Non-coherent Multi-Layer Constellaions for Unequal Error Protection - PowerPoint PPT Presentation

Non-coherent Multi-Layer Constellaions for Unequal Error Protection Speaker: Karim G. Seddik 1 in collaboration with Kareem M. Attiah 2 , Ramy H. Gohary 3 , and Halim Yanikomergolu 3 1 American University in Cairo 2 Alexandria University 3 Carlenton University ICC, May 2017 Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 1 / 21

Table of Contents Overview 1 Aim Set Partitioning How to Measure the Distance between Constellation Points? 2 Design Approach 3 Proposed Scheme Illustration Selection Criterion of Step Parameter t 4 Geometric Mean Approximation Polynomial Regression Examples Simplified Decoder 5 Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 2 / 21

Aim (1) Layered Coding; whereby layers exhibit varying sensitivity against channel errors. Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 3 / 21

Aim (1) Layered Coding; whereby layers exhibit varying sensitivity against channel errors. Transmitter chooses one of possible N supersymbols from the set S = { s 1 , . . . , s N } . Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 3 / 21

Aim (1) Layered Coding; whereby layers exhibit varying sensitivity against channel errors. Transmitter chooses one of possible N supersymbols from the set S = { s 1 , . . . , s N } . For simplicity, assume L = 2. Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 3 / 21

Aim (1) Layered Coding; whereby layers exhibit varying sensitivity against channel errors. Transmitter chooses one of possible N supersymbols from the set S = { s 1 , . . . , s N } . For simplicity, assume L = 2. Define the sets S 1 = { s 1 N 1 } and S 2 = { s 2 N 2 } , where the set S 1 ( S 2 ) 1 , . . . , s 1 1 , . . . , s 2 contains the more (less) protected subsymbols. Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 3 / 21

Aim (1) Layered Coding; whereby layers exhibit varying sensitivity against channel errors. Transmitter chooses one of possible N supersymbols from the set S = { s 1 , . . . , s N } . For simplicity, assume L = 2. Define the sets S 1 = { s 1 N 1 } and S 2 = { s 2 N 2 } , where the set S 1 ( S 2 ) 1 , . . . , s 1 1 , . . . , s 2 contains the more (less) protected subsymbols. � s 1 i , s 2 � The chosen supersymbol s i is mapped onto the pair by some bijective i function f : S → S 1 × S 2 . Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 3 / 21

Aim (2) Non-Coherent Signaling Tx Rx M N Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 4 / 21

Aim (2) Non-Coherent Signaling Tx Rx M N Channel state information (CSI) is unknown at both transmitter and receiver. Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 4 / 21

Aim (2) Non-Coherent Signaling Tx Rx M N Channel state information (CSI) is unknown at both transmitter and receiver. Transmission takes place over T symbol durations. Channel coefficients are assumed constant during that interval. We only consider T > M . Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 4 / 21

Aim (2) Non-Coherent Signaling Tx Rx M N Channel state information (CSI) is unknown at both transmitter and receiver. Transmission takes place over T symbol durations. Channel coefficients are assumed constant during that interval. We only consider T > M . Transmitted symbols are represented by points on the Grassmannian manifold G T , M ( C ). Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 4 / 21

Set Partitioning Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 5 / 21

Set Partitioning The constellation is decomposed into disjoint subsets. Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 5 / 21

Set Partitioning 1 S j 1 s i More important symbols encode subsets. Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 5 / 21

Set Partitioning 1 S j 2 S 4 1 2 2 s i S 1 S 3 2 S 2 Less important symbols encode points within a chosen subset. Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 5 / 21

Set Partitioning d 1 ( d 2 ) controls the error probability of the more (less) protected symbols. Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 5 / 21

Table of Contents Overview 1 Aim Set Partitioning How to Measure the Distance between Constellation Points? 2 Design Approach 3 Proposed Scheme Illustration Selection Criterion of Step Parameter t 4 Geometric Mean Approximation Polynomial Regression Examples Simplified Decoder 5 Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 6 / 21

How to Measure the Distance between Constellation Points? For unitary signaling, the asymptotic error probability of mistaking X i for X j under GLRT detection is given by 1 � 2 MN − 1 � P ij = ( d cp ( X i , X j ) γ T ) − MN M MN , MN � 1 � M M sin 2 θ k � d cp ( X i , X j ) = . k =1 { θ i } M i =1 are the principle angles between the subspaces associated with X i and X j . 1 M. Brehler and M. K. Varanasi, Asymptotic error probability analysis of quadratic receivers in rayleigh-fading channels with applications to a unified analysis of coherent and noncoherent space-time receivers, IEEE Transactions on Information Theory. 2 I. Kammoun, A. M. Cipriano, and J. C. Belfiore, Non-coherent codes over the grassmannian, IEEE Transactions on Wireless Communications. Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 7 / 21

How to Measure the Distance between Constellation Points? For unitary signaling, the asymptotic error probability of mistaking X i for X j under GLRT detection is given by 1 � 2 MN − 1 � P ij = ( d cp ( X i , X j ) γ T ) − MN M MN , MN � 1 � M M sin 2 θ k � d cp ( X i , X j ) = . k =1 { θ i } M i =1 are the principle angles between the subspaces associated with X i and X j . The Chordal Product Distance 2 (not a true distance!) can be employed to define the coding gain of the more (less) protected symbols. d 1 = min i � = j d cp (Ω i , Ω j ) , d 2 = min X , Y ∈C d cp ( X , Y ) . 1 M. Brehler and M. K. Varanasi, Asymptotic error probability analysis of quadratic receivers in rayleigh-fading channels with applications to a unified analysis of coherent and noncoherent space-time receivers, IEEE Transactions on Information Theory. 2 I. Kammoun, A. M. Cipriano, and J. C. Belfiore, Non-coherent codes over the grassmannian, IEEE Transactions on Wireless Communications. Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 7 / 21

How to Measure the Distance between Constellation Points? A related quantity is the Chordal Distance M sin 2 θ k . � d c ( X i , X j ) = k =1 Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 8 / 21

How to Measure the Distance between Constellation Points? A related quantity is the Chordal Distance M sin 2 θ k . � d c ( X i , X j ) = k =1 The AM-GM inequality governs the relationship between the two metrics d cp ( X i , X j ) ≤ 1 M d c ( X i , X j ) . Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 8 / 21

How to Measure the Distance between Constellation Points? A related quantity is the Chordal Distance M sin 2 θ k . � d c ( X i , X j ) = k =1 The AM-GM inequality governs the relationship between the two metrics d cp ( X i , X j ) ≤ 1 M d c ( X i , X j ) . Our design approach is entirely based on the more intuitive chordal distance. However, the connection between both metrics is essential since the coding gains are expressed in terms of the product chordal distance. Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 8 / 21

Table of Contents Overview 1 Aim Set Partitioning How to Measure the Distance between Constellation Points? 2 Design Approach 3 Proposed Scheme Illustration Selection Criterion of Step Parameter t 4 Geometric Mean Approximation Polynomial Regression Examples Simplified Decoder 5 Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 9 / 21

Design Approach (1) 3 For an initial layer ξ 1 , we define a sequence of L − 1 children layers ξ 2 , ξ 3 , . . . , ξ L . 3 K. M. Attiah, K. Seddik, R. H. Gohary, and H. Yanikomeroglu, A systematic design approach for non-coherent grassmannian constellations, ISIT, 2016. Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 10 / 21