Minimization of Quadratic Forms in Wireless Communications

Transcription

1 Minimization of Quadratic Forms in Wireless Communications Ralf R. Müller Department of Electronics & Telecommunications Norwegian University of Science & Technology, Trondheim, Norway Dongning Guo Department of Electrical Engineering & Computer Science Northwestern University, Evanston, IL, U.S.A. Aris L. Moustakas Physics Department National & Capodistrian University of Athens, Greece

2 Introduction Let with x C K and J C K K. E := K min x X x Jx Ì ÈÖÓ Ð Ñ Example : X = {x : x x = K} = E = minλ(j) for Wishart matrix [ α] 2 + Example 2: X = {x : x 2 = } K =??? for Wishart matrix [ α π ] 2 + Example 3: X = {x : x 2 = } K =??? for Wishart matrix [ πα 2 ] 2 +

3 Introduction Let with x C K and J C K K. E := K min x X x Jx Ì ÈÖÓ Ð Ñ Example : X = {x : x x = K} = E = minλ(j) for Wishart matrix [ α] 2 + Example 2: X = {x : x 2 = } K =??? for Wishart matrix [ α π ] 2 + Example 3: X = {x : x 2 = } K =??? for Wishart matrix [ πα 2 ] 2 +

4 Introduction Let with x C K and J C K K. E := K min x X x Jx Ì ÈÖÓ Ð Ñ Example : X = {x : x x = K} = E = minλ(j) for Wishart matrix [ α] 2 + Example 2: X = {x : x 2 = } K =??? for Wishart matrix [ α π ] 2 + Example 3: X = {x : x 2 = } K =??? for Wishart matrix [ πα 2 ] 2 +

5 Introduction Let with x C K and J C K K. E := K min x X x Jx Ì ÈÖÓ Ð Ñ Example : X = {x : x x = K} = E = minλ(j) for Wishart matrix [ α] 2 + Example 2: X = {x : x 2 = } K =??? for Wishart matrix [ α π ] 2 + Example 3: X = {x : x 2 = } K =??? for Wishart matrix [ πα 2 ] 2 +

6 Introduction Let with x C K and J C K K. E := K min x X x Jx Ì ÈÖÓ Ð Ñ Example : X = {x : x x = K} = E = minλ(j) for Wishart matrix [ α] 2 + Example 2: X = {x : x 2 = } K =??? for Wishart matrix [ α π ] 2 + Example 3: X = {x : x 2 = } K =??? for Wishart matrix [ πα 2 ] 2 +

7 Introduction Let with x C K and J C K K. E := K min x X x Jx Ì ÈÖÓ Ð Ñ Example : X = {x : x x = K} = E = minλ(j) for Wishart matrix [ α] 2 + Example 2: X = {x : x 2 = } K =??? for Wishart matrix [ α π ] 2 + Example 3: X = {x : x 2 = } K =??? for Wishart matrix [ πα 2 ] 2 +

8 Introduction Let with x C K and J C K K. E := K min x X x Jx Ì ÈÖÓ Ð Ñ Example : Example 2: X = {x : x x = K} = E = minλ(j) X = {x : x 2 = } K =??? for Wigner matrix 2 for Wigner matrix 2 π Example 3: X = {x : x 2 = } K =??? for Wigner matrix π

9 Introduction 2 Ï ÖØ Å ØÖ Ü E x x = K x 2 = x 2 = : K = 5, K/N

10 Application 3 The Gaussian Vector Channel Let the received vector be given by where t is the transmitted vector r = Ht + n n is uncorrelated (white) Gaussian noise H is a coupling matrix accounting for crosstalk In many applications, e.g. antenna arrays, code-division multiple-access, the coupling matrix is modelled as a random matrix with independent identically distributed entries (i.i.d. model). Crosstalk can be processed either at receiver or transmitter

11 Application 4 Processing at Transmitter If the transmitter is a base-station and the receiver is a hand-held device one would prefer to have the complexity at the transmitter. E.g. let the transmitted vector be where x is the data to be sent. t = H (HH ) x Then, r = x + n. No crosstalk anymore due to channel inversion.

12 Application 5 Problems of Simple Channel Inversion Channel inversion implies a significant power amplification, i.e. x ( HH ) x > x x. In particular, let α = K N ; the entries of H are i.i.d. with variance /N. Then, for fixed aspect ratio α with probability. x ( HH ) x lim K x x = α

13 Application 6 Tomlinson-Harashima Precoding Tomlinson 7, Harashima & Miyakawa 72 Choose that representation that gives the smallest transmit power.

14 Application 6 Tomlinson-Harashima Precoding Tomlinson 7, Harashima & Miyakawa 72 Choose that representation that gives the smallest transmit power.

15 Application 6 Tomlinson-Harashima Precoding Tomlinson 7; Harashima & Miyakawa 72 Choose that representation that gives the smallest transmit power.

16 Application 6 Tomlinson-Harashima Precoding Tomlinson 7, Harashima & Miyakawa 72 Instead of representing the logical 0 by +, we present it by any element of the set {..., 7, 3, +, +5,...} = 4Z +. Correspondingly, the logical is represented by any element of the set 4Z. Choose that representation that gives the smallest transmit power.

17 Application 7 Generalized TH Precoding Let B 0 and B denote the sets presenting 0 and, resp. Let (s, s 2,s 3,...,s K ) denote the data to be transmitted. Then, the transmitted energy per data symbol is given by with and E = K min x X x Jx X = B s B s2 B sk J = (HH ).

18 Replica Calculations 8 Zero Temperature Formulation Quadratic programming is the problem of finding the zero temperature limit (ground state energy) of a quadratic Hamiltonian. The transmitted power is written as a zero temperature limit E = lim β βk log x X lim β lim K E J e βk Tr(Jxx ) βk log x X e βk Tr(Jxx ) with β denoting temperature.

19 Replica Calculations 8 Zero Temperature Formulation Quadratic programming is the problem of finding the zero temperature limit (ground state energy) of a quadratic Hamiltonian. The transmitted power is written as a zero temperature limit E = lim β βk log x X lim β lim K E J e βk Tr(Jxx ) βk log x X e βk Tr(Jxx ) with β denoting temperature.

20 Replica Calculations 9 We want Free Fourier Transform lim K K E J log x X e βk Tr(Jxx ). We know lim K K log E J e K Tr JP = n a= λ a (P) 0 R J ( w)dw. We would like to exchange expectation and logarithm: E log X = lim X n 0 n log E X n. X

21 Replica Calculations 9 We want Free Fourier Transform lim K K E J log x X e βk Tr(Jxx ). We know lim K K log E J e K Tr JP = n a= λ a (P) 0 R J ( w)dw. We would like to exchange expectation and logarithm: E log X = lim X n 0 n log E X n. X

22 Replica Calculations 9 We want Free Fourier Transform lim K K E J log x X e βk Tr(Jxx ). We know lim K K log E J e K Tr JP = n a= λ a (P) 0 R J ( w)dw. We would like to exchange expectation and logarithm: E log X = lim X n 0 n log E X n. X

23 Replica Calculations 0 We want lim K with K E J log x X Replica Continuity e βk Tr(Jxx ) = lim K lim n 0 = lim K lim n 0 = lim K lim n 0 = lim n 0 n Q := nk log E J nk log E J ( x X n a= x a X nk log E J n a= E Q n x a x a. a= βλ a (Q) 0 x X e βk Tr(Jxx ) ) n ( e βk Tr Jx a x ) a x n X R J ( w)dw ( K Tr Jβ n ) x a x a e a=

24 Replica Calculations 0 We want lim K with K E J log x X Replica Continuity e βk Tr(Jxx ) = lim K lim n 0 = lim K lim n 0 = lim K lim n 0 = lim n 0 n Q := nk log E J nk log E J ( n x X a= x a X nk log E J n a= E Q n x a x a. a= βλ a (Q) 0 x X e βk Tr(Jxx ) ) n ( e βk Tr Jx a x a ) x n X R J ( w)dw ( K Tr Jβ n ) x a x a e a=

25 Replica Calculations 0 We want lim K with K E J log x X Replica Continuity e βk Tr(Jxx ) = lim K lim n 0 = lim K lim n 0 = lim K lim n 0 = lim n 0 n Q := nk log E J nk log E J ( n x X a= x a X nk log E J n a= E Q n x a x a. a= βλ a (Q) 0 x X e βk Tr(Jxx ) ) n ( e βk Tr Jx a x a ) x n X R J ( w)dw ( K Tr Jβ n ) x a x a e a=

26 Replica Calculations 0 We want lim K with K E J log x X Replica Continuity e βk Tr(Jxx ) = lim K lim n 0 = lim K lim n 0 = lim K lim n 0 = lim n 0 n nk log E J nk log E J ( x X n nk log E J n a= E Q Q ab := K x ax b. βλ a (Q) 0 a= x a X x X e βk Tr(Jxx ) ) n ( e βk Tr Jx a x ) a x n X R J ( w)dw ( K Tr Jβ n ) x a x a e a=

27 Replica Calculations Replica Symmetry Q := q + χ β q q q q q + χ... β q q q q... q + χ β q q q q q + χ β with some macroscopic parameters q and χ. This is the most critical step. In general, the structure of Q is more complicated. Generalizations are called replica symmetry breaking (RSB).

28 Replica Calculations 2 Main Result Let P(s) denote the limit of the empirical distribution of the information symbols s, s 2,...,s K as K. Let q and χ be the simultaneous solutions to 2 q = argmin z 2qR ( χ) 2xR( χ) Dz dp(s) x B s z χ = argmin 2qR ( χ) 2xR( χ) zdz dp(s) 2qR ( χ) x B s where Dz = exp( z 2 /2)dz/ 2π, R( ) is the R-transform of the limiting eigenvalue spectrum of J, and 0 < χ <. Then, replica symmetry implies as K. K min x X x Jx q χ χr( χ)

29 Replica Calculations 3 Some R-Transforms I : R(w) = HH : R(w) = αw (HH ) : Marchenko-Pastur (MP) law R(w) = α ( α) 2 4αw 2αw U + U : R(w) = + + 4w 2 w inv. MP

30 Examples 4 Inv. MP with Odd Integer Lattice (TH Precoding) Let J = (HH ) and χ < : E = 20 c 2 + L α+ α i=2 L πe i=2 ( (c 2 ci i c2 +c i i )Q ( (c i c i )exp 2αE ) (c i +c i )2 4αE ) E [db] L =, 2, 3, 6, α

31 Examples 5 0 Convex Relaxation E [db] ÓÒÚº Ö Ð Ü Ø ÓÒ B 0 = [+; + ) B = ( ; ] Both sets are convex. Convex optimization, but small gains α

32 Examples 6 Odd Integer Quadrature Lattice

33 Examples 6 Odd Integer Quadrature Lattice Same energy per bit E b = E log 2 S in both cases.

34 Examples 7 Complex TH Precoding L =, 2, 3, 6, 00 E b [db] E b = E 2 = 4 3 for L α

35 Examples 8 Complex Convex Relaxation... allows for convex programming.

36 Examples 9 Complex Convex Relaxation (cont d) E α... achieves part of the gain of TH precoding.

37 An Invariance Result of Inverse MP Kernels 20 Complex Semi-Discrete Set The imaginary part is purely used to reduce transmit energy.

38 An Invariance Result of Inverse MP Kernels 2 3 Complex Semi-Discrete Set (solid lines) 2.5 E b [db] 2.5 E b = 4 3 L =, 2, 3, 6 for L α Convex opt. for L =.

39 An Invariance Result of Inverse MP Kernels 22 A Fake Gain The inverse MP kernel has the following property: Let H and H be random matrices of size K N and K N respectively, with K > K and with i.i.d. entries of zero mean and variance /N. Then, min x X x (HH ) x K min x X C K K x (H H ) x K 0. The redundant symbols serve no purpose.

40 Open Problems 23 Ï ÒØ lim K K log E A,B e K TrAPBP = f {R A ( ),R B ( ),..., }. Rigorous or Hand-Waving

41 Why We Need Analytical Results in Engineering 24 ÓÚ Ö Ò ÒØ Ñ ØØ Ö What happens if the MP-law has a mass point at zero (K > N)? Can we precode without interference? The precoder produces lim argmin ǫ 0 x X The received signal becomes x (HH + ǫi) x K r = lim ǫ 0 HH (HH + ǫi) x + n. If the energy is finite, there is no interference.

42 Why We Need Analytical Results in Engineering 24 ÓÚ Ö Ò ÒØ Ñ ØØ Ö What happens if the MP-law has a mass point at zero (K > N)? Can we precode without interference? The precoder produces lim argmin ǫ 0 x X The received signal becomes x (HH + ǫi) x K r = lim ǫ 0 HH (HH + ǫi) x + n. If the energy is finite, there is no interference.

43 Why We Need Analytical Results in Engineering 24 ÓÚ Ö Ò ÒØ Ñ ØØ Ö What happens if the MP-law has a mass point at zero (K > N)? Can we precode without interference? The precoder produces lim argmin ǫ 0 x X The received signal becomes x (HH + ǫi) x K r = lim ǫ 0 HH (HH + ǫi) x + n. If the energy is finite, there is no interference.