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

Non-coherent Multi-Layer Constellaions for Unequal Error Protection

Speaker: Karim G. Seddik1 in collaboration with Kareem M. Attiah2, Ramy H. Gohary3, and Halim Yanikomergolu3

1American University in Cairo 2Alexandria University 3Carlenton University

ICC, May 2017

Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 1 / 21

Table of Contents

1

Overview Aim Set Partitioning

2

How to Measure the Distance between Constellation Points?

3

Design Approach Proposed Scheme Illustration

4

Selection Criterion of Step Parameter t Geometric Mean Approximation Polynomial Regression Examples

5

Simplified Decoder

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 = {s1, . . . , sN }.

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 = {s1, . . . , sN }. 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 = {s1, . . . , sN }. For simplicity, assume L = 2. Define the sets S1 = {s1

1, . . . , s1 N1} and S2 = {s2 1, . . . , s2 N2}, where the set S1 (S2)

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 = {s1, . . . , sN }. For simplicity, assume L = 2. Define the sets S1 = {s1

1, . . . , s1 N1} and S2 = {s2 1, . . . , s2 N2}, where the set S1 (S2)

contains the more (less) protected subsymbols. The chosen supersymbol si is mapped onto the pair

• s1

i , s2 i

• by some bijective

function f : S → S1 × S2.

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 GT,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

si

1

Sj

1

More important symbols encode subsets.

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

Set Partitioning

si

1

Sj

1

S4

2

S3

2

S2

2

S1

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

d1 (d2) 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

1

Overview Aim Set Partitioning

2

How to Measure the Distance between Constellation Points?

3

Design Approach Proposed Scheme Illustration

4

Selection Criterion of Step Parameter t Geometric Mean Approximation Polynomial Regression Examples

5

Simplified Decoder

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 Xi for Xj under GLRT detection is given by1 Pij = (dcp(Xi, Xj)γT)−MNMMN 2MN − 1 MN

• ,

dcp(Xi, Xj) = M

• k=1

sin2 θk 1

M

. {θi}M

i=1 are the principle angles between the subspaces associated with Xi and Xj.

• 1M. 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.

• 2I. 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 Xi for Xj under GLRT detection is given by1 Pij = (dcp(Xi, Xj)γT)−MNMMN 2MN − 1 MN

• ,

dcp(Xi, Xj) = M

• k=1

sin2 θk 1

M

. {θi}M

i=1 are the principle angles between the subspaces associated with Xi and Xj.

The Chordal Product Distance2 (not a true distance!) can be employed to define the coding gain of the more (less) protected symbols. d1 = min

i=j dcp(Ωi, Ωj),

d2 = min

X,Y∈Cdcp(X, Y).

• 1M. 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.

• 2I. 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 dc(Xi, Xj) =

M

• k=1

sin2 θk.

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 dc(Xi, Xj) =

M

• k=1

sin2 θk. The AM-GM inequality governs the relationship between the two metrics dcp(Xi, Xj) ≤ 1 M dc(Xi, Xj).

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 dc(Xi, Xj) =

M

• k=1

sin2 θk. The AM-GM inequality governs the relationship between the two metrics dcp(Xi, Xj) ≤ 1 M dc(Xi, Xj). 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

1

Overview Aim Set Partitioning

2

How to Measure the Distance between Constellation Points?

3

Design Approach Proposed Scheme Illustration

4

Selection Criterion of Step Parameter t Geometric Mean Approximation Polynomial Regression Examples

5

Simplified Decoder

Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 9 / 21

Design Approach (1)3

For an initial layer ξ1, we define a sequence of L − 1 children layers ξ2, ξ3, . . . , ξL.

• 3K. 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

Design Approach (1)3

For an initial layer ξ1, we define a sequence of L − 1 children layers ξ2, ξ3, . . . , ξL. Starting at X = X(0), the motion of a point traveling along some direction B is given by X(t) = [X

X⊥]

• V cos Σt

U sin Σt

• .

Where B = UΣV† is the SVD of B, X⊥ is the orthogonal complement of X and t is a step parameter.

X(0) B X(0) X(t)

• 3K. 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

Design Approach (1)3

For an initial layer ξ1, we define a sequence of L − 1 children layers ξ2, ξ3, . . . , ξL. Starting at X = X(0), the motion of a point traveling along some direction B is given by X(t) = [X

X⊥]

• V cos Σt

U sin Σt

• .

Where B = UΣV† is the SVD of B, X⊥ is the orthogonal complement of X and t is a step parameter.

X(0) B X(0) X(t)

Elements of ξi are found by transitioning along K geodesics emanating from each member in the ith parent layer.

• 3K. 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

Design Approach (2)

Elements of ξi are found by transitioning along K geodesics emanating from each member in the ith parent layer. ξi =

• [Xp

Xp⊥]

• Vk cos Σkti

Uk sin Σkti

• ∀Xp ∈ Ci, ∀k = 1, . . . , K
• .

Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 11 / 21

Design Approach (2)

Elements of ξi are found by transitioning along K geodesics emanating from each member in the ith parent layer. ξi =

• [Xp

Xp⊥]

• Vk cos Σkti

Uk sin Σkti

• ∀Xp ∈ Ci, ∀k = 1, . . . , K
• .

The ith parent layer is formed by augmenting all existing points at depth i Ci =

i−1

• j=1

ξj,

Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 11 / 21

Design Approach (2)

Elements of ξi are found by transitioning along K geodesics emanating from each member in the ith parent layer. ξi =

• [Xp

Xp⊥]

• Vk cos Σkti

Uk sin Σkti

• ∀Xp ∈ Ci, ∀k = 1, . . . , K
• .

The ith parent layer is formed by augmenting all existing points at depth i Ci =

i−1

• j=1

ξj, The union set of all ξi forms the constellation.

Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 11 / 21

Design Approach (2)

Elements of ξi are found by transitioning along K geodesics emanating from each member in the ith parent layer. ξi =

• [Xp

Xp⊥]

• Vk cos Σkti

Uk sin Σkti

• ∀Xp ∈ Ci, ∀k = 1, . . . , K
• .

The ith parent layer is formed by augmenting all existing points at depth i Ci =

i−1

• j=1

ξj, The union set of all ξi forms the constellation. The size of the constellation is given by |C| = N′(K + 1)L−1, where N′ is the initial constellation size and K is the number of geodesic directions.

Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 11 / 21

Design Approach (3)

Initial constellation.

Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 12 / 21

Design Approach (3)

Geodesic directions ensure maximal spacing.

Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 12 / 21

Design Approach (3)

Children points are found along the geodesic directions.

Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 12 / 21

Design Approach (3)

The process can be further repeated to add more points.

Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 12 / 21

Table of Contents

1

Overview Aim Set Partitioning

2

How to Measure the Distance between Constellation Points?

3

Design Approach Proposed Scheme Illustration

4

Selection Criterion of Step Parameter t Geometric Mean Approximation Polynomial Regression Examples

5

Simplified Decoder

Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 13 / 21

Selecting t to Ensure a Desired Coding Gain

The step parameter t controls how farther away are children points from their respective parents.

Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 14 / 21

Selecting t to Ensure a Desired Coding Gain

The step parameter t controls how farther away are children points from their respective parents. With careful selection of this parameter, a specific coding gain can be ensured for the more protected layer over the Equal Error Protection scheme.

Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 14 / 21

Selecting t to Ensure a Desired Coding Gain

The step parameter t controls how farther away are children points from their respective parents. With careful selection of this parameter, a specific coding gain can be ensured for the more protected layer over the Equal Error Protection scheme. We propose two ways of doing this:

1

Geometric Mean Approximation.

2

Polynomial Regression.

Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 14 / 21

Geometric Mean Approximation

A lower bound on minimum subset distance d1 is established in terms of β = sin t. d1 1 M (dpack − 2 √ Mβ)2 − M − 1 2M2 (dpack − 2 √ Mβ)4, where dpack = M(T−M)

T

• N′

N′−1.

Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 15 / 21

Geometric Mean Approximation

A lower bound on minimum subset distance d1 is established in terms of β = sin t. d1 1 M (dpack − 2 √ Mβ)2 − M − 1 2M2 (dpack − 2 √ Mβ)4, where dpack = M(T−M)

T

• N′

N′−1.

Setting the desired coding gain to be equal to the RHS guarantees that the actual coding gain is at least as large as the desired value.

Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 15 / 21

Geometric Mean Approximation

A lower bound on minimum subset distance d1 is established in terms of β = sin t. d1 1 M (dpack − 2 √ Mβ)2 − M − 1 2M2 (dpack − 2 √ Mβ)4, where dpack = M(T−M)

T

• N′

N′−1.

Setting the desired coding gain to be equal to the RHS guarantees that the actual coding gain is at least as large as the desired value. This bound is rather crude when the constellation size is large.

Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 15 / 21

Polynomial Regression

Assume d1 =

n

• i=1

aiβi.

Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 16 / 21

Polynomial Regression

Assume d1 =

n

• i=1

aiβi. With enough observations (l ≥ n), we can estimate the unknown coefficients ai.

Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 16 / 21

Polynomial Regression

Assume d1 =

n

• i=1

aiβi. With enough observations (l ≥ n), we can estimate the unknown coefficients ai. Performance analysis suggests that this is the more reliable approach.

Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 16 / 21

Performance Analysis

R = 1.5 bpcu, 33% of data is important

4 6 8 10 12 14 16 18 20 10

−5

10

−4

10

−3

10

−2

10

−1

10 SNR [dB] Symbol Error Rate EEP UEP − PR ξ1 UEP − GMA ξ1 UEP − PR ξ2 UEP − GMA ξ2

Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 17 / 21

Performance Analysis

R = 2 bpcu, 50% of data is important

4 6 8 10 12 14 16 18 20 10

−4

10

−3

10

−2

10

−1

10 SNR [dB] Symbol Error Rate EEP UEP − PR ξ1 UEP − PR ξ2

Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 18 / 21

Table of Contents

1

Overview Aim Set Partitioning

2

How to Measure the Distance between Constellation Points?

3

Design Approach Proposed Scheme Illustration

4

Selection Criterion of Step Parameter t Geometric Mean Approximation Polynomial Regression Examples

5

Simplified Decoder

Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 19 / 21

Simplified Decoder

The decoder initially examines all constellation points X ∈ ξ1.

Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 20 / 21

Simplified Decoder

The decoder initially examines all constellation points X ∈ ξ1. The decoder then investigates the children of the q largest ML metric. The process is repeated for L > 2.

Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 20 / 21

Simplified Decoder

The decoder initially examines all constellation points X ∈ ξ1. The decoder then investigates the children of the q largest ML metric. The process is repeated for L > 2. A decision is made in favor of some X, if X maximizes the ML metric over all examined points.

Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 20 / 21

Simplified Decoder

The decoder initially examines all constellation points X ∈ ξ1. The decoder then investigates the children of the q largest ML metric. The process is repeated for L > 2. A decision is made in favor of some X, if X maximizes the ML metric over all examined points. A reduction factor of ( q

K )L−1 is realized in the number of operations needed.

Attiah, Seddik, Gohary, and Yanikomeroglu Design of Grassmannian Codes for UEP May 23, 2017 20 / 21

Analysis of Simplified Decoder

K = 15.

4 6 8 10 12 14 16 18 20 10

−4

10

−3

10

−2

10

−1

10 SNR [dB] Symbol Error Rate ξ1 − Simplified(q = 1) ξ1 − Simplified(q = 2) ξ1 − GLRT ξ2 − Simplified(q = 1) ξ2 − Simplified(q = 2) ξ2 − GLRT

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