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MIMO Maximum Likelihood Soft Demodulation Based on Dimension Reduction Jungwon Lee∗ , Ji-Woong Choi? , Hui-Ling Lou† ∗
Mobile Solutions Lab, Samsung US R&D Center, San Diego, CA ? DGIST (Daegu Gyeongbuk Institute of Sci. and Tech.), Korea † Marvell Semiconductor, Santa Clara, CA GLOBECOM 2010 Dec. 8, 2010
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Outline
•
Introduction
•
System Model
•
Problem Formulation
•
Dimension Reduction Approach
•
Complexity Analysis
•
Simulation Results
•
Conclusions 2 / 17
Introduction
Spatial multiplexing multi-input multi-output (MIMO) systems - Can increase the data rate linearly with the number of antennas. - Need to employ a receiver that eectively handles the interference among the multiple spatial streams. • Receivers for spatial multiplexing MIMO - Hard detectors •
• Equalizers: linear equalizer (LE) and decision feedback equalizer
(DFE)
• Maximum likelihood (ML) hard detectors
- Soft demodulators
• Soft demodulators using an equalizer output • MIMO ML soft demodulators: soft version of MIMO ML hard
detectors
•
We present a novel approach for MIMO ML soft demodulation. - In practical wireless systems, soft demodulators are almost exclusively used. - MIMO ML soft demodulation produces best performance in general. 3 / 17
System Model (1)
•
Spatial multiplexing MIMO transmitter and receiver - NT transmit antennas - NR receive antennas - NS spatial streams with NS ≤ min{NR , NT }
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System Model (2)
•
Receive signal model y = Hx + z
- y ∈ C NR ×1 : receive signal vector - H ∈ C NR ×NS : eective channel matrix including MIMO precoding • Known at the receiver.
- x ∈ C NS ×1 : transmit symbol vector - z ∈ C NR ×1 : circularly symmetric complex Gaussian random noise vector f (z) =
1
π NR
exp −||z||2
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Problem Formulation- MIMO ML Hard Detection
•
MIMO ML hard detection - Finding the transmit symbol vector that is most likely transmitted: ˆ = arg max f (y|x) x x∈X
• X : set of all possible M NS transmit symbol vector candidates with the modulation order of M • f (y|x) = N1R exp −||y − Hx||2 π
- Equivalent to nding the transmit symbol vector that minimizes the Euclidean distance (ED) ||y − Hx||2 between y and Hx: ˆ = arg min ||y − Hx||2 x x∈X
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Problem Formulation- MIMO ML Soft Demodulation
•
MIMO ML soft demodulation - Calculating log likelihood ratio (LLR) of each coded bit bs,n , the n-th bit of the s-th stream: L (bs,n ) = log
P {y|bs,n = 1} P {y|bs,n = 0}
- Almost the same as calculating approximate LLR derived using Max-Log-MAP approximation: ˜ (bs,n ) , min ky − Hxk2 − min ky − Hxk2 L (0)
x∈Xs,n
(1)
x∈Xs,n
(b) • Xs,n : set of transmit symbol vector candidates with bs,n =b
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Problem Formulation- Hard vs Soft •
MIMO ML hard detection ˆ = arg min ||y − Hx||2 x x∈X
- There exist low-complexity algorithms that try to nd the optimum x without calculating all the EDs for all transmit symbol vectors. • Sphere decoding, M-algorithm, K-best, etc.
•
MIMO ML soft demodulation
˜ (bs,n ) , min ky − Hxk2 − min ky − Hxk2 L (0)
x∈Xs,n
(1)
x∈Xs,n
- Looks similar to the hard detection problem. - Much more di cult in reality.
• The search space for the transmit symbol vector with the minimum (b) ED is limited to Xs,n . (0) (1) • Partitioning transmit symbol vector candidates into Xs,n and Xs,n is all dierent depending on s and n. 8 / 17
Dimension Reduction Approach (1) •
We propose a dimension reduction approach for soft demodulation. - Central idea: reduce the dimension for soft demodulation.
•
Assume that we are interested in calculating the LLRs only for the last NSso streams and we don't care about the rst NSha streams, where NSha + NSso = NS .
•
Partition the transmit symbol vector and the channel matrix: y = [Hha Hso ]
xha xso
+z
= Hha xha + Hso xso + z.
- xso ∈ C NS ×1 so and xha ∈ C NS ×1 . ha so NR ×NS - H ∈C and Hha ∈ C NR ×NS . so
ha
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Dimension Reduction Approach (2)
•
LLR for the n-th bit of the s-th stream for NS − NSso + 1 ≤ s ≤ NS : ˜ (bs,n ) = L
min ky − Hxk2 − min ky − Hxk2 (0)
(1)
x∈Xs,n
x∈Xs,n
=
ha
so,(0)
min so,(1)
xso ∈Xs,n so so - yha (xso ), y−H x
xha xso
xha ∈X ha
(b) - Xs,n =
ha ha 2
min ky (x ) − H x k
min xso ∈Xs,n
−
so
ha
so
ha ha 2
min ky (x ) − H x k
xha ∈X ha
ha so,(b) x ∈ X ha , xso ∈ Xs,n
.
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Dimension Reduction Approach (3) •
Dimension reduction soft demodulation for the last NSso streams 1 For each xso , use an e cient MIMO ML hard detector to nd the transmit symbol subvector xˆ ha (xso ) with the minimum ED without actually calculating all EDs: ˆ ha (xso ) = arg min kyha (xso ) − Hha xha k2 . x xha ∈X ha
2 Calculate the corresponding minimum ED: ˆ ha (xso )k2 . D(xso ) = kyha (xso ) − Hha x
3 After calculating the minimum EDs for all xso , calculate the LLR: ˜ (bs,n ) = L
min so,(0)
xso ∈Xs,n
•
D(xso ) −
min
D(xso ).
so,(1)
xso ∈Xs,n
LLR for other than the last NSso streams can be calculated by rearranging the transmit symbol vector and solving the same problem. 11 / 17
Complexity Analysis •
Complexity measure - Average number of visited nodes in a tree search
•
Complexity comparison NS
Dimension reduction Exhaustive
4
406 (NSso =1) 2,098 (NSso =2) 8,465 (NSso =3) 69,905
8 6,344 (NSso =1) 29,618 (NSso =2) 1.34×106 (NSso =4) 4.58×109
- M = 16 - Hard detector used for dimension reduction approach: sphere decoder •
To reduce the complexity further, a suboptimal hard detector can be used instead of an optimal hard detector. 12 / 17
Simulation Results (1)
•
Simulation Parameters - General setting: WiMAX (IEEE 802.16) radio conformance test - Convolutional turbo code with code rate of 1/2 - Vehicular-A 60 km/h - High antenna correlation model with 4 transmit and 4 receive antennas - 4 spatial streams
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Simulation Results (2)
for dimension reduction soft demodulation (DRSD) (Resulting in the lowest complexity.) • 16 QAM for all streams • NSso = 1
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Conclusions
•
We proposed a dimension reduction soft demodulator (DRSD). - A DRSD reduces the dimension of the search space for soft demodulation. - A DRSD relies on a hard detector for the rest of the dimension.
•
A DRSD with an optimal hard detector signi cantly reduces complexity compared to the exhaustive ML method.
•
A DRSD with a suboptimal hard detector achieves further complexity reduction with little performance degradation.
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Appendix- Additional Simulation Results (1)
• NS = 2, NSso = 1,
and 64 QAM for all streams
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Appendix- Additional Simulation Results (2)
• NS = 4, NSso = 1,
and 4 QAM for all streams
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