Differential PSK (DPSK) and Its Performance

Transcription

1 Differetial PSK (DPSK) ad Its Performace st ( ) = At ( )cos(2 πft+ θ( t) + θ) c A(t) = amplitude iformatio θ(t) = phase iformatio f c = the carrier frequecy θ = the phase shift itroduced by the chael I coheret commuicatios, θ is kow to receivers I o-coheret commuicatios, θ is ukow to receivers ad assumed to be a radom variable distributed uiformly over (-π, π)

2 Coheret Detectio of Differetially Ecoded M-PSK Sigals I coheret commuicatios, phase estimatio is required. The receiver usually derives its frequecy ad phase demodulatio refereces from a carrier sychroizatio loop. The sychroizatio loop may itroduce a phase ambiguity where ˆθ is the estimate acquired by the receiver through the sychroizatio loop. For M-PSK sigals, the phase ambiguity Φmay take ay value i Thus ay value of Φ other tha will cause a correctly detected phase to be erroeously mistake for oe of the other possible phases, eve i the absece of oise. 2π 2 π( M) {,, } M M φ= θˆθ Solutio: differetial ecodig while maitaiig coheret detectio

3 Differetially Ecoded M-PSK Sigals Istead of ecodig iformatio ito absolute phases, the iformatio is ow ecoded usig phase differeces betwee successive sigal trasmissio. Example: mapped to phase, mapped to phase π Biary sequece {a } Absolute phase sequece { θ } π π π π Differetially ecoded Phase sequece The actually trasmitted Phase sequece π π Iitial value The iformatio sequece {a } is ow carried by phase differece i {θ }.

4 Differetially Ecoded M-PSK Sigals Biary sequece {a } Absolute phase sequece { θ } Differetially ecoded Phase sequece The actually trasmitted Phase sequece π π π π π π Iitial value Assume the phase ambiguity Φ itroduced by the sychroizatio loop is costat durig successive sigal itervals. I the absece of oise, the receiver would covert the received phase sequece ito +Φ, π+φ, +Φ, +Φ, +Φ, π+φ, +Φ. The received sequece would be the differetially decoded ito π, π,,, π, π. The we get the origial biary sequece () back.

5 Differetially Ecoded M-PSK Sigals Example: M=4. Assume that the followig Gray mappig is used Biary sequece absolute phase π/2 π 3π/2 Biary sequece {a } Absolute phase sequece { θ } π/2 π π/2 3π/2 π Differetially ecoded π/2 3π/2 3π/2 3π/2 π/2 phase sequece {θ } Estimated phase sequece (i the absece of oise) θˆ π/2+φ 3π/2+Φ 3π/2+Φ +Φ 3π/2+Φ π/2+φ Differetially decoded sequece π π/2 3π/2 π a Decoded biary sequece ˆ Φ є {, π/2, π, 3π/2} is the phase ambiguity

6 The Structure of Differetial Ecoder ad Decoder ˆ { } θ Modulo 2π additio { θ } {θ } Delay T Figure a. A differetial ecoder Delay T Figure b. A differetial decoder - ˆ { } θ { θ } = the absolute phase sequece {θ } = the trasmitted phase sequece A digital comm. system that employ differetial ecodig of the iform. Symbols ad coheret detectio of successive differetially ecoded M-PSK sigals is called a differetially ecoded coheret M-PSK system.

7 Optimal Coheret Receiver for Differetially Ecoded M-PSK 2Es rt ( ) = cos(2 πft c+ θ+ θ) T T t ( + ) T, =, ±, ± 2, θ= phase shift itroduced by the chael θˆ= estimate of θ provided by carrier sych. loop φ= θ θˆ = the phase ambiguity θ = θθ, { θ } represets the iform. sequece r(t) X 2 cos(2 πft+ θˆ ) 2 si(2 πft+ θˆ ) X c c ( + ) T T 9 Carrier Syc. loop ( + ) T T dt dt r,c η r,s = ta r r s, c, ˆ { } θ Modulo 2π additio - Delay T ˆ { } θ Select the smallest η -β η -β η -β M- Subtractor

8 Performace Let P deote Pr{ mˆ = m m= m } i receiver fro M-PSK. π E s i P = - e x p { M } dα α E 2 π π s M 2 π N s i s = e e rg y /s y m b o l. i i the optimal coheret Oe ca show that the prob. of correct decisio i the optimal coheret receiver for differetially ecoded M-PSK is P c M i= 2 i M 2 e c i i= Sice P < P for i, it follows that M P P= P= P P = i < PP= P= P c o i C coheret M-PSK i= The symbol error P e > P e coheret M-PSK M 2 2 i i= P= P P e M 2 2 em PSK em PSK i i= = 2 P ( P ) P

9 Differetial PSK (DPSK) The phase shift is treated as a radom variable distributed uiformly over (-π, π), ad o phase estimatio is provided at the receiver. We are ow cocered with o-coheret commuicatios The phase shift θ is the same durig two cosecutive sigal trasmissio itervals [(-)T, T) ad [T, (+)T). The iformatio phase sequece { θ } is still differetially ecoded as before. The trasmitted sigal s(t) i the iterval [(-)T, (+)T] is 2Es cos(2 πft c + θ + θ), (-)T t<t T s( t) = 2Es cos(2 πft c + θ+ θ), T t<(+)t T θ = θ - θ -. θ ad θ - are idepedet. Both of them take values uiformly over 2π 2 π( M),, M M { θ } ad {θ } are i.i.d. sequeces

10 Differetial PSK (DPSK) The received waveform is the r(t)=s(t)+(t) To derive the optimal receiver, the observatio iterval should be take as (-)T t <(+)T sice θis the same i this iterval. We have four orthoormal basis fuctios: 2 cos 2 πft c ( -) T t T φ (t)= T T< t ( + ) T 2 si 2 πft c ( -) T t T φ2 (t)= T T< t ( + ) T ( -) T t T φ3 (t)= 2 cos 2 πft c T< t ( + ) T T ( -) T t T φ4 (t)= 2 si 2 πft c T< t ( + ) T T

11 Differetial PSK (DPSK) X ( + ) T T dt r,c Waveform chael r(t) X 2 cos 2 πft c T 9 Local Oscillator 2 si 2 πft c T ( + ) T T dt Delay T Delay T r -,c r -,s r,s Vector chael r = E cos( θ + θ) + η, r = E si( θ + θ) + η c, s c, s, s s, η, η, η, η are i.i.d c, s,, c, s Each of them N(, N /2) We estimate θ from the ew observatio (r,c, r,s, r -,c, r -,s )

12 Differetial PSK (DPSK) 2πi Let βi =, i M. M Give θ = β, θ,ad θ i Pr (, r, r, cr, θ = β, θ θ ) c, s,, s i, 2 2 [( rjc, Es cos( θj+ θ )] + [( rj, s Es si( θj+ θ )] j= exp 2 N = ( πn ) 2 E s j θ = c exp re + r cos( θα) N ) c does ot deped o θ, θ - ad θ 2) r = r,c + jr,s ad r - =r -,c +jr -,s 3) αis give by r e -jθ + r - e -jθ -= r e -jθ + r - e jα

13 Differetial PSK (DPSK) θ is distributed uiformly over (-π, π) ad θ - tales values uiformly 2π 2 π ( M ) over {,,, } M M P( r, r, r, r θ = β, θ ) c, s,, c, s i, π 2 Es j θ = c exp{ re r cos( θ α)} dθ 2π + π N 2 E 2 E = ci ( re + r ) = ci ( re + r ) s j θ s j βi N N 2π The above formula is idepedet of θ π xcosy where I (x)= e dy is modified Bessel fuctio π 2 E P( r, r, r, r θ = β ) = ci ( re + r ) s j βi c, s,, c, s i N -

14 Differetial PSK (DPSK) Applyig the MAP rule, we ow choose ˆ θ as β iff k P( r, r, r, r θ = β ) = max P( r, r, r, r θ = β ) c, s,, c, s k c, s,, c, s i i 2 E 2 E I ( re + r ) = max I ( re + r ) s j βk s j βi N i N ( x>, I ( x) is a icreasig fuctio) j βk j βi re + r = max re + r r r cos( η η β ) = max[ r r cos( η η β )] k i i η η β = mi η η β k i i where r = r e ad r = r e jη i jη

15 Optimal Receiver for M-DPSK r(t) X 2 cos 2 πft c T 9 Local Oscillator ( + ) T T dt r,c η = ta r r s, c, X 2 si 2 πft c T ( + ) T T dt r,s ψ-β θˆ Select the smallest ψ -β ψ -β M- Subtractor ψ - + Delay T Differece from the receiver for differetially ecoded M-PSK?

16 Performace of M-DPSK ) 2 N N(, ~,, ) si( ) cos( ) si( ) cos( ) cos( 4 4,,,,,,,,,,,,,, x s c s c s s s c s c s s s c s c s c I E r E r E r E E r + + = + + = = = + + = η η η η η θ θ η θ θ η θ θ θ η θ θ θ η θ θ Oe way to calculate the performace of M-DPSK is to fid first the pdf of phase differece c s c s r r r r,,,, ta ad, ta = = = η η η η ψ

17 Performace of M-DPSK θ (r,c, r,s ) oise vector (η,c, η,s ) ψ (r -,c, r -,s ) oise vector (η -,c, η -,s ) θ - + θ It follows that the pdf of ψ is idepedet of the agle θ - + θ. It ca be show that the coditioal pdf of ψ give θ satisfies the followig equatio 2 P( ψ ψ ψ θ ) = f ( ) d 2 ψ ψ ψ ψ θ ψ F θ ( ψ2 ) F ( ) θ ψ + ψ< θ < ψ 2 = F θ ( ψ2 ) F ( ) or θ ψ ψ> θ θ> ψ2 E exp{ [ cos( θψ)cos t]} si( θψ) F θ ( ψ) = dt 4π cos( θψ)cost s π 2 N π 2

18 = = = + + = = + = = = cos cos ]} cos 4 cos ( exp{ 4 si cos cos ]} cos 4 cos ( exp{ 4 si cos cos ]} cos 4 cos ( exp{ 4 si ) ( ) ( ) ( ) ˆ ( π π π π π π β β π π π π π π π π π π π π π β π β β θ π β ψ π β β θ β θ dt t M t N E M P dt t M t N E M P dt t M t N E M M F M F M M P P s e s c s i i i i i i i i i Performace of M-DPSK We the get

19 Special case M=2 P b =P e =/2e -E b/n For large M, M-DPSK requires 3dB more Eb/N tha coheret M-PSK

20 Compariso of Digital Modulatio Methods To make a fair compariso amog differet digital modulatio methods, oe must ivestigate the tradeoff amog the badwidth efficiecy, bit error probability, ad bit SNR. The badwidth efficiecy of a modulatio method is defied as the bit rate (R b ) to badwidth (W) ratio R b /W The badwidth is calculated from the power spectral desity fuctio of the trasmitted waveform, which i tur depeds o the sigal set used.

21 Compariso of Digital Modulatio Methods Assume that the optimal sigal set is used. Theorem 3.7.: The maximum umber N of dimesio of the sigal source spaed by a set of sigals of duratio of T ad badwidth W grows liearly with time T ad badwidth W, respectively: N=KWT where K is a costat aroud 2. The value of K depeds o specific defiitios of badwidth. We ormally choose K=2.

22 Examples PAM sigals: N= => W=/(2T) (sigle sidebad) The badwidth efficiecy of M-ary PAM is Rb W log M / T = = 2 log M bits/secod/hz / 2T M-PSK: N=2 => W=/T Rb log M / T = = log M bits/secod/hz W / T M 2 -QAM: N=2 => W=/T Rb W 2 log M / T = = 2 log M bits/secod/hz / 2T M-FSK: N=M => W=M/(2T) Rb log M / T 2log M = = bits/secod/hz W M / 2T M As M approaches ifiity, R b /W goes to

23 Compariso of Digital Modulatio Methods I the case of PAM, QAM, ad PSK, icreasig M results i a higher bit rate-to-badwidth ratio R/W. However, the cost of achievig the higher data rate is a icrease i the SNR per bit. Therefore, these modulatio methods are appropriate for chaels that are badlimited, where we desire R/W> ad where there is sufficiet high SNR to support icreases i M. Example: Telephoe chaels

24 Compariso of Digital Modulatio Methods M-ary orthogoal sigals yields R/W. As M icreases, R/W decreases. However, the SNR/bit required to achieve a give error probability ( -5 ) decreases as M icreases. Cosequetly, M-ary orthogoal sigals are appropriate for powerlimited chaels that have sufficietly large badwidth to accommodate a large umber of sigals. As M goes to ifiity, Pe approaches zero provided that SNR/bit -.6 db

25 Digital Commuicatio Block Diagram 数字通信系统框图 Iformatio source Source ecoder Chael ecoder Modulator Text, speech, Audio, image, video Natural redudacy removal Cotrolled redudacy isertio oise Chael User Source decoder Chael decoder Demodulator Source codig Chael codig Discrete chael Fig. 数字通信系统框图

26 Chael Capacity ad Coded Modulatio

27 Chael Models ad Chael Capacity Biary Symmetric Chaels {a } BPSK Modulator s(t) AWGN Chael Demodulator ad detector ˆ { } a Assume that a = or, i.i.d, ad symbol ad are equally likely. The 2E b P ( ˆ b= P a a ) = Q N q q -q -q Equivalet diagram

28 Biary Symmetric Chaels P( aa ˆ ˆ aˆ aa a ) = [ p ] 2 2 b For ay fixed bit SNR E b /N, the block error prob. goes to as approaches ifiity. I order to trasmit iformatio reliably over the BSC, oe has to employ chael ecoders ad decoders. {a } Chael ecoder BSC Chael decoder ˆ { } a A importat questio: with the use of chael ecoders ad decoders, how may umber of bits of iformatio ca be reliably trasmitted over the BSC? The aswer is the chael capacity of the BSC.

29 Chael Capacity of the BSC C = max I( X ; Y ) X X is the iput radom variable to the BSC, Y is the correspodig output. The maximizatio is take over all possible iput radom variable X. For BSC, C = H(q) with the maximum achieved by the equally likely iput X.

30 Chael Capacity of the BSC {a } Chael BPSK ecoder Modulator s(t) AWGN Chael Demodulator ad detector Chael decoder ˆ { } a The cocateatio of the chael ecoder ad BPSK modulator is a biary coded modulatio scheme ad gives biary coded sigals. The BPSK detector is ot optimal ay more sice coded sigals are correlated ad the BPSK makes a hard decisio ( or ) bit by bit. To improve the performace, we may cosider a detector that outputs Q>2 outputs. That is, a detector quatizes the demodulator output ito Q>2 outputs (or o quatizatio). I this case, we say that the detector has made a soft decisio. With a detector makig a soft decisio, we get a equivalet discrete chael with two iputs ad Q>2 possible outputs.

31 Discrete Memoryless Chaels m P(y m ) Y m Y m M- P(y Q- m M- ) Y Q- A geeral discrete chael With ay M-ary modulator ad a soft (or hard) decisio detector, we get a equivalet discrete chael with M iputs ad Q M possible outputs. The maximum umber of bits of iformatio that ca be reliably trasmitted over the chael is give by chael capacity C= max I( X ; Y ) X

32 Discrete Iput, Cotiuous-Output Chaels {a } PAM Modulator s(t) AWGN Demodulator {y } Chael The demodulator coverts the received waveform ito scalar vectors, o estimatio is made at this poit. The correspodig composite chael is the equivalet to the scalar chael Y = X + X takes values i the amplitude set of the PAM modulator is a Gaussia radom variable with ~ N(, N /2) Chael capacity C= max I( X ; Y ) X The modulator is give, but coded PAM sigal ad their receiver icludig the detector ad chael decoder are left ope.

33 Bad-limited, power-limited AWGN chaels Assume the chael is bad limited ad power limited r(t) = x(t) + (t) x(t) is the iput waveform bad-limited to W ad power-limited to P (t) = the AWGN with S (f) = N /2. To compute the chael capacity, we first covert the waveform chael ito discrete time Gaussia chael. + xt ( ) = xgt ( ) 2W = si 2πwt x= x( ), g( t) = sic2wt= 2W 2πwt This is equivalet to projectig x(t) ito the space spaed by the orthoormal basis { 2 ( )} + Wg t : 2W + x= 2 w xt ( ) g( t - ) dt 2W

34 x { } + /2W xt ( ) = xgt ( ) 2W /2W = -W W -W W + i Bad-limited, power-limited AWGN chaels (t) is i.i.d ad each i is Gaussia, i ~N(, N W) Sample at r= x+ {r } t=, =, ±, ± 2, 2W Sice x(t) is power-limited to P, each sample x is the eergy limited to P. The chael capacity C N per use of the discrete-time Gaussia chael is C N = max {I(X ; r : X is a iput with E x P P 2 [ ] } = log( + ) 2 NW Durig T secods, we have 2WT samples, ad the discrete-time chael is used 2WT times. Thus, the chael capacity C i bits per secod is P C= 2W CN= W log( + ) bits/secod NW

35 Badwidth Efficiecy versus Bit SNR C P EC b = log( + ) = log( + ) W NW NW Eb = N C / W C / W 2 P= Eergy/secod= EC b C/W E b /N bit SNR C/W badwidth efficiecy -.6 E b /N db. Shao limit Ideal diagram of badwidth efficiecy versus bit SNR

36 Badwidth Efficiecy versus Bit SNR ) Let W, we have C C / W Eb 2 N = P N log e bits/secod = lim = l 2. 6dB C / W> C / W 2) I order to achieve essetially error free trasmissio, E b /N must be -.6 db (Shao limit) 3) A trasmissio rate R b < C is achievable, ay rate R b >C is ot achievable (Shao oisy chael codig theorem) 4) The proof of oise chael codig theorem ivolves the wellkow radom codig argumet

37 Achievig Chael Capacity with Orthogoal Sigals P e C T ( Rb ) 2 i b = 2 T ( C Rb ) i 2 2 if R C /4 2 2 if C /4 R b C The error probability ca be made arbitrarily small as T goes to ifiity (M goes to ifiity for a fixed bit rate R b = log M/T ). For a power-limited AWGN chael with ubouded badwidth orthogoal sigalig (or M-FSK) is asymptotically optimal i the sese that the correspodig (symbol) error probability goes to o expoetially as M goes to ifiity if the bit rate R b = log M/T is less tha the chael capacity.

38 Coded Modulatio-A Probabilistic Approach Although orthogoal sigalig is asymptotically optimal, there is a cosiderable gap betwee the performace of practical ucoded commuicatio systems ad the optimal performace theoretically achievable give by the oisy chael codig theorem. To reduce the gap, oe must resort to coded modulatio. Algebraic approach (specific codig desig techiques) Probabilistic approach (aalysis of the performace of a geeral class of coded sigals)

39 Radom Codig Based o M-ary Biary Coded Sigals Objective Desig a biary code C cosistig of M biary codewords C i =(C i,c i2,, C i ), i M- The correspodig coded sigals {s i (t)} gives good error prob. performace 2εc si ( t) = (2Cij ) cos 2 πft c, j, t [( j -) Tc, jtc] T s = [(2C ),,(2C )] i i i c The ecodig bit rate is R b bits/secod Blocks of k bits are ecoded at a time ito oe of the M waveforms { s ( )} M i t i=, M = 2 k = 2 R bt Assume R b </T c, T=T c. This implies that k <, ad k M 2 ( k ) T ( DRb ) = = 2 = 2, D= / T 2 2 b ε c

40 Radom Codig Based o M-ary Biary Coded Sigals We cosider the esemble of (2 ) M distict ways i which we ca select M codewords from the total 2 possible biary sequeces. Associated with each of the (2 ) M biary codes, there is a coded commuicatio system. {a } Chael Code C BPSK Modulator { si ( t)} AWGN Chael Optimal Receiver ˆ { } a {a } Chael BPSK Code C 2 Modulator 2 { si ( t)} AWGN Chael Optimal Receiver ˆ { } a {a } Chael BPSK Code C 2 M Modulator M 2 { si ( t)} AWGN Chael Optimal Receiver ˆ { } a

41 Radom Codig Based o M-ary Biary Coded Sigals j Istead of evaluatig the error prob. of each coded system idividually, we compute the average error prob. of the 2 M coded commuicatio systems: P M P {{ s ( t)}} 2 j e = M e i 2 j= This is equivalet to choosig a biary code from the set of all possible code C j, j 2 M, ad evaluate the average error prob. E[ P ( C)], where P( C= Cj) =, j 2 M 2 M 2 j e= e = e i j= P E[ P ( C)] P {{ s ( t)}} P( C j ) Applyig the uio boud, we get M e i P {{ s ( t)}} 2 εd( C, C ) Pe {{ S ( t)}} Q( ) = Q M M j j j j j si ( t) sl ( t) c i l i M i l, l i 2N M = = il, 2N i l j where C deotes the ith codeword of the jth codebook C. i C j e

42 Radom Codig Based o M-ary Biary Coded Sigals 2 x j j 2 j c i l Sice Qx ( ) e for ay x>, Pe {{ si ( t)}} exp{ } M il, N i l Pe 2 M 2 M M j= il, i l j j c i l M il, 2 N i l il, i l j j εcdc ( i, Cl ) exp{ } N εdc (, C ) = [ exp{ }] M εcdc ( i, Cl ) = E[exp{ }] M N εdc (, C ) where C i ad C l are the ith ad lth codeword of the radom codebook C respectively.

43 Radom Codig Based o M-ary Biary Coded Sigals C i ad C l are idepedet, each takig values uiformly over the set (,) P( d ( Ci, Cl ) = d ) = 2 d εcd ( Ci, Cl ) E[exp{ }] = 2 e N d d= ε = + e = + e c c N N 2 2 [ log( )] C N where log( ),, ε dε N εc N R TTDR T ( DRR ) P ( M)2 [ log( + e )] < 2 = 2 e ε b b R= + e D= = TD T c c

44 Radom Codig Based o M-ary Biary Coded Sigals Sice is the average error prob. of the 2 M coded comm. Systems, P e There exists at least oe biary code C j such that the correspodig error prob. P e (C j )<2 -T(DR -Rb) < 2 -(R -Rc) where R c = k/ = R b /D is referred to as the code rate. Whe R b <DR (or R c <R ), oe ca always desig a biary coded system so that the block error prob. goes to expoetially fast as the block legth approached ifiity.

45 Radom Codig Based o M-ary Biary Coded Sigals If oe selects a code at radom, the this code is good with a very high probability. The probability that the error probability of the radomly chose code P αp e e is less tha /α. As the dimesio is large eough, good codes are abudat. But the implemetatio complexity is high. The rate R is called the cutoff rate. bits/dimesio R db -5 5 Ec/N SNR per dimesio

46 Review of Fiite Fields

47 Group Structures Defiitio A group is a set G together with a biary operatio * o G such that the followig three properties hold: ) * is associative, that is, for ay a,b,c G, (a*b)*c=a*(b*c) 3) For each a G, there exists a iverse elemet a - G such that a*a - = e 2) There is a idetity or (uity) elemet e i G such that for all a G, a*e = e*a = a Sometimes, we deote the group as a triple (G, *, e). If the group also satisfies 4) for all a, b a * b = b * a The the group is called abelia or commutative.

48 Example Let Z, the set cosistig of all itegers Q, the set of all ratioal umbers + ad are ordiary additio ad multiplicatio. The (Z, +, ), (Q, +, ), (Q*,, ) are all groups where Q* is the all ozero ratioal umbers. Furthermore, they are abelia How about (Z*,, )?

49 Let be a positive iteger (>) ad Z represet the set of remaider of all itegers o divisio, i.e., Z = {,, 2,, -} We defie a+b ad ab the ordiary sum ad product of a ad b reduced by modulo respectively. Propositio (a)(z, +, ) forms a group (b)(z p *,., ) forms a group for ay prime p Let Z * = { a Z a }

50 Defiitio 2 A multiplicative group b=a i is said to be cyclic if there is a elemet a G such that for ay b G there is some iteger i with b = a i. Such ad elemet is called the geerator of the cyclic group, ad we write G = <a> G a a 2

51 Examples (Z 3 *,,) is a cyclic group with geerator 2. Z 3 = {, 2} = <2>, 2 =, 2 2 = mod 3 (Z 7 *,,) is a cyclic group with geerator 3. 3 = 3, 3 2 = 2, 3 3 = 6, 3 4 = 4, 3 5 =5, 3 6 = mod 7 (Z 5 *,,) is a cyclic group with geerator 2. 2 = 2, 2 2 = 4, 2 3 = 3, 2 4 = mod 7 Every elemet i Z 5 * ca be writte i a power of 2

52 Fiite Groups Defiitio 3 A group G is called fiite if it cotais a fiite umber of elemets. The umber of elemets i G is called the order of G, deoted as G

53 Rigs Defiitio 4 A rig (R,+, ) is a set R, together with two biary operatios, deoted by + ad such that ) R is a abelia group with respective to + 2) is associative, that is ( a b) c = a (b c ) for all a, b, c R. 3) The distributive law holds: a (b + c ) = a b + a c (b + c ) a = b a + c a (Z, +, ) ad (Q, +, ) are rigs (Z, +, ) forms a rig, called residue class rig modulo (Z 4, +, ) is a rig.

54 Fields Let (F,+, ) be a rig, ad let F* = { a F a } The set of ozero elemets i F. Defiitio 5 A field is a rig (F,+, ) such that F* together with the multiplicatio forms a abelia group. (Q,+, ), (R,+, ), ad (C,+, ) are fields where R is the set of all real umbers, ad C is the set of all complex umbers

55 Fiite Fields Defiitio 6 A fiite field F is a field that cotais a fiite umber q of elemets. This umber is called the order of the field. F is also called a Galois field, deoted by GF(q). Propositio 2 Let P be a prime, the (Z p,+, ) is fiite field with order p. This field is deoted as GF(p). The additio ad multiplicatio are carried out modulo p. Examples: GF(2) ad GF(3) o page 266

56 Polyomials Let R be a arbitrary rig. A polyomial over R is a expressio of the form f(x) = a + a x + a x where is a oegative iteger, the coefficiets a i are elemets of R. x is a symbol ot belogig to R, called a idetermiat over R. f(x) is said to to be irreducible if f(x) caot be factored ito a product of lower degree polyomials over R x 2 +x+ is irreducible i GF(2)

57 Costructio of GF(2 ) ad GF(q ) Step Selcet, a positive iteger ad p a prime. Step 2 Choose that f(x) is a irreducible polyomial over GF(p) of degree. Step 3 We agree that αis a elemet that satisfies f(α) =. Let GF(p ) = {a + a α + a α a i GF(p) }

58 Costructio of GF(2 ) ad GF(p ) For two elemets g(α) ad h(α) i GF(P ), g(α)=a + a α+ + a - α - h(α)=b + b α+ + b - α - Defie two operatios: a) Additio g(α)+ h(α)= a +b +(a +b )α+ +(a - +b - ) α - b) Multiplicatio g(α) h(α)= r(α) where r(x) is the remaider of g(x)h(x) divided by f(x) Theorem: The set GF(p ) together with the two operatios above forms a fiite field, ad the order of the field is p

59 Example: Let p=2 ad f(x) = +x+x 3. The f(x) is irreducible over GF(2). Let α be a root of f(x). That is, f(α)=. The fiite field GF(2 3 ) is defied by GF(2 3 )={a + a α + a 2 α 2 a i GF(2) } As a 3-tupe = = = = = = = = As a polyomial α α 2 + α α + α 2 + α + α 2 + α 2 α 7 = As a power of α = = =α =α 2 =α 3 =α 4 =α 5 =α 6

60 Primitive Elemets ad Primitive Polyomials Fact: For ay fiite field F, its mutliplcative field F*, the set of ozero elemets i F is cyclic. Defiitio: A geerator of the cyclic group GF(p )* is called a primitive elemet of GF(p ). A polyomial has a primitive elemet as zero is called a primitive polyomial Example: x 3 +x+ is primitive polyormial over GF(2) x 7 - = (x+)(x 3 +x+)(x 3 +x 2 +)