Let be a. SBTC codeword matrix with. L-PSK symbol constellation and. â« Previous super-orthogonal codes: â SOSTTC [Jafarkhani, 03'] : â SOSTBC [Lee, 04'] :.
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Expanded Super-Orthogonal Space-Time Block Codes with Systematic Construction Yuan-Wen Ting, Wei-De Wu, and Chi-chao Chao
Institute of Commun. Eng., National Tsing Hua University, Hsinchu, Taiwan
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Outlines
Lab UWB Project Status Introduction Generic Codeword Format of Expanded SuperOrthogonal Space-Time Block Codes ( ESOSTBCs ) Systematic Construction of ESOSTBC Codebooks Simulation Results Conclusion
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Lab UWB Project Status
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Under-Sampled UWB
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Results
[ Wu, ISIT’06 ]
[Chiu, ICC’05] [Chan, VTC’05] [Ting, NST’06]
[Wu’07, to be submitted]
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Next Step
A more generalized framework of What if not restricted to OFDM ? under-sampled communications !
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Introduction
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MISO Analogy in Under-Sampled UWB Mt × K subcarriers
under-sampling
K subcarriers
Enable the rich applications of space-time coding to under-sampled UWB communications
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Application of Space-Time Block Codes
Space-time block codes (STBCs) are attractive choices:
For 2x1 MISO channel, the Alamouti code is optimal Full diversity and simplified maximum-likelihood (ML) decoding from the orthogonal codeword structure
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Spectral Efficiency Issue for STBC
The maximal rate of a square STBC fall off exponentially with the number of transmit antennas [Su et al, 03’].
Possible solutions to improve the spectral efficiency:
Expanded symbol constellation
Performance degradation and implementation difficulties
Expanded codebook size
Super-orthogonal space-time codes, e.g., SOSTTC [Jafarkhani, 03’] and SOSTBC [Lee, 04’] Higher spectral efficiency without expanding symbol constellation. Besides, simplified ML decoding can be preserved. Can we do better than the previous works ?
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Signal Model and Considered Metrics
Signal Model:
ˆ 0) Y m×t = H m×n S(x n×t + N m×t ˆ 0 )is an n × t ESOSTBC codeword with where S(x L-PSK symbol entries and governed by x = {x, x+ } with x+ the additional freedoms by the codeword expansion.
we will consider ∆H,min , ∆p,min and ∆E,minwhich correspond to the minimum rank, rank minimum product of positive eigenvalues and minimum trace of all pairwise codeword distance matrices of the form: D(x01 , x02 )
£ ¤£ ¤H 0 0 0 0 ˆ ˆ ˆ ˆ = S(x1 ) − S(x2 ) S(x1 ) − S(x2 ) .
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Generic Codeword Format of Expanded ( ESOSTBCs )
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Expanded Codeword Formats
Let S n (x) be a n × n SBTC codeword matrix with L-PSK symbol constellation and w ≡ 2π/L.
Previous super-orthogonal codes:
ˆ 2 (x0 ) = LS 2 (x) SOSTTC [Jafarkhani, 03’] : S 0 ˆ (x ) = S 4 (x)RP S 4 SOSTBC [Lee, 04’] : where L and R are diagonal matrices with diagonal entries of the form ej2πwθ , and P is a permutation matrix.
Proposed:
ˆ n (x0 ) = P 1 LS n (x)RP 2 S
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ESOSTBCs by Phase Rotations Only
A recursive construction for square STBCs: ¸ ∙ ∗ −xr+1 I 2r−1 S 2r−1 (x1 , x2 , . . . , xr ) S 2r (x1 , x2 , . . . , xr+1 ) = xr+1 I 2r−1 SH 2r −1 (x1 , x2 , . . . , xr )
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Remarks
⎛
1 ⎜0 ˆ S4 = ⎜ ⎝0 0
0 1 0 0
⎞
⎛
0 0 ⎜ 0 0 ⎟ ⎟ S 4 (x1 , x2 , x3 ) ⎜ ⎝ 1 0 ⎠ 0 ejwθ1
⎞
1 0 0 0 0 ejwθ2 0 0 ⎟ ⎟ jwθ3 0 ⎠ 0 0 e 0 0 0 ejwθ4
The total number of additional degrees of freedom in n × n ESOSTBCs with phase rotations only :
1 + (22 − 1) + (23 − 1) + . . . + (2r − 1) = 2(2r − 1) − r
which increases about linearly with n (= 2r ).
Nevertheless, as the number of transmit antennas are limited in practice, we will focus on 4 × 4ESOSTBCs in the following discussions.
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Extension by Permutations ¢ ¡ P = eσ(1) eσ(2) eσ(3) eσ(4) will be abbreviated by (σ(1) σ(2) σ(3) σ(4)). For example, the abbreviation P = (e4 e3 e2 e1 ) is (4 3 2 1).
Selected permutation set: n P=
(1 2 3 4) , (4 3 2 1) , (2 1 4 3) , (3 4 1 2)
o
for
Can ensure that the minimum rank of ESOSTBC codewords with the same expanded phases equals to 4. Can matched the structure of ⎛
x1 ⎜ x2 S 4 (x1 , x2 , x3 ) = ⎜ ⎝ x3 0
−x∗2 x∗1 0 −x3
−x∗3 0 x∗1 x2
⎞
0 x∗3 ⎟ ⎟ ∗ ⎠ −x2 x1
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Generic 4 × 4 ESOSTBC Codeword Format
Property: P L(θ1 ) S 4 (x1 , x2 , x3 ) R(θ2 , θ3 , θ4 ) = L(θ10 ) S 4 (x01 , x02 , x03 ) R(θ20 , θ30 , θ40 ) P
Implication: Generic 4 × 4 ESOSTBC codeword format can be given by L(θ1 )S 4 (x1 , x2 , x3 )R(θ2 , θ3 , θ4 )P .
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ESOSTBC Codebook Representation
Let P 0 = (1 2 3 4), P 1 = (4 3 2 1), P 2 = (2 1 4 3), and P 3 = (3 4 1 2) and denote θ(i) = (θ1(i) , θ2(i) , θ3(i) , θ4(i) ).Then, a ESOSTBC codebook can be represented by
SOSTBC as a special case of ESOSTBC:
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Systematic Construction of ESOSTBC Codebooks
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Observations:
SOSTBC can guarantee minimum rank of 2.
Exhaustive search is not a good idea
How can we obtain a ESOSTBC codebook with better metric? The product space of all degrees of freedom can be very large.
Proposed two-phase strategy:
First find the phase conditions so that the minimum rank of the codeword difference matrix of any pair of codeword matrices can be larger than 3. Construct the codebook form valid phase parameters. Benefit: confined complexity in each phase
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Phase Conditions
Desiring
³ min0 rank L(θ1 )S 4 (x)R(θ2 , θ3 , θ4 )P
x, x
´
−L(θ10 )S 4 (x0 )R(θ20 , θ30 , θ40 )P 0 ≥ 3
is equivalent to³ requiring
´
min0 rank S 4 (x) − L(δ1 )S 4 (x )R(δ2 , δ3 , δ4 )P˜ ≥ 3 x,x
0
where P˜ = P 0 P −1 = P −1 P 0 ∈ {P 0 , P 1 , P 2 , P 3 } and (δ1 , δ2 , δ3 , δ4 ) can be determined according to
It suffices to find conditions on the phase differences!
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Codebook Construction
Can the enlarged codebook have minimum rank ≥ 3?
Repeat the procedure until the codebook cannot grow any further.
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ESOSTBC Codebook
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Simulation Results
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Decoding Algorithm
Can apply simplified ML decoding in the computation of T (s|s+ ).
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High Spectral Efficiency
QPSK ESOSTBCs and 8PSK ESOSTBCs in 4 × 1 MIMO systems
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Improved Performance
QPSK ESOSTBCs and SOSTBCs in 4 × 1 MIMO systems
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Cont’d
8-PSK ESOSTBCs and SOSTBCs in 4 × 1 MIMO systems
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Conclusion
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Summary
Generic ESOSTBC codeword format provides additional degrees of freedom for high spectral efficiency and improved performance metrics
Systematic codebook construction to obtain good codes at low complexity
