2011 IEEE Information Theory Workshop
Quantized Compute and Forward:
A Low-Complexity Architecture for Distributed
Antenna Systems
Song-Nam Hong and Giuseppe Caire
EE Department, University of Southern California, Los Angeles CA
Paraty, Brazil, October 17-20, 2011
Outline
• Motivation
• Quantized Compute and Forward (QCoF)
• Distributed Antenna Systems (uplink)
• Scheduling and Multiuser Diversity
• Conclusions
1
Many small dumb antenna sites
2
• Power consumption of macrocell base stations is one of the major operating
costs in cellular communications.
• Conventional macrocell base stations are powered 24/7 at ∼ 100% (always
enough active users in a macrocell).
• Trend: move towards much more dense deployment in order to reduce
distance and exploit locality.
• Two ways: 1) user deployed (femtocells); 2) Distributed Antenna Systems
(DAS).
• In this talk: a low-complexity scheme for DAS uplink.
3
Distributed Antenna System
4
System model
UT
h11
z1
+
AT
z2
UT
UT
+
h�k
AT
R0
CP
z3
+
AT
+
y`(i) = H`x(i) + z`(i),
R0
R0
z4
UT
R0
AT
` = 1, . . . , L
5
Wyner model with rate-constrained backhaul
Well-known and well-understood problem:
• Sanderovich, Somekh, Poor, Shamai, IT 2009 Compress and Forward (CF),
Decode and Forward (DF).
• Application of Avestimehr, Diggavi, Tse, IT 2011 Quantize Remap and
Forward.
• Nazer, Sanderovich, Gastpar, Shamai, ISIT 2009 Compute and Forward
(CoF) with possible rate splitting and superposition.
In this work:
• Simplified low-complexity version of CoF: Quantized CoF.
• LDPC design for QCoF.
• Impact of scheduling and multiuser diversity on DAS with CoF/QCoF.
6
Quantized Compute and Forward (real channels)
• Let p be a prime integer and κ ∈ R+. We consider the two nested onedimensional lattices
Λs = {x = κpz : z ∈ Z},
Λc = {x = κz : z ∈ Z}.
• PAM constellation set S , Λc ∩ Vs. Natural embedding g : Zp → R.
• Modulation mapping m : Zp → S, such that v = m(u) , [κg(u)] mod Λs.
• Demodulation mapping, u = m−1(v) , g −1([v/κ] mod pZ), with v ∈ S.
• A linear code C over Zp with block length n, dimension k and rate R =
(in bit/symbol) is the set of codewords
k
n
log(p)
C = {c = wG : w ∈ Zkp }
7
Encoding
• Each user generates a codeword ck = (ck (1), . . . , ck (n)) as ck = wk G (same
code).
• Let xk = (xk (1), . . . , xk (n)) denote the transmitted signal of user k.
• The i-th component of user k signal is given by
xk (i) = [m(ck (i)) + dk (i)]
mod Λs
where (dk (1), . . . , dk (n)) are i.i.d. dithering sequences ∼ Uniform(Vs), known
at the receivers.
• Transmit power: SNR , E[|xk (i)|2] = κ2p2/12.
8
Receiver processing at each AT
• For i = 1, . . . , n, the signal received at AT ` is given by
y`(i) = H`x(i) + z`(i)
where x(i) = (x1(i), . . . , xK (i))T.
• The matrix H` has dimensions M × K (some columns may be zero, for not
fully connected network).
• The goal of AT ` is to recover a linear combination c` =
transmitted users’ codewords.
L
q`,k ck of the
9
• The receiver selects the integer coefficients vector a` = (a`,1, ..., a`,K )T ∈ ZK
and computes
"
u`(i) = m−1
QΛc
αT` y`(i) −
L
X
!#
a`,k d`(i)
!
mod Λs ,
i = 1, . . . , n.
k=1
• Letting u` = (u`(1), . . . , u`(n)), we have that
u=
K
M
q`,k ck ⊕ z̃`,
`=1
with q`,k = g −1([a`,k ] mod pZ), and where z̃` is a discrete additive noise
vector.
10
Analog non-linear mapping with quantization
scalar quantizer
y(i)
α
×
remove dither
u(i)
+
sawtooth transformation
11
Decoding
• Each AT ` sends the sequence u` to the CP over the backhaul link (rate
requirement: R0 ≥ log p).
LK
• Each AT decodes locally the superposition codeword c` = `=1 q`,k ck from
b ` over the backhaul
u`, and sends the corresponding information message w
link (rate requirement: R0 ≥ nk log p).
• By linearity, if no decoding error occurs, we have
b` =
w
K
M
q`,k wk
`=1
12
• Low-complexity decoding at the CP: if the system matrix Q = [q`,k ] has rank
K over Zp, then the individual messages can be obtained by solving
b1
w
w1
 ..  = Q  .. 
bL
w
wK




• Remark: if M > 1, we can obtain more linear combinations from the same
AT (eventually, we can serve up to M L users).
• Computation rate: supremum of R =
large n at each AT `.
k
n
log p such that Pe → 0 for sufficiently
• Theorem: For given H`, a` and α`, Rcomp(H`, a`, α`) = log p − H(Z̃`).
13
Minimizing the noise variance
• Omitting indices ` and i for simplicity, consider:
h
v = QΛc α y −
T
K
X
ak dk
i
mod Λs
k=1
h
= QΛc α Hx + α z −
T
T
L
X
ak dk
i
mod Λs
k=1
h
= QΛc
K
X
k=1
ak m(ck ) + (αTH − aT)x +αTz
{z
}
|
i
mod Λs
ξ
• The non-integer error term ξ is statistically independent of the m(ck )’s and
has variance
T
2
2
σξ = SNR H α − a + kαk2.
14
• The components of the discrete noise z̃ are distributed as
Z̃ = m−1 ([QΛc (ξ)]
mod Λs) .
• Instead of minimizing H(Z̃) (difficult), we minimize σξ2 (easy).
15
• For given a, the optimal α is given by
h
−1
α = HH + SNR
• This results in
σξ2 = aT
T
h
I
i−1
Ha
i−1
SNR−1I + HTH
a
h
−1
• Letting L denote the lower triangular Cholesky factor of SNR
the variance minimization can be written as
T 2
minimize L a ,
I + HTH
i−1
,
a ∈ ZK , a 6= 0
• This can be solved by LLL and Schnorr-Euchner “sphere” enumeration, by
discarding the all zero point.
16
Code design for QCoF
• We need to design good codes over Zp, for the additive noise discrete
channel u = c ⊕ z̃.
• Additive noise channels are symmetric: linear codes are capacity achieving.
• Options: Polar Codes, LDPC, IRA .....
• We used a family of protograph-based IRA codes (details omitted).
17
√
Coding performance (H = [1, 0.75, − 2])
4.5
CoF
QCoF, p=251
QCoF, p=17
QCoF, p=7
QCoF, p=3
RA codes over Z
4
3.5
17
Computation Rates
3
R=4/5
RA codes over Z7
RA codes over Z3
2.5
R=2/3
IRA codes over Z7
R=4/5
2
R=1/2
R=2/3
1.5
R=1/2
R=4/5
1
0.5
0
15
R=2/3
R=1/2
20
25
30
35
40
45
50
SNR [dB]
Fig. 2.
√
Computation rates for three-user Gaussian MAC with coefficients h = [1, 0.75, − 2] and ADC level, p = 3, p = 7, 18
p = 17, or 255.
DAS performance on the symmetric Wyner model
2
Rate per User
1.5
1
0.5
CF
DF
CoF
QCoF, p=251
QCoF with PA, p=251
QCoF, p=7
QCoF with PA, p=7
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
a
SNRFig.
= 3.15SNR
dB,
R0 Achievable
= 2 bit/symbol.
= 15dB.
rates per user as a function of the inter-cell interference level γ, for finite backhaul capacity
R0 = 2 bits and ADC level, p = 7 or 251.
19
DAS performance: multiuser diversity
• In a real wireless network the channel coefficients h`,k may be all non-zero,
but some of them may be small.
• We can select the users to be served, and the coefficients to be treated as
zero.
• If SNR|hv`,k |2 < 1, we do not include UT k in the QCoF of AT ` and include
the signal of this users as additional noise.
• We add users one by one (greedy) until the computation rate increases and
the system matrix Q is full rank.
• The choice of the a` vectors is decentralized: if no full-rank matrix Q can be
found, we declare zero rate.
• For time-varying channels, this is akin an outage event (we consider ergodic
rates, e.g., obtained by retransmission).
20
DAS multiuser diversity: Wyner model with fading
3.5
3
Rate per User
2.5
2
1.5
1
Upper Bound
CF
CoF, Greedy
0.5
QCoF, Random
QCoF, Greedy
QCoF, Random
0
5
10
15
SNR [dB]
20
25
γ = 0.7, R0 = 3 bit/symbol, L = 5, K = 25, for QCoF, p = 7.
Fig. 4.
Achievable rates per user as a function of SNR, for finite backhaul capacity R0 = 3 bits, interference level γ = 0.7,
ADC level, p = 7, L = 5, and K = 25
21
DAS multiuser diversity: Wyner model with fading and M = 2
110
100
90
M=1, b=2
M=2, b=2, CASE I
M=2, b=2, CASE II
M=1, b=8
M=2, b=8, CASE I
M=2, b=8, CASE II
System Rates
80
70
60
50
40
30
20
10
15
20
25
30
35
SNR [dB]
γ = 0.7, L = 16, CoF only (or QCoF with sufficiently large p).
Fig. 6.
Achievable system rates as a function of SNR, for finite backhaul capacity R0 = 6 bits, interference level γ = 0.7,
and L = 16.
22
DAS multiuser diversity: treating weak links as zero
35
Greedy
Greedy w/ Preprocessing
30
25
System Rates
20
15
10
5
0
5
10
15
20
25
30
SNR [dB]
Fig. 8.
Achievable system rates for scheduling algorithms in a DAS with L = 8, K = 10, and R0 = 5.
K = 10, L = 8, distance dependent pathloss, fading and shadowing.
23
DAS multiuser diversity: number of users
70
L=8
L=16
60
System Rates
50
40
30
20
10
0
50
100
150
200
250
300
350
400
450
Number of UTs
SNR = 20 dB, distance dependent pathloss, fading and shadowing.
Fig. 9.
SNR = 20dB. Achievable system rates of CoF in a DAS with R0 = 5.
24
Conclusions
• Very simple quantized version of CoF.
• Easy and efficient LDPC/IRA code design (Polar Codes could be naturally
used too).
• Attractive scheme for DAS, in conjunction with user scheduling and selection.
• User selection (multiuser diversity) very different from the conventional
wisdom (choose strong channels).
• Somehow ad-hoc approach to avoid the very weak link bottleneck.
• We have a similar scheme (Reverse QCoF) for the downlink (on-going
comparisons with respect to more conventional distributed beamforming
techniques).
25