m = (m 1,..., m M ) with integers m 1... m N 0. We denote by U (m) the corresponding irreducible representations.

30 Representation theory of unitary groups Representation theory: recap ρ : G V, a homomorphism from a group G to a group of invertible matrices V. Example : GL(M)- complex M M invertible matrices, U(M)- its subgroup of unitary matrices. ρ is irreducible if it has no non-trivial decomposition. The irreducible polynomial representations of U(M) are parametrized by m = (m 1,..., m M ) with integers m 1... m N 0. We denote by U (m) the corresponding irreducible representations.

31 Representation theory of unitary groups Dimension and orthogonality Dimension of an irreducible representation : dimension of its invariant subspace. For U(M) d m = [ M i=1 1 ] ( 1) M(M 1)/2 (k) (M i)! where k i = m i i + M. For irreducible representations U (m) and U (m ), (U (m) ) ij (U (m ) ) kl du = δ mm δ ikδ jl d m an orthogonality property we use later. Remember that we need to evaluate p(uωv )dudv

32 Representation theory of unitary groups Dimension and orthogonality Dimension of an irreducible representation : dimension of its invariant subspace. For U(M) d m = [ M i=1 1 ] ( 1) M(M 1)/2 (k) (M i)! where k i = m i i + M. For irreducible representations U (m) and U (m ), (U (m) ) ij (U (m ) ) kl du = δ mm δ ikδ jl d m an orthogonality property we use later. Remember that we need to evaluate p(uωv )dudv

33 Representation theory of unitary groups Dimension and orthogonality Dimension of an irreducible representation : dimension of its invariant subspace. For U(M) d m = [ M i=1 1 ] ( 1) M(M 1)/2 (k) (M i)! where k i = m i i + M. For irreducible representations U (m) and U (m ), (U (m) ) ij (U (m ) ) kl du = δ mm δ ikδ jl d m an orthogonality property we use later. Remember that we need to evaluate p(uωv )dudv

34 Representation theory of unitary groups Character of representation Character : trace of the representation, χ(g) = tr(ρ(g)). A (m) is the m representation of A, a d m dimensional matrix. Character of irreducible representations : χ m (A) = tr[a (m) ] = det ( a m j+m j) i (a 1,..., a m ) where a i are the eigenvalues of A. Character expansion of exponential : exp(t tr(a)) = m α m (t)χ m (A) α m (t)- coefficient of each character in the expansion.

35 Representation theory of unitary groups Character of representation Character : trace of the representation, χ(g) = tr(ρ(g)). A (m) is the m representation of A, a d m dimensional matrix. Character of irreducible representations : χ m (A) = tr[a (m) ] = det ( a m j+m j) i (a 1,..., a m ) where a i are the eigenvalues of A. Character expansion of exponential : exp(t tr(a)) = m α m (t)χ m (A) α m (t)- coefficient of each character in the expansion.

36 Representation theory of unitary groups Character of representation Character : trace of the representation, χ(g) = tr(ρ(g)). A (m) is the m representation of A, a d m dimensional matrix. Character of irreducible representations : χ m (A) = tr[a (m) ] = det ( a m j+m j) i (a 1,..., a m ) where a i are the eigenvalues of A. Character expansion of exponential : exp(t tr(a)) = m α m (t)χ m (A) α m (t)- coefficient of each character in the expansion.

37 Representation theory of unitary groups Character of representation Character : trace of the representation, χ(g) = tr(ρ(g)). A (m) is the m representation of A, a d m dimensional matrix. Character of irreducible representations : χ m (A) = tr[a (m) ] = det ( a m j+m j) i (a 1,..., a m ) where a i are the eigenvalues of A. Character expansion of exponential : exp(t tr(a)) = m α m (t)χ m (A) α m (t)- coefficient of each character in the expansion.

38 Computation of the m.g.f. Outline 1 Introduction 2 MIMO working model 3 Eigenvalue computations 4 Representation theory of unitary groups 5 Computation of the m.g.f.

39 Computation of the m.g.f. Semi-correlated channels Recall that, p(g) = c(t)etr( T 1 G G). Define Λ = diag(λ) = Ω 2. Therefore, p(uωv )dudv = etr( ΛU T 1 U)dU = α m ( 1)χ m (ΛU T 1 U)dU = = m m α m ( 1)tr(Λ (m) U m (T (m) ) 1 U (m) )du m α m ( 1) χ m (T 1 )χ m (Λ) d m

40 Computation of the m.g.f. Semi-correlated channels Recall that, p(g) = c(t)etr( T 1 G G). Define Λ = diag(λ) = Ω 2. Therefore, p(uωv )dudv = etr( ΛU T 1 U)dU = α m ( 1)χ m (ΛU T 1 U)dU = = m m α m ( 1)tr(Λ (m) U m (T (m) ) 1 U (m) )du m α m ( 1) χ m (T 1 )χ m (Λ) d m

41 Computation of the m.g.f. Semi-correlated channels Recall that, p(g) = c(t)etr( T 1 G G). Define Λ = diag(λ) = Ω 2. Therefore, p(uωv )dudv = etr( ΛU T 1 U)dU = α m ( 1)χ m (ΛU T 1 U)dU = = m m α m ( 1)tr(Λ (m) U m (T (m) ) 1 U (m) )du m α m ( 1) χ m (T 1 )χ m (Λ) d m

42 Computation of the m.g.f. Cauchy-Binet summation Let Υ = diag(τ), the eigenvalues of T 1. = m [ n t i=1 k 1 >...k n t 0 [ n t τi nr i=1 ( 1) m i (m i i + n t )! [ n t i=1 ( 1) k i k i! ] det(e τ i λ j ) (τ) (λ) ] det(τ m j j+n t i ] k det(τ j i ) det(λ k j (τ) (λ) ) det(λ m j j+n t i ) (τ) (λ) i ), (k i = m i i + n t ) Recall the Cauchy-Binet formula : det(a kj i ) det(bk j i ) w(k i ) = det(w(a i b j )) k 1 >...k n t 0 where W(z) = i=0 w(i)zi.

43 Computation of the m.g.f. Cauchy-Binet summation Let Υ = diag(τ), the eigenvalues of T 1. = m [ n t i=1 k 1 >...k n t 0 [ n t τi nr i=1 ( 1) m i (m i i + n t )! [ n t i=1 ( 1) k i k i! ] det(e τ i λ j ) (τ) (λ) ] det(τ m j j+n t i ] k det(τ j i ) det(λ k j (τ) (λ) ) det(λ m j j+n t i ) (τ) (λ) i ), (k i = m i i + n t ) Recall the Cauchy-Binet formula : det(a kj i ) det(bk j i ) w(k i ) = det(w(a i b j )) k 1 >...k n t 0 where W(z) = i=0 w(i)zi.

44 Computation of the m.g.f. Cauchy-Binet summation Let Υ = diag(τ), the eigenvalues of T 1. = m [ n t i=1 k 1 >...k n t 0 [ n t τi nr i=1 ( 1) m i (m i i + n t )! [ n t i=1 ( 1) k i k i! ] det(e τ i λ j ) (τ) (λ) ] det(τ m j j+n t i ] k det(τ j i ) det(λ k j (τ) (λ) ) det(λ m j j+n t i ) (τ) (λ) i ), (k i = m i i + n t ) Recall the Cauchy-Binet formula : det(a kj i ) det(bk j i ) w(k i ) = det(w(a i b j )) k 1 >...k n t 0 where W(z) = i=0 w(i)zi.

45 Computation of the m.g.f. Continuation Computation of m.g.f. : g(z)... [(1 + λi ) z λi M N ] (λ) (τ) n t = ( 1) nt(nt 1)/2 det L τj nr z (τ) j=1 τ n r i det(e τ iλ j ) where L z,ij is a confluent hypergeometric function of τ i s. The case of non-zero mean, uncorrelated channels follows similarly. Difficulty arises with fully correlated channels.

46 Computation of the m.g.f. Continuation Computation of m.g.f. : g(z)... [(1 + λi ) z λi M N ] (λ) (τ) n t = ( 1) nt(nt 1)/2 det L τj nr z (τ) j=1 τ n r i det(e τ iλ j ) where L z,ij is a confluent hypergeometric function of τ i s. The case of non-zero mean, uncorrelated channels follows similarly. Difficulty arises with fully correlated channels.

47 Computation of the m.g.f. Continuation Computation of m.g.f. : g(z)... [(1 + λi ) z λi M N ] (λ) (τ) n t = ( 1) nt(nt 1)/2 det L τj nr z (τ) j=1 τ n r i det(e τ iλ j ) where L z,ij is a confluent hypergeometric function of τ i s. The case of non-zero mean, uncorrelated channels follows similarly. Difficulty arises with fully correlated channels.

48 Computation of the m.g.f. Real matrices and orthogonal groups Remember we had the nice result etr( ΛU T 1 U)dU = m α m ( 1) χ m (T 1 )χ m (Λ) d m Do we have analogous results for the orthogonal matrices?