Communcation over interference channels Dustin Cartwright 1 February - PowerPoint PPT Presentation

Communcation over interference channels Dustin Cartwright 1 February 24, 2011 1 work in progress with Guy Bresler and David Tse

Multiple-input multiple-output channel transmitter receiver ◮ The transmitter sends a signal v ∈ C N by transmitting across N = 3 antennas.

Multiple-input multiple-output channel transmitter receiver ◮ The transmitter sends a signal v ∈ C N by transmitting across N = 3 antennas. ◮ The receiver detects Hv ∈ C N across its N antennas, where each entry of the H ∈ C N × N depends on the signal path.

Multiple-input multiple-output channel transmitter receiver ◮ The transmitter sends a signal v ∈ C N by transmitting across N = 3 antennas. ◮ The receiver detects Hv ∈ C N across its N antennas, where each entry of the H ∈ C N × N depends on the signal path. ◮ If H is known and invertible, then the receiver can reconstruct the message v .

Multiple users of the same channel receiver 1 transmitter 1 H 11 ◮ K transmitter-receiver pairs using the same channel. H 12 ◮ Determined by K 2 channel matrices H ij of size N × N . receiver 2 transmitter 2 H 21 H 22

Multiple users of the same channel receiver 1 transmitter 1 H 11 ◮ K transmitter-receiver pairs using the same channel. H 12 ◮ Determined by K 2 channel matrices H ij of size N × N . receiver 2 transmitter 2 H 21 ◮ Reciever 1 only cares about transmitter 1’s message, etc. H 22

Strategies for interference alignment ◮ Each transmitter has a subspace V j ⊂ C N to transmit in. ◮ Each receiver has a subspace U i ⊂ C N and only pays attention to its signal modulo U i .

Strategies for interference alignment ◮ Each transmitter has a subspace V j ⊂ C N to transmit in. ◮ Each receiver has a subspace U i ⊂ C N and only pays attention to its signal modulo U i . In order for this to work, we need: ◮ For i � = j , H ij V j ⊂ U j . ◮ ( H ii V i ) ∩ U i = ∅ .

Strategies for interference alignment ◮ Each transmitter has a subspace V j ⊂ C N to transmit in. ◮ Each receiver has a subspace U i ⊂ C N and only pays attention to its signal modulo U i . In order for this to work, we need: ◮ For i � = j , H ij V j ⊂ U j . ◮ ( H ii V i ) ∩ U i = ∅ . If each H ii is generic, the second condition is satisfied automatically.

Questions ◮ For which N , K , and ( d 1 , . . . , d K ) will generic channel matrices have a solution?

Questions ◮ For which N , K , and ( d 1 , . . . , d K ) will generic channel matrices have a solution? ◮ What is the information capacity of this channel?

Questions ◮ For which N , K , and ( d 1 , . . . , d K ) will generic channel matrices have a solution? ◮ What is the information capacity of this channel? ◮ How to parametrize spaces of solution strategies?

Incidence correspondence K C N × N � K ( K − 1) � � × Gr( d i , N ) × Gr( N − d i , N ) i =1 Subvariety of those ( H 12 , . . . , H K − 1 , K , V 1 , . . . , V K , U 1 , . . . , U K ) such that H ij V j ⊂ U i for 1 ≤ i � = j ≤ K

Incidence correspondence K C N × N � K ( K − 1) � � × Gr( d i , N ) × Gr( N − d i , N ) i =1 Subvariety of those ( H 12 , . . . , H K − 1 , K , V 1 , . . . , V K , U 1 , . . . , U K ) such that H ij V j ⊂ U i for 1 ≤ i � = j ≤ K This is a vector bundle over a product of Grassmannians.

Incidence correspondence K C N × N � K ( K − 1) � � × Gr( d i , N ) × Gr( N − d i , N ) i =1 Subvariety of those ( H 12 , . . . , H K − 1 , K , V 1 , . . . , V K , U 1 , . . . , U K ) such that H ij V j ⊂ U i for 1 ≤ i � = j ≤ K This is a vector bundle over a product of Grassmannians. Question C N × N � K ( K − 1) surjective? � Is the projection onto

Existence of solutions Theorem Assume that d = d 1 = · · · = d K and K ≥ 3 . Then a generic set of channel matrices has a solution if and only if 2 N ≥ d ( K + 1) . If so, the dimension of the solution variety is � � dK 2 N − d ( K + 1)

Existence of solutions Theorem Assume that d = d 1 = · · · = d K and K ≥ 3 . Then a generic set of channel matrices has a solution if and only if 2 N ≥ d ( K + 1) . If so, the dimension of the solution variety is � � dK 2 N − d ( K + 1) For non-constant d i , we have the necessary conditions: d i + d j ≤ N for all i , j � � 2 d i ( N − d i ) ≥ for all subsets S ⊂ { 1 , . . . , K } d i d j i ∈ S i � = j ∈ S

K = 3 The threshold case for feasibility is ( d 1 , d 2 , d 3 ) = ( d , d , N − d ) , where d 1 ≤ N / 2. =

K = 3 The threshold case for feasibility is ( d 1 , d 2 , d 3 ) = ( d , d , N − d ) , where d 1 ≤ N / 2. = ◮ After change of coordinates, can assume that all but one channel matrix is the identity.

K = 3 The threshold case for feasibility is ( d 1 , d 2 , d 3 ) = ( d , d , N − d ) , where d 1 ≤ N / 2. = ◮ After change of coordinates, can assume that all but one channel matrix is the identity. ◮ For dimension reasons, inclusions become equalities: V 1 = U 3 = V 2 ⊂ U 1 = V 3 = U 2 ⊃ H 21 V 1

An eigenvector-like problem Given generic N × N matrix H , find ◮ V ⊂ C N , subspace of dimension d ◮ U ⊂ C N , subspace of dimension e = N − d such that V ⊂ U and HV ⊂ U

An eigenvector-like problem Given generic N × N matrix H , find ◮ V ⊂ C N , subspace of dimension d ◮ U ⊂ C N , subspace of dimension e = N − d such that V ⊂ U and HV ⊂ U ◮ For d = e = 1, this is equivalent to V = U being the span of an eigenvector.

An eigenvector-like problem Given generic N × N matrix H , find ◮ V ⊂ C N , subspace of dimension d ◮ U ⊂ C N , subspace of dimension e = N − d such that V ⊂ U and HV ⊂ U ◮ For d = e = 1, this is equivalent to V = U being the span of an eigenvector. ◮ More generally, for d = e > 1, take V = U to be spanned by � N � d eigenvectors. In particular, solutions. d

An eigenvector-like problem Given generic N × N matrix H , find ◮ V ⊂ C N , subspace of dimension d ◮ U ⊂ C N , subspace of dimension e = N − d such that V ⊂ U and HV ⊂ U ◮ For d = e = 1, this is equivalent to V = U being the span of an eigenvector. ◮ More generally, for d = e > 1, take V = U to be spanned by � N � d eigenvectors. In particular, solutions. d ◮ For e > d , variety of solutions of dimension � � N − ( e − d ) ( e − d )

Parametrizing the solution variety Recall: Want to find V , U such that V ⊂ U and HV ⊂ U . � � d ◮ Set ℓ := e − d ◮ Choose S ⊂ C N of dimension d − ℓ ( e − d ).

Parametrizing the solution variety Recall: Want to find V , U such that V ⊂ U and HV ⊂ U . � � d ◮ Set ℓ := e − d ◮ Choose S ⊂ C N of dimension d − ℓ ( e − d ). ◮ Choose S + HS ⊂ T ⊂ C N of dimension e − ℓ ( e − d ).

Parametrizing the solution variety Recall: Want to find V , U such that V ⊂ U and HV ⊂ U . � � d ◮ Set ℓ := e − d ◮ Choose S ⊂ C N of dimension d − ℓ ( e − d ). ◮ Choose S + HS ⊂ T ⊂ C N of dimension e − ℓ ( e − d ). ◮ Set U = S + T + . . . H ℓ − 1 T V = T + . . . + H ℓ T

Parametrizing the solution variety Recall: Want to find V , U such that V ⊂ U and HV ⊂ U . � � d ◮ Set ℓ := e − d ◮ Choose S ⊂ C N of dimension d − ℓ ( e − d ). ◮ Choose S + HS ⊂ T ⊂ C N of dimension e − ℓ ( e − d ). ◮ Set U = S + T + . . . H ℓ − 1 T V = T + . . . + H ℓ T Structure of whole variety seems complicated: when e = d + 1, then it is the toric variety for the Minkowski sum of hypersimplices ∆ e , N + ∆ N , d .

Numbers of solutions Return to K ≥ 3 d = d 1 = . . . = d k Zero-dimensional when N = d ( K +1) . The number of solutions is: 2 K d 3 4 5 6 7 1 2 - 216 - 1,975,560 2 6 3700 388,407,960 3 20 - - 4 70 . . . . . . � 2 d � d d

Number of solutions when d = 1 Assume d = d 1 = · · · d K = 1 and 2 N = K + 1. Degenerate each H ij to a rank 1 matrix: H ij V j ⊂ U j ⇐ ⇒ V j ⊂ ker H ij or U j ⊃ im H ij

Number of solutions when d = 1 Assume d = d 1 = · · · d K = 1 and 2 N = K + 1. Degenerate each H ij to a rank 1 matrix: H ij V j ⊂ U j ⇐ ⇒ V j ⊂ ker H ij or U j ⊃ im H ij Theorem Number of solutions = number of balanced orientations of the graph G G has edges t j − s i whenever i � = j. Balanced orientation means that in degree ( v ) = out degree ( v ) = K − 1 2 for all vertices v of G.

Further questions ◮ If the d i are not necessarily all equal, when does a feasible strategy exist?

Further questions ◮ If the d i are not necessarily all equal, when does a feasible strategy exist? ◮ Can we parametrize the solution variety in more cases?

Further questions ◮ If the d i are not necessarily all equal, when does a feasible strategy exist? ◮ Can we parametrize the solution variety in more cases? ◮ What if the receivers and transmitters have different numbers of antennas?

Further questions ◮ If the d i are not necessarily all equal, when does a feasible strategy exist? ◮ Can we parametrize the solution variety in more cases? ◮ What if the receivers and transmitters have different numbers of antennas? ◮ What if the channel matrices have the form ˜  0 · · · 0  H ij ˜ 0 0 H ij   H ij =  ?  . .  ... . .   . .  ˜ · · · 0 0 H ij