these ghouwayel

7/31/2019 These Ghouwayel

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tel00354490,version1

20Jan2009
http://hal.archives-ouvertes.fr/http://tel.archives-ouvertes.fr/tel-00354490/fr/


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CG(q)

CG(Ft)CG(Ft)

CG(2m)


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GF(2m

)GF(2m)

GF(2m)GF(2m)

GF(Ft)


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GF(Ft)GF(Ft)

GF(Ft)GF(2m)

2n + 12n + 1

Ft)

Ft

GF(2m)


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GF(2m)

GF(2m)

GF(2m)

GF(24)


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C

GF(2m)


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CG(Ft) Ft t

GF(Ft)

CG(Ft)

CG(Ft))

CG(Ft) CG(2n)

CG(2n)


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Filtrage

RF

Transp.

RF/FI

Filtrage

FI

Transp.

FI/FI

Filtrage

FI

Dmodu-

lateur

Conv.

A/N TNS

RF IF Bande

de base

a) Architecture super-htrodyne

Filtrage

RF

ConversionA/N

(large bande)TNS

RF RF

b) Architecture de Radio Logicielle idale

Filtrage

RF

Transp.

RF/FI

Filtrage

FI

Conversion

A/NTNS

RF FI

c) Architecture de Radio Logicielle Restreinte (FI Basse Frquence)

FiltrageRF

DmodulationRF/Bande

de base

ConversionA/N

TNS

RF

d) Architecture de Radio Logicielle Restreinte (Conversion directe)

Bande

de base


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2n

C

CG(q)C = ( 0, 1,..., N1)

= ( 0, 1, ..., N1)

Vk =

N1i=0

j 2ikN i = 0, ..., 1


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=1

exp( 2/ ) C

exp( 2/ ) = ( 0, 1,..., N1) CG q

j =N1i=0

ij i i = 1N

N1j=0

ij j,

= 0,..., 1C

CG(q)

C CG(q)

C CG(q)

Vk =N1

i=0j 2ik

N i, j =N1

i=0 ij

i,

exp( 2/ )

CG(q)

CG(q)

C(n, k)k 2t

tn

C


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CG(2m) 2m 1 2m2m

CG(Ft) Ft22

t+ 1

CG(Ft) CG(2m)CG(Ft)

CG(Ft)

CG(Ft)

Ft

CG(Ft)

GF(2m)CG(Ft)

n nn = 22

t

log(n)log(n) n2

n = 16


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CG(Ft)

CG(Ft)

CG(Ft) CG(2m)

CG(17)

CG(16)

0 2 4 6 8 10 1210

7

106

105

104

103

102

101

100

Eb/N0

BitErrorRate(BER)

(i) Uncoded

(ii) RS(16,12)(F.E)

(iii) RS(16,12)(44)

(iv) RS(16,12)(54)

(v) RS(15,11)

0.16 dB

CG(17) CG(16)


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CG(Ft)CG(2m) m + 1

C CG(Ft)

C

C

O(N2)O(N log N) N


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C

CG(Ft)

CG(Ft)

C CG(Ft)

r

Nw

i

Ft

A = an1an2...a0

A =n1i=0

2iai.

2n + 1 A

A < 2n + 12n mod (2n + 1) = 2n (2n + 1) = 1,

2n + 1

A mod (2n + 1) = (A mod 2n A div 2n) mod (2n + 1),

A mod 2n

A div 2n

n2n + 1


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2n1

2n + 1

A BA = A 1 B = B 1

S = S 1

S = A + B + 1,

(2n + 1)

(A + B + 1) mod (2n + 1) =

(A + B) mod 2n A + B 2n(A + B + 1) mod 2n A + B < 2n

= (A + B + cout) mod 2n.

2n + 1

2n + 1

2n + 1


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2n + 1

n + 1

2n + 1

2n + 1

(x + y) mod (2n + 1) =

(x + y) mod 2n 0 x + y < 2n(x + y) mod 2n + 2n 1 2n < x + y 2n+12n x = 2n y = 0

x y = 2n .

(x + y) mod (2n + 1) = S2nS2 + S2n2

n,

S2

S2 = [S2n+1S2n...S

20 ] = [S

1n1...S

10 ] + (2

n 1)(S1n+1 S1n),

S1

S1 = [S1n+1S1n...S10 ] = x + y.

2n+1

CG(Ft) C

2n + 1

(x y) mod (2n + 1) =

2n x = 2n y = 0

(x + y + 1 + Sn) mod 2n

DM

CG(Ft) C


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0+

2n

+

n+1 bits

(x+y) mod (2n +1)

2n

n-1 bits

n bits

n+1 bits

x yn+1 bits

S1

S2

Optional pipeline stage

(a) Proposed modulo 2n+1 adder (b) Proposed reconfigurable adder

DM

1

1

0

1

0

0

1

0

1

2n + 1

n+1 bits

n+1 bits

2n

n bits

x y

1 0

1

0...,,,1 nns

+

2n

+

DM

1


(a) Proposed modulo 2n+1 subtracter (b) Proposed reconfigurable subtracter

1

0

1

0

0

1

0

1

2n + 1

2n + 1

2n + 1

2n + 1


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(n + 1) (n + 1)

2n + 1

n 8

O(22n n) O(2n n)2n + 1

n(n+1)(n+1)

Z2n+1 = {a | 1 a 2n}2n

x y = (cL cH) mod (2n + 1) =

(cL cH) mod 2n cH cL(cL cH + 1) mod 2n cH > cL,

Ft

cL =

n1i=0 pi2

i

et cH =

n1i=0 pn+i2

i

,

n n

(n + 1) (n + 1)

P = P2n22n + 2n

n1

i=0

pn+i2i +

n1

i=0

pi2i,

2n + 1

x y =

(cL + c

H + 2) mod 2n P2n = 1 P2n = 0 c

L + c

H + 1 < 2n ,

(cL + c

H + 1) mod 2n

cL =n1

i=0 pi2i cH =

n1i=0 pn+i2

i

xy n

n

2n + 1


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''

Lc

''

Hc

Modulo 2n adder

(xy)mod(2n+1)

1=0

=0

p2n

n bits

(n+1) * (n+1) multiplier

x

y X


2n + 1

2n + 1 xiy

2n + 1

(n + 1) (n + 1)

2n + 1 (n+ 1)(n+ 1)2n +1

2n + 1

(n + 1) (n + 1)Z2n+1 = {a | 1 a 2n}

Z2n+1 = {0, 1, ..., 2n}

C CG(Ft)1 0

nc

2

t

+ 1


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Lc

Hc

Modulo 2n adder

(xy)mod(2n+1)

1


x

y

X


n bits

n bits

(2n + 1)

n

DM

Bi

Br

X

X

-

i

X

X+

Pr

Pi

mux 1

mux 2

mux 3

mux 5

mux 4

ncbits

n+1=2t+1 bits

nc

nc

nc

Wi

Wr1

0

0

1

1

0

0

1

1

0

CG(Ft)

Pr =

(Br Wr Bi Wi)(Br i), i = {0, 1, ..., Ft12 1}

Pi = (Br Wi + Bi Wr).

DM

DM = 1C


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DM Ft

P1r P2r 2

t + 1CG(Ft)

X

X

X

+

+-

Ft-1

+

+

Ft-1

+

Wr

Wi

Pipeline stage

nc bits

nc bits

nc bits

X

DM

nc bits

n+1=2t +1 bits

n

n

nc

i

1

0

0

1

1

0

0

1

1

00

1

1

0

1

0

0

1

0

1

0

1

1

0

1

0

Bi

Br

Ai

Ar

1

rP

1

iP

2

rP

2

iP

mux 1


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N/2 logN N

O(N/2 logN)

logN N/2

log N

logN

N

C CG(Ft)

log Nstages

Global Control Unit(GCU)

Global Control Unit

(GCU)

Stage 1 Stage 2 Stage (log N)Data input Data output


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RPE

Data out

Stage Architecture

1

0

Stage Control Unit(SCU)

iB

AGU

RAMs

rW

iW

i

ROMs

AGU

Data in

= 1T C 106


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W = ej2/N

n 2nn

N log NN/2

1/2

CG(Ft)

nc = 13

CG(257)

Q

Q

Q

1/2

1/2

r

NW


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nc nCG(Ft)

nc nc ncn n n

= 1T C 106V = 590 V = 313 V = 168

R = 713 R = 375 R = 202

W = ej2/N

n

N = 64

nc nc n

n = 9 64N = 256

N = 64


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nc

nc

SQNR = 10 log(E[

|S(k)2

|]

E[|N(k)2|] ),


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E[|S(k)2|] E[|N(k)2|]S(k) N(k) = S(k) Sf(k)

13

CG(2m)

C CG(Ft) CG(2m)

CG(2

m

)CG(2m)

CG(2m)

Input data Output data

FFT-GF2

Mux

DMFFT

TM


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CG(2m)

TMFFTInput data Output data

TM

CommonOperator

Tc/2 Tc/2

Op1 task

Tc

Op2 task

Tc

Op1 task

Tc

Op2 task

Operator1

Tc

Operator2

Tc

Tc

N2


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CG(2m)

CG(2m) m

CG(2m) {f1, f2, f4, f8}CG(24)

CG(2m)

{f5, f10} CG(24)

5 XOR gatesimplemented

according to thematirx S

Eight GF(2n)multipliers

(The multiplicandsare given

by the vector V2 )

10 XOR gatesimplementedaccording tothe matrix Q

4 XOR gates, 2 GF(2n) multipliers)

Cell1

Cell2

Cell15

ROM memory

(contains thecoefficientsof the matrix A)

Stage 1 Stage 2 Stage 3 Stage 4

0f

Principal unit

1f

2f

4f

8f

7f

14f

13f

11f

5f

10f

Additional unit

0f

0F

1F

14F

CG(2m)


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CG(2m)

CGNc

CG(24) TcTc

Tc

CG(25) TcTc

Tc

CG(26) TcTc

Tc

CG(27) Tc Tc

Tc

CG(28) Tc

Tc

Tc

Tc

Tc CG(2m)

CG(2m)

CG(2m)

Nc CG(24)

N = 15

3Tc

6Tc

C


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CG(2m)

Partial Product genearation

Partial Product reduction

Carry-propagate adder Polynomial reduction

1 0

8 bits 8 bits

8 words by 8 bits

2 words by 16 bits 15 bits

16 88

0

16

m

S

CG(2m)2m 2 m

CG(2m)

CG(2m) mm

Sm

i

C CG(Ft) CG(2m)


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8 bits

7

8

9

10

11

12

13

14

XOR

XOR

XOR

XOR

0P

1P

2P3P4P5P6P7P8P9P10P11P12P13P14P

XOR

XORXOR

ROM

m

L1

L2

L3L4

XOR

XOR

6

m = 6 7 8

CG(Ft) Ft

CG(2m)

C

CG(Ft)


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C CG(Ft)CG(2m) CG(2m)

C CG(Ft) CG(2m)


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C

C C

C GF(2m

)

C

C

GF(2m)

2m

1

C


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Ft Ft GF(Ft)C

C

GF(Ft)GF(Ft)

GF(2m)

GF(Ft)

C

C

GF(2m)

C GF(Ft)GF(2m)


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Informationsource

Sourceencoder Encryptor

Modulator RF Front End

RF Front EndDemodulatorInformationSourcedecoder

DecryptorChanneldecoder

Timing & synchronisation

Detector

DACChannelencoder

ADC


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LNA

LO

LNA AGC

LO

LO90

AGC ADC

DSP

AGC ADC

HPA

LO LO

LO90

AGC DAC

DSP

AGC DAC

AGC

ADC

DSP

LNA

DACAMP

Duplexer


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LNA LO90

ADC

DSP

ADC

HPA LO90

DAC

DSP

DAC

AFE

ADC DSPData

LO

DFE

I/Q downconversion

Sample rateconversion

Channelization

RF IF

LNA


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fs/2

fs


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Standard UMTS IS95 DECT WI-FIGSM

Function HMI Equalization Synchro ModulationCoding

Estimation Single FDM

Multichannel OFDM

channel per

channelFilterbank

FFT

CIC

Butterfly

Mapping

MACCordic

LUTZ-1

Deconvolution

FIR

MulDiv

Bloc coding


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Equalization Multichannel OFDM

FIR

FFT

Channel per

Channel

Mapping

LUT MACZ-1

Butterfly

CIC

Cordic

Filter Bank

1000/ 500

500 / 1000

15 / 20

x 180

x 6

x 4

x 100

x 4

a

a

a

1 / 1 15 / 810 / 5 4/ 21 / 1

x 2 x 2

x 2

x 180

x 180

x 1

x 2

x 1

x 2


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C0(t)


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CI,2 Wd

DPDCH3

CI,3 Wd

DPDCH5

CI,1 Wd

(DPDCH1)

CQ,2

Wd

DPDCH2

CQ,3

Wd

DPDCH4

CQ,4

Wd

DPDCH6

CQ,1 Wc

(DPCCH)

I

Q

CSCTMBIT2SYMBOL

PRECODER NRZ

OFF

ON

OFF

ON

appr. GMSK

/4 - DQPSK

QPSK

dual QPSK

b dz C0 , BT=0.3

35.0,cos

22.0,cos


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Z-1

Z-1

Z-1

Z-1

-g r-1

x(n)

y(n)

-g 0 -g 1 -g 2 -g 3

h r-1h 0 h 1 h 2 h 3 h r

H(x) =h0 + h1x + ... + hrx

r

g0 + g1x + ... + gr1xr1 + xr.


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Z -1 Z -1 Z -1 Z -1

-g r-1

x(n)

y(n)

-g 0 -g 1 -g 2 -g 3

h r-1h 0 h 1 h 2 h 3 h r

Switch

Switch

N2 N log2NN


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Butterfly

FIR

FFT

Equalization

Filterbank

Butterfly

MAC

LUT

FFT

Filterbank

Multichannel OFDM

channelchannel

Filterbank

FFT

CIC

Cordic

Butterfly

FIR

MAC

LUTZ-1

Multichannel OFDMEqualization

MAC

LUTZ-1


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GF(2m)

GF(2m)

GF(2m)

GF(2m)

GF(Ft)

GF(Ft)GF(Ft)


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GF(Ft)

GF(2m

)

GF(Ft)


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Informationsource

Sourceencoder

Channelencoder

Modulator

DemodulatorChannel

decoder

Source

decoderDestination

channel

u v

r


20Jan2009


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&


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c = a b

a (b c) = (a b) c.

a e = e a = a.

a b = b a = e.G

G Ga b a b = b a.

+


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+

a(bc) = (ab)c.

a(b + c) = ab + ac, (b + c)a = ba + ca.

ab = ba a b R

(a + b)c = ac + bc

F FF

GF(4) GF(2)

F F V

V

F VV


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Vv1 v2 c

c(v1 + v2) = cv1 + cv2.

v v v c1 c2

(c1 + c2)v = c1v + c2v.

c1 c2

(c1c2)v = c1(c2v).

F (a1, a2,...,an)n F

n FFn

n Rn

n Cn

GF

Z qq q

a + b = (a + b) mod(q),

a.b = (a.b) mod(q).

a q a = a mod(q) a b

q q a = b + mqm

q q


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F

f(x) = fn1xn1 + fn2x

n2 + ... + f1x + f0,

x fn1,...,f0 Ffi i x

fn11

f(x) deg f(x)fn1

GF(q)

GF(q)[x]GF(q)

F[x]F F[x]

p(x)F p(x)

p(x)p(x) F[x]/p(x)

p(x) = x3+1 GF(2) GF(2)p(x) GF(2)[x]/x3+1 {0, 1, x , x+1, x2, x2+1, x2+x, x2+

x + 1}p(x)

p(x)

m GF(q)qm

GF(q) m qm

qm

GF(4) GF(2)p(x) = x2+ x +1

{0, 1, x , x + 1}GF(2)


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GF(q)

GF(5) 21 = 2 22 = 4 23 = 3 24 = 1GF(5) 4 GF(5)

2 4

GF(16)

C GF(q) C(n, k)n k

GF(q)n kn k

C + C

ci cj n k

dij dijt

dmin 2t + 1.

C GF(q)n

k nC qk

C(n, k)G k

kk

= .

3 5


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G = 1 0 0 0 11 1 0 0 1

0 0 1 1 0

.= [ 1 0 1 ]

= =

1 0 1

1 0 0 0 1

1 1 0 0 1

0 0 1 1 0

= 1 0 1 1 1 .

C GF(q)n k

C C

CGF(q)n C C n k

C nC

T = 0

CC n k n GF(q)

T = 0

T

= 0

= [ ]

k k k n k

k k

GF(q)

GF

= (c0, c1,...,cn1) C C = (cn1, c0,...,cn2) C


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(x)

= [mk1 ... m1 m0] (x) = mk1xk1 + ... + m1x + m0,

(x)(x) = (x) (x),

(x)(x) (x) (x) n 1 k 1 n k

(x)

(x)

(x) = xn1 (x) + (x)

(x) (x)

(x) (x) = xn1 (x) + (x) xn1 (x) = (x) (x) (x)

(x) (x) (x) xn1 (x)

(x)GF(q = 2)GF(2m)

GF(2) GF(2m)GF(2m)

GF

GF(q)GF(q)


20Jan2009


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f = (f0, f1,...,fn1)

F = (F0, F1,...,Fn1)

Fk =n1i=0

fiej2ik

n , k = 0,...,n 1,

j =1 exp(j2/n) nth

GF(q) n nthexp(j2/n)

f = (f0, f1,...,fn1)

GF(q) n fF = (F0, F1,...,Fn1)

Fj =n1i=0

fiij, j = 0,...,n 1,

f F

fi =1

n

n1j=0

Fjij, i = 0,...,n 1.

i 0, 1,...,n 1f

j 0, 1,...,N 1 F

w

n ej2n n GF(q)

w n q1 m n qm 1n GF(qm)

f n GF(q)GF(qm) n GF(qm)

f FGF

f GF(q) F GF(qm)qm 1 GF(q)

n n = qm 1w = GF(qm)

GF(q)

af + bf

aF + bF

(fiil) F(j+l)mod n


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GF(2m)

(f(il)mod n) Fj jl

fi =1n

n1j=0 Fj

ij, i = 0,...,n 1n = 1 + 1 + ... + 1 n

ei =n1

l=0 f(il)mod n gl, i = 0, 1,...,n 1Ej = Fj Gj, j = 0, 1,...,n 1

f(x) =n1

l=0 fixi j Fj = 0

GF(2m)

GF(2m)

f(x) fi i n

1 GF(q)

f(x) = f0 + f1x + ..., fn1xn1,

n qm 1 m GF(qm)n f = {f0, f1,...,fn1}GF(qm) f = {F0, F1,...,Fn1}

Fj =n1i=0

fiij.

n n = qm

1 n

GF(qm)f(x)

F(x) = F0 + F1x + ..., Fn1xn1.

f(x) F(x)f(x) j Fj F(x)

i i fi

GF(q)(x) GF(q) n k k


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(x) n 1(x) = (x) (x)

ci =n1k=0

mkgik.

Cj = MjGj .

(x) = (x j0)(x j0+1)...(x j0+i)...(x j0+d2),

d j0

s

112

2

110...0...00...

00 ntj

t

jCCCCCC

Frequency domain

110

... nccccTime domain

IFFT

Encoding


20Jan2009


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GF(2m)

GF(2m)

n GF(q) n q 1GF(q) n

d 1 {j0, j0 + 1,...,j0 + d 2}GF(q)

n = q 1

dmin = n k + 1

kGF(2m)

GF

RS(n, k) d 1j = j0, j1,...,j0 + d 2

n d + 1GF(q)

RS(n, k)k = n d + 1 j0 j0 = 1

(x) = (x )(x 2)...(x 2t).2t

g(x) = (x )(x 2) = x2 + 4x + 3

(x) = 6 x4 + x3 + 4x2 + x + 1.

(x)

= [1 0 0 4 1 6]


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= I F F T ( ) = [1 0 5 6]

m(x)xnk = x2 x2 (x) g(x)

v(x)

= [3 2 1 4 1 6]

Z-1 Z-1

Message xn-km(x)S2

S1

c(x)

3

4

(x) = x2 + 4x + 3

(n k)(x)

k 3, 2, 1, , 4, 1, 6

(x)x2

k (x)

GF(2m)

= +

Rj = Cj + Ej j =0, 1,...,n 1 2t j0 j0 + 2t 1

Cjj0, j0 + 1,...,j0 + 2t 1

Sj = Rj+j01 = Ej+j01 j = 1,..., 2t.


20Jan2009


81/202

GF(2m)

j0 = 1 Sj = Rj2t

e(x) = en1xn1 + en2x

n2 + ... + e1x + e0.

2t

Sj = (j) = (j) + (j) = (j) j = 1, ..., 2t.

t t

t1 0 t1 ti1, i2,...,it1

e(x) = eit1xit1 + ... + ei2x

i2 + ei1xi1

eil l S1

S1 = e() = eit1it1 + ... + ei2

i2 + ei1i1.

2t 1 2,...,2teil Yl

il Xl2t

S1 = Yt1Xt1 + Yt11Xt11 + ... + Y1X1

S2 = Yt1X2t1 + Yt11X

2t11 + ... + Y1X

21

S2t = Yt1X2tt1 + Yt11X

2tt11 + ... + Y1X

2t1

t1

(x) = t1xt1 + t11x

t11 + ... + 1x + 1

X1l l =1,...,t1

(x) = (1 xX1)(1 xX2)...(1 xXt1).(x)

(x)YlX

j+t1l x = X

1l

YlXj+t1l (t1X

t1l + t11X

(t11)l + ... + 1X

1l + 1) = 0,


20Jan2009


82/202

Yl(t1Xjl + t11X

j+1l + ... + 1X

j+t11l + X

j+t1l ) = 0

l j l = 1l = t1 j

t1l=1

Yl(t1Xjl + t11X

j+1l + ... + 1X

j+t11l + X

j+t1l ) = 0

t1

t1l=1

YlXjl + t11

t1l=1

YlXj+1l + ... + 1

t1l=1

YlXj+t11l +

t1l=1

YlXj+t1l = 0,

Xl

t1Sj + t11Sj+1 + ... + 1Sj+t11 + Sj+t1 = 0

Sj+t1 = (t1Sj + t11Sj+1 + ... + 1Sj+t11),1 j 2t t1

Sk = t1

j=1jSkj mod n k = t1 + 1, ..., 2t1.

t

Sk = t

j=1

j Skj mod n k = t + 1, ..., 2t.

t1 t1

t1 t1 t1t1 t

S1 S2 ... St1

S2 S3 ... St1+1

St1 St1+1 ... S2t11

t1

t11

1

=

St1+1St1+2

S2t1

.


20Jan2009


83/202

GF(2m)

Mt1 =

S1 S2 ... S t1S2 S3 ... S t1+1

St1 St1+1 ... S 2t11

.

t1 Mt Mt1

t1

Mt1 (x)

(x)

Xl Yl Yl l = 1,...,t1t1

X =

X1 X2 ... X t1X21 X

22 ... X

2t1

Xt11 Xt12 ... X

t1t1

det(X) = (X1 X2...Xt1)det

1 1 ... 1X1 X2 ... X t1

Xt111 Xt112 ... X

t11t1

.

t1(X1, X2,...,Xt1) Yl

Y1

Y2

Yt1

=

X1 . . . X t1

X2 . . . X 2t1

Xt11 . . . X t1t1

1

S1

S2

St1

.

tt t t3 t


20Jan2009


84/202

(x) 2t

((x), t1)2t

Sk

Sk = Lk1

j=1k1j Skj k = 1, ..., 2t,

Lk SkSk

k = Sk Sk = Sk Lk1j=1

k1j Skj .

k ((x), Lk) ((x), Lk)

((x), Lk)

S1, S2,...,S2t(0)(x) = 1 B(0)(x) = 1 L0 = 0

k = 1,..., 2t (2t)(x)

k =n1j=0

k1j Skj,

Lk = k(k Lk1) + (1 k)Lk1,(k)(x) = (k1)(x) kxBk1(x),

B(k)(x) = 1k k(k1)(x) + (1

k)xB

k1(x),

k = 1 k = 0 2Lk1 k 1 k = 0 (2t(x), L2t)S1, S2,...,S2t

2t(x) 2t n 2t

k k = 2t + 1 nSk = Sk


20Jan2009


85/202

GF(2m)

1

0

n

j

jkjS

R=FFT(r)

r

)(

)(

)1(

1

)(

)(

1 xB

x

x

x

xB

x

1)()( xBx

?

??0

?

12 kL

kk

SS

1

tk 2

LkL 0

jjj SRC

No

Yes

No

Yes

No

0, kLRS jj

Recovered codeword

1 kk

nk

k > 2t

(x)

tj=1

jij = 1 i = 1, 2,...,n 1.

(n i)th

tj=0

jij = 0 i = 1, 2,...,n 1,

0 = 1 n(x) (x)


20Jan2009


86/202

FFTBuffer

register computationRecursiveextension

)(xIFFT

+

-

Data

Time-domainencoder

(using division circuit)

Recovereddata

Channel

Frequency-domain decoder

FFTBuffer

register computationRecursiveextension

)(x +

-

Data

Recovered

data

Channel

Frequency-domain decoder

Load data

symbols IFFT

Frequency-domain encoder

Pad obligatory

symbols

(x)(x) j =

tj=0 j

ij = 0 (n i)thF F T((x))

(x) (x)

(x) = (x)S(x) mod(x2t+1),

S(x) =2t

j=1

Sjxj.

(x) (x)


20Jan2009


87/202

Chien search

With FFT

1

0

n

j

jkjS

R=FFT(r)

)(

)(

)1(

1

)(

)(

1 xB

x

x

x

xB

x

1)()( xBx

?0

?

12 kL

1LkL 0

)(mod)()()(12 txxxSx

No

No

0, kLRS jj

?

tk 2

No

Yes

jj

jxx1'

)(

0)()(

)('

i

i

ii

ii Ifrc

0)( iii Ifrc

Yes

Yes

r

Recovered codeword

1 kk

b(x) B(x) x i

i

r = c + ec e

(0)i = 1


20Jan2009


88/202

b(0)i = 1 i L0 = 0 r = 1, ..., 2t

(2t)i i = 0,...,n 1

k =n1i=0

ik(k1i ri,

Lk = k(k Lk1) + (1 k)Lk1,(k)i =

(k1)i kibk1i ,

b(k)i =

1k k

(k1)i + (1 k)ibk1i ,

k = 1 k

= 0 2Lk1

k

1 k = 0

2ti = 0

ei = 0ci = ri si,

si

1

0

n

i

ii

iks

i

i

i

k

i

k

i

i

bb

)1(

1

1

0 kL

?

?

?

0

?

12 kL

ir

riiss

1

tk 2

LkL 0

iiisrc

No

Yes

No

Yes

No

1...,,0, nibrsiiii

r

1 kk

nk

Recovered codeword


20Jan2009


89/202

(x) B(x)2t k t

2t 6t2

n2 t2

g(x)2, 3...,2t1 d 2

2, 3, ..., 2t 1GF(q)

GF(q) ce(q 1) ce

qq 1

(n, k) g(x)

k c0, c1,...,cn1 n

= n 1


20Jan2009


90/202

1'210...0...0

ndCCCC

edge frequency

(2m) time-domain symbols

IFFT

Encoding

ec

ec

ec

d -2 successive zeros

Check frequencies

(2m-1) spectral components

1'10......

nccc

ec

edge symbol

ce =n1

i=0ci

i,

c0, c1,...,cn1 n = n 1

ce, c0, c1,...,cn1 dmin = nk + 1 =q k + 1

GF(2m)

2m 1 m2

GF(2m)

C 2mGF(2m)


20Jan2009


91/202

GF(Ft)

2m

GF(2m)

GF(Ft)

GF(Ft) Ft

GF(Ft)

GF(Ft)

GF(Ft)

Ft = 22t + 1,

Ft t

Ft t = 0, 1, 2, 3, 4


20Jan2009


92/202

t 5F0 F1 F2 F3 F4

t = 5 F5 = 4294967297 = (641) (6700417)

2 n n

n = 1.

n 2 2 n = 2t+1 22

t+1= 1 n = 2t+1

=

2 = 2

2

2n+28

2 2t+4 GF(Ft)2t+4

2t+4

F2 = 17

GF(F2)

n

1n

2n

3n

4n

6n

2 8 4 4 36 n = Ft 1 = 16 316 616 1

6 =

2 62 = 2 3 6GF(17) n = 0 15

GF(17) = {0, 1, 2, ..., 22t}

= 3 = 3


20Jan2009


93/202

Ft

Ft

GF(Ft)

GF(Ft)

GF(2m) GF(Ft)

GF(2m) GF(Ft)

GF(Ft)22

t

(Ft)2t GF(Ft)

22t GF(Ft)2t

22t 1

22t

2t

2t + 1GF(Ft) 2

t + 1


20Jan2009


94/202

GF(Ft)

GF(2n)

GF(Ft) Ft2

GF(Ft)

22t

2t

22t

22t

GF(Ft)2t

22t

GF(Ft)

GF(2m)GF(Ft)

n

2

log n n/2


20Jan2009


95/202

nlog n

Phase 3Phase 1

Forney

algorithmBerlekamp

algorithm

Phase 2

2tcycles8tcycles 3 cycles+ +

ncomputatio

xx )(),('

Phase 3

Syndromecomputation

Phase 1Chien

search

Forney

algorithmBerlekampalgorithm

Phase 2

n cycles 2tcycles8tcycles n cycles 3 cycles+ +

ncomputatio

xx )(),('

Syndrome

Computation

with FNT

log n cycles

Chien

Search

with FNT

log n cycles

With FNT

GF(Ft)

GF(Ft)GF(2m)

EbN0

EbN0

c

GF(17)

GF(17)

GF(16)

EbN0

2t + 1


20Jan2009


96/202

0 2 4 6 8 10 1210

7

106

105

104

103

102

101

100

Eb/N0

BitErrorRate(BER)

(i) Uncoded(ii) RS(16,12)(F.E)

(iii) RS(16,12)(44)

(iv) RS(16,12)(54)

(v) RS(15,11)

0.16 dB

GF(17) GF(16)

22t

= 16EbN0

(Eb/N0)

22t

22t

2t2t + 1 22

t

GF(17)

GF(16)

= 10log(1/R1) 10log(1/R2),R1 R2

= 1 0log(15/11) 10log(68/48) = 0.16


20Jan2009


97/202

0 2 4 6 8 10 1210

7

106

105

104

103

102

101

10

0

Eb/N0

FrameErrorRate(FER)

(i) uncoded

(ii) RS(16,12)(F.E)

(iii) RS(16,12)(44)

(iv) RS(16,12)(54)

(v) RS(15,11)

GF(17) GF(16)

(tc) R1 R2

Pi =1

2erfc

Ri

Eb

N0, i = 1, 2.

GF(Ft)


20Jan2009


98/202

0 1 2 3 4 5 6 7 810

4

103

102

101

100

Eb/N0 (dB)

Channelerrorprobability

RS(15,11)RS(16,12)RS(255,223)

RS(256,224)

C GF(Ft)

GF(2m)tel00354490,version1

20Jan2009


99/202

Ft

GF(Ft)


20Jan2009


100/202

Nf(n) n = 0, 1,...,N 1

F(k) =N1n=0

f(n)WknN , k = 0, 1,...,N 1

WknN = ej2/N j =

1 WknN

N2 Nlog2N

rN/r r logrN

logrNr

2

F(k) =

N/21n=0

f(2n)WknN/2 + WkN

N/21n=0

f(2n + 1)WknN/2, k = 0, 1,...,N/2 1


20Jan2009


101/202

F(N/2+k) =

N/21n=0

f(2n)WknN/2WkNN/21

n=0

f(2n+1)WknN/2, k = 0, 1,...,N/21.

XkXN/4+k XN/2+k X3N/4+k k = 0, 1,...,N/4 1

r

N

k+N = k, k+N/2 = k

W


20Jan2009


102/202

r

Nw

i

GF(Ft)Ft

Ft


20Jan2009


103/202

n A = (an1, an2,...,a0) B = (bn1, bn2,...,b0)cin (n + 1)

n S = (sn1, sn2,...,s0) cout

ai {0, 1, 2}ai = ai,c + ai,s ai,c ai,s {0, 1} n

S C

n

n(xn1, xn2..., x0)

(yn1, yn2,...,y0) n

y0 = x0y1 = x1 x0

yn2 = xn2 xn3 x1 x0yn1 = xn1 xn2 x1 x0

y0 = x0yi = xi yi1; i = 1, 2,...,n 1

gi

pi(2n 1)


20Jan2009


104/202

2n + 1

2n 1

X X = X 10 0

(2n + 1) n

(2n + 1)

n A = an1an2...a0

A =n1i=0

2iai.

A (2n + 1)

A < 2n + 1

2n mod (2n + 1) = 2n (2n + 1) = 1,2n + 1

A mod (2n + 1) = (A mod 2n A div 2n) mod (2n + 1),A mod 2n A div 2n n

2n + 1 (2n + 1)n n

2n + 1

M M(A + B) mod (M) = (A mod M + B mod M) mod M,

(A.B) mod (M) = (A mod M).(B mod M) mod M.

A = an1an2...a1a0 B = bn1bn2...b1b0 n S =sn1sn2...s1s0gi = ai.bi pi = ai + bi +

ci = gi +

i1j=0

(

ik=j+1

pk)gj +

ik=0

pkcin,


20Jan2009


105/202

cin sn1sn2...s1s0 S

si = hi ci1 hi = ai bi

(gm, pm) (gk, pk) = (gm +pm.gk, pm.pk),

ci = Gi Gicin = 0

(Gi, Pi) =

(g0, p0) i = 0

(gi, pi) (Gi1, Pi1) i n 1.

Parallel-prefix carry computation cin

cout

a0 b0a1 b1an-2 bn-2an-1 bn-1aibi

gi hi pi

hi ci-1

sis0s1sn-2sn-1

A B A = A

1 B = B

1

A + B = S

(A + 1) + (B + 1) = S + 1

A + B + 1 = S,

(2n + 1)

(A + B + 1) mod (2n + 1) =

(A + B) mod 2n A + B 2n(A + B + 1) mod 2n A + B < 2n

= (A

+ B

+ cout) mod 2n

.


20Jan2009


106/202

(2n + 1)

cin = cout

(A + B + 1) mod (2n + 1) = (A + B + cout) mod 2n.

n

2n

(2n1)

n

m m(2n + 1)

n > 8

(2n + 1)

A = 6B = 4 C = 5 A = 5 = 1012 B

= 3 = 0112C = 4 = 1002

2n + 1


20Jan2009


107/202

Parallel-prefix carry computation

cout

a0 b0a1 b1an-2 bn-2an-1 bn-1

Extra stage

s0s1sn-2sn-1

gi gi-1

g p

pi-1pi

2n + 1

A= 101

B= 011

+

S 1 000+cout 0

Correct result 000(indicating the value 1)

C= 100

B= 011

+

S 0 111+cout 1

result indicating 000real zero

2n + 1


20Jan2009


108/202

aibi

a0b0

an-1 b n-1

(2

n

+ 1)

2n + 1 2n

(n + 1)(2n + 1) x y

(x + y + 1) mod (2n + 1) =2n x = 2n y = 2n

(x + y) mod 2n + cout 0 x + y < 2n+1.

x y (n + 2)

(x + y + 1) mod (2n + 1) = (x + y) mod 2n + sn+12n + sn+1 sn,

sn n

2n + 1

2n + 1

(x + y) mod (2n + 1) =

(x + y) mod 2n 0 x + y < 2n((x + y) mod 2n + (2n 1)) mod 2n 2n < x + y 2n+12n x + y = 2n.


20Jan2009


109/202

n+1 bits

n+1 bitsn+1 bits

n bits

n+1 bits

x y

(x+y+1) mod (2n +1)


n bits

n+1 bits

x y

n+1 bits

2n

1 0

(a)

Mostsig

nif

ican

tbit

(x+y+1)mod (2n+1)

(b)

sn+1, , 0

2n + 1

(x + y) mod (2n + 1) = S2nS2 + S2n2

n,

S2

S2 = [S2

n+1

S2

n

...S2

0

] = [S1

n

1

...S1

0

] + (2n

1)(S1

n+1 S1

n

),

S1

S1 = [S1n+1S1n...S

10 ] = x + y.

GF(Ft = 2n + 1) 0 x, y 2n

0 x + y 2n+1.

x + y = 2n+1 x = y = 2n)

S1 = 2n+1(i.e. S1n+1 = 1, S1i = 0 f or = 0, ..., ),

S2 = 0 + 2n 1 S2n = 0 2n 1x + y = 2n x = 0 y = 2n x = 2n y = 0)

S1n = 1, S1n+1 = 0,

(S1n S1n+1 = 1), S2 = 2n + 2n 1 = 2n+1 1.

S2n = 1 2nS2n = 1


20Jan2009


110/202

r

NW

+

a

c

b

d

+

+

real

ima

a

c

+

b

d

real

ima

2n < x + y < 2n+1

S1 = 0.2n+1 + S1n2n + ... + S10 ,

2n mod (2n + 1) = (1 + 2n + 1) mod (2n + 1) = (1) mod (2n + 1).2n (1)

2n = 2n 1

2n + 2n 1 < S2 = x + y + 2n 1 < 2n+1 + 2n 1,

2n+1 S2 < 3 2n 1,S2n = 0 (x + y + 2

n 1) mod 2n

0 x + y < 2n

S1n+1 = S1n = S2n = 0 and (x + y) mod 2n+1 = x + y.

2n + 1


20Jan2009


111/202

0+

2n

+

n+1 bits

(x+y) mod (2n +1)

2n

n-1 bits

n bits

n+1 bits

x yn+1 bits

S1

S2


(a) Proposed modulo 2n+1 adder (b) Proposed reconfigurable adder

DM

1

1

0

1

0

0

1

0

1

2n + 1

2n + 1

(x y) mod (2n + 1) =

2n x = 2n y = 0

(x + y + 1 + Sn) mod 2n

n+1 bits

n+1 bits

2n

n bits

x y

1 0

1

0...,,,1 nns

+

2n

+

DM

1


(a) Proposed modulo 2n+1 subtracter (b) Proposed reconfigurable subtracter

1

0

1

0

0

1

0

1

2n + 1

x

y (n + 1) GF(Ft) y xS = (x + y + 1) (n + 2) x y


20Jan2009


112/202

0 x + y 2n+1

2n 1 y 2n+1 1,

2n 1 + 1 x y = x + y + 1 2n + 2n+1 1 + 12n x + y + 1 3 2n.

x y = x + y + 1 2n+1, Sn+1 = 1, Sn = 0x + y + 1

x y = x + y + 1 < 2n+1, Sn+1 = 0, Sn = 1x + y + 1 + 1

(x = 2n and y = 0) = Sn+1 = Sn = 12n

2n + 1

2n + 1

Ft)

2n + 1

2n + 12n + 1

(n + 1) (n + 1)


20Jan2009


113/202

(2

n

+ 1)

n 22n nn 8

xy mod (2n + 1) =

x + y

2

2

x y2

2mod (2n + 1)

= ((x + y) (x y)) mod (2n + 1)(x + y) (x y)

O(22n n) O(2n n)p

(p 1)

(2n + 1)(2n + 1)

(n + 1) (n + 1)

P mod (2n + 1) = (P mod 2n P div 2n) mod (2n + 1).

Z2n+1 Z2n+1 = {a Z2n+1 | gcd(a, 2n + 1) = 1}

(2n + 1) (2n + 1) a 1 (a)

a (2n + 1) Z2n+1

(2n + 1) = 2n Z2n = {a | 0 a (2n 1)}Z2n+1 = {a | 1 a 2n} 0

2n

2n + 1 Z2n+12n (1) (2n + 1)

x

y = (cL

cH) mod (2n + 1) = (cL cH) mod 2

n cH cL(cL cH + 1) mod 2n cH > cL,


20Jan2009


114/202

cL cH x y

cL =n1i=0

pi2i and cH =

n1i=0

pn+i2i.

x yx = 0 y = 0

cL cH

cL =

(2n + 1 x) mod 2n y ,(2n + 1 y) mod 2n x ,

xy mod 2n

and cH =

0 y ,

0 x ,

xy div 2n

.

x y

x y =

(cL cH) mod 2n cH cL,(cL cH + 1) mod 2n cH > cL,

n n(2n + 1)

cL

cH

cL =

0 x = 0 y = 0,0 x = 0 y = 0,1 x = 0 y = 0,

xy mod 2n

and cH =

0 x = 0 y = 0,0 x = 0 y = 0,1 x = 0 y = 0,

xy div 2n

(cL cH) mod 2n = (cL cH + 2n) mod 2n = (cL + cH + 1),

(cL cH + 1) mod 2n = (cL + cH + 2) mod 2n,cL c

H

cL cH cL + 2n cH 2n cL + cH + 1 2n.

cL c

H

cL + c

H + 1

x y =

(cL + c

H + 1) mod 2n cL + c

H + 1 2n,(cL cH + 1) mod 2n cL + cH + 1 < 2n

xy


20Jan2009


115/202

(n + 1) (n + 1) n nx = 2

n

y = 2n

2n OR2n

x = 1 y = 1

P = xy P

P = P2n22n + 2n

n1i=0

pn+i2i +

n1i=0

pi2i,

x y =

(c

L + c

H + 2) mod 2n P2n = 1 P2n = 0 c

L + c

H + 1 < 2n ,(cL + c

H + 1) mod 2n

cL =n1

i=0 pi2i cH =

n1i=0 pn+i2

i

''

Lc

''

Hc

Modulo 2n adder

(xy)mod

(2n+1)

1=0

=0

p2n

n bits


x

yX


2n + 1

(2n + 1) xiy

2n +1

(2n + 1)

Z

x y = (n1

i=0

(xi y 2i) mod (2n + 1)) mod (2n + 1)


20Jan2009


116/202

x y = (n + 2 +n1i=0

P Pi) mod (2n + 1),

P Pi = xi.yni1...y0yn1...yni + xi.0...0 1...1.

0...0 1...1 n i i(2n + 1)

(2n + 1)

2n

+1 (n + 1) (n + 1)

(n + 1) (n + 1)

(n + 1) (n + 1)

Z2n+1 Z2n+1

2n + 1(n + 1) (n + 1)

2n

+ 1

x y =

(cL + cH + 2) mod 2n cL + cH + 1 < 2

n

(cL + cH + 1) mod 2n ,

x y Z2n+1 = {0, 1, ..., 2n}

x yP2n

i, i {0, 1, ..., Ft12 1}


20Jan2009


117/202

Lc

Hc

Modulo 2n adder

(xy)mod(2n+1)

1


x

y

X


n bits

n bits

(2n + 1)

Ft 2 F t12 = Ft 1 = 1 mod (Ft) 2n + 1

2n +1DM

2n + 1

DM 1 0nc

GF(Ft) 2t + 1 n + 1 = (2t + 1) nc

Br i

muxPr

Br Bi Wr Wi

Pr

Pi

Pr =

(Br Wr Bi Wi)(Br i), i = {0, 1,..., Ft12 1}

Pi = (Br Wi + Bi Wr).

nc DM DM


20Jan2009


118/202

n

DM

Bi

Br

X

X

-

i

X

X+

Pr

Pi

mux 1

mux 2

mux 3

mux 5

mux4

nc bits

n+1=2t+1 bits

nc

nc

nc

Wi

Wr1

0

0

1

1

0

0

1

1

0

WrDM

DM 0 GF(Ft)Ft

Ar Br Ai Bin + 1 = 2t + 1

nc i

P1r P2r n

D M F F T


20Jan2009


119/202

X

X

X

+

+-

Ft-1

+

+

Ft-1

+

Wr

Wi

Pipeline stage

nc bits

nc

bits

nc bits

X

DM

nc bits

n+1=2t +1 bits

n

n

nc

i

1

0

0

1

1

0

0

1

1

00

1

1

0

1

0

0

1

0

1

0

1

1

0

1

0

Bi

Br

Ai

Ar

1

rP

1

iP

2

rP

2

iP

mux 1

N N2

N

N

N


20Jan2009


120/202

log Nstages

Global Control Unit

(GCU)

Global Control Unit

(GCU)

Stage 1 Stage 2 Stage (log N)Data input Data output

DM1 0 DM

mN = 2m

ncnw

t n = 2t + 1GF(Ft)


20Jan2009


121/202

1

1

RPE

Data out

Stage Architecture

1

0

Stage Control Unit(SCU)

iB

AGU

RAMs

rW

iW

i

ROMs

AGU

Data in

i i 1 1

Wr Wi i


20Jan2009


122/202


20Jan2009


123/202

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

r

NW

r

NW

4

0

r

NW

r

NW

4

r

NW

4

r

NW

4

2

4

6

2

6

4

0

r

NW

r

NW

2

0

1

7

6

r

NW

3

4

5

Bij

Bij,1 Bi

j,2 i

i + 1


20Jan2009


124/202

t=18

t=14


t=1

t=2

1

1B1

2B

1

3B

1

4B

1

5B

1

6B

1

7B

1

8B

2

1B

2

2B

2

3B

2

4B

2

5B

2

6B

2

7B

2

8B

3

1B

3

3B

3

2B

3

4B

3

5B

3

7B

3

6B

3

8B

4

1B4

5B

4

2B

4

6B4

3B

4

7B

4

4B4

8B

t=3

t=4

t=5

t=6

t=7

t=8

t=9

t=10

t=11

t=12

t=13

t=15

t=16

),(1

1,2

1

1,1BB

),(1

2,2

1

2,1BB

),(1

1,4

1

1,3BB

),(1

2,4

1

2,3BB

),(1

1,6

1

1,5 BB

),(1

2,6

1

2,5BB

),(1

1,8

1

1,7BB

),(1

2,8

1

2,7BB

),(2

1,3

2

1,1BB

),(2

2,3

2

2,1BB

),(2

1,4

2

1,2 BB

),(2

2,4

2

2,2BB

),(2

1,7

2

1,5BB

),(2

2,7

2

2,5BB

),(2

1,8

2

1,6 BB

),(2

2,8

2

2,6BB

),(3

1,5

3

1,1BB

),(3

2,5

3

2,1BB

),(3

1,6

3

1,2BB

),(

3

2,6

3

2,2 BB),(

3

1,7

3

1,3BB

),(3

2,7

3

2,3 BB

),(3

1,8

3

1,4BB

),(3

2,8

3

2,4BB

t=17

i

jB ijB 1,

i

jB 2,

i: denotes the stage number

j: denotesthe Butterfly number

in the stage i

Stage i

B21,1 B21,2

B21 B31 B

33

B31 B33 B

23,1 B

23,2

B22,1 B22,2 B

23,1 B

23,2

1018

iBi1j,1 i 1

Bi1j,1Bi1j,2

Bij,2Bij,1

i i + 1


20Jan2009


125/202

t=18

t=14


t=1

t=2

11B

1

2B

1

3B

1

4B

1

5B

1

6B

1

7B

1

8B

2

1B

2

3B

2

5B

2

7B

2

2B

2

4B

2

6B

2

8B

3

1B

3

5B

3

3B

3

7B

3

2B

3

6B

3

4B

3

8B

4

1B

4

5B

4

3B

4

7B

4

2B

4

6B

4

4B

4

8B

t=3

t=4

t=5

t=6

t=7

t=8

t=9

t=10

t=11

t=12

t=13

t=15

t=16

),( 1 1,21

1,1 BB

),(1

2,2

1

2,1 BB

),(1

1,4

1

1,3 BB

),(1

2,4

1

2,3 BB

),(1

1,6

1

1,5 BB

),(1

2,6

1

2,5 BB

),(1

1,8

1

1,7BB

),(1

2,8

1

2,7BB

),(2

1,3

2

1,1 BB

),( 22,3

2

2,1BB

),( 21,42

1,2 BB

),(2

2,4

2

2,2 BB

),( 21,7

2

1,5 BB

),(2

2,7

2

2,5 BB

),( 21,8

2

1,6BB

),( 2 2,82

2,6 BB

),( 31,53

1,1 BB

),( 32,5

3

2,1BB

),(3

1,6

3

1,2 BB

),(3

2,6

3

2,2 BB

),( 31,73

1,3 BB

),(3

2,7

3

2,3 BB

),(3

1,8

3

1,4 BB

),(3

2,8

3

2,4 BB

t=17

GF(Ft)

ii 1

1


20Jan2009


126/202

1T C106

18bit18bittel00354490,version1

20Jan2009


127/202

Carry_in

Shared_arith_in reg_chain_in

carry_out

Shared_arith_outReg_chain_out

Combinatorial Output

Register Output

Combinatorial Output

Register Output

Full

Adder

Full

Adder

Combinatorial

Logicr

Reg 0

Reg 1

D Q

D Q

ALUT 1

ALUT 2

The combinatoriallogic is adaptivelydividedbetweenthe two ALUTs

ALM

2n + 1

Ft

n = 5, 9, 17 Ft

n

(Ft)


20Jan2009


128/202

(Ft)n = 2t + 1

(Ft)GF(Ft) V

R

n = 2t + 1

(Ft)

= 1TC

106 V = 8276 V = 4490 V = 2245R = 10832 R = 6300 R = 3420

Ft


20Jan2009


129/202

GF(Ft

nc

WP

Bi+1j,1 = Bi

j,1 + WrN B

ij,2

Bi+1j,2 = Bi

j,1 WrN Bij,2,Bi+1j,1 B

i+1j,2

max(|Bij,1|, |Bij,2|) max(|Bi+1j,1 |, |Bi+1j,2 |) 2 max(|Bij,1|, |Bij,2|).12

2 22b16 b

b 2b b2 22b12

b

GF(Ft)GF(Ft)

10 16


20Jan2009


130/202

Q

Q

Q

1/2

1/2

r

NW

GF(Ft) F

tt = 0, 1, 2, 3

GF(Ft) GF(Ft = 257)16 256

GF(257) F4 = 65537

nc n C GF(Ft)n = 9 n = 17

GF(Ft = 28 + 1 = 257) GF(Ft = 216 + 1 = 65537)

RV nc

4 8

N = 64


20Jan2009


131/202

nc nc ncn n n

= 1

T C 106

V = 590 V = 313 V = 168

R = 713 R = 375 R = 202

C lock~ clkctrl

D in_i[0]

D in_i[1]

D in_i[2]

D in_i[3]

D in_i[4]

D in_i[5]

D in_i[6]

D in_i[7]

D in_i[8]

D in_i[9]

D in_i[10]

D in_i[11]

D in_i[12]

D in_r[0]

D in_r[1]

D in_r[2]

D in_r[3]

D in_r[4]

D in_r[5]

D in_r[6]

D in_r[7]

D in_r[8]

D in_r[9]

D in_r[10]

D in_r[11]

D in_r[12]

D M

Enable_in

R eset

R eset~ clkctr l

S tart_in

~ GN D

dout_i [0]

dout_i [1]

dout_i [2]

dout_i [3]

dout_i [4]

dout_i [5]

dout_i [6]

dout_i [7]

dout_i [8]

dout_i [9]

dout_i [10]

dout_i [11]

dout_i [12]

dout_r[0]

dout_r[1]

dout_r[2]

dout_r[3]

dout_r[4]

dout_r[5]

dout_r[6]

dout_r[7]

dout_r[8]

dout_r[9]

dout_r[10]

dout_r[11]

dout_r[12]

ena_out

start_out

C OM BOU T

LC ELL_C OM B (0000000000000000)

I N C L K O U T C L K

C LKBU F

I N C L K O U T C L K

C LKBU F

D M F F T :in s t

~ G N D

C lo c k ~ c lk c trl

R e s e t~ c lk c trl

C lo c k

R e s e t

D M

S ta rt_ in

E n a b le _ in

D in _ r[1 2 ]

D in _ r[1 1 ]

D in _ r[1 0 ]

D in _ r[9 ]

D in _ r[8 ]

D in _ r[7 ]

D in _ r[6 ]

D in _ r[5 ]

D in _ r[4 ]

D in _ r[3 ]

D in _ r[2 ]

D in _ r[1 ]D in _ r[0 ]

D in _ i[7 ]

D in _ i[6 ]

D in _ i[5 ]

D in _ i[4 ]

D in _ i[3 ]

D in _ i[2 ]

D in _ i[1 ]

D in _ i[0 ]

D in _ i[1 2 ]

D in _ i[1 1 ]

D in _ i[1 0 ]

D in _ i[9 ]

D in _ i[8 ]

S ta rt_ o u t

E n a b le _ o u t

D o u t_ i[1 2 ]

D o u t_ i[1 1 ]

D o u t_ i[1 0 ]

D o u t_ i[9 ]

D o u t_ i[8 ]

D o u t_ i[7 ]

D o u t_ i[6 ]

D o u t_ i[5 ]

D o u t_ i[4 ]

D o u t_ i[3 ]

D o u t_ i[2 ]

D o u t_ i[1 ]

D o u t_ i[0 ]

D o u t_ r[1 2 ]

D o u t_ r[1 1 ]

D o u t_ r[1 0 ]

D o u t_ r[9 ]

D o u t_ r[8 ]

D o u t_ r[7 ]

D o u t_ r[6 ]

D o u t_ r[5 ]D o u t_ r[4 ]

D o u t_ r[3 ]

D o u t_ r[2 ]

D o u t_ r[1 ]

D o u t_ r[0 ]DMFFT

GF(257) 9

R V


20Jan2009


132/202

Clock~clkctrl

Din_i[0]

Din_i[1]

Din_i[2]

Din_i[3]Din_i[4]

Din_i[5]

Din_i[6]

Din_i[7]

Din_i[8]

Din_i[9]

Din_i[10]

Din_i[11]

Din_i[12]

Din_r[9]

Din_r[10]

Din_r[11]

Din_r[12]

Enable_in

IN1

IN2

IN3

IN4

IN5

IN6

IN7

IN8

IN9

IN10

IN11

IN12

IN13

IN14

IN15

IN16

IN17

IN18

IN19

IN20

IN21

IN22

IN23

IN24

IN25

IN26

IN27

IN28

IN29

IN30

IN31

IN32

IN33

IN34

IN35

IN36

IN37

IN38

IN39

IN40

IN41

IN42

IN43

IN44

Reset~clkctrl

~GND

dout_i[0]~142

dout_i[1]~141

dout_i[2]~140

dout_i[3]~139

dout_i[4]~138

dout_i[5]~137

dout_i[6]~136

dout_i[7]~135

dout_i[8]~134

dout_i[9]~133

dout_i[10]~132

dout_i[11]~131

dout_i[12]~130

dout_r[0]~142

dout_r[1]~141

dout_r[2]~140

dout_r[3]~139

dout_r[4]~138

dout_r[5]~137

dout_r[6]~136

dout_r[7]~135

dout_r[8]~134

dout_r[9]~133

dout_r[10]~132

dout_r[11]~131

dout_r[12]~130

ena_out

OUT1

OUT2

OUT3

OUT4

OUT5

OUT6

OUT7

OUT8

OUT9

OUT10

OUT11

OUT12

OUT13

OUT14

OUT15

OUT16

OUT17

OUT18

OUT19

OUT20

OUT21

OUT22

OUT23

OUT24

OUT25

OUT26

OUT27

OUT28

OUT29

OUT30

OUT31

OUT32

start_out

Clock~clkctrl

Din_r[0]

Din_r[1]

Din_r[2]

Din_r[3]

Din_r[4]

Din_r[5]

Din_r[6]

Din_r[7]

Din_r[8]

Enable_in

IN1

IN2

IN3

IN4

IN5

IN6

IN7

IN8

IN9

IN10

IN11

IN12

IN13

IN14

IN15

IN16

IN17

IN18

IN19

IN20

IN21

IN22

IN23

IN24

IN25

IN26

IN27

IN28

IN29

IN30

IN31

IN32

Reset~clkctrl

Start_in

~GND

dout[0]~98

dout[1]~97

dout[2]~96

dout[3]~95

dout[4]~94

dout[5]~93

dout[6]~92

dout[7]~91

dout[8]~90

ena_out~DUPLICATE

OUT1

OUT2

OUT3

OUT4

OUT5

OUT6

OUT7

OUT8

OUT9

OUT10

OUT11

OUT12

OUT13

OUT14

OUT15

OUT16

OUT17

OUT18

OUT19

OUT20

OUT21

OUT22

OUT23

OUT24

OUT25

OUT26

OUT27

OUT28

OUT29

OUT30

OUT31

OUT32

OUT33

OUT34

OUT35

OUT36

OUT37

OUT38

OUT39

OUT40

OUT41

OUT42

OUT43

OUT44

start_out

COMBOUT

LCELL_COMB(0000000000000000)

I N CL K O U TC L K

CLKBUF

I N CL K O U TC L K

CLKBUF

f f t : inst

~GND

Clock~clkct rl

Reset~clkct rl

Clock

Reset

Enable_in

Start_in

Din_i [12]

Din_i [11]

Din_i [9]

Din_i [8]

Din_i [7]

Din_i [6]

Din_i [5]Din_i [4]

Din_i [3]

Din_i [2]

Din_i [1]

Din_i [0]

Din_r[12]

Din_r[11]

Din_r[10]

Din_r[9]

Din_r[8]

Din_r[7]

Din_r[6]

Din_r[5]

Din_r[4]

Din_r[3]

Din_r[2]

Din_r[1]

Din_r[0]

Start_out

Enable_out

St rt_out

Ena_out

Dout [8]

Dout [7]

Dout [6]

Dout [5]

Dout [4]

Dout [2]

Dout [1]

Dout [0]

Dout_i [12]

Dout_i [11]

Dout_i [10]

Dout_i [9]

Dout_i [8]

Dout_i [7]

Dout_i [6]

Dout_i [5]

Dout_i [4]

Dout_i [3]

Dout_i [2]

Dout_i [1]

Dout_i [0]

Dout_r[12]

Dout_r[11]

Dout_r[10]

Dout_r[9]

Dout_r[8]

Dout_r[7]

Dout_r[6]

Dout_r[5]

Dout_r[4]

Dout_r[3]

Dout_r[2]

Dout_r[1]

Dout_r[0]

Din_i [10]

FNT: inst1

Dout [3]

FNT FFT

nc

nc = n = 9


20Jan2009


133/202

nc = 16 7

nc

Floating-point FFT

Fixed-point FFT

SQNRcomputation

SQNR-

+x(n) Q xq(n)

SQNR = 10 log( E[|S(k)2

|]E[|N(k)2|] ),


20Jan2009


134/202

E[|S(k)2|] E[|N(k)2|]

9 10 11 12 13 14 15 160

10

0

0

0

0

0

0

0

nb of bits

N=64

ALUTs Gain (%)

SQNR=10log(E[|S(k)|2]/E[|N(k)|

2] (dB)

9 10 11 12 13 14 15 160

10

20

30

40

50

60

70

nb of bits

N=256

ALUTs Gain (%)

SQNR=10log(E[|S(k)|2]/E[|N(k)|

2] (dB)

N N


20Jan2009


135/202

GF(Ft)

GF(2m)GF(2m)


20Jan2009


136/202


20Jan2009


137/202

GF(2m)

GF(2m)

GF(Ft)

GF(2m) GF(2m)

GF(Ft)

GF(2m)


20Jan2009


138/202

GF(2m)

GF(Ft)

GF(2m) GF(Ft)Ft

GF(2m) GF(Ft)GF(Ft)

C

GF(2m)


20Jan2009


139/202

Input data Output data

FFT-GF2

Mux

DMFFT

TM

TMFFTInput data Output data

TM

C GF(Ft) GF(2m)


20Jan2009


140/202

GF(2m)

TcTc

Common

Operator

Tc/2 Tc/2

Op1 task

Tc

Op2 task

Tc

Op1 task

Tc

Op2 task

Operator1

Tc

Operator2

Tc

Tc

n

t = n Tc

t = (n/2) Tc t

t


20Jan2009


141/202

Stage i

din_r1

1

0

Stage i-1

1

0

Stage i+1

RAM: 1-PortRPE

FFT or FNT computation time:t = nTc

Stage iStage i-1 Stage i+1

RAM: 3-Port

1

0

1

0

RPE

FFT or FNT computation time:t = (n/2)Tc

t

t = (n/4) Tc


20Jan2009


142/202

GF(2m)

GF(2m)

O(n2) O( 14n(log n)2) n

GF(24) = 1, 5m (n + 1)Tc Tc

GF(2m)

(n 255)

t t2 tt

t = (n + 1)Tc


20Jan2009


143/202

GF(2m)

L(y) =

i

Liy2i, Li GF(2m).

L(y) L(a + b) = L(a) + L(b)

GF(2m

) = (0, 1,...,m1)

x =m1i=0

xii, then L(x) =m1i=0

xiL(i),

xi GF(2) Cks n = 2m 1GF(2)

C0 = {0},Ck1 = {k1, k12, k122,...,k12m11},

Ckl = {kl, kl2, kl22

,...,kl2ml1

},ks ks2ms mod n

f(x) =n1i=0

fixi, fi GF(2m)

f(x) =l

i=0

Li(xki),

Li(y) =

mi1j=0

fki2jmod n y2j

.


20Jan2009


144/202

GF(2m)

0 i < n f(x)i ks2

j

mod nf0 L0(x

0) L0(y) = f0y

f(x)

f = f0, f1,...,fn1 n | (2m 1) n GF(2m)f Fj

Fj =n1

i=0fi

ij, j = 0,...,n 1,

n GF(2m) f(x)

f(x) =n1i=0

fixi, i = 0,...,n 1.

f(x)

f(j ) =l

i=0Li(

jki).

ki mi mi|mGF(2mi) (ki)j GF(2mi)

i = (i,0,...,i,mi1)

jki =

mi1s=0

aijsi,s, aijs GF(2).

Li(i,s) =

mi1p=0

2p

i,sfki2p,

Fj = f(j) =

li=0

mi1s=0

aijsLi(i,s)

=l

i=0

mi1s=0

aijs(

mi1p=0

2p

i,sfki2p),


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j [0, n 1]

,

F = (F0, F1,...,Fn1)T f = (f0, fk1 , fk12, fk122,...,fk12m11,...,fkl , fkl2,...,fkl2ml1)

T

Fj fi Aaijs GF(2) L 2pi,s

mi L

L f GF(2m) iL

mi

L fA S = Lf L

(l + 1)f (l + 1)

Lf

,

Q l + 1B P

GF(2m)

GF(24)GF(2m)


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GF(2m)

GF(2m) Cks

GF(2m)GF

GFGF

GF

GF(23) C{1} C{3}GF(24) C{1} C{2} C{4}GF(25) C{1} C{5}GF(26) C{1} C{2} C{3} C{6}GF(27) C{1} C{7}GF(28) C{1} C{2} C{4} C{8}n C{m} n m

GF GF(2m)GF(24)

C0 = {f0}C1 = {f1, f2, f4, f8}C3 = {f3, f6, f12, f9}C5 = {f5, f10}C7 =

{f7, f14, f13, f11

}


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fi C1, C3, C7 (, 2, 4, 8)

= 3

GF(24

) = + 8

C5 (, 2) = 5

f(i) i = 0,...,n 1 f(j) = li=0 Li(jki)aijs A GF(2

4) l k0 k1k2 k3 k4 f(

1) = L0(0) + L1() + L2(

3) + L3(5) + L4(

7)= L0(1) + L1() + L1(

8) + L2() + L3() + L4() + L4(2) + L4(

4)aijs

f(i)

Li(y) i = 0,...,l Li(y)

fi

GF(24)S = Lf

S = Lf = Q

R Ti

.

P fi

=

1 0 1 1 0 0 0 0 1

1 0 1 0 1 0 0 1 01 1 0 1 0 0 1 0 0

1 1 0 0 1 1 0 0 0

1 1 1 1

1 1 0 0

0 0 1 1

0 1 0 1

1 0 1 0

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

8

4

2

.

1 0 0 0

1 0 1 0

1 0 1 0

1 0 0 1

1 1 0 0

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

f1

f2f4

f8

C5

F = ALf = AS

GF(24)(C0, C1, C3, C5, C7)

(C1, C3, C7)


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GF(2m)

C5

fi(C1 C3 C7)

fiV1 = P fi V1

V2 = (RTi )

GF(24)V1 V2

[V] = [V1].[V2]V2

Q VQ V

S

F = A

Cp0 Cp1 Cp3 Cp5 Cp7T

Cp0 = f0 Cp1 Cp3 Cp5 Cp7C1 C3 C5 C7 F

F1 Cpi

A Cp5F1

A

C5 (Cp5)f0 F1

GF(24)


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4

S

Q

5 XOR gatesimplemented

according to thematirx P

Eight GF(2n

)multipliers

(The multiplicandsare given

by the vector V2 )

10 XOR gatesimplementedaccording tothe matrix Q

4 XOR gates, 2 GF(2n) multipliers)

Cell1

Cell2

Cell15

ROM memory(contains thecoefficientsof the matrix A)


0f

Principal unit

1f

2f

4f

8f

7f

14f

13f

11f

5f

10f

Additional unit

0f

0F

1F

14F

GF

C5

A

t t n NsNc Tc Nc

y x x x


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GF(2m)

4

5

10

3

9

12

6

5

10

ControlUnit

ROM1

ROM15

0f

1f

2f

4f

8f

5f

10f

7f

14f

13f

11f

0F

1F

14F

13F

6F

7F

Shiftregisters

Pipeline stage


Additional stage

Ns(R)

R = nt fc fc =1

Tc

8

GF(2m) 12m(m+1)

2m + 2(mC2 1) mC2

GF(28) mC2 = 4

Nc.n

cgn

cg


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GF(24)

Tc 2m + 2(mC21) = 8 + 2 = 10

t = 6 Tc

2m+2(mC21)GF(2m) 12m(m + 1)

m

t = (n + 1)Tc

t

=t8 Tc

GF(2m) GF(2m)

GFNs Nc

GF(24) TcTc

Tc

GF(25) TcTc

Tc

GF(26) TcTc

Tc

GF(27) TcTc

Tc

GF(28) Tc

Tc

Tc

Tc

(Tc =1

fc)


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GF(2m)

GF(24)

Nc Tc

t = 3 Tc

t = 6 Tc

these ghouwayel

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