Sequential and Parallel Algorithms for the Eneralized Maximum Subarray Problem
Dynamic Subarray Architecture for Wideband Hybrid ... · Dynamic Subarray Architecture for Wideband...
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Dynamic Subarray Architecture for Wideband Hybrid Precoding in Millimeter Wave MIMO Systems
Sungwoo Park, Ahmed Alkhateeb, and Robert W. Heath Jr. Wireless Networking and Communications Group Department of Electrical and Computer Engineering The University of Texas at Austin
This work is supported in part by the National Science Foundation under Grant No. 1319556, and by a gift from Huawei Technologies.
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Digital precoding vs. Hybrid precoding
Digitalbasebandprecoding(Conven4onalMIMO)
Analog RF
precoder
Digital baseband precoder
RF chain
RF chain
RF chain
. . .
Hybridanalog/digitalprecoding
NRF < NTX
FRF FBB
. . .
[1] O. El Ayach et el., “Spatially sparse precoding in millimeter wave MIMO systems,” IEEE Trans. on Wireless Comm., Mar. 2014. [2] A. Alkhateeb et el, “MIMO precoding and combining solutions for millimeter-wave systems,” IEEE Comm. Mag., Dec. 2014. [3] W. Ni et el., “Hybrid block diagonalization for massive multiuser MIMO systems,” IEEE Trans. on Comm., Jan 2016.
. . .
Digital baseband precoder
RF chain
RF chain
RF chain
RF chain
RF chain
. . .
. . .
FFD
. . .
RF chains (r=1,…,NRF)
antennas (n=1,…,NTX)
streams (s=1,…,S)
FHB = FRF FBB
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Digital precoding vs. Hybrid precoding
Digitalbasebandprecoding(Conven4onalMIMO)
Analog RF
precoder
Digital baseband precoder
RF chain
RF chain
RF chain
. . .
Hybridanalog/digitalprecoding
NRF < NTX
FRF FBB
. . .
[1] O. El Ayach et el., “Spatially sparse precoding in millimeter wave MIMO systems,” IEEE Trans. on Wireless Comm., Mar. 2014. [2] A. Alkhateeb et el, “MIMO precoding and combining solutions for millimeter-wave systems,” IEEE Comm. Mag., Dec. 2014. [2] W. Ni et el., “Hybrid block diagonalization for massive multiuser MIMO systems,” IEEE Trans. on Comm., Jan 2016.
. . .
Digital baseband precoder
RF chain
RF chain
RF chain
RF chain
RF chain
. . .
. . .
FFD
. . .
RF chains (r=1,…,NRF)
antennas (n=1,…,NRF)
streams (s=1,…,S)
FHB = FRF FBB
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Hybrid precoding: Narrowband vs. Wideband
Narrowbandhybridprecoding
Analog RF
precoder
Digital baseband precoder
RF chain
RF chain
RF chain
. . .
. . .
WidebandHybridprecoding(MIMO-OFDM)
. . .
. . .
. . .
IFFT & RF chain
IFFT & RF chain
IFFT & RF chain . . .
. . .
. . .
. . .
Analog RF
precoder
subcarrier (k=1,…,K)
FBB[2]
FBB[K]
FBB FRF FRF
. . .
Digital baseband precoder
FBB[1]
FHB = FRF FBB FHB[k]= FRF FBB[k] for k=1,…,K per-subcarrier & frequency-domain wideband & time-domain
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Hybrid precoding: Narrowband vs. Wideband
Narrowbandhybridprecoding
Analog RF
precoder
Digital baseband precoder
RF chain
RF chain
RF chain
. . .
. . .
WidebandHybridprecoding(MIMO-OFDM)
. . .
. . .
. . .
IFFT & RF chain
IFFT & RF chain
IFFT & RF chain . . .
. . .
. . .
. . .
Analog RF
precoder
subcarrier (k=1,…,K)
FBB[2]
FBB[K]
FBB FRF FRF
. . .
Digital baseband precoder
FBB[1]
FHB = FRF FBB FHB[k]= FRF FBB[k] for k=1,…,K
Constraints on FRF
1) NRF < NTX 2) Composed of phase shifters 3) Common for all subcarriers
Constraints on FRF
1) NRF < NTX 2) Composed of phase shifters
per-subcarrier & frequency-domain wideband & time-domain
�[FRF]m,n = ej✓m,n
�
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Hybrid structure – Fully-connected vs. Partially-connected
Fully-connected structure Partially-connected (subarray) structure
FRF =⇥fRF,1 fRF,2 · · · fRF,NRF
⇤
Analog RF precoder
RF chain
. . . . . .
. . . . . .
. . . . . .
RF chain
NRF NTX
. . . . . .
NTX
NTX
FRF =
2
66664
fRF,S1 0 · · · 0
0 fRF,S2 0...
... 0. . . 0
0 · · · 0 fRF,SNRF
3
77775
SNRF
S1
. . .
Analog RF precoder
RF chain
. . .
. . . . . .
. . .
RF chain
NRF
. . . . . .
. . . . . .
Nsub =NTX
NRF
Nsub =NTX
NRF
NRFNTX phase shifters and NTX adders are necessary. NTX phase shifters are necessary.
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arg max
{FRF,FBB(1),··· ,FBB(K)}
KX
k=1
log det
✓I+
1
N0
H(k)FRF
FBB
(k)F⇤BB
(k)F⇤RF
H⇤(k)
◆
s.t.KX
k=1
||FRF
FBB
(k)||2F
Ptot
(Eq. 1)
(Eq. 2)
arg max
{FRF,ˆFBB(1),··· ,ˆFBB(K)}
KX
k=1
log det
✓I+
1
N0
He↵
(k)ˆFBB
(k)ˆF⇤BB
(k)H⇤e↵
(k)
◆
s.t.KX
k=1
||ˆFBB
(k)||2F
Ptot
He↵(k) = H(k)FRF(F⇤RFFRF)
� 12
F̂BB(k) = (F⇤RFFRF)
12 FBB(k)
Using
If FRF is given, the optimum solution of can be found by using a conventional SVD scheme with respect to the effective channel at each subcarrier.
F̂BB 1( ),!, F̂BB K( ){ }
equivalent
Goal: Maximize the sum of per-subcarrier mutual information Constraints
on FRF
1) NRF < NTX 2) Composed of phase shifters 3) Common for all subcarriers
WB hybrid precoding design - Problem formulation (1/2)
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The above problem is a non-convex optimization problem. Instead, let us maximize the sum of squared singular values of effective channels ( = maximize the sum of effective SNR per subcarrier & stream)
argmax
FRF
KX
k=1
SX
s=1
�2s
�H (k)FRF(F
⇤RFFRF)
� 12�
(Eq. 3)
(Eq. 4)
Once the optimum FBB(k) for each k is determined given FRF, the optimization problem is only with respect to FRF.
argmax
FRF
KX
k=1
SX
s=1
log
1 +
�2s
�H (k)FRF(F
⇤RFFRF)
� 12
�ps,k
N0
!
equivalent
approximated (relaxation)
ps,k: water-filling power control
WB hybrid precoding design - Problem formulation (2/2)
Solution
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Hybrid precoding solution – Fully-connected case
F?RF = argmax
FRF
KX
k=1
SX
s=1
�2s
�H[k]FRF(F
⇤RFFRF)
� 12�
Optimization problem
F?RF = [VR]1:NRF
A
where A is an arbitrary invertible matrix and
R =1
K
KX
k=1
H⇤[k]H[k] = VR⇤RV⇤R
Solution w/ phase shifter constraint
F̂?RF = arg min
X,|[X]m,n|=1kX� F?
RFk2F = ]F?
RF
where is a matrix with ]F?RF [F̂?
RF]m,n = ej]([F?RF]m,n)
Analog RF precoder
RF chain
. . . . . .
. . . . . .
. . . . . .
RF chain
NRF NTX
. . . . . .
NTX
NTX
Solution
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Hybrid precoding solution – Partially-connected case
H[k] =⇥HS1 [k] HS2 [k] · · · HSNRF
[k]⇤
RSr =
1
K
KX
k=1
H⇤Sr[k]HSr [k], for r = 1, · · · , NRF
S1 = {1, · · · , Nsub}S2 = {Nsub + 1, · · · , 2Nsub}
...
SNRF = {(NRF � 1)Nsub + 1, · · · , NRFNsub}
F?RF = argmax
FRF
KX
k=1
SX
s=1
�2s
�H[k]FRF(F
⇤RFFRF)
� 12�
Optimization problem
FRF =
2
64fRF,S1 · · · 0
.... . .
...0 · · · fRF,SNRF
3
75
where αr is an arbitrary nonzero complex number, is the dominant eigenvector of , and Subarray partitioning vRSr ,1 RSr
f?RF,Sr= ↵rvRSr ,1, for r = 1, · · · , NRF
Solution w/ phase shifter constraint à the same as the fully-connected case
SNRF
S1
. . .
Analog RF precoder
RF chain
. . .
. . . . . .
. . .
RF chain
NRF
. . . . . .
. . . . . .
Nsub =NTX
NRF
Nsub =NTX
NRF
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Pre-codin
g (BB) (freq-
domain)
Streams Antennas
Pre-codin
g (BB) (freq-
domain)
FFT & DAC
FFT & DAC
FFT & DAC
FFT & DAC
RF
. . .
FBB[2]
BB
FBB[1]
RF chains
FBB[K ]RFF
Comparison between fully-connected vs. partially-connected Fully-connected structure Partially-connected (subarray) structure
Pre-codin
g (BB) (freq-
domain)
Streams Antennas
Pre-codin
g (BB) (freq-
domain)
FFT & DAC
FFT & DAC
FFT & DAC
FFT & DAC
. . .
FBB[2]
BB
FBB[1]
RF chains
FBB[K ]AF
RF
RF
RF
RF
RFF
Analog RF precoder
RF chain
. . . . . .
. . . . . .
. . . . . .
RF chain
NRF NTX
. . . . . .
NTX
NTX
max
FRF
KX
k=1
SX
s=1
�2s
�H [k]FRF(F
⇤RFFRF)
� 12�= K
NRFX
r=1
�r (R) max
FRF
KX
k=1
SX
s=1
�2s
�H [k]FRF(F
⇤RFFRF)
� 12�= K
NRFX
r=1
�1 (RSr )
λi(A) : i-th largest singular value of A
Analog RF precoder
RF chain
. . .
. . . . . .
. . .
RF chain
NRF
. . . . . .
. . . . . .
Nsub =NTX
NRF
Nsub =NTX
NRF
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Pre-codin
g (BB) (freq-
domain)
Streams Antennas
Pre-codin
g (BB) (freq-
domain)
FFT & DAC
FFT & DAC
FFT & DAC
FFT & DAC
RF
. . .
FBB[2]
BB
FBB[1]
RF chains
FBB[K ]RFF
Comparison between fully-connected vs. partially-connected Fully-connected structure Partially-connected (subarray) structure
Analog RF precoder
RF chain
. . . . . .
. . . . . .
. . . . . .
RF chain
NRF NTX
. . . . . .
NTX
NTX
max
FRF
KX
k=1
SX
s=1
�2s
�H [k]FRF(F
⇤RFFRF)
� 12�= K
NRFX
r=1
�r (R) max
FRF
KX
k=1
SX
s=1
�2s
�H [k]FRF(F
⇤RFFRF)
� 12�= K
NRFX
r=1
�1 (RSr )
Depends on and partitioning Depends on R =
1
K
KX
k=1
H⇤[k]H[k]
!(S1, ...,SNRF)R
λs(A) : s-th largest singular value of A
Pre-codin
g (BB) (freq-
domain)
Streams Antennas
Pre-codin
g (BB) (freq-
domain)
FFT & DAC
FFT & DAC
FFT & DAC
FFT & DAC
. . .
FBB[2]
BB
FBB[1]
RF chains
FBB[K ]AF
RF
RF
RF
RF
RFF
Analog RF precoder
RF chain
. . .
. . . . . .
. . .
RF chain
NRF
. . . . . .
. . . . . .
Nsub =NTX
NRF
Nsub =NTX
NRF
Which is the best sub-array structure (best partitioning) ?
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Adjacent type Interlaced type AF
Analogprecoding
Analogprecoding
Analogprecoding
Analogprecoding Analog
recodingAnalogrecodingAnalog
recodingAnalogprecoding
Analogrecoding
Analogprecoding
Analogrecoding
Analogprecoding
Uniform Linear Array (ULA) case
The best subarray structure depends on the R matrix. è We propose a dynamic subarray structure that adapts to the R matrix.
S1 = {1, 2, 3}S2 = {4, 5, 6}S3 = {7, 8, 9}
S1 = {1, 4, 7}S2 = {2, 5, 8}S3 = {3, 6, 9}
1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9
S1 = {1, 2, 3, 4}S2 = {5, 6, 7, 8}S3 = {9, 10, 11, 12}S4 = {13, 14, 15, 16}
S1 = {1, 5, 9, 13}S2 = {2, 6, 10, 14}S3 = {3, 7, 11, 15}S4 = {4, 8, 12, 16}
S1 = {1, 2, 5, 6}S2 = {3, 4, 7, 8}S3 = {9, 10, 13, 14}S4 = {11, 12, 15, 16}
Horizontal type Vertical type Squared type
Uniform Planar Array (UPA) case
Dynamic subarray - Problem formulation
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Partition a set of NTX antennas into NRF subsets - cardinality of each subset : variable (non-empty)
What to do
Goal (optimum criterion)
Optimal solution
* Optimum solution: Exhaustive search - # of total cases: Stirling number of the second kind
1
(NRF)!
NRFX
k=0
(�1)NRF�k
✓NRF
k
◆kNTX
Ex) {NTX=16, NRF=4} à 1.7×108
{S?r }
NRF
r=1 =arg max
S1,...,SNRF
NRFX
r=1
�1 (RSr )
s.t.
NRF[
r=1
Sr = {1, · · · , NTX} , Si \ Sj = ; for i 6= j, |Sr| > 0 8r
Analog RF precoder
RF chain
. . . . . .
. . .
RF chain
NRF
. . . . . .
. . . . . .
NTX
Dynamic (mapper)
1) Needs to check too many cases 2) Needs to calculate the singular value at every case
Analog RF precoder
RF chain
. . . . . .
. . .
RF chain
NRF
. . . . . .
NTX
NRF
NTX
NRF
Fixed
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Approximation of the largest singular value
Definition of the approximate largest singular value
�̂1 (RS) ,1
|S|
|S|X
i=1
|S|X
j=1
|[RS ]i,j | =1
|S|X
i2S
X
j2S|[R]i,j |
�1,LB (RS) �1 (RS) �1,UB (RS)
�1,LB (RS) = m+s
(|S|� 1)12
�1,UB (RS) = m+ s(|S|� 1)12
m =Tr(RS)
|S| , s =
✓Tr(R2
S)
|S| �m2
◆ 12
[4] H. Wolkowicz and G. P. Styan, “Bounds for eigenvalues using traces,” Linear Algebra and Its Applications, pp. 471-506, 1980
�1,LB (RS) �̂1 (RS) �1,UB (RS)
* Lower & upper bound on the largest singular value (exact value)
Property of the approximate largest singular value
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Dynamic subarray partitioning algorithm (proposed)
f (S, n, r) ,(
0, if |S| = 0 or {n = NRF and r = 0}1|S|
Pi2S
Pj2S |[R]i,j |, otherwise.
Complexity: O(NTX2 logNTX)
Complexity: O(NTX2NRF)
Total complexity: O( NTX
2 max(NRF,logNTX) )
Complexity: O(NTX2NRF)
è Polynomial time !
Dynamic subarray
partitioning algorithm
R, NRF, NTX inpu
t ou
tput
S1, ...,SNRF
Channel model for simulations
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Azimuth domain Elevation domain
range of cluster angles [-ΔAZ, ΔAZ]
bore- sight
range of cluster angles [-ΔEL, ΔEL]
bore- sight
… …
… …
Cluster 1
Cluster Ncluster
…
Nsubray subrays (w/ angle spread 2o)
a (�p, ✓p) =h1 ej
2⇡d� sin(�p) · · · ej
2⇡d� (NANT�1) sin(�p)
iT
a (�p, ✓p) =h1 · · · ej
2⇡d� (m sin(�p) sin(✓p)+n cos(✓p)) · · · ej
2⇡d� ((NW�1) sin(�p) sin(✓p)+(NH�1) cos(✓p))
iTfor ULA (uniform linear array) for UPA (uniform planar array)
H[k] =NclusterX
c=1
NsubrayX
r=1
↵c,r!⌧c,r [k]aR(�R,c,r, ✓R,c,r)a⇤T(�T,c,r, ✓T,c,r) !⌧p [k] =
D�1X
d=0
p(dTs � ⌧p)e� j2⇡kd
K
* Distribution of AoAs and AoDs : Uniform in a confined range TX RX
18
Simulation results
-10 -5 0 5 10 15 20SNR (dB)
0
5
10
15
20
25
30
35
40
Spec
tral e
ffici
ency
(bps
/Hz)
Digital basebandHybrid, fully-connectedHybrid, subarray, dynamic (proposed)Hybrid, subarray, fixed (vertical)Hybrid, subarray, fixed (horizontal)Hybrid, subarray, fixed (squared)Hybrid, subarray, fixed (interlaced)
4.5 5 5.518
18.5
19
- TX: 64 antennas (UPA), 8 RF chains - RX: 4 antennas (UPA), 4 RF chains - CH: 8 clusters 10 subrays (ΔAZ=180°, ΔEL=90°)
16 64 144 256Number of transmit antennas
0
5
10
15
20
25
30
35
Spect
ral e
ffic
iency
(bps/
Hz)
Digital basebandHybrid, fully-connectedHybrid, subarray, dynamic (proposed)Hybrid, subarray, fixed (vertical)Hybrid, subarray, fixed (horizontal)Hybrid, subarray, fixed (squared)Hybrid, subarray, fixed (interlaced)
- TX: NTX antennas (UPA), 4 RF chains - RX: 4 antennas (UPA), 4 RF chains - CH: 8 clusters 10 subrays (ΔAZ=180°, ΔEL=90°), SNR 10 dB
- solid: w/o phase shifter constraint ( ) - dashed: w/ phase shifter constraint ( )
F?RF
]F?RF
Conclusions
19
We derived closed-form solutions for wideband hybrid precoding.
Fully connected structure Partially-connected structure (Subarray structure)
We proposed a dynamic subarray structure based on spatial channel covariance.
20
Thank you !