Lattices System Model Diversity Analysis for Unitary Precoded Integer-Forcing Optimal Design of Full-Diversity Unitary Precoders Simulation Results Conclusions Full Diversity Unitary Precoded Integer-Forcing Amin Sakzad Clayton School of IT Joint work with Emanuele Viterbo May 2015 Amin Sakzad Full Diversity Unitary Precoded Integer-Forcing
Lattices System Model Diversity Analysis for Unitary Precoded Integer-Forcing Optimal Design of Full-Diversity Unitary Precoders Simulation Results Conclusions Lattices 1 System Model 2 Diversity Analysis for Unitary Precoded Integer-Forcing 3 Optimal Design of Full-Diversity Unitary Precoders 4 Simulation Results 5 Conclusions 6 Amin Sakzad Full Diversity Unitary Precoded Integer-Forcing
Lattices System Model Diversity Analysis for Unitary Precoded Integer-Forcing Optimal Design of Full-Diversity Unitary Precoders Simulation Results Conclusions Definitions A lattice is a discrete additive subgroup of R n . For example Z 2 in R 2 . Amin Sakzad Full Diversity Unitary Precoded Integer-Forcing
Lattices System Model Diversity Analysis for Unitary Precoded Integer-Forcing Optimal Design of Full-Diversity Unitary Precoders Simulation Results Conclusions Definitions A lattice is a discrete additive subgroup of R n . For example Z 2 in R 2 . Every lattice does have a bases and every lattice point is an integer linear combinations of bases vectors. Amin Sakzad Full Diversity Unitary Precoded Integer-Forcing
Lattices System Model Diversity Analysis for Unitary Precoded Integer-Forcing Optimal Design of Full-Diversity Unitary Precoders Simulation Results Conclusions Definitions A lattice is a discrete additive subgroup of R n . For example Z 2 in R 2 . Every lattice does have a bases and every lattice point is an integer linear combinations of bases vectors. A lattice Λ can be represented with a generator matrix G by stacking its n -dimensional bases vectors as rows of G . Amin Sakzad Full Diversity Unitary Precoded Integer-Forcing
Lattices System Model Diversity Analysis for Unitary Precoded Integer-Forcing Optimal Design of Full-Diversity Unitary Precoders Simulation Results Conclusions Successive minimas For an n -dimensional lattice Λ G , we define the m -th successive minima, for 1 ≤ m ≤ n as ǫ m (Λ G ) � inf { r : dim ( span (Λ G ∩ B r ( 0 ))) ≥ m } . The m -th successive minima of Λ G is the infimum of the numbers r such that there are m independent vectors of Λ G in B r ( 0 ) . Amin Sakzad Full Diversity Unitary Precoded Integer-Forcing
Lattices System Model Diversity Analysis for Unitary Precoded Integer-Forcing Optimal Design of Full-Diversity Unitary Precoders Simulation Results Conclusions Successive minimas For an n -dimensional lattice Λ G , we define the m -th successive minima, for 1 ≤ m ≤ n as ǫ m (Λ G ) � inf { r : dim ( span (Λ G ∩ B r ( 0 ))) ≥ m } . The m -th successive minima of Λ G is the infimum of the numbers r such that there are m independent vectors of Λ G in B r ( 0 ) . The quantity ǫ 1 is also called the minimum distance of Λ G . Amin Sakzad Full Diversity Unitary Precoded Integer-Forcing
Lattices System Model Diversity Analysis for Unitary Precoded Integer-Forcing Optimal Design of Full-Diversity Unitary Precoders Simulation Results Conclusions Full-diversity lattices and minimum product distance An n -dimensional lattice Λ G is called full-diversity if for all disjoint x , y ∈ Λ G , the number of elements in { m : [ x ] m � = [ y ] m } be exactly n . Amin Sakzad Full Diversity Unitary Precoded Integer-Forcing
Lattices System Model Diversity Analysis for Unitary Precoded Integer-Forcing Optimal Design of Full-Diversity Unitary Precoders Simulation Results Conclusions Full-diversity lattices and minimum product distance An n -dimensional lattice Λ G is called full-diversity if for all disjoint x , y ∈ Λ G , the number of elements in { m : [ x ] m � = [ y ] m } be exactly n . The minimum product distance of a full-diversity lattice Λ G is denoted by d p, min (Λ G ) and is defined by: � d p, min (Λ G ) � min | [ x ] m | . 0 � = x ∈ Λ G m Amin Sakzad Full Diversity Unitary Precoded Integer-Forcing
Lattices System Model Diversity Analysis for Unitary Precoded Integer-Forcing Optimal Design of Full-Diversity Unitary Precoders Simulation Results Conclusions Lattice Codes For any point x ∈ Λ the Voronoi cell V ( x ) is � � � k α m ℓ m : � v − x � ≤ � v − y � , ∀ y ∈ Λ , α m ∈ C v = . m =1 Amin Sakzad Full Diversity Unitary Precoded Integer-Forcing
Lattices System Model Diversity Analysis for Unitary Precoded Integer-Forcing Optimal Design of Full-Diversity Unitary Precoders Simulation Results Conclusions Lattice Codes For any point x ∈ Λ the Voronoi cell V ( x ) is � � � k α m ℓ m : � v − x � ≤ � v − y � , ∀ y ∈ Λ , α m ∈ C v = . m =1 A lattice code C ⊆ Λ is a finite set of points of Λ . Amin Sakzad Full Diversity Unitary Precoded Integer-Forcing
Lattices System Model Diversity Analysis for Unitary Precoded Integer-Forcing Optimal Design of Full-Diversity Unitary Precoders Simulation Results Conclusions Lattice Codes For any point x ∈ Λ the Voronoi cell V ( x ) is � � � k α m ℓ m : � v − x � ≤ � v − y � , ∀ y ∈ Λ , α m ∈ C v = . m =1 A lattice code C ⊆ Λ is a finite set of points of Λ . A subset Λ ′ ⊆ Λ is called a sublattice if Λ ′ is a lattice itself. Amin Sakzad Full Diversity Unitary Precoded Integer-Forcing
Lattices System Model Diversity Analysis for Unitary Precoded Integer-Forcing Optimal Design of Full-Diversity Unitary Precoders Simulation Results Conclusions Lattice Codes For any point x ∈ Λ the Voronoi cell V ( x ) is � � � k α m ℓ m : � v − x � ≤ � v − y � , ∀ y ∈ Λ , α m ∈ C v = . m =1 A lattice code C ⊆ Λ is a finite set of points of Λ . A subset Λ ′ ⊆ Λ is called a sublattice if Λ ′ is a lattice itself. Given a sublattice Λ ′ , we define the quotient Λ / Λ ′ as a lattice code. The notions of coding lattice and shaping lattice. Amin Sakzad Full Diversity Unitary Precoded Integer-Forcing
Lattices System Model Diversity Analysis for Unitary Precoded Integer-Forcing Optimal Design of Full-Diversity Unitary Precoders Simulation Results Conclusions Figure: A full-diversity non-vanishing minimum product distance lattice with its bases vectors. Amin Sakzad Full Diversity Unitary Precoded Integer-Forcing
Lattices System Model Diversity Analysis for Unitary Precoded Integer-Forcing Optimal Design of Full-Diversity Unitary Precoders Simulation Results Conclusions MIMO Channel Model 1 We consider a quasi-static flat-fading n × n MIMO channel as above Figure with both CSIT and CSIR. Amin Sakzad Full Diversity Unitary Precoded Integer-Forcing
Lattices System Model Diversity Analysis for Unitary Precoded Integer-Forcing Optimal Design of Full-Diversity Unitary Precoders Simulation Results Conclusions MIMO Channel Model 1 We consider a quasi-static flat-fading n × n MIMO channel as above Figure with both CSIT and CSIR. The channel matrix is H ∈ C n × n with entries distributed independently and identically as CN (0 , 1) . Amin Sakzad Full Diversity Unitary Precoded Integer-Forcing
Lattices System Model Diversity Analysis for Unitary Precoded Integer-Forcing Optimal Design of Full-Diversity Unitary Precoders Simulation Results Conclusions MIMO Channel Model 1 We consider a quasi-static flat-fading n × n MIMO channel as above Figure with both CSIT and CSIR. The channel matrix is H ∈ C n × n with entries distributed independently and identically as CN (0 , 1) . An n -layer lattice coding scheme is used. For 1 ≤ m ≤ n , the m -th layer is equipped with a lattice encoder E : R k → Λ / Λ ′ ⊂ C n s m �→ x m . Amin Sakzad Full Diversity Unitary Precoded Integer-Forcing
Lattices System Model Diversity Analysis for Unitary Precoded Integer-Forcing Optimal Design of Full-Diversity Unitary Precoders Simulation Results Conclusions Figure: System model block diagram. Amin Sakzad Full Diversity Unitary Precoded Integer-Forcing
Lattices System Model Diversity Analysis for Unitary Precoded Integer-Forcing Optimal Design of Full-Diversity Unitary Precoders Simulation Results Conclusions MIMO Channel Model 2 Let WΣV h be the singular value decomposition (SVD) of H , i.e. W , V ∈ C n × n are two unitary matrices, Σ is a diagonal matrix given by Σ = diag ( σ 1 , . . . , σ n ) for which σ 1 ≥ · · · ≥ σ n . Amin Sakzad Full Diversity Unitary Precoded Integer-Forcing
Lattices System Model Diversity Analysis for Unitary Precoded Integer-Forcing Optimal Design of Full-Diversity Unitary Precoders Simulation Results Conclusions MIMO Channel Model 2 Let WΣV h be the singular value decomposition (SVD) of H , i.e. W , V ∈ C n × n are two unitary matrices, Σ is a diagonal matrix given by Σ = diag ( σ 1 , . . . , σ n ) for which σ 1 ≥ · · · ≥ σ n . A unitary precoder matrix U = VP is then employed at the transmitter where P ∈ C n × n is a unitary matrix. Amin Sakzad Full Diversity Unitary Precoded Integer-Forcing
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