An Adaptive Modulation Scheme for Two-user Fading MAC with Quantized Fade State Feedback -Sudipta Kundu & B.Sundar Rajan

1. An Adaptive Modulation Scheme for Two-user Fading
MAC with Quantized Fade State Feedback
Sudipta Kundu and B. Sundar Rajan
Department of Electrical Communication Engineering
Indian Institute Of Science, Bangalore
IEEE PIMRC’12
Sydney, Australia,
September 9-12, 2012
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2. Outline of Presentation
1 Introduction
2 Contributions
3 Channel Quantization
4 Adaptive Modulation Scheme
5 Concluding Remarks and Future Work
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3. Outline
1 Introduction
2 Contributions
3 Channel Quantization
4 Adaptive Modulation Scheme
5 Concluding Remarks and Future Work
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4. System Model
User-1 z ∼ CN (0, σ 2)
h1
x1 ∈ S 1
√ √
y= P h 1 x1 + P h 2 x2 + z
User-2
h2
x2 ∈ S 2
Two-user fading MAC with Gaussian noise.
The symbols of the users are jointly decoded at the destination as follows
√ √
(s1 , s2 ) = arg
ˆ ˆ min |y − ( Ph1 s1 + Ph2 s2 )|2 .
(s1 ,s2 )∈S1 ×S2
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5. Motivation - Example : With No CSI
Consider the case when there is no additive noise at the destination, and
both users use QPSK signal sets at the input.
√
Let h1 = 1∠0 and h2 = 2∠ π .4
2
1
3 (4, 1)
OR
4 (2, 4)
QPSK Signal Set
Sum Constellation
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6. Fade State
User-1 z ∼ CN (0, σ 2)
h1
x1 ∈ S 1
√ √
y= P h 1 x1 + P h 2 x2 + z
User-2
h2
x2 ∈ S 2
Two-user fading MAC with Gaussian noise.
Can be viewed as a single user AWGN channel with symbols chosen from
√ √ √
Ssum = Ph1 S1 + Ph2 S2 = Ph1 (S1 + γe jθ S2 ),
Seff
h2
where γ = h1
and θ = ∠ h2 . (fade state)
h 1
Assume S1 = S2 = S, where S is a symmetric M-PSK signal set, with M = 2λ , λ being a
positive integer.
Assume γ ≥ 1.
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7. Fade State
User-1 z ∼ CN (0, σ 2)
h1
x1 ∈ S 1
√ √
y= P h 1 x1 + P h 2 x2 + z
User-2
h2
x2 ∈ S 2
Two-user fading MAC with Gaussian noise.
Can be viewed as a single user AWGN channel with symbols chosen from
√ √ √
Ssum = Ph1 S1 + Ph2 S2 = Ph1 (S1 + γe jθ S2 ),
Seff
h2
where γ = h1
and θ = ∠ h2 . (fade state)
h 1
Assume S1 = S2 = S, where S is a symmetric M-PSK signal set, with M = 2λ , λ being a
positive integer.
Assume γ ≥ 1.
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8. Outline
1 Introduction
2 Contributions
3 Channel Quantization
4 Adaptive Modulation Scheme
5 Concluding Remarks and Future Work
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9. Contributions
Quantization of (Γ, Θ) plane when both the users use M-PSK signal
sets
Adaptive Modulation Scheme based on constellation rotation which
satisfies a minimum distance guarantee δ in Seff (γ, θ)
Optimal angles for rotation in closed form
Upper bound on δ
Performance gains
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10. Outline
1 Introduction
2 Contributions
3 Channel Quantization
4 Adaptive Modulation Scheme
5 Concluding Remarks and Future Work
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11. Distance Distribution in Effective Constellation
Distance between (s1 , s2 )sum , (s1 , s2 )sum ∈ Ssum given by
√
d(s1 ,s2 )sum ↔(s ,s )sum = P|h1 ||(s1 − s1 ) + γe jθ (s2 − s2 )|
1 2
√
= P|h1 |d(s1 ,s2 )↔(s1 ,s2 ) ,
where (s1 , s2 ), (s1 , s2 ) ∈ Seff and
d(s1 ,s2 )↔(s1 ,s2 ) = |(s1 − s1 ) + γe jθ (s2 − s2 )|
.
(s1 − s1 )
If γe jθ = − , then d(s1 ,s2 )↔(s1 ,s2 ) = 0 .
(s2 − s2 )
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12. Distance Distribution in Effective Constellation
Distance between (s1 , s2 )sum , (s1 , s2 )sum ∈ Ssum given by
√
d(s1 ,s2 )sum ↔(s ,s )sum = P|h1 ||(s1 − s1 ) + γe jθ (s2 − s2 )|
1 2
√
= P|h1 |d(s1 ,s2 )↔(s1 ,s2 ) ,
where (s1 , s2 ), (s1 , s2 ) ∈ Seff and
d(s1 ,s2 )↔(s1 ,s2 ) = |(s1 − s1 ) + γe jθ (s2 − s2 )|
.
(s1 − s1 )
If γe jθ = − , then d(s1 ,s2 )↔(s1 ,s2 ) = 0 .
(s2 − s2 )
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13. Singular Fade States
Definition (Singular Fade State) 1.5
A fade state (γ, θ) is said to be a 1
singular fade state if |Seff | < M 2 .
0.5
s1 − s1 0
γe jθ = −
s2 − s2 −0.5
Depends on the signal sets being used −1
by the users!
−1.5
−1.5 −1 −0.5 0 0.5 1 1.5
H = Set of all singular fade states.
Singular fade states for QPSK case.
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14. Observations
Minimum distance is Seff , dmin (γ, θ) ≤ dmin (S).
When both the users use M-PSK signal sets,
2π
Distance distribution in Seff is periodic in θ with period .
M
If (γ , θ ) is a singular fade state, then there exists singular fade states
at (γ , θ + p 2π ), where 1 ≤ p ≤ M − 1.
M
To study distance profile in Seff , sufficient to consider only the case
π
when 0 ≤ θ ≤ .
M
π
Sufficient to obtain quantization only for wedge [0, M ] .
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18. Distance Class
6
3 3
6
5
2
7
2 2
7 5
8 2
1 8 1 10
3 1
1
10 3
0 0
9
9 4 11
−1 11 14 −1 4
14
13 12
−2 12 15 −2
13
15
−3 16 −3
16
−3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3
(γ, θ) = (1.8, 25◦ ) (γ, θ) = (2.1, 7◦ )
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19. Definitions
Definition (Distance Class)
2
A distance class denoted by C, is a subset of Seff (γ, θ), which contains the pairs
of the form {(s1 , s2 ), (s1 , s2 )}, (s1 , s2 ) = (s1 , s2 ) where (s1 , s2 ) and (s1 , s2 ) denote
the complex points in Seff , such that the distance between the two elements of a
pair is same for all pairs in C and this property holds for all values of γ and θ,
though the value of the distance depends on (γ, θ).
16
For the QPSK case, = 120 possible pairwise distances in Seff (γ, θ) is
2
partitioned into just 20 distance classes.
¯
For a given S, C = set of the all distance classes for it
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20. Definitions
Definition (Class Distance Function)
Associated with every distance class C is a class distance function
dC (γ, θ) : (Γ, Θ) → R, which gives the value of the distance between the
two elements of a pair in C for any (γ, θ).
Definition
The region corresponding to distance class C, R(C) is the region on the
(Γ, Θ) plane for which the class distance function dC (γ, θ), gives the
minimum distance in Seff .
π
RW (C) = R(C) ∩ wedge [0, ].
M
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21. Channel Quantization for QPSK Signal Sets
(γ, θ) = (2, 14◦ ).
√
Step 1 The singular fade states lying in the wedge [0, π/4] are ( 2, π/4) and
(1, 0).
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22. Channel Quantization for QPSK Signal Sets
(γ, θ) = (2, 14◦ ).
√
Step 2 At ( 2, π/4), the class distance function
dCk (γ, θ) = 2γ 2 + 4 − 4γ cos θ − 4γ sin θ reduces to zero.
2
1
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23. Channel Quantization for QPSK Signal Sets
(γ, θ) = (2, 14◦ ).
Step 2 For the singular fade state (1, 0) the class distance functions that
reduce to zero are as follows
dCk (γ, θ) = 2γ 2 + 2 − 4γ cos θ
2
2
dCk (γ, θ) = 4γ 2 + 4 − 8γ cos θ = 2dCk (γ, θ) ≥ dCk (γ, θ).
2 2 2
3 2 2
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24. Channel Quantization for QPSK Signal Sets
To find the region RW (Ck1 ) we need to find (γ, θ) where
2 2
dCk (γ, θ) ≤ dCk (γ, θ),
1 2
2 2
dCk (γ, θ) ≤ dmin (S).
1
The pairwise boundaries are as follows
2 2
dCk (γ, θ) = dCk (γ, θ) ⇒ γ sin θ = 1/2
1 2
2
dCk (γ, θ) = dmin (S) = 2 ⇒ (γ cos θ − 1)2 + (γ sin θ − 1)2 = 1.
2
1
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25. Channel Quantization for QPSK Signal Sets
To find the region RW (Ck1 ) we need to find (γ, θ) where
2 2
dCk (γ, θ) ≤ dCk (γ, θ),
1 2
2 2
≤ dCk (γ, θ) dmin (S).
1
The pairwise boundaries are as follows
2 2
dCk (γ, θ) = dCk (γ, θ) ⇒ γ sin θ = 1/2
1 2
dCk (γ, θ) = dmin (S) = 2 ⇒ (γ cos θ − 1)2 + (γ sin θ − 1)2 = 1.
2 2
1
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26. Channel Quantization for QPSK Signal Sets
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28. Outline
1 Introduction
2 Contributions
3 Channel Quantization
4 Adaptive Modulation Scheme
5 Concluding Remarks and Future Work
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29. Violation Circles
Goal: To provide a minimum distance guarantee of δ in Seff .
We have dCki (γ, θ) = |s2,i − s2,i ||γe jθ − γi e jθi | .
Avoid fade states for which dCki (γ, θ) < δ, 1 ≤ i ≤ NW i.e.,
δ
|γe jθ − γi e jθi | < , for 1 ≤ i ≤ NW .
|s2,i − s2,i |
Above equation represents circular regions on (Γ, Θ) plane with center
δ
at (γi , θi ) and radius ρ(γi , θi ) = |s2,i −s | . (Violation Circles)
2,i
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30. Example: Violation Circles for QPSK case
Class representative for
C
distance class√ k1 is (3,√
5).
Therefore, ρ( 2, π ) = 2.
4
The√ violation circle centered
at ( 2, π ) has radius ( √2 ).
4
δ
Similarly, the violation circle
centered at (1, 0) has radius
δ
√ .
2
Violation circles for QPSK signal sets
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31. Constellation Rotation Scheme
For fade states inside the violation circles : users adapt their
transmission in order to meet the minimum distance requirement
For fade states outside the violation circles : no adaptation required
Adaptation via rotation of constellation of one user relative to the
other, without varying transmit power
S + γe jθ {e jα S} = S + γe j(θ+α) S
i.e., fade state (γ, θ) is transformed to (γ, θ + α) after rotation.
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32. Optimal Angle of Rotation
Definition
An optimal rotation angle, for a violation circle with center at singular
fade state (γi , θi ) , 1 ≤ i ≤ NW is that angle of rotation which maximizes
the minimum distance in Seff for the same transmit power, when fade
state (γ, θ) = (γi , θi ).
Find the optimal phase θi,opt for each (γi , θi ).
Optimal phase corresponds to the phase of one of the point of
intersections of the arc γ = γi with the pairwise boundaries between
the regions within the wedge.
Optimal rotation angle αi,opt transforms fade state from (γi , θi ) to
(γi , θi,opt ).
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33. Optimal Rotation Angles for the QPSK case
Optimal rotation angles for the QPSK case
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34. Upper bound on δ
P is transfered to P’ after rotation, which still lies
γ sin θ within the violation circle and thus does not satisfy
2 the minimum distance guarantee.
Initial position of fade state (γ, θ)
P
Effective shifted
position of the
1
fade state
after rotation.
P’
γ cos θ
0 1 2
Required: For each i, 1 ≤ i ≤ NW ,
ρ(γi , θi ) + ρ(γj , θj ) ≤ d(γi ,θi,opt )↔(γj ,θj ) , for all 1 ≤ j ≤ NW .
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35. Upper bound on δ
P is transfered to P’ after rotation, which still lies
γ sin θ within the violation circle and thus does not satisfy
2 the minimum distance guarantee.
Initial position of fade state (γ, θ)
P
Effective shifted
position of the
1
fade state
after rotation.
P’
γ cos θ
0 1 2
Required: For each i, 1 ≤ i ≤ NW ,
ρ(γi , θi ) + ρ(γj , θj ) ≤ d(γi ,θi,opt )↔(γj ,θj ) , for all 1 ≤ j ≤ NW .
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36. Example: QPSK case
The radii of the violation circles are
√ δ
ρ(1, 0◦ ) = ρ( 2, 45◦ ) = √
2
d(√2,45◦ )↔(√2,20.7◦ ) = d(1,0◦ )↔(√2,20.7◦ ) ≈ 0.5936
d(√2,45◦ )↔(1,30◦ ) = d(1,0◦ )↔(1,30◦ ) ≈ 0.5176.
To avoid overlap we must have
δ
2( √ ) ≤ min{0.5936, 0.5176} =⇒ δ ≤ 0.365.
2
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37. Summary
The destination checks if the fade state lies within any of the NW
violation circles.
The location of the fade state - whether in any of the NW violation
circles or outside is indicated to the users by a feedback of
log2 (NW + 1) bits. The destination also indicates which of the two
ratios h1 or h1 is used for quantization (i.e., for γ >= 1) using a
h2
h2
single bit feedback.
(Total feedback overhead = log2 (NW + 1) + 1 bits per quasistatic interval.)
h2
One of the two users (User-2 if ratio computed is h1 and User-1
otherwise), using the feedback they receive, rotates its constellation
by the optimal angles.
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38. Simulation Results: QPSK
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39. Outline
1 Introduction
2 Contributions
3 Channel Quantization
4 Adaptive Modulation Scheme
5 Concluding Remarks and Future Work
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40. Conclusions and Future Work
Conclusions:
Channel quantization for M-PSK signal sets
Identified violation circles which result in violating minimum distance
guarantee
Obtained optimal angles of rotation in closed form
Presented simulation results to show the extent of performance
improvement by the proposed scheme
Future Work:
Coding across time
More than two users
With QAM signal sets
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41. Thank You!
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42. Channel Quantization for 8-PSK Signal Sets
Quantization of the wedge [0, π/8] for 8-PSK signal sets.
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43. Sudipta Kundu & B. Sundar Rajan (IISc) IEEE PIMRC’12 September 9-12, 2012 2/5

44. Optimal Angle of Rotation
Lemma
Let dCki (γ, θ) be the class distance function which reduces to zero at the
singular fade state (γi , θi ) ∈ HW . For a fixed γ0 and (γ0 , θ) lying within
the wedge [0, π/M], the value of dCki (γ0 , θ) increases as the difference
|θi − θ| increases.
(γi, θi)
γ = γ0
1
Diagram illustrates the above lemma.
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45. Procedure to find the Optimal Phase
π
Step 1 If there are no such points of intersections then θi,opt = | M − θi |.
Step 2 If there are L such points of intersections, calculate the phase of each of
these points of intersection θl,intersect , 1 ≤ l ≤ L by solving the equation
2 2
dCk (γ, θ) = dCk (γ, θ)|γ=γi .
a,l b,l
Then compute the minimum distances corresponding to the point of
intersection (γi , θl,intersect ) as
2
dmin (γi , θl,intersect ) = dCk (γ, θ)|γ=γi ,θ=θl,intersect .
a,l
Choose l = arg max1≤l≤L dmin (γi , θl,intersect ), and θi,opt = θl ,intersect .
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46. Simulation Results: 8-PSK
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