from the golden code to perfect space-time block...

From the Golden Code to

Perfect Space-Time Block Codes

joint work withJean-Claude Belfiore and Ghaya Rekaya, ENST, Paris, France

Emanuele Viterbo, Politecnico di Torino, Torino, Italy

Algo & LMA Seminar

November 18th, 2004

Frederique Oggier

Institut de Mathematiques Bernoulli

Ecole Polytechnique Federale de Lausanne

[email protected]

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From the Golden code to Perfect Space-Time Block Codes • November 18th, 2004

Outline

I Channel model and code design criteria

I The Golden code

. a shaping constraint

. a discrete minimum determinant

. performance of the Golden code

I The Perfect Space-Time codes

. definition

. constructions of perfect Space-Time codes

. performance of perfect Space-Time codes

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The 2 × 2 MIMO channel

x x3 4

x1

x2

22h

12h

21h

11h

X H Y=HX+Z

y1

y2

y y3 4

I X : 2 × 2 matrix codeword from a space-time code

C =

{

X =

(

x1 x2

x3 x4

)

|x1, x2, x3, x4 ∈ C

}

the xi may belong to a signal constellation S with |S| = m (e.g. PSK, QAM) or be functionsof the information symbols taken from S.

I H : 2 × 2 channel matrix is a complex Gaussian matrix with independent, zero mean, entries.

I Z: 2 × 2 complex Gaussian noise matrix.3


Code design criteria

I To achieve the maximum diversity we need full rank codeword differences, i.e.,

rank(X1 − X2) = 2 ∀ X1 6= X2 ∈ C

I To maximize the throughput, we use full rate codes, i.e., the four symbols x1, x2, x3, x4 in thecodewords are functions of four symbols independently chosen from S and yield |C| = m4

distinct codewords.

I Note that Alamouti’s code X =

(

x1 −x∗2

x2 x∗1

)

is full rank but not full rate in a 2 × 2 MIMO.

I The asymptotic coding gain is given by the minimum determinant of Cδmin (C) = min

X1 6=X2∈C| det(X1 − X2)|2

I A good code will attempt to maximize δmin (C).

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2 × 2 Space–Time codes

I Let K = Q(θ) = {a + bθ | a, b ∈ Q(i)}. It is a quadratic extension of Q(i).

I We define the infinite lattice code C∞ as

C∞ =

{[

x1 x2

x3 x4

]

=

[

a + bθ c + dθ

γ(c + dθ) a + bθ

]

: a, b, c, d ∈ Z[i], γ ∈ C

}

I C∞ is a linear code, i.e., X1 + X2 ∈ C∞ for all X1,X2 ∈ C∞

I The finite code C is obtained by limiting the information symbols to a, b, c, d ∈ S ⊂ Z[i].

I The signal constellation S is a 2b–QAM, with in-phase and quadrature components equal to±1,±3, . . . and b bits per symbol.

I The minimum determinant of C∞ is given by

δmin (C∞) = minX1 6=X2∈C∞

| det(X1 − X2)|2 = min06=X∈C∞

| det(X)|2

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The Golden code: first ingredients

I The Golden code is related to the Golden number θ = 1+√

52 , one of the roots of θ2 − θ − 1 = 0

(θ = 1−√

52 is the other one)

I Consider K = Q(i,√

5) = {a + bθ|a, b ∈ Q(i)} as a relative quadratic extension of Q(i).

I The determinant of X ∈ C∞ is

det(X) = (a + bθ)(a + bθ) − γ(c + dθ)(c + dθ)

= NK/Q(i)(a + bθ) − γNK/Q(i)(c + dθ).

I For an element z = a + bθ ∈ K, the relative norm is

NK/Q(i)(z) = (a + bθ)(a + bθ) = a2 + ab − b2.

I Thus0 = det(X) ⇐⇒ γ = NK/Q(i)(a + bθ)/NK/Q(i)(c + dθ) ⇐⇒ γ = NK/Q(i)(z)

for an element z = a + bθ ∈ K.

I Choose γ which is not a norm.6


The Golden code: a Space-Time lattice code

I A complex lattice Λ is given by its generator matrix:

Λ = {Mv | v ∈ Z[i]n}

I Note that X ∈ C∞ can be written

X = diag(

M

[

a

b

])

+ diag(

M

[

c

d

])

·[

0 1

γ 0

]

=

[

a + bθ c + dθ

γ(c + dθ) a + bθ

]

,

where

M =

[

1 θ

1 θ

]

.

I We add a structure of lattice on each layer.

I The Golden code is found as a sublattice of the ST lattice code in order to obtain a “cubic”shaped constellation, which guarantees no shaping loss.

I In particular, the vectors (x1, x2, x3, x4) should belong to a rotated version of the complexlattice Z[i]4 and the finite constellation should be contained in a rotated hypercube.

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The Golden code: a Space-Time lattice code (2)

I We define CI ⊂ C∞ by restricting

x1, x2, x3, x4 ∈ I = (α)Z[i][√

5], α = 1 + i − iθ,

where Z[i][√

5] = {a + b√

5 | a, b ∈ Z[i]}, so that we get a rotated MZ[i]2 complex lattice oneach layer, hence a rotated Z[i]4, when the code matrices are vectorized.

I Codewords of the Golden code CI are given by

X = diag(

M

[

a

b

])

+ diag(

M

[

c

d

])

·[

0 1

γ 0

]

=1√5

[

α(a + bθ) α(c + dθ)

γα(c + dθ) α(a + bθ)

]

where a, b, c, d ∈ Z[i], α = 1 + i(1 − θ), and

M =

[

α αθ

α αθ

]

.

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Choosing γ for the Golden code

I The determinant of a codeword X is now

det(X) =1

5

(

NK/Q(i)(z1) − γNK/Q(i)(z2))

∀ z1 = α(a + bθ) ∈ I, z2 = α(c + dθ) ∈ I.

I Choose a γ which is not norm of elements of K to guarantee non-zero determinants.

I Choose a γ ∈ Z[i], so that

det(X) ∈ 1

dZ[i] for all X.

This guarantees discrete non-vanishing determinants as the spectral efficiency increases.

I Choose |γ| = 1, to guarantee that the same average energy is transmitted from each antennaat each channel use. This limits the choice to γ = ±1,±i.

I We choose γ = i, since we can prove that it is not a norm of an element of K = Q(i,√

5).

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The minimum determinant of the Golden code

I Let us relate the minimum determinant to the algebraic structure of the code. We have

det(X) =1

5NK/Q(i)(α) ·

(

NK/Q(i)(a + bθ) − γNK/Q(i)(c + dθ))

∀ a, b, c, d ∈ Z[i]

I As the second term in the above equation only takes values in Z[i] and its minimum modu-lus is equal to 1 (take a = 1 and b = c = d = 0), we conclude that the Golden code has

δmin(CI) =1

25|NK/Q(i)(α)|2 =

1

25|2 + i|2 =

1

5

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Some determinants for the codewords of CI

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Performance of the Golden Code

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The Golden code from a cyclic division algebra

I Recall that a codeword of the Golden code is of the form

X =

[

x1 0

0 x2

]

+

[

x3 0

0 x4

]

·[

0 1

γ 0

]

=

[

x1 x3

γx4 x2

]

where x1, x2, x3, x4 ∈ K and γ is not an algebraic norm of any element of K.

I This is the form in which we can represent the elements of a cyclic division algebra A of degree2 over K.

I Division algebras guarantee that det(X) 6= 0 for all codewords.

I Isomorphic versions of the Golden code were independently derived by [Yao, Wornell, 2003]and by [Dayal, Varanasi, 2003] by analytic optimization.

I Our algebraic construction enables to generalize to some larger n × n systems.

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Perfect Space-Time codes

We define a perfect Space-Time code to be a n × n linear dispersion block code that

I is full rate (codewords contain n2 symbols from 2b-QAM or 2b-HEX)

I has minimum non zero determinant which is constant for increasing spectral efficiency(non-vanishing determinant)

I gives a cubic constellation shaping

I is constructed from cyclic division algebras.

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Cyclic Algebras: linear full-rate codes

I Elements in a cyclic algebra A of degree n can be represented by matrices of the form

x0 x1 . . . xn−1

γσ(xn−1) σ(x0) . . . σ(xn−2)... ...

γσn−1(x1) γσn−1(x2) . . . σn−1(x0)

.

with xi ∈ K.

I The n2 information symbols u`,k are encoded into codewords by

x` =

n−1∑

k=0

u`,kνk, ` = 0, . . . , n − 1

where {νk}n−1k=0 is a basis of K.

I Information symbols u`,k considered are QAM (Z[i]) and HEX (Z[j]). Thus

Q(i) ⊂ K or Q(j) ⊂ K

where j is a third root of unity.

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Cyclic Division Algebras: the rank criterion

I Cyclic Division Algebras: these are cyclic algebras where all matrices are invertible.

I Recall that a cyclic algebra depends on a parameter γ.

I Proposition.If γ and its powers γ2, . . . , γn−1 are not a norm, then the cyclic algebra A is a division algebra.

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Cyclic Division Algebras: the lattice structure

I We restrict the elements of A to those where xi ∈ I, I ⊂ K. We thus get a STBC of the form

CI =

x0 x1 . . . xn−1

γσ(xn−1) σ(x0) . . . σ(xn−2)... ...

γσn−1(x1) γσn−1(x2) . . . σn−1(x0)

| xi ∈ I ⊆ K, i = 0, . . . , n − 1

.

I The n2 information symbols u`,k ∈ Z[i] or Z[j] are encoded into codewords by

x` =

n−1∑

k=0

u`,kνk ` = 0, . . . , n − 1

where {νk}n−1k=0 is a basis of I.

I We choose I ⊆ K so that the signal constellation on each layer is a finite subset of the rotatedversions of the lattices Z2n or An

2 (the hexagonal lattice).

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Cyclic Division Algebras: choosing γ

I Choose γ such that none of its power is a norm to get a non-zero determinant.

I Choose γ ∈ Z[i] or Z[j] so that we get discrete non-vanishing determinants. That is

det(X) ∈ 1

dZ[i] or

1

dZ[j].

I Choose |γ| = 1, to guarantee that the same average energy is transmitted from each antennaat each channel use. This limits the choice to

γ ∈ {±1,±i} ⊂ Z[i] or γ ∈ {1, j, j2,−1,−j, j2} ⊂ Z[j].

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Cyclic Division Algebras: the minimum determinant

I det(X) is called the reduced norm of X.

I Thanks to the property of the reduced norm, we have

δmin(CI) ≥1

dN(I)

where N(I) ∈ N is called the norm of I.

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Perfect Space-Time codes: summary

Take A a cyclic division algebra.

I cyclic algebras: yields a full-rate linear code

I cyclic division algebras: fullfills the rank criterion

I the restriction to I: good shaping, and thus energy efficient codes

I the choice of γ: same average energy transmitted from each antenna

I discrete minimum determinant, which is non-vanishing.

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All the perfect ST codes

I There exists an entire family of 2× 2 perfect ST codes based on the fields K = Q(i,√

p) wherep ≡ 5 (8) is a prime. The other codes have δmin(C∞) = 1/p, hence the Golden code is the bestone.

I The 4 × 4 perfect ST code is found from K = Q(i, 2 cos(2π/15)) and γ = i. We find therotated complex lattice Z[i]4 with the principal ideal generated by α = 1 − 3i + iθ2 and getδmin = 1/1125.

I The 3 × 3 perfect ST code requires HEX symbols since it uses as base field Q(j) and K =

Q(j, 2 cos(2π/7)). We find a rotated version of the lattice A32 using the principal ideal gener-

ated by α = 1 + j + θ and get δmin = 1/49.

I The 6 × 6 perfect ST code requires HEX symbols since it uses as base field Q(j) and K =

Q(j, 2 cos(π/14)). We find a rotated version of the lattice A62 using a non principal ideal factor

of (7)OK and get 2−67−5 ≤ δmin ≤ 2−67−4.

I We prove that these are the only existing perfect ST codes.

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Performance of some 2 × 2 perfect ST codes

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Performance of the 3 × 3 perfect ST code

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Performance of the 4 × 4 perfect ST code

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Conclusions

I It has been recently shown that codes constructed from cyclic division algebras with cen-ter F = Q(i) or Q(j) with non-vanishing minimum determinant achieve the Diversity vsMultiplexing Gain tradeoff. [P. Elia, K.R. Kumar, S.A. Pawar, V.V. Kumar, Hsiao-feng Lu,2004]

I They also propose some STBC algebraic constructions with non-vanishing minimum deter-minant, that are not perfect since they do not satisfy the cubic shaping condition.

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Some references

I M. O. Damen, A. Tewfik, and J.-C. Belfiore, “A construction of a space-time code based onthe theory of numbers,” IEEE Trans. Inform. Theory, vol. 48, no. 3, pp. 753–760, March 2002.

I H. El Gamal and M. O. Damen, “Universal space-time coding,” IEEE Trans. Inform. Theory,vol. 49, no. 5, pp. 1097–1119, May 2003.

I J.-C. Belfiore and G. Rekaya, “Quaternionic lattices for space-time coding,” in Proceedings ofthe Information Theory Workshop, Paris, March 31–April 4, 2003.

I B. A. Sethuraman, B. S. Rajan, and V. Shashidhar, “Full-diversity, high-rate space-time blockcodes from division algebras,” IEEE Trans. Inform. Theory, vol. 49, pp. 2596–2616, October2003.

I J. Boutros, E. Viterbo, C. Rastello, and J.-C. Belfiore, “Good Lattice Constellations for bothRayleigh fading and Gaussian channels,” IEEE Trans. Inform. Theory, vol. 42, no. 2, pp. 502–518, March 1996.

I H. Cohn, Advanced Number Theory, Dover Publications, Inc. New York, 1980.

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