Jointly Optimum Power and Signature Sequence
Allocation for Fading CDMA
Onur Kaya
Sennur Ulukus
Joint Power and Sequence Allocation – K ≤ N
Introduction
• Dynamic resource allocation – transmit powers, bandwidth, time slots; or in general
waveforms – to combat fading and improve capacity.
• CDMA (Vector MAC): allocate transmit powers and signature sequences to users.
K
r = ∑ pi hi bi si + n
i =1
• Power control only: maximize ergodic sum capacity subject to average power
constraints
– Fading channels, waterfilling in time, users treat each other as noise,
– More power to better channel states; no power to very poor channel states.
• Signature sequence allocation only: find sum-capacity maximizing set of sequences
(waveforms) for a given set of (fixed) power constraints, and no fading.
– Notion of oversized/non-oversized users according to power constraints,
– Orthogonal sequences to oversized, GWBE sequences to non-oversized users.
Optimum Power Allocation: K = 4, N = 3
• Optimal signature sequences constitute an orthogonal set for any power alloc’n.
• Problem reduces to K independent single user Goldsmith-Varaiya problems, i.e.,
• We consider a CDMA system with perfect CSI at the transmitters.
• Then, both powers and sequences can be chosen as functions of channel states.
• First, fix an arbitrary valid power allocation over the fading states.
h3
X
2
X
h2
o
h1
o
• For each fixed allocation, find the sequences that maximize the sum capacity at each
state h.
h3
X
X
o
o
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1
0.5
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0
0
0.1
h2
h1
o
o
• Define the signature sequence optimized sum capacity at h
Copt (h, p(h)) = max Csum (h, p(h),S(h))
S (h )
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+
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⎛ 1 σ ⎞
pk (h) = ⎜ −
⎟ , i = 1,..., K
⎝ λ k hk ⎠
2
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0.8
0.9
h
0.1
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h1
Power Allocation for User 2, h =h =0.9
Power Allocation for User 1, h =h =0.9
3
0.8
h2
h1
2
• Channel non-adaptive sequence selection performs as well as any channel
adaptive selection.
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1
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1
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• For a given power control policy P(h), let L(h) and L(h) be sets of oversized
and non-oversized users respectively, for a given h.
• Define D=diag(p1h1, … , pKhK). Optimum signature sequences satisfy,
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0
Theorem (Number of simultaneously transmitting users): Let K(h) be a subset
of {1,…,K}, such that∀i∈ K(h), pi*(h) > 0, where pi*(h) is the maximizer of
Eh[Copt(h,p(h))]. Then, with probability 1, |K(h)| ≤ N.
⎧⎛ 1 σ 2 ⎞
⎪⎜ −
⎟ , i ∈ K (h)
pi (h) = ⎨⎝ λ i hi ⎠
⎪
otherwise
⎩0,
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h
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SDS T si (h) = µi (h)si (h)
⎡
⎛
∑ pi (h)hi ⎞⎟ ⎤⎥
p (h)h ⎞
⎛
Eh ⎢ ∑ log ⎜1 + i 2 i ⎟ + ( N − | L(h) |) log ⎜ 1 + 2 i∈L (h )
⎜
σ
σ
( N − | L(h) |) ⎠⎟ ⎥⎦
⎢⎣ i∈L (h )
⎝
⎠
⎝
0
0.8
0.8
2
• At most N users transmit: assign orthogonal sequences to those users.
• Optimum power allocation is similar to single user waterfilling
X
1.5
1
0.5
• The sequence optimized ergodic sum-capacity is then
X
2
1.5
• Concave maximization over an affine set of constraints, using KKT conditions,
⎧ ∑ j∈L (h ) p j h j
, i ∈ L (h)
⎪
µi (h) = ⎨ N − | L(h) |
⎪
i ∈ L(h)
⎩ pi hi ,
X
o
o
2.5
2.5
⎡K
p (h)h ⎞ ⎤
⎛
max Eh ⎢ ∑ log ⎜1 + i 2 i ⎟ ⎥
p (h )
σ
⎝
⎠⎦
⎣ i =1
s.t. Eh [ pi (h) ] = pi , i = 1,..., K
Joint Power and Sequence Allocation – K > N
Joint Power and Sequence Allocation
Power Allocation for User 2
Power Allocation for User 1, h3=h4=0.4
1
h2
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h
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h
1
Iterative Power and Sequence Optimization
• Instead of simultaneously solving for all powers, which in turn requires solving for
λi, we propose the following one-user-at-a-time algorithm:
repeat
for i = 1 to K and for all h
-find oversized users
-compute signature sequences for all users
-update ith user’s power using waterfilling keeping other powers fixed
end
until p(h) converges.
Convergence of the Iterative Algorithm
• The algorithm corresponds to iteration of the best sequence-only update for all users
and best power-only update for one user, so sum capacity values are non-decreasing.
• The sum capacity is bounded from above, so this algorithm converges to a limit.
+
⎛ n1 satisfies
⎞ KKT conditions.
σ 2 the
• The fixed point pn+1
pk (h)
k = 1,..., K
−
(h) == ⎜p (h)
⎟ ,
−1
• Algorithm converges to jointly
k s k A k s k ⎠power and signature sequence allocation.
⎝ λk hoptimum
Convergence of Sum Capacity for Different Sequence Sets, K>N
Convergence of Sum Capacity for Different Sequence Sets, K=N=3
1
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• Here, a channel adaptive allocation of orthogonal sequences is necessary.
• Define γ i = hi / λi, and let γ [i] be the order statistics for γ i s, and let for given h
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sum
γ [1] ≥ … ≥ γ [n] > σ2 ≥ γ [n+1] ≥ … ≥ γ [K+1] = 0
0.85
C
• Then we can optimize only over power control policies, using optimum sequences
computed for each policy.
Csum
0.66
0.64
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max E h ⎣⎡Copt (h, p(h)) ⎦⎤
p (h )
s.t. E h [ pi (h) ] = pi , i = 1, ..., K
pi (h) ≥ 0 ∀h, i = 1, ..., K
• If n ≤ N, the users with highest n γ i s transmit with powers pi*(h).
• If n > N, by Theorem, the users with highest N γ i’s transmit with positive
powers.
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Fixed Orthogonal Sequences
Fixed Random Sequences
Adaptive Optimal Sequences
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0.56
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0
2
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10
Number of iterations
12
Adaptive Optimum Sequences
Fixed GWBE Sequences
Fixed Random Sequences
0.75
14
16
18
0
5
10
15
Number of iterations
20
25