Feasibility Conditions for Interference Alignment Cenk M. Yetis (Istanbul Tech. Univ., Turkey) Tiangao Gou, Syed A. Jafar (Univ. of California Irvine, USA) Ahmet H. Kayran (Istanbul Tech. Univ., Turkey)
December 1, 2009
Interference Network
Figure: 4-user interference network
Interference Network
Figure: 4-user interference network
Interference Network
Figure: 4-user interference network
Interference Network
Figure: 4-user interference network
Interference Network
Figure: 4-user interference network
Linear Interference Alignment in Signal Space [1] [1]
[3] [3]
v1 x 1
H[13] v1 x1
H[11] [2] [2]
v1 x 1
H[12]
[1] [1]
H[11] v1 x1
[2] [2]
H[12] v1 x1 [4] [4] H[14] v1 x1
H[13] H[14]
[3] [3]
v1 x 1 [4] [4]
v1 x 1
Figure: Interference alignment in signal space for a 4-user interference network
Linear Interference Alignment in Signal Space [1] [1]
v1 x 1
[1] [1]
H[11] v1 x1
H[11] [2] [2]
v1 x 1
[4] [4]
H[12] H[13]
H[14] v1 x1 [2] [2]
H[12] v1 x1 [3] [3] H[13] v1 x1
H[14]
[3] [3]
v1 x 1 [4] [4]
v1 x 1
Figure: Interference alignment in signal space for a 4-user interference network
Linear Interference Alignment in Signal Space [j]
vn : Transmit Beamforming Vectors for aligning interference [1] [1]
v1 x 1
[1] [1]
H[11] v1 x1
H[11] [2] [2]
v1 x 1
[4] [4]
H[12] H[13]
H[14] v1 x1 [2] [2]
H[12] v1 x1 [3] [3] H[13] v1 x1
H[14]
[3] [3]
v1 x 1 [4] [4]
v1 x 1
Figure: Interference alignment in signal space for a 4-user interference network
Linear Interference Alignment in Signal Space [j]
vn : Transmit Beamforming Vectors for aligning interference [1] [1]
v1 x 1
[1]
H[11] [2] [2]
v1 x 1
u1
[1] [1]
H[11] v1 x1 [4] [4]
H[12] H[13] H[14]
H[14] v1 x1 [2] [2]
[2]
u1
H[12] v1 x1 [3] [3] H[13] v1 x1
[3]
u1 [3] [3]
v1 x 1
[4]
u1 [4] [4]
v1 x 1
[j]
un : Receive Beamforming Vectors for nulling interference
Figure: Interference alignment in signal space for a 4-user interference network
Linear Interference Alignment in Signal Space [j]
vn : Transmit Beamforming Vectors for aligning interference [1] [1]
v1 x 1
[1]
H[11] [2] [2]
v1 x 1
u1
[1] [1]
H[11] v1 x1 [4] [4]
H[12] H[13] H[14]
H[14] v1 x1 [2] [2]
[2]
u1
[3]
u1
H[12] v1 x1 [3] [3] H[13] v1 x1
Interference alignment
↓ Unselfish scheme
[3] [3] v1 x 1 [4]
u1 [4] [4]
v1 x 1
[j]
un : Receive Beamforming Vectors for nulling interference
Figure: Interference alignment in signal space for a 4-user interference network
Linear Interference Alignment in Signal Space [1] [1]
v1 x 1
High SNR
H[11] [2] [2]
v1 x 1
H[12] H[13] H[14]
[3] [3]
v1 x 1 [4] [4]
v1 x 1
Figure: Interference alignment in signal space for a 4-user interference network
Linear Interference Alignment in Signal Space [1] [1]
v1 x 1
High SNR
H[11] [2] [2]
v1 x 1
DoF (Degrees of Freedom)
H[12] H[13] H[14]
[3] [3]
v1 x 1
4 DoF
[4] [4]
v1 x 1
Figure: Interference alignment in signal space for a 4-user interference network
Constant vs. Time-Varying Channel I
We assume constant channel GeneralMIMO channel (no structure): ··· ··· H[kj] = ... . . . ... ···
···
M×N
Constant vs. Time-Varying Channel I
We assume constant channel GeneralMIMO channel (no structure): ··· ··· H[kj] = ... . . . ... ···
I
···
M×N
Time-varying channel Diagonal MIMO channel(diagonal structure): .. 0 . . [kj] ¯ .. H = .. . 0 M ×M n
Mn : symbol extension length
n
Constant vs. Time-Varying Channel I
We assume constant channel GeneralMIMO channel (no structure): ··· ··· H[kj] = ... . . . ... ···
I
···
M×N
Time-varying channel Diagonal MIMO channel(diagonal structure): [j] .. X (Mn (t − 1) + 1) 0 . X [j] (Mn (t − 1) + 2) .. ¯ [kj] = ¯ [j] = H X . .. . .. [j] (M t) . 0 X n M ×M n
Mn : symbol extension length
n
Mn ×1
Important Parameters Important parameters for the feasibility of interference alignment are
M [1]
N [1]
d [1] M [2] d [2] M [i] d [i] M [K ] d [K ]
N [2]
.. .
.. .
.. .
.. .
N [i]
N [K ]
Figure: K -user MIMO interference network
Important Parameters Important parameters for the feasibility of interference alignment are I
# of users, K
M [1]
N [1]
d [1] M [2] d [2] M [i] d [i] M [K ] d [K ]
N [2]
.. .
.. .
.. .
.. .
N [i]
N [K ]
Figure: K -user MIMO interference network
Important Parameters Important parameters for the feasibility of interference alignment are I
# of users, K
I
# of antennas of each user, M [i] and N [i] M [1]
N [1]
d [1] M [2] d [2] M [i] d [i] M [K ] d [K ]
N [2]
.. .
.. .
.. .
.. .
N [i]
N [K ]
Figure: K -user MIMO interference network
Important Parameters Important parameters for the feasibility of interference alignment are I
# of users, K
I
# of antennas of each user, M [i] and N [i]
I
# of beams (DoF) of each user, d [i] M [1]
N [1]
d [1] M [2] d [2] M [i] d [i] M [K ] d [K ]
N [2]
.. .
.. .
.. .
.. .
N [i]
N [K ]
Figure: K -user MIMO interference network
Feasibility Question
M [1] = 2 d [1] = 1 M [2] = 2 d [2] = 1 M [3] = 2 d [3] = 1 M [4] = 2 d [4] = 1
N [1] = 3 N [2] = 3 N [3] = 3 N [4] = 3
Figure: (2 × 3, 1)4 symmetric network
Feasibility Question
M [1] = 2 d [1] = 1 M [2] = 2 d [2] = 1 M [3] = 2 d [3] = 1 M [4] = 2 d [4] = 1
N [1] = 3 N [2] = 3 N [3] = 3 N [4] = 3
Figure: (2 × 3, 1)4 symmetric network
M [1] = 2 d [1] = 1 M [2] = 2 d [2] = 1 M [3] = 2 d [3] = 1 M [4] = 2 d [4] = 1
N [1] = 2 N [2] = 3 N [3] = 3 N [4] = 3
Figure: (2 × 2, 1)(2 × 3, 1)3 asymmetric network
Feasibility Question
M [1]
N [1]
d [1] M [2] d [2] M
[i]
d [i] M [K ] d [K ]
N [2]
.. . .. .
?
.. . .. .
N [i]
N [K ]
Figure: K -user MIMO interference network
Feasibility of Interference Alignment [j]
[1] [1] H[11] v1 x1
[4] [4] H[14] v1 x1 [12] [2] [2] H v1 x 1 [3] [3] H[13] v1 x1
[j]
vn /un : nth beamforming vector of the j th transmitter/receiver H[kj] : Channel between the j th transmitter and the k th receiver [j]
xn : nth data signal of the j th transmitter
Feasibility of Interference Alignment [1] [1]
H[11] v1 x1 [4] [4] H[14] v1 x1 [12] [2] [2] H v1 x 1 [3] [3] H[13] v1 x1
Interference alignment
Feasibility of Interference Alignment [1] [1]
H[11] v1 x1 [4] [4] H[14] v1 x1 [12] [2] [2] H v1 x 1 [3] [3] H[13] v1 x1
[1]†
[4]
[1]†
[3]
[1]†
[2]
u1 H[14] v1 = 0 u1 H[13] v1 = 0 u1 H[12] v1 = 0
Interference alignment
Feasibility of Interference Alignment [1] [1]
H[11] v1 x1 [4] [4] H[14] v1 x1 [12] [2] [2] H v1 x 1 [3] [3] H[13] v1 x1
[1]†
[4]
[1]†
[3]
[1]†
[2]
[4]†
[1]
[4]†
[2]
[4]†
[3]
u1 H[14] v1 = 0 Receiver 1 u1 H[13] v1 = 0
.. .
u1 H[12] v1 = 0 .. . u1 H[41] v1 = 0
Receiver 4 u1 H[42] v1 = 0
u1 H[43] v1 = 0
Interference alignment l Feasibility of interference alignment
Feasibility of interference alignment (similar equations for other receiver nodes)
Feasibility of Interference Alignment [1] [1]
H[11] v1 x1 [4] [4] H[14] v1 x1 [12] [2] [2] H v1 x 1 [3] [3] H[13] v1 x1
[1]†
[4]
[1]†
[3]
[1]†
[2]
[4]†
[1]
[4]†
[2]
[4]†
[3]
u1 H[14] v1 = 0 Receiver 1 u1 H[13] v1 = 0
.. .
u1 H[12] v1 = 0 .. . u1 H[41] v1 = 0
Receiver 4 u1 H[42] v1 = 0
u1 H[43] v1 = 0
Interference alignment l Feasibility of interference alignment
Feasibility of interference alignment (similar equations for other receiver nodes)
Solvability of Multivariate Polynomials [k ]†
[j]
um H[kj]vn = 0, j 6= k, j, k ∈ {1, 2, · · · , K } ∀n ∈ {1, 2, ..., d [j] } and ∀m ∈ {1, 2, ..., d [k ] }
Solvability of Multivariate Polynomials [k ]†
[j]
um H[kj]vn = 0, j 6= k, j, k ∈ {1, 2, · · · , K } ∀n ∈ {1, 2, ..., d [j] } and ∀m ∈ {1, 2, ..., d [k ] } Unknowns Coefficients
: Beamforming vectors : Channel gains
Solvability of Multivariate Polynomials [k ]†
[j]
um H[kj]vn = 0, j 6= k, j, k ∈ {1, 2, · · · , K } ∀n ∈ {1, 2, ..., d [j] } and ∀m ∈ {1, 2, ..., d [k ] } Unknowns Coefficients
: Beamforming vectors : Channel gains
Solvability of multivariate polynomials
Solvability of Multivariate Polynomials [k ]†
[j]
um H[kj]vn = 0, j 6= k, j, k ∈ {1, 2, · · · , K } ∀n ∈ {1, 2, ..., d [j] } and ∀m ∈ {1, 2, ..., d [k ] } Unknowns Coefficients
: Beamforming vectors : Channel gains
Solvability of multivariate polynomials ↓ Algebraic geometry
Solvability of Multivariate Polynomials [k ]†
[j]
um H[kj]vn = 0, j 6= k, j, k ∈ {1, 2, · · · , K } ∀n ∈ {1, 2, ..., d [j] } and ∀m ∈ {1, 2, ..., d [k ] } Unknowns Coefficients
: Beamforming vectors : Channel gains
Solvability of multivariate polynomials ↓ Algebraic geometry ↓ Bezout’s and Bernshtein’s theorems
Proper system Bezout’s theorem: A generic system is solvable iff N e ≤ Nv Ne : Total # of equations Nv : Total # of variables
Proper system Bezout’s theorem: A generic system is solvable iff N e ≤ Nv Ne : Total # of equations Nv : Total # of variables
Definition (Proper system) For all subsets of equations, # of variables ≥ # of equations
Proper system Bezout’s theorem: A generic system is solvable iff N e ≤ Nv Ne : Total # of equations Nv : Total # of variables
Definition (Proper system) For all subsets of equations, # of variables ≥ # of equations
E1 E2 E3 E4
Proper system Bezout’s theorem: A generic system is solvable iff N e ≤ Nv Ne : Total # of equations Nv : Total # of variables
Definition (Proper system) For all subsets of equations, # of variables ≥ # of equations
E1
V1
E2
V2
E3
V3
E4
V4
Proper system Bezout’s theorem: A generic system is solvable iff N e ≤ Nv Ne : Total # of equations Nv : Total # of variables
Definition (Proper system) For all subsets of equations, # of variables ≥ # of equations
E1
V1
E2
V2
E3
V3
E4
V4
Subsets of equations: {E1}· · · {E4} {E1,E2},{E1,E3}· · · {E1,E2,E3},{E1,E2,E4}· · · {E1,E2,E3,E4}
Proper system Bezout’s theorem: A generic system is solvable iff N e ≤ Nv Ne : Total # of equations Nv : Total # of variables
Definition (Proper system) For all subsets of equations, # of variables ≥ # of equations
E1
V1
E2
V2
E3
V3
E4
V4
Subsets of equations: {E1}· · · {E4} {E1,E2},{E1,E3}· · · {E1,E2,E3},{E1,E2,E4}· · · {E1,E2,E3,E4} For subset {E1,E2}: V 1 + V 2 ≥ |{E1, E2}| = 2 For subset {E1,E2,E3,E4}: V 1 + · · · + V 4 ≥ |{E1, E2, E3, E4}| = 4 Nv ≥ N e
Details of Proper System (Counting # of Equations)- 1 [1] [1]
v1 x 1
[2] [2]
v1 x 1
[3] [3]
v1 x 1
[4] [4]
v1 x 1
Details of Proper System (Counting # of Equations)- 1 [1] [1]
v1 x 1
H[11] [2] [2] v1 x 1
H[12] H[13] H[14]
[3] [3]
v1 x 1
[4] [4]
v1 x 1
Details of Proper System (Counting # of Equations)- 1 [1] [1]
v1 x 1
H[11] [2] [2] v1 x 1
H[12] H[13] H[14]
[3] [3]
v1 x 1
[4] [4]
v1 x 1
NE = 3
Details of Proper System (Counting # of Equations)- 1 [1] [1]
v1 x 1
H[11] [2] [2] v1 x 1
NE = 3
H[12] H[13]
NE = 3
H[14]
NE = 3 [3] [3] v1 x 1
NE = 3 [4] [4]
v1 x 1
Details of Proper System (Counting # of Equations)- 1 [1] [1]
v1 x 1
H[11] [2] [2] v1 x 1
NE = 3
H[12] H[13]
NE = 3
H[14]
NE = 3 [3] [3] v1 x 1
NE = 3 [4] [4]
v1 x 1
Ne =
X
d [k ]d [j], K , {1, 2, · · · , K }
k ,j∈K k 6=j
Details of Proper System (Counting # of Variables) - 2 [k ] [k ]
[k ]
V[k ] = [v1 v2 · · · vd [K ] ] T [k ] = span(V[k ] ) = {v : ∃a ∈ Cd
[k ] ×1
, v = V[k ]a}
T [k ] = {v : ∃a ∈ Cd
[k ] ×1
, v = V[k ]B−1 Ba}
= span(V[k ] B−1 )
Details of Proper System (Counting # of Variables) - 2 [k ] [k ]
[k ]
V[k ] = [v1 v2 · · · vd [K ] ] T [k ] = span(V[k ] ) = {v : ∃a ∈ Cd
[k ] ×1
, v = V[k ]a}
T [k ] = {v : ∃a ∈ Cd
[k ] ×1
, v = V[k ]B−1 Ba}
= span(V[k ] B−1 ) Choose Bd [k ] ×d [k ] by deleting the bottom M [k ] − d [k ] rows of ˜ [k ] = V[k ] B−1 : V[k ] . Then, V # " I [k ] d [k ] ˜ = V ] ] ] ¯ [k ¯ [k ¯ [k ¯ [k[k] ] v v v ··· v 1 2 3 d
Id [k ] : d [k ] × d [k ] identity matrix ] [k ] [k ] − d [k ] × 1 vectors. ¯ [k v n , ∀n ∈ {1, 2, ..., d }: M
Details of Proper System (Counting # of Variables) - 2 [k ] [k ]
[k ]
V[k ] = [v1 v2 · · · vd [K ] ] T [k ] = span(V[k ] ) = {v : ∃a ∈ Cd
[k ] ×1
, v = V[k ]a}
T [k ] = {v : ∃a ∈ Cd
[k ] ×1
, v = V[k ]B−1 Ba}
= span(V[k ] B−1 ) Choose Bd [k ] ×d [k ] by deleting the bottom M [k ] − d [k ] rows of ˜ [k ] = V[k ] B−1 : V[k ] . Then, V # " I [k ] d [k ] ˜ = V ] ] ] ¯ [k ¯ [k ¯ [k ¯ [k[k] ] v v v ··· v 1 2 3 d
Id [k ] : d [k ] × d [k ] identity matrix ] [k ] [k ] − d [k ] × 1 vectors. ¯ [k v n , ∀n ∈ {1, 2, ..., d }: M
Details of Proper System (Counting # of Variables) - 3 Example [k ] [k ]
V[k ] = [v1 v2 ]: M [k ] = 4 d [k ] = 2
a e b f V= c g d h
Details of Proper System (Counting # of Variables) - 3 Example [k ] [k ]
V[k ] = [v1 v2 ]: M [k ] = 4 d [k ] = 2
a e b f V= c g d h
Linear operations
˜ [k ] V
1 0 0 1 = i k j l
[k ]
[k ]
e.g., v1 ← (1/a)v1
i, j, k, l : f (a, b, · · · , h)
Details of Proper System (Counting # of Variables) - 3 Example [k ] [k ]
V[k ] = [v1 v2 ]: M [k ] = 4 d [k ] = 2
a e 4 variables b f V= c g d h
Linear operations
˜ [k ] V [k ]
˜1 + bv ˜2 = av
[k ]
] ˜ [k ˜ [k ] = ev 1 + f v2
v1 v2
[k ]
[k ]
[k ]
[k ]
e.g., v1 ← (1/a)v1
2 variables 1 0 0 1 i, j, k, l : f (a, b, · · · , h) = i k j l
[k ]
[k ]
˜ n are also basis If vn are basis vectors, then v vectors of the same subspace
Details of Proper System (Counting # of Variables) - 3 a e b f V= c g d h
Example [k ] [k ]
V[k ] = [v1 v2 ]: M [k ] = 4 d [k ] = 2
Linear operations
˜ [k ] V
Nv =
1 0 0 1 = i k j l
K X
k =1
[k ]
[k ]
e.g., v1 ← (1/a)v1
i, j, k, l : f (a, b, · · · , h)
d [k ] M [k ] + N [k ] − 2d [k ]
Symmetric Systems (Simplification) - 1
Theorem
A symmetric system is proper if and only if Nv ≥ Ne ⇒ M + N − (K + 1)d ≥ 0 Nv : Total # of variables Ne : Total # of equations
Symmetric Systems (Simplification) - 1
Theorem
A symmetric system is proper if and only if Nv ≥ Ne ⇒ M + N − (K + 1)d ≥ 0 Nv : Total # of variables Ne : Total # of equations
Example
(2 × 3, 1)4 : M + N − (K + 1)d = 2 + 3 − (5) = 0 → Proper (1 × 2, 1)3 : M + N − (K + 1)d = 1 + 2 − (4) < 0 → Improper
Symmetric Systems (Upper bound) - 2 Corollary The DoF of a proper symmetric system, which is normalized by a single user’s DoF in the absence of interference, is upper bounded by: dK max(M, N) d ≤1+ − min(M, N) min(M, N) min(M, N)
Symmetric Systems (Upper bound) - 2 Corollary The DoF of a proper symmetric system, which is normalized by a single user’s DoF in the absence of interference, is upper bounded by: dK max(M, N) d ≤1+ − min(M, N) min(M, N) min(M, N) For M = N:
dK d ≤2− M M
Symmetric Systems (Upper bound) - 2 Corollary The DoF of a proper symmetric system, which is normalized by a single user’s DoF in the absence of interference, is upper bounded by: dK max(M, N) d ≤1+ − min(M, N) min(M, N) min(M, N) For M = N:
dK d ≤2− M M
General MIMO channel (No structure) Total DoF Single user DoF = 2
−→
Diagonal MIMO channel (Time-varying) Total DoF Single user DoF = K /2
Symmetric Systems (Group) - 3 Corollary If (M × N, d)K system is proper (improper) then so is the K (M + 1) × (N − 1), d system as long as d ≤ min(M, N − 1). Similarly, if the (M × N, d)K system is proper (improper) then so K is the (M − 1) × (N + 1), d system as long as d ≤ min(M − 1, N) M
N
M d ≤ min(M, N) M
M
N
.. .
.. .
.. .
.. .
N
N
Figure: K -user MIMO interference network
Symmetric Systems (Group) - 3 Corollary If (M × N, d)K system is proper (improper) then so is the K (M + 1) × (N − 1), d system as long as d ≤ min(M, N − 1). Similarly, if the (M × N, d)K system is proper (improper) then so K is the (M − 1) × (N + 1), d system as long as d ≤ min(M − 1, N) M −1
N+1
M −1 d ≤ min(M − 1, N) M −1
M −1
N+1
.. .
.. .
.. .
.. .
N+1
N+1
Figure: K -user MIMO interference network
Symmetric Systems (Group) - 3 Corollary If (M × N, d)K system is proper (improper) then so is the K (M + 1) × (N − 1), d system as long as d ≤ min(M, N − 1). Similarly, if the (M × N, d)K system is proper (improper) then so K is the (M − 1) × (N + 1), d system as long as d ≤ min(M − 1, N) M+1
N−1
M+1 d ≤ min(M, N − 1) M+1
M+1
N−1
.. .
.. .
.. .
.. .
N−1
N−1
Figure: K -user MIMO interference network
Symmetric Systems (Group) - 4
Example (4 × 1, 1)4 : Zero-forcing suffices
Symmetric Systems (Group) - 4
Example (4 × 1, 1)4 : Zero-forcing suffices (3 × 2, 1)4
Symmetric Systems (Group) - 4
Example (4 × 1, 1)4 : Zero-forcing suffices (3 × 2, 1)4 , (2 × 3, 1)4 and (1 × 4, 1)4 are in the same group
Symmetric Systems (Group) - 4
Example (4 × 1, 1)4 : Zero-forcing suffices (3 × 2, 1)4 , (2 × 3, 1)4 and (1 × 4, 1)4 are in the same group
Example (1 × 3, 1)3 , (2 × 2, 1)3 , and (3 × 1, 1)3 are in the same group
Asymmetric Systems (Simplification) - 1
Theorem
An asymmetric system is improper if Nv < N e ⇔
K X
k =1
K X d [k ] M [k ] + N [k ] − 2d [k ] < d [k ]d [j] k ,j∈K k 6=j
Asymmetric Systems (Simplification) - 1
Theorem
An asymmetric system is improper if Nv < N e ⇔
K X
k =1
K X d [k ] M [k ] + N [k ] − 2d [k ] < d [k ]d [j] k ,j∈K k 6=j
Example (2 × 3, 1)4 : Rigorous proof by using Bernshtein’s theorem
Asymmetric Systems (Simplification) - 1
Theorem
An asymmetric system is improper if Nv < N e ⇔
K X
k =1
K X d [k ] M [k ] + N [k ] − 2d [k ] < d [k ]d [j] k ,j∈K k 6=j
Example (2 × 3, 1)4 : Rigorous proof by using Bernshtein’s theorem (2 × 2, 1)(2 × 3, 1)3 : Nv = 11 < Ne = 12 → Improper
Asymmetric Systems (Bottleneck) - 2
Bottleneck equations: Equations with the fewest # of variables (i.e., the equations involving the fewest number of transmitter and receiver antennas)
Asymmetric Systems (Bottleneck) - 2
Bottleneck equations: Equations with the fewest # of variables (i.e., the equations involving the fewest number of transmitter and receiver antennas)
Example (2 × 1, 1)2 : Zero-forcing suffices (2 × 1, 1)(1 × 2, 1): M [1] = 2 d [1] = 1
N [1] = 1
M [2] = 1 d [2] = 1
N [2] = 2
Asymmetric Systems (Bottleneck) - 2
Bottleneck equations: Equations with the fewest # of variables (i.e., the equations involving the fewest number of transmitter and receiver antennas)
Example (2 × 1, 1)2 : Zero-forcing suffices (2 × 1, 1)(1 × 2, 1): M [1] = 2 d [1] = 1
N [1] = 1
M [2] = 1 [2]
N [2] = 2
d
# of variables: M [2] − d [2] = 1 − 1 = 0
=1
# of variables: N [1] − d [1] = 1 − 1 = 0
0 variable and 1 equation
Multi-Beam Cases [1] [1]
v1 x 1
[2] [2]
v1 x 1
[3] [3]
User with multi-beam
v2 x 2
[3] [3]
v1 x 1
[4] [4]
v1 x 1
Multi-Beam Cases [1] [1]
v1 x 1
[1] [1]
H[11] v1 x1 H
[11] [4] [4]
[2] [2] v1 x 1
H[14] v1 x1
H[12]
[2] [2]
H[12] v1 x1
H[13]
User with multi-beam
[3] [3] v2 x 2
H
[3] [3]
H[13] v1 x1
[14]
[3] [3]
H[13] v2 x2
[3] [3]
v1 x 1
[4] [4]
v1 x 1
[1]
[4]
[1]
[3]
[1]
[3]
[1]
[2]
u1 H[14]v1 = 0 u1 H[13]v1 = 0 u1 H[13]v2 = 0 u1 H[12]v1 = 0
Summary (Backward) - 1 I
Feasibility of interference alignment → Solvability of multivariate polynomial system
Summary (Backward) - 1 I
I
Feasibility of interference alignment → Solvability of multivariate polynomial system Proper system definition (inspired from Bezout’s theorem) I
Counting the # of variables (eliminating superfluous variables)
Summary (Backward) - 1 I
I
Feasibility of interference alignment → Solvability of multivariate polynomial system Proper system definition (inspired from Bezout’s theorem) I
I
Counting the # of variables (eliminating superfluous variables)
Simplifications of proper system condition for symmetric and asymmetric systems I I
Symmetric system: Nv ≥ Ne → Proper Asymmetric system: I I
Nv < Ne → Improper Bottleneck equations
Summary (Backward) - 1 I
I
Feasibility of interference alignment → Solvability of multivariate polynomial system Proper system definition (inspired from Bezout’s theorem) I
I
Counting the # of variables (eliminating superfluous variables)
Simplifications of proper system condition for symmetric and asymmetric systems I I
Symmetric system: Nv ≥ Ne → Proper Asymmetric system: I I
I
Nv < Ne → Improper Bottleneck equations
Upper bound for symmetric systems
Summary (Backward) - 1 I
I
Feasibility of interference alignment → Solvability of multivariate polynomial system Proper system definition (inspired from Bezout’s theorem) I
I
Counting the # of variables (eliminating superfluous variables)
Simplifications of proper system condition for symmetric and asymmetric systems I I
Symmetric system: Nv ≥ Ne → Proper Asymmetric system: I I
Nv < Ne → Improper Bottleneck equations
I
Upper bound for symmetric systems
I
Grouping symmetric systems
Summary (Forward) - 2 I
For single beam cases: Proper systems are almost surely feasible and improper systems are not
Summary (Forward) - 2 I
For single beam cases: Proper systems are almost surely feasible and improper systems are not I
Rigorous proofs by using Bernshtein’s theorem I I
(2 × 3, 1)4 , (2 × 3, 1)2 (3 × 2, 1)2 , and (2 × 2, 1)3 (3 × 5, 1) All cases where Ne > Nv
Summary (Forward) - 2 I
For single beam cases: Proper systems are almost surely feasible and improper systems are not I
Rigorous proofs by using Bernshtein’s theorem I I
I
(2 × 3, 1)4 , (2 × 3, 1)2 (3 × 2, 1)2 , and (2 × 2, 1)3 (3 × 5, 1) All cases where Ne > Nv
New closed form solutions: (2 × 3, 1)2 (3 × 2, 1)2 and (2 × 3, 1)4
Summary (Forward) - 2 I
For single beam cases: Proper systems are almost surely feasible and improper systems are not I
Rigorous proofs by using Bernshtein’s theorem I I
I
I
(2 × 3, 1)4 , (2 × 3, 1)2 (3 × 2, 1)2 , and (2 × 2, 1)3 (3 × 5, 1) All cases where Ne > Nv
New closed form solutions: (2 × 3, 1)2 (3 × 2, 1)2 and (2 × 3, 1)4 Various numerical results
Summary (Forward) - 2 I
For single beam cases: Proper systems are almost surely feasible and improper systems are not I
Rigorous proofs by using Bernshtein’s theorem I I
I
I
I
(2 × 3, 1)4 , (2 × 3, 1)2 (3 × 2, 1)2 , and (2 × 2, 1)3 (3 × 5, 1) All cases where Ne > Nv
New closed form solutions: (2 × 3, 1)2 (3 × 2, 1)2 and (2 × 3, 1)4 Various numerical results
For multi-beam cases:
Summary (Forward) - 2 I
For single beam cases: Proper systems are almost surely feasible and improper systems are not I
Rigorous proofs by using Bernshtein’s theorem I I
I
I
I
(2 × 3, 1)4 , (2 × 3, 1)2 (3 × 2, 1)2 , and (2 × 2, 1)3 (3 × 5, 1) All cases where Ne > Nv
New closed form solutions: (2 × 3, 1)2 (3 × 2, 1)2 and (2 × 3, 1)4 Various numerical results
For multi-beam cases: I
Include standard information theoretic outer bounds → Strengthen the connection between proper and feasible systems
Summary (Forward) - 2 I
For single beam cases: Proper systems are almost surely feasible and improper systems are not I
Rigorous proofs by using Bernshtein’s theorem I I
I
I
I
(2 × 3, 1)4 , (2 × 3, 1)2 (3 × 2, 1)2 , and (2 × 2, 1)3 (3 × 5, 1) All cases where Ne > Nv
New closed form solutions: (2 × 3, 1)2 (3 × 2, 1)2 and (2 × 3, 1)4 Various numerical results
For multi-beam cases: I
I
Include standard information theoretic outer bounds → Strengthen the connection between proper and feasible systems If the system is improper, then it is infeasible I
Various numerical results
Summary (Forward) - 2 I
For single beam cases: Proper systems are almost surely feasible and improper systems are not I
Rigorous proofs by using Bernshtein’s theorem I I
I
I
I
Proper → Feasible Improper → Infeasible
(2 × 3, 1)4 , (2 × 3, 1)2 (3 × 2, 1)2 , and (2 × 2, 1)3 (3 × 5, 1) All cases where Ne > Nv
New closed form solutions: (2 × 3, 1)2 (3 × 2, 1)2 and (2 × 3, 1)4 Various numerical results
For multi-beam cases: I
I
Include standard information theoretic outer bounds → Strengthen the connection between proper and feasible systems If the system is improper, then it is infeasible I
Various numerical results
Summary (Forward) - 2 I
For single beam cases: Proper systems are almost surely feasible and improper systems are not I
Rigorous proofs by using Bernshtein’s theorem I I
I
I
I
Proper → Feasible Improper → Infeasible
(2 × 3, 1)4 , (2 × 3, 1)2 (3 × 2, 1)2 , and (2 × 2, 1)3 (3 × 5, 1) All cases where Ne > Nv
New closed form solutions: (2 × 3, 1)2 (3 × 2, 1)2 and (2 × 3, 1)4 Various numerical results
For multi-beam cases: I
I
Include standard information theoretic outer bounds → Strengthen the connection between proper and feasible systems If the system is improper, then it is infeasible I
Various numerical results
Improper → Infeasible Proper + Inf. Th. → Feasible
Bezout’s and Bernshtein’s Theorems (Overview) - 1 I
Both provide # of solutions
Bezout’s and Bernshtein’s Theorems (Overview) - 1 I
Both provide # of solutions → Prove solvability indirectly
Bezout’s and Bernshtein’s Theorems (Overview) - 1 I I
Both provide # of solutions → Prove solvability indirectly Generic system
Bezout’s and Bernshtein’s Theorems (Overview) - 1 I I
Both provide # of solutions → Prove solvability indirectly Generic system I
Independent random coefficients
Bezout’s and Bernshtein’s Theorems (Overview) - 1 I I
Both provide # of solutions → Prove solvability indirectly Generic system I I
Independent random coefficients Bezout’s theorem: Dense polynomials
Bezout’s and Bernshtein’s Theorems (Overview) - 1 I I
Both provide # of solutions → Prove solvability indirectly Generic system I I I
Independent random coefficients Bezout’s theorem: Dense polynomials Bernshtein’s theorem: Sparse polynomials
Bezout’s and Bernshtein’s Theorems (Overview) - 1 I I
Both provide # of solutions → Prove solvability indirectly Generic system I I I
Independent random coefficients Bezout’s theorem: Dense polynomials Bernshtein’s theorem: Sparse polynomials
Dense polynomial: f1 = c11 x13 + c12 x23 + c13 x12 x2 + c14 x1 x22 + c15 x12 + c16 x22 + c17 x1 x2 + c18 x1 + c19 x2 + c110 deg(f1 ) = 3
Bezout’s and Bernshtein’s Theorems (Overview) - 1 I I
Both provide # of solutions → Prove solvability indirectly Generic system I I I
Independent random coefficients Bezout’s theorem: Dense polynomials Bernshtein’s theorem: Sparse polynomials
Dense polynomial: f1 = c11 x13 + c12 x23 + c13 x12 x2 + c14 x1 x22 + c15 x12 + c16 x22 + c17 x1 x2 + c18 x1 + c19 x2 + c110 deg(f1 ) = 3 Sparse polynomial: f1 = c11 x1 x22 + c12 x12 + c13 x22 + c13 deg(f1 ) = 3
Bezout’s and Bernshtein’s Theorems (Overview) - 1 I I
Both provide # of solutions → Prove solvability indirectly Generic system I I I
I
Independent random coefficients Bezout’s theorem: Dense polynomials Bernshtein’s theorem: Sparse polynomials
Bernshtein generalizes Bezout’s theorem
Dense polynomial: f1 = c11 x13 + c12 x23 + c13 x12 x2 + c14 x1 x22 + c15 x12 + c16 x22 + c17 x1 x2 + c18 x1 + c19 x2 + c110 deg(f1 ) = 3 Sparse polynomial: f1 = c11 x1 x22 + c12 x12 + c13 x22 + c13 deg(f1 ) = 3
Bezout’s and Bernshtein’s Theorems (# of Solutions) 2 Theorem (Bezout’s Theorem - specialized) # of solutions for generic systems = deg(f 1 )deg(f2 ) · · · deg(fn )
Bezout’s and Bernshtein’s Theorems (# of Solutions) 2 Theorem (Bezout’s Theorem - specialized) # of solutions for generic systems = deg(f 1 )deg(f2 ) · · · deg(fn )
Theorem (Bernshtein’s Theorem - specialized) # of solutions for generic systems=Mixed volume of Newton polytopes, MV(P1 , · · · , Pn ), Pi : Newton polytope of function fi
Bezout’s and Bernshtein’s Theorems (# of Solutions) 2 Theorem (Bezout’s Theorem - specialized) # of solutions for generic systems = deg(f 1 )deg(f2 ) · · · deg(fn )
Theorem (Bernshtein’s Theorem - specialized) # of solutions for generic systems=Mixed volume of Newton polytopes, MV(P1 , · · · , Pn ), Pi : Newton polytope of function fi
MV = 0 → Not solvable
MV 6= 0 → Solvable
Rigorous Connections
Example (2 × 3, 1)4 and (2 × 3, 1)2 (3 × 2, 1)2 : Feasible because mixed volumes are 9 and 8, respectively → Proper
Rigorous Connections
Example (2 × 3, 1)4 and (2 × 3, 1)2 (3 × 2, 1)2 : Feasible because mixed volumes are 9 and 8, respectively → Proper
Example (2 × 2, 1)3 (3 × 5, 1): Infeasible because mixed volume is 0 → Improper
Rigorous Connections
Example (2 × 3, 1)4 and (2 × 3, 1)2 (3 × 2, 1)2 : Feasible because mixed volumes are 9 and 8, respectively → Proper
Example (2 × 2, 1)3 (3 × 5, 1): Infeasible because mixed volume is 0 → Improper Mixed volume computation is #P-complete
New Closed Form Solutions
I
Asymmetric system (2 × 3, 1)2 (3 × 2, 1)2
I
Symmetric system (2 × 3, 1)4
Numerical Results (Interference percentage) - 1 Interference percentage (Interference leakage):
pk =
[k ] dP j=1
λj [Q[k ] ]
Tr[Q[k ] ]
λj : Smallest eigenvalue of a matrix Tr: Trace of a matrix Q[k ] : Interference covariance matrix at the k th receiver Q[k ] =
K X P [j] [kj] [j] [j]† [kj]† H V V H d [j]
j=1,j6=k
P [j] : The transmit power of the j th transmitter
Numerical Results (Interference percentage) - 1 Interference percentage (Interference leakage):
pk =
[k ] dP j=1
Interference leakage = 0 at every node λj [
Q[k ]
l
]
Feasible
Tr[Q[k ] ]
l Interference alignment achievable
λj : Smallest eigenvalue of a matrix Tr: Trace of a matrix Q[k ] : Interference covariance matrix at the k th receiver Q[k ] =
K X P [j] [kj] [j] [j]† [kj]† H V V H d [j]
j=1,j6=k
P [j] : The transmit power of the j th transmitter
Numerical Results (Simulation) - 2 60 2
2
(2x3,1) (3x2,1) , DoF=4 4
(2x3,1) , DoF=4 (6x4,2)4, DoF=8, Equivalent system of (5x5,2)4 50
4
(5x5,2) , DoF=8 4
4
(4x6,2) , DoF=8, Equivalent system of (5x5,2)
Interference percentage
40
30
20
10
0 DoF
DoF+1
DoF+2 Total number of beams in the 4−user MIMO interference networks
DoF+3
DoF+4
General Outer Bounds (Multi-beam cases) - 1 I
Point-to-point MIMO channel: DoF=min(M, N)
I
2-user MIMO interference channel (M [1] , M [2] and N [1] , N [2] ): DoF=min M [1] + M [2] , N [1] + N [2] , max(M [1] , N [2] ), max(M [2] , N [1] )
General Outer Bounds (Multi-beam cases) - 1 I
Point-to-point MIMO channel: DoF=min(M, N)
I
2-user MIMO interference channel (M [1] , M [2] and N [1] , N [2] ): DoF=min M [1] + M [2] , N [1] + N [2] , max(M [1] , N [2] ), max(M [2] , N [1] )
I
General DoF outer bounds for a K -user MIMO interference network: d [i] ≤ min(M [i] , N [i])
(1) d [i]+d [j] ≤ min M [i] +M [j], N [i] +N [j], max(M [i] , N [j] ), max(M [j] , N [i] ) (2) for all i, j ∈ K
General Outer Bounds (Multi-beam cases) - 1 I
Point-to-point MIMO channel: DoF=min(M, N)
I
2-user MIMO interference channel (M [1] , M [2] and N [1] , N [2] ): DoF=min M [1] + M [2] , N [1] + N [2] , max(M [1] , N [2] ), max(M [2] , N [1] )
I
General DoF outer bounds for a K -user MIMO interference network: d [i] ≤ min(M [i] , N [i])
(1) d [i]+d [j] ≤ min M [i] +M [j], N [i] +N [j], max(M [i] , N [j] ), max(M [j] , N [i] ) (2) for all i, j ∈ K
Example (3 × 3, 2)2 : Proper condition → OK General outer bound (2) → NOK
General Outer Bounds (Cooperative) - 2
Example (3 × 4, 2)(1 × 3, 1)(10 × 4, 2): Proper condition+General outer bounds → OK
General Outer Bounds (Cooperative) - 2
Example (3 × 4, 2)(1 × 3, 1)(10 × 4, 2): Proper condition+General outer bounds → OK Cooperative (1st and 2nd users), (4 × 7, 3)(10 × 4, 2): General outer bound (2) → NOK
General Outer Bounds (Cooperative) - 2
Example (3 × 4, 2)(1 × 3, 1)(10 × 4, 2): → Infeasible Proper condition+General outer bounds → OK Cooperative (1st and 2nd users), (4 × 7, 3)(10 × 4, 2): General outer bound (2) → NOK
General Outer Bounds (Cooperative) - 3
General and cooperative outerbounds → OK Feasible or infeasible? → Use proper system condition
General Outer Bounds (Cooperative) - 3
General and cooperative outerbounds → OK Feasible or infeasible? → Use proper system condition
Example
(5 × 5, 2)4 : Ne = 48 → Test (248 − 1) subsets Use simplification: Nv ≥ Ne → Proper Nv ≥ Ne ⇒ M + N − (K + 1)d ≥ 0 M + N − (K + 1)d = 5 + 5 − 10 = 0 → Proper Use grouping: (2 × 8, 2)4 , (3 × 7, 2)4 , (4 × 6, 2)4 , (5 × 5, 2)4 , (6 × 4, 2)4 , (7 × 3, 2)4 , and (8 × 2, 2)4 are in the same group
General Outer Bounds (Cooperative) - 3
General and cooperative outerbounds → OK Feasible or infeasible? → Use proper system condition
Example
(5 × 5, 2)4 : Ne = 48 → Test (248 − 1) subsets Use simplification: Nv ≥ Ne → Proper Nv ≥ Ne ⇒ M + N − (K + 1)d ≥ 0 M + N − (K + 1)d = 5 + 5 − 10 = 0 → Proper Use grouping: (2 × 8, 2)4 , (3 × 7, 2)4 , (4 × 6, 2)4 , (5 × 5, 2)4 , (6 × 4, 2)4 , (7 × 3, 2)4 , and (8 × 2, 2)4 are in the same group
Example (5 × 5, 3)(5 × 5, 2)3 : Ne = 60 → Test (260 − 1) subsets Use simplification: Nv < Ne → Improper Nv = 48 < Ne = 60 → Improper
Conclusion I
Feasibility of interference alignment → Solvability of multivariate polynomial system
Conclusion I
I
Feasibility of interference alignment → Solvability of multivariate polynomial system Proper system definition I
Counting the # of variables & equations
Conclusion I
I
Feasibility of interference alignment → Solvability of multivariate polynomial system Proper system definition I
I
Counting the # of variables & equations
For single beam cases: Proper systems are almost surely feasible and improper systems are not I I I
Rigorous proofs New closed form solutions Various numerical results
Conclusion I
I
Feasibility of interference alignment → Solvability of multivariate polynomial system Proper system definition I
I
For single beam cases: Proper systems are almost surely feasible and improper systems are not I I I
I
Counting the # of variables & equations
Rigorous proofs New closed form solutions Various numerical results
For multi-beam cases: I
I
Include standard information theoretic outer bounds → Strengthen the connection between feasible and proper systems If the system is improper, it is infeasible I
Various numerical results
Bezout’s and Bernshtein’s Theorems (Mixed Volume) 3 For each polynomial equation fi : Newton polytope = convex hull of a support set, P i = Conv(Ai )
Example fi = ci1 x1 + ci2 x1 x2 + ci3 ai1 = (1, 0), ai2 = (1, 1), and ai3 = (0, 0) Ai = {ai1 , ai2 , ai3 } ai2 = (1, 1)
P1 ai3 = (0, 0)
ai1 = (1, 0)
MV(P1 , P2 ) = −Vol(P1 ) − Vol(P2 ) + Vol(PS ) PS = P1 + P2 (Minkowski sum)
Bezout’s and Bernshtein’s Theorems (Mixed Volume) 3 For each polynomial equation fi : Newton polytope = convex hull of a support set, P i = Conv(Ai )
Example fi = ci1 x1 + ci2 x1 x2 + ci3 ai1 = (1, 0), ai2 = (1, 1), and ai3 = (0, 0) Ai = {ai1 , ai2 , ai3 } ai2 = (1, 1)
P1 ai3 = (0, 0)
ai1 = (1, 0)
MV(P1 , P2 ) = −Vol(P1 ) − Vol(P2 ) + Vol(PS ) PS = P1 + P2 (Minkowski sum)
Bezout’s and Bernshtein’s Theorems (Mixed Volume) 4 Example f1 = c11 x1 x22 + c12 x12 + c13 x22 + c13 f2 = c21 x13 x2 + c22 x24 + c23 x1 x2 The support sets of f1 and f2 are A1 = {(1, 2), (2, 0), (0, 2), (0, 0)} and A2 = {(3, 1), (0, 4), (1, 1)}, respectively
Bezout’s and Bernshtein’s Theorems (Mixed Volume) 4 Example f1 = c11 x1 x22 + c12 x12 + c13 x22 + c13 f2 = c21 x13 x2 + c22 x24 + c23 x1 x2 The support sets of f1 and f2 are A1 = {(1, 2), (2, 0), (0, 2), (0, 0)} and A2 = {(3, 1), (0, 4), (1, 1)}, respectively Minkowski sum: AS = {(4, 3), (1, 6), (2, 3), (5, 1), (2, 4), (3, 1), (3, 3), (0, 6), (1, 3), (3, 1), (0, 4), (1, 1)}
Bezout’s and Bernshtein’s Theorems (Mixed Volume) 4 Example f1 = c11 x1 x22 + c12 x12 + c13 x22 + c13 f2 = c21 x13 x2 + c22 x24 + c23 x1 x2 The support sets of f1 and f2 are A1 = {(1, 2), (2, 0), (0, 2), (0, 0)} and A2 = {(3, 1), (0, 4), (1, 1)}, respectively Minkowski sum: AS = {(4, 3), (1, 6), (2, 3), (5, 1), (2, 4), (3, 1), (3, 3), (0, 6), (1, 3), (3, 1), (0, 4), (1, 1)}
6 5 4 3 2 1 0
PS = P1 + P2
P1
P2
0 1 2 3 4 5
