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: xxia@ee.udel.edu & xianggen@gmail.com URL：http://www.ee.udel.edu/~xxia

数学与电子工程

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

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)

特 拉 华 州 第 二 小 州

纽约市

费城

华盛顿DC

特拉华州

巴尔的摩市

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

秋天的校园

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.

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.

Evans Hall Home of ECE Department 电子与计算机工程系

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

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。。。

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

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

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

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

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.

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

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

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

Finite Fields: A Perfect Application

1011

11

11

10

00

11

10

？=

1110

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

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

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

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

Liang-Xia’02

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 +=

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

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

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

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

General Size

CHC

Y - CA

H

Y - CA

For L=2 transmit antennas:

k=p=2

Rate R=k/p=1

12

21

2xx

xxL

Its proof is given in next slide.

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

For 8 transmit antennas:

– k=p=8

– Rate R=k/p=1

12345678

21436587

34127856

43218765

56781234

65872143

78563412

87654321

8

xxxxxxxx

xxxxxxxx

xxxxxxxx

xxxxxxxx

xxxxxxxx

xxxxxxxx

xxxxxxxx

xxxxxxxx

L

• 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

The basic problem for real orthogonal designs or real space-time

block codes for PAM signals is solved.

12

2

12

21

1

xx

xx

xx

xx

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

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

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?

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

• (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.

Closed Form for COD Self-Similarity

Construct COD Bn+2, Bn+1 from Bn

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,

A Theorem (Lu-Fu-Xia’05)

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

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

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

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

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

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

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

)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

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

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

• 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

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

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

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

Thank You