University of Tehran Department of Electrical and Computer Engineering
Electrical & Computer Engineeringxxia/Math_EE.pdf · Xiang-Gen Xia (. 夏香根) . Department of...
Transcript of Electrical & Computer Engineeringxxia/Math_EE.pdf · Xiang-Gen Xia (. 夏香根) . Department of...
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Xiang-Gen Xia (夏香根)
Department of Electrical and Computer Engineering University of Delaware
特拉华大学 Newark, DE 19716, USA
Email: [email protected] & [email protected] URL:http://www.ee.udel.edu/~xxia
数学与电子工程
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提纲 我的单位 电子工程里的数学 数字通信里的数学 初等数学及其应用 高等数学及其应用 正交设计在现代通信里的应用
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美国特拉华州 (State of Delaware) 特拉华大学 (University of Delaware) 电子与计算机工程系 (Department of Electrical and Computer Engineering)
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特 拉 华 州 第 二 小 州
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纽约市
费城
华盛顿DC
特拉华州
巴尔的摩市
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Our Location at the Center of the East Coast of the United States
University of Delaware 特拉华大学1743年起,是全美最老的大学之一 比普林斯顿大学还老
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秋天的校园
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Famous Alumni • Joe Biden, Vice President of the
USA.
• Chris Christie, Governor of New Jersey and potential presidential candidate.
• Joe Flacco, NFL Super Bowl MVP (most valuable player).
• Xin Wang, builder of RenRen Net (人人网)
• Wayne Westerman, inventor of multi-touch interface.
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Famous Faculty • Dave Farber, Internet pioneer. Pioneer’s Circle of Internet Hall of Fame 网络先驱者名人墙
• Dave Mills, Internet pioneer and
inventor of the Network Time Protocol.
• Richard Heck, 2010 Nobel Prize in Chemistry.
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Evans Hall Home of ECE Department 电子与计算机工程系
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Innovating in leading tech sectors FingerWorks, a company started by Electrical and Computer Engineering Professor John Elias and UD alumnus Wayne Westerman, developed the key technology in the iPhone’s multi-touch interface.
“The iPhone would not have been possible without the engineering solutions of Professors John Elias and Wayne Westerman of the University of Delaware who developed multi-touch sensing capabilities” --- Steve Jobs’ biography
2005年Apple公司买了 FingerWorks公司后 2007年才有iPhone
而真正的智能手机又是从iPhone开始的
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从我在美国过去28年的生活来看,其它方面没有改进(衣,食,住,行),只有通信等电子产品改进了老百姓的日常生活。 在所有电子产品里,由于芯片速度的增加,通信/计算机改变得最大
智能手机的出现改变了人们的日常生活。。。
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Electrical Engineering
Communications Control Computers
Mathematics
Signal Processing
Communications Signal Processing
Radar and Sonar Signal Processing
BiomedicalSignal Processing
Devices Algorithms
Physics Chemistry
Radar/Sonar imaging
Fourier transform
Medical imaging
Radon transform and Fourier transform
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Communications and Math
Analog Communications1G
Digital Communications2G, 3G, 4G 5G
Complex analysisDifferential equationsLinear algebra
∑ −n
n nTtps )(
Transmitter Receiver
How to design these signalsto be transmitted
Modern algebraCombinatoricsGeometry Algebraic geometryNumber theoryAlgebraic number theory
How to design themwaveforms
Real analysisFunctional analysisHarmonic analysisNumerical analysis
A hot topic in 5G now
How to receive them
Probability theoryStatistics Linear algebra
It is YOU to design both of transmitter and receiver: Math plays a perfect andtruly useful role here
摸拟通信 数字通信
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初等数学及现代通信
圆周上等分点之和等于零
∑ = 0
Discrete Fourier Transform (DFT)
10
),(1
0
2
−≤≤
=∑−
=
Nk
kNeN
n
Nknj
δπ
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初等数学及现代通信
• DFT 与OFDM
• 现在的 无线通信:4G/LTE 和 WiFi
)DFT()DFT()DFT( x(n)h(n)x(n)h(n) •=⊗
• 宽带信道:线性卷积
)()1()1()1()()0()( LnxLhnxhnxhny −−++−+=
• 加和去CP 后 线性卷积变成循环卷积
)()()( nxnhny ⊗=
)()()( kXkHkY =
DFT
OFDM
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10?,1
0
22
−≤≤=∑−
=
NkeN
n
Nknj π
问题:
请看我写的 Discrete Chirp-Fourier Transform (DCFT) IEEE Trans. on Signal Processing, Nov. 2000.
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SpeechImageVideo
compression
01101
binary to complex mapping
digitaltoanalog
x[n] x(t)
SpeechImageVideo
decompression
01101
binary to complex demapping
analogtodigital
y[n] y(t)
transmitter
receiver
error correctioncoding
000111111000111
error correctiondecoding
010110111000101
complex value to complex value channely = x + w
binary to binarychannel
Digital Communication System Block Diagram
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o Real data, such as speech/image/video, are collected and converted to binary sequences00110100111000110101100110
o Binary sequences are mapped to complex numbers
o Received signal (the step of waveforms is skipped):y = x + w
where x is a transmitted value and takes one of 1, i, -1, -i called a signal constellation (QPSK ) and w is the noise
o How to decide what x is and what x represents? The half of the minimum distance between any two constellation points
on the complex plane is the tolerable level of noise
o A signal constellation design: to find a finite set of a fixed number of complex numbers with a fixed sum of all the norms such that its minimum distance is maximized. This is related to sphere packing: how to design 6 or more points is still open and
it is conjectured that the equilateral triangular lattice points are optimal (this was shown asymptotically)
Digital Modulation and Demodulation
00
10
11
01 y Received Value y
x is iand thusrepresents01
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Error Correction Coding
o How to correct binary errors? This leads to error correction coding The simplest error correction coding: repetition code
1111; 0000Assumed 111 is transmitted but 101 is occurred at the receiver
111101111: also compare the distances with the two codewords 111 and 000
find the one that is the closest to the received 101000
The distance between binary codewords is called Hamming distance The decoding is also the minimum distance decoding This simple code can correct one error but needs to expand three times
NOT a good code (code rate 1/3)o In practice, one prefers to simple encoding/decoding This leads to linear codes
x=Gs where s is a binary information vector, G is a binary encoding matrix called generator matrix, and x is a binary codeword
o Hamming code: input 4 bits, output 7 bits (code rate is 4/7)minimum Hamming distance 3correct one bit error
o Can we do better?? (The above arithmetics are over the binary field)
1110110110111000010000100001
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Finite Fields: A Perfect Application
1011
11
11
10
00
11
10
?=
1110
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Reed-Solomon Codes (RS Codes)
o Let α be a primitive element of GF(2𝑚𝑚) for a positive integer m (it is a non-zero binary vector of size m: for example, [0,1,0, … , 0]𝑇𝑇) o An (n,k) RS code (1960) has the following generator matrix with n=2𝑚𝑚 − 1
partial Vandermonde matrix
Its minimum Hamming distance is n-k+1 that is optimal for (n, k) linear codes o RS codes are used in all computer memory and hard drivers, and also in many other communications systems: One of the most useful and famous error correction codes o Reed received his Ph.D. in mathematics and was a USC professor (passed away in 2012) He has another famous code: Reed-Muller codes, where the concept of majority decoding was first used
=
−
−
−
−
)1(
)1(33
)1(22
1
1
111
knn
k
k
k
G
αα
αααααα
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Multiple Antennas (多天线系统)
o What we talked before is for single antenna:
o What to do for multiple antennas ? 4G, 5G, …
x(t)
Instead of designing a set of finite complex scalarvalues, such as, 1, i, -1, -i, to maximize its minimum distance, we need to design a set of matrices called a space-time code (STC) such that its minimum absolute value of the determinates of the difference matrices of any two distinguished matrices in the set is maximized, when the total energy is fixed:
The goal is to
When N=2 and L<6, the optimal 2 by 2 STC can be easily obtained by using 2 by 2 orthogonal matrices with spherical packing pointsWhen L=6, it does not hold anymore
}forwithmatricesare:,,{)( 2110 jiNXNNXXXXXLX
FijiL ≠=×≠= −
}10:)det(min{)( −≤≠≤−= LjiXXd jiLX
}max{ )(LXd
x1(t)
x2(t)
xN(t)
.
.
.
time/时间
spac
e/空间
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00 01 11 10 1 j -1 -j
Every time slot
1001
−
−10
01
−
−0110
0110Every two time slots
One Tx antenna
Two Tx antennas
1st time slot 2nd time slot
1st Tx antenna
2nd Tx antenna
Bits to complex number mapping
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Optimal 2 by 2 Code of Six 2 by 2 Unitary Matrices
o Let
121
2
2),2/arccos(23/)1(,8/31,222/5
θπθθ −=−=
−=−=+−=
adabdad
22/
522/
4
32
10
21 ,
,
,
IeXIeX
babbbbba
Xbabbbbba
X
babbbbba
Xbabb
bbbaX
θθ ii
iiii
iiii
iiii
iiii
−==
−−−−−+−
=
+−++−−−
=
−−+−++−
=
+−−−−−−
=
Wang-Xia’04
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Liang-Xia’02
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Another Way to Construct Space-Time Codes
Binary bits are first mapped to complex valued symbols xi and these xi are embeddedinto an N by N matrix : Example, the well-known Alamouti code:
∈
−
== S2112
21 ,scalars:**
xxxxxx
XC
o Use orthogonal designs (compositions of quadratic forms) This leads to orthogonal space-time codes
o Use cyclic division algebra This leads to non-vanishing determinate codes Heavily involve with algebraic number theory, such as cyclotomic fields
2122
22
1 andanyfor)( xxIxxXX H +=
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Alamouti code from 2 by 2 orthogonal design
Alamouti Code for 2 Transmit Antennas
(1998)
** 12
21
xx
xx
Information bits
are mapped to
complex symbols
x1 and x2
encoder
** 12
21
xx
xx
-x2* x1
x1* x2
It is an option in 3G
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Alamouti Scheme: Fast ML
Decoding and Full Diversity
Signal Model:
Y=CA+W,
where
is a signal constellation, for example
S21
12
21,:
**xx
xx
xxC C
},1{ jS
S
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Alamouti Code: Fast ML Decoding
ML decoding is to minimize
Orthogonality:
for any values x1 and x2.
The cross term x1x2 can be canceled and x1 and x2 can be separated:
x1 and x2 can be decoded separately:
The decoding complexity is reduced from to
}{}{}{
)}(){(|||| 2
CCAAtrYCACAYtrYYtr
CAYCAYtrCAY
HHHHHH
HF
22
22
1 )|||(| IxxCCH
)()(|||| 2211
2 xfxfCAY F
)(minand)(minmin 2211),( 21
221
xfxfSxSxSxx
2|| S
||2 S i.e., complex symbol-wise decoding
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For any two different matrices
Their difference matrix is also orthogonal
Because of the orthogonality, B has full rank
Alamouti Code: Full Rank Property
),(),(~
),(~~
,),(
2121
*
1
*
2
21
21*
1
*
2
21
21
yyxxCC
yy
yyyyCC
xx
xxxxCC
),(*)(*)(
)~
,( 2211
1122
2211yxyxC
yxyx
yxyxCCB
22
222
11 )|||(|)~
,())~
,(( IyxyxCCBCCB H
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General Size
CHC
Y - CA
H
Y - CA
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For L=2 transmit antennas:
k=p=2
Rate R=k/p=1
12
21
2xx
xxL
Its proof is given in next slide.
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Proof of rate : pkeiR .,.,1
.so,spaceldimensionain
vectorstindependenlinearlymostatare
Theret.independenlinearlyare1size
of,,...,2,1,)(vectorsthatprovesThis
.,...,2,1,00)()(
Thus.)()(
,ofityorthogonalthetoDue.ofcolumns
firsttheare)(where,)()(
:)(,ofcolumnfirst
the.matricesconstantrealreal
are,,...,2,1,where,Let
1
11
1
2
11
1
1
11
1
1
pkp
p
p
kiA
kixCC
xCC
CA
AxAC
CC
Considernp
kiAxAC
i
i
T
k
i
i
T
i
i
k
i
ii
i
k
i
ii
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For 8 transmit antennas:
– k=p=8
– Rate R=k/p=1
12345678
21436587
34127856
43218765
56781234
65872143
78563412
87654321
8
xxxxxxxx
xxxxxxxx
xxxxxxxx
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xxxxxxxx
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xxxxxxxx
L
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• A1, A2,…, An of size and B1, B2,…, Bk of size are two
families of Hurwitz matrices if and only if the following two C are real
orthogonal designs where
and Two different representations
• There are n square Hurwitz matrices A1, A2,…, An of size by
using Clifford algebra with k=p rate=1
• There are p Hurwitz matrices B1, B2,…, Bk of size with
kp np
kk xBxBC 11
xx nAAC ,,1 T
kxx ],,[ 1 x
pp
)( pn np
)( pn
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The basic problem for real orthogonal designs or real space-time
block codes for PAM signals is solved.
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12
2
12
21
1
xx
xx
xx
xx
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Amicable family: family of matrices of the
same size forms an
amicable family if
is a
complex orthogonal design iff
{Ai,Bi,i=1,…,k} is an Amicable design.
ts BBAA ,,,,, 11
tjsiABBAiii
tji
sji
BBBB
AAAAii
tjsiIBBAAi
i
H
jj
H
i
i
H
jj
H
i
i
H
jj
H
i
j
H
ji
H
i
1,1,)(
,1
,1
,0
,0)(
1,1,)(
k
i
iiiik xBxAxxC1
1 )Im()Re(),,( i
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However the other representation
does not work!
Use a different representation
where
T
kk
nk
xxxx
AAxxC
))Im(),Re(,),Im(),(Re(where
,,,),,(
11
11
x
xx
xxxx nn BABA ,,11
T
kxx ),,( 1 x
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Two Questions
Can a non-square p by n complex
orthogonal design have rate 1, i.e., k=p,
when n>2? If not, what is the bound?
How to construct rate over ½ complex
orthogonal designs?
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Rate Upper Bounds for Complex Orthogonal
Designs
Liang-Xia’03 showed that their symbol rates, k/p, is strictly less than 1 for more than 2 transmit antennas.
H.Wang-Xia’03 showed that their symbol rates, k/p, can not be above ¾ when n>2, and conjectured that their symbol rates are upper bounded by
Su-Xia’03 first showed that ¾ holds for n>2 when no linear processing is allowed.
Liang’03 showed that this conjecture holds when no linear processing is allowed.
H.Wang-Xia’03showed that for a p by n generalized complex orthogonal design, the rate is upper bounded by 4/5 when n>2.
22
12
n
n
p
k
For 3 and 4
transmit antennas
Rates 4
3
p
k
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• (Wang-Xia’03) The above rate upper bounds hold for a
finite QAM (excluding PSK or PAM) signal constellation.
• (Liang 2003, Su-Xia-Liu 2004, Lu-Fu-Xia 2005)
constructed complex orthogonal designs with the above rates
and the constructions by Lu-Fu-Xia 2005 have closed-
forms.
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Closed Form for COD Self-Similarity
Construct COD Bn+2, Bn+1 from Bn
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Construction Unites for n=2k-1
nn d is variablescomplex nonzero of number COD, np : nB
nn BB as riablescomplex va ofset same vector,1p : n
nn BB as riablescomplex va ofset same vector,1q :ˆn1,
nm,n
n,n
d v
BQQ
variablescomplex nonzero of number COD, nq : nm,nm, 0
m,n
n,n
Qa
BQQ
s riablescomplex va ofset same
vector,1q : 0n1,-mnm,
m,n
n,n
Qa
BQQ
s riablescomplex va ofset same
ˆˆ COD, 1q :ˆ0n1,mnm,
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A Theorem (Lu-Fu-Xia’05)
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Orthogonality among Units
)()1()(
)()(
iBjB
jBiB
n
k
n
nn12 where
)(ˆ)(ˆ
)()(k- n
iBjB
jBiB
nn
nn
)()(
)()(
,1
,1
iBjQ
jQiB
nn
nn
)(ˆ)(ˆ
)()(
,,
,,
iQjQ
jQiQ
nmnm
nmnm
)(ˆ)(
)()(
,,1
,1,
iQjQ
jQiQ
nmnm
nmnm
)(ˆ)(
)()(
,1
,1
iBjQ
jQiB
nn
nn
)(ˆ)(
)()(ˆ
,2
,2
iBjQ
jQiB
nn
nn
)(ˆ)(
)()(ˆ
,1,1
,1,1
iQjQ
jQiQ
nmnm
nmnm
All the above matrices are COD
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Orthogonal Property among Units
Bn(i) has the same structure of Bn, but the indices of nonzero complex
variables in Bn(i) are from (i-1)dn+1 to idn, where dn is the number of
nonzero complex variables in Bn.
123
*
2
*
13
*
3
*
12
*
3
*
21
0
0
0
0
xxx
xxx
xxx
xxx
B4 with d4 =3 B4( i )
1)1(32)1(33)1(3
*
2)1(3
*
1)1(33)1(3
*
3)1(3
*
1)1(32)1(3
*
3)1(3
*
2)1(31)1(3
0
0
0
0
iii
iii
iii
iii
xxx
xxx
xxx
xxx
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Inductive Construction for n+1 and
n+2
)1()1()2(
)2()1(1
n
k
n
nn
nBB
BBB
)2(ˆ)3(ˆ)4(
)1()1()4()3(
)4()1()1()2(
)3()2()1(
,1
,1
,1
2
nnn
n
k
nn
nn
k
n
nnn
n
BBQ
BQB
QBB
BBB
B
)1(ˆ
)2(
)3(
)4()1( ,1
2
n
n
n
n
k
n
B
B
B
Q
B
)4(ˆ
)3(ˆ
)2(ˆ
)1()1(
ˆ
,1
2
n
n
n
n
k
n
Q
B
B
B
B
)2(ˆ)3(ˆ)4(
)1(ˆ)4()3(
)4()1(ˆ)2(
)3()2()1(
,,,1
,1,1,
,1,1,
,,,1
2,
nmnmnm
nmnmnm
nmnmnm
nmnmnm
nm
QQQ
QQQ
QQQ
QQQ
Q
)4(
)3(
)2(
)1(
,1
,
,
,1
2,
nm
nm
nm
nm
nm
Q
Q
Q
Q
Q
)4(ˆ
)3(ˆ
)2(ˆ
)1(ˆ
ˆ
,1
,
,
,1
2,
nm
nm
nm
nm
nm
Q
Q
Q
Q
Q
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Orthogonality for New Units
)()1()(
)()(
iBjB
jBiB
n
k
n
nn
COD
)()1()(
)()(
22
22
iBjB
jBiB
n
k
n
nn
?
)()1()(
)()(
22
22
iBjB
jBiB
n
k
n
nn
)1(ˆ)1()6(ˆ)7(ˆ)8(
)2()1()5()1()8()7(
)3()1()8()5()1()6(
)4()7()6()5(
)5(ˆ)2(ˆ)3(ˆ)4(
)6()1()1()4()3(
)7()4()1()1()2(
)8()1()3()2()1(
1
,1
,1
1
,1
,1
,1
,1
,1
,1
n
k
nnn
n
k
n
k
nn
n
k
nn
k
n
nnnn
nnnn
nn
k
nn
nnn
k
n
n
k
nnn
BBBQ
BBQB
BQBB
QBBB
BBBQ
BBQB
BQBB
QBBB
COD
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Rate Formula
nn
nmnmnmnm
nnn
nn
nmnmnmnm
nnn
dv
mvvvv
vvv
pq
mqqqq
qqq
,0
,1,,12,
,1,02,0
,0
,1,,12,
,1,02,0
0,2
3
0,2
3
)!1()!(
)!12(
)!()!1(
])1(
[)!2(
12,
12,
mkmk
kv
mkmk
k
mmkk
q
km
km
k
k
q
vR
k
k
k2
1
12,0
12,0
12
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Design Examples
)2(ˆ)3(ˆ)4(
)1()1()4()3(
)4()1()1()2(
)3()2()1(
111,1
11,11
1,111
111
3
BBQ
BQB
QBB
BBB
Bk
k
11 xB
23
*
13
*
12
*
3
*
21
0
0
0
xx
xx
xx
xxx
*
11 xB 11ˆ xB 01,1 Q 01,1 Q 1,1Q̂
*
12
*
21
11
11
2)1()1()2(
)2()1(
xx
xx
BB
BBB
k
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Design Examples
156
246
345
654
423
513
612
321
33
33
4
0
0
0
0
0
0
0
0
)1()1()2(
)2()1(
xxx
xxx
xxx
xxx
xxx
xxx
xxx
xxx
BB
BBB
k
COD for 4 antennas
p = 6, d = 8, R = 3/4
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)2(ˆ)3(ˆ)4(
)1()1()4()3(
)4()1()1()2(
)3()2()1(
333,1
33,13
3,133
333
5
BBQ
BQB
QBB
BBB
Bk
k
COD for 5 antennas
p = 15, d = 10, R = 2/3
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33 than ' COD size (half)smaller one exists thereodd, isk if 1,-2k n nn BB
)1(ˆ)2(ˆ)3(ˆ)4(
)2()1()4()3(
)3()4()1()2(
)4()3()2()1(
'
,1
,1
,1
,1
3
nnnn
nnnn
nnnn
nnnn
n
BBBQ
BBQB
BQBB
QBBB
B
332
1' nn pp
Smaller Size COD for n=4l
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Smaller Size COD for n=4l
123
213
312
321
1111,1
111,11
11,111
1,1111
4
0
0
0
0
)1(ˆ)2(ˆ)3(ˆ)4(
)2()1()4()3(
)3()4()1()2(
)4()3()2()1(
'
xxx
xxx
xxx
xxx
BBBQ
BBQB
BQBB
QBBB
B
COD from our design for 4 antennas: d = 3, p = 4, R = 3/4
----- coincides with the existing one
Liang’s and Su-Xia-Liu’s: d = 6, p = 8, R = 3/4
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• A design example for n=8
transmit antennas.
• In this case, d=35, p=56.
• Rate =d/p=5/8
• This construction is inductive
for all n with closed-forms
• Liang’s and Su-Xia-Liu’s:
d = 70, p = 112, R = 5/8
• These constructions do
not have closed-forms
and computer-aid or
manual help is needed
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COD Construction Comparison
d p d p Rate=d/p
1 1 1 1 1 1
2 2 2 2 2 1
3 3 4 3 4 3 / 4
4 6 8 3 4 3 / 4
5 1 0 1 5 1 0 1 5 4 / 6
6 2 0 3 0 2 0 3 0 4 / 6
7 3 5 5 6 3 5 5 6 5 / 8
8 7 0 1 1 2 3 5 5 6 5 / 8
9 1 2 6 2 1 0 1 2 6 2 1 0 6 / 1 0
1 0 2 5 2 4 2 0 2 5 2 4 2 0 6 / 1 0
1 1 4 6 2 7 9 2 4 6 2 7 9 2 7 / 1 2
1 2 9 2 4 1 5 8 4 4 6 2 7 9 2 7 / 1 2
1 3 1 7 1 6 3 0 0 3 1 7 1 6 3 0 0 3 8 / 1 4
1 4 3 4 3 2 6 0 0 6 3 4 3 2 6 0 0 6 8 / 1 4
Liang &Su-Xia-Liu Lu-Fu-Xian
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结束语 ·
Binary field Finite fields Complex number field Quaternionic numbers Octonionic numbers Norm identities (Composition formulas)
yxyx •=•
四元数体 八元数体
代数:
分析: yxyx •≤•
Perfect application at the transmitter side
Optimal receiver When dot is inner product, it is the Schwarz inequality matched filter (匹配滤波器) When dot is addition, it is the triangular inequality When dot is multiplication, it is the norm inequality
Counting: 数准了就是代数,数不准就是分析 根本就数不清楚 是几何拓扑
Algebraic number fields
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结束语·
电子工程中有材料和算法两大块, 而算法就是应用数学
纵观过去几十年,人们在生活上最大的变化就是在通信上的变化
数学在通信里的应用起着非常重要的作用
所有的数学都是有 用的,你不知道哪天就会用上
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Thank You