Adaptable Viterbi detector for a decomposed CPM model over rings of integers

C.J.A.Levita, M.Benaissa and I.J.Wassel1

Ahstract: A novel, prtmtical and flexible way of iniplcnicnting continuous phase modulation (CPM) using i i decomposed model iind ii Viterbi dctcctor is presented. The dccomposcd CPM niodulator, coml”’ising ii coiili~iiioiis phase encoder (CI’E) and 21 i~ictnoi-ylcss modulator (MM), is implementcd and can he concatenaled with it ring convolutional encoder (CE) siicli that tlic detector may he rciidily adapted 10 Llnc ring of intcgcrs over which Llic CEiCPE opcratcs. A partial-response CPM syslcni is siniulalcd over additivc while Gaussian noisc (AWGN) and Rayleigh flat kiding (RFF) channels, and results, i n terms of Lhe bit error rate (UER) against the signal to noisc ratio (SNK), are prcscntcd, discussed and cvalualcd.

1 Introduction

The iisc o f coiivolution~il cncodcrs based on a ring of i i itc- gers was first suggested by Baldini, Pessoa and Arantes [ I ] , and Massey and Mittelholzer [2], and cotitintied by Farrell and Baldini [D]. Building on t l ic work or Rinioldi [4], Yang and Taylor [SI and Rinioldi and Li [61 presented optimal ring convolutional cncodcrs (binary, qiiaternary and octal) for trcllis coded niodulation (TCM) schemes using continu- ous phase Frcqucncy shift keying (CPFSK), also known as contitinoits phase moduliiLion (CPM). CPM waveforms ai-c characterised by having a constant envelope and continu- ous phase. Opliinxil convolutional codes liave been found, which maximise the minimum Euclidcan disliincc bctwccn codewords for a given transmitter and rcceivcr coinplcxity [s, 61.

The work in the referenced papers l i as hccn primarily concerned with building a suitable continuous pliasc niodu- lalor and also dclcrtnining which codes will achieve oplinliil Euclidcan distances for a given modulator. This paper presents a syslcm in which tlic pci-fortn~incc or sclienic~ based on CPM a i d ring convoli~tional cncodcrs ciin hc lcstcd for practical applications. A dccoinposcd modcl lor CPM comprising ii continuous pliiisc encoder (CPE) and a nicmorylcss ~nodulator (M M) is iinpletnicnLcd. The input Lo the modulator has M diVercnt levels, and hence it can bc called M-ary CPM. A dcmodulalor based around a Vitcrbi soft decision dctcctor is iilso iniplcmcntcd. This dcmodula- tor is flexible in that it can adapt readily to the ring of intc- gers, Z,,, sclcctcd for the CPM sclicmc. For the systems studied here, tlie modulation index h = IiM. Siniulation

results arc presented for both tlic addilivc white Gaussian iioisc (AWGN) clniinncl and the Rayleigh Hat fading (RI’F) channel. The complete system IUS hccn validated by using the AWGN channel, whcrc the rcsulls obtained coinpire fiivourably with available dala iiml with the theoretical B I 3 i bounds (a fiitiction or &/Nil) for the schemes consid- crcd.

2 Decomposed CPM model (CPE+MM)

The uscfulncss OS CPM in digital mobile systems stems honi lhc facl that it provides a good lradc-on‘ bctwccn power and spectrum elliciency, for a given bit error rate. Its constant cnvclopc permits the use of iionliiicar class C amplilicrs, which arc 2d11 to DdR more power cllicicnt than the class A o r AB power anipliticrs demanded by niodulai- tion sclicincs with a non-constant carrier ;iinplitndc. Bcaiiisc Ihc atrlicr CPM rcprcsentation [7, XI docs not pro- duce a time inwiriant phase trellis, Rinioldi [4] proposed a dccotnposcd CPM model in which the transinillctl physical phase trellis is tilted to makc it more anicnable to Vitcrbi decoding. This approach is now detailed.

A gcncral rcprcscnlalion of an M a y continuous PSK (Cl’Sl<) modulator is given by

2E,9 s( t , ( I j = & c : o s ( 2 T f i l L + </5(t,Zj + y o ) (1)

where & is tlie symbol energy (J), 7’ is the symbol period (s), .fo is the carrier frequency (I-lz), ((t, a ) is tlie phase carrying the information vector ( I = {U ( , } , U,, E {*I, 4, ,.., -t(M ~~ I)) , and

The data-caring phasc Kt, o) at the nth symbol interval is dcfincd by

is tlie iirhitrary conslanL phase oKsct.

.j(t;a) = 2 T h &>q(/ , - n-r) rrT 5 t < nT -I- T

( 2 ) 7 L Z O

where the p;u.ainclcr h is rcfcrrcd lo as the modulation index. For priicticwl purposes, 11 = KIP, wlicrc K and P arc relatively prime positivc inlcgcrs (is., they have no coininon factors). The funclion y ( t ) is known a s the phase shaping Fnnction and is ohtained by integrating anolicr

137

[unction g(/), which is naturally ailled the l'rcqucncy shiip- ing function. Tlic present work utillscs a raised cosine ( I C ) pulse shape for g(t), which is normalised such that JZ ,g(t)dr = [7]. Generalising, this pulsc shaping fainily is called LRC, whcn L = 1 the CI'M scheme is called full- rcsponse [7], and when L > I it is called partial response [XI, as a consequcncc of the interfercncc induced hetween tlic transmitted symbols. This is known as intersymbol intcrrcr- clice (ISI).

Defining a iicw carrier ficqucncyJ; =.fo - ((A4 - I)h12T), eqn. I can be rewritten a s

s(t , i i) = ,cos(2nf1t+?i,(t,n)+ipo) (3) E where the new data-carlying tilted phase is

( M ~ 1)ha lb ( t ,U) = @ ( t , Z ) + '1' t TIT 5 t < rr?' + T

This corl-c.spond? to shifting thc inilial .fo carrier by

Now, dcnoling

(4)

(A4 - I)/rRT Hz or tilting the phase by ( M ~ I)hdT rad.

a modified inrorinatioii symbol sequence U = {U?,) can bc dclincd, where U,, t (0 , 1, ..., hf - 1). So, I)(/, a ) can be writlen as qdl, U ) for the intervril nTa t < n 7 + 7: If we let I = z + nT, this time interval becomcs 0 5 z < T and $(t, N )

becomes $(c + n7; U ) . From eqn. 5 wc gct

n, = 2 1 ~ ~ (lvr - 1) ( 6 ) Rcprcscnting tlic expression for i/(c + nT, U ) modulo 2n

gives (scc the Appendix), n - L

$(r + n,T, U ) = mod2, iiiodp 2ah U, [ [ i=o I I ,.-I

+ ~ ? T / I , 2 1 i , , - , q ( ~ + i ~ ) + w(r) L=O

0 5 7 5 7' (7) The w h a or W(T) = nh(M - 1 ) [ ( d q - 2Z::dy(~ + i7)

+ (I- ~ I)] does not depend upon the dalii ami is coilstan1 [(./2)h(M - I)(L - I), Tor LRC] Tor the interval considered. The data-depcndcnl only term represcnls the sum mod,, or all input symbols Crom time interval 0 till n - L. The data and time-dependent Lcrms contain the most rcccnt input symbol (U),) and the past 1, - I inputs (U,,-, ... U,, still present in the CPM mcmory, and are responsiblc lor the phase change (via y(c t in) currently hcing output by the modulator. This the CPM can be split into two parts. Firstly, tlie MM that just accepts data and generates the required phase waveform, and secondly the CPE, which comprises a sum mod, of past symbols plus the memory elcincnls that encode tlie data such that tlie phase changes at thc modulator output will iilways be continuous. The CPE output can be viewed a s being the data vector

' ? x.,., X ! c , 2

v, = lllod,. rg U%] ( 8 )

The vcclor X , has L + 1 symbols and call be split into two sub-vectors: A',,,,, with L symbols (L$) helonging to the

13x

ring Z,w, and X,,,, with the symbol V,, belonging to the ring Z,. Furthermore, U, can bc represented in radix-P form, i.e. U, = $?{Uii P'<dV-.i, with /cM some positive integer and M = FA,. 'A siniplc example is to make IM = P (kM = 1). Thus the CPE will hive (at any limc) input lines and L . k , t 1 output lines, cadi carrying modulo-P digits. In this case, the cntirc CPE will operate over thc same ring or integers Z,, making it naturally suitcd to be preceded by convolutional encoders also operating over the ring Z,.

CPM (CPFSK) ............. ~ ........................................... X.

Fig. 1 CPM , i ~ ~ , , , ~ , ~ j . s ' , ~ ~ , " ~ , i : C P I : .I- M M

Fig. 1 shows the proposed CPM decomposition into ii CPE and MM. From eqns. 7 and 8, thc MM physical (tilled) phase li,(t + nT, U ) can be written a s

1 1, I

?(T, X T L ) = moda, 2ahV, ,+4nh~U, - iq (7 + i T ) [ i=o and eqn. 3 is transfomied into

,s(T, X q L ) = cos ( a s f i ( ~ + ' i i ~ / l ) + ~ ( r : X , , ) + 7 ~ O )

O S T i T (9) E

wlicrc

3 Implementation

3.1 The CPM modulator For the purpose or system evaluation, 2RC schemes with h = 112 (A4 = P = 2) and h = 114 (A4 = P = 4) are imple- mcnlcd using the baseband decomposed CPM model shown in Fig. 2. The CPE has two mcmory (delay) cells. Oiie cell stores the previous transmittcd symbol and the other Lhc sum (mod P) of all thc prcviously transinitled symbols. Note that each output waverorin per symbol period has eight samples in this example. Consequently, there is a multirate up-sample hnction block hetwccn the input symbol U,, and each filter. Because lhe phase response to each inpul symbol lasts for two symbol inter- vals (L = 2), the previous symbol will interfere with the current one. The phase shaping function is implenicntcd using two finite impulse rcsponsc (FIR) digital filtcrs.

is an arbitrary constant phasc offset.

3.2 The adaptable Viterbi detector The Viterhi algorithm has been employed to racilitate max- imum likelihood dccoding of convolutional lrellis codes. It works by assigning probahilities (or metrim) to the states and branchcs in the receiver trellis with the aim ofdclccting the most likely received sequcncc of dala. A general over- view or the Vitcrbi detector employcd here is shown in Fig. 3.

When a signal is reccivcd (corrupted by channel impair- ments), the Euclidean distaiiccs between it and all the possi- blc waveforms (held in the branch waveform table) over one symbol interval are computcrl. These dislanccs (or branch metrics) serve a s a probability that a given wave- form1symbol was actually sent. But these branch natrics are no1 uscd in isolation. They arc used in conjunction with thc state (or cumulative) mctrics associated wilh the states of the trellis that each of the branches dcparts from, which themselves represent tlic cumulative metrics of all the previ- ously selected branches. In this way, the Euclidean dislanw:

-~b, t , , t ,~~~i . . vx 147, NO. J, J W , ~ a n n ICE

Fig2

transition table

table

update

paths table waveform branch survivor search SPT

update state metric8

table metrics

state-transitionibranch waveform table address = C x P + input1 BWT I STT current state input I OUtPUt I OUtPUt

" I n I wn n I his" n

I I compute branch metiics (EM)

state metrics table 1

state cum. metric 0 SMO cum~lative

2 3 SM3

suiviuor path table previous states previous states

timeinstant - nT\(n+ l )T (n+2lT ("+SIT 7 J

betwcen the entire received sequence and the lnost likcly path through the trellis can bc ca1cul;ited on a synibol-by- symbol basis, considerably reducing the decoding coniplcx- ily. Thus, when the survivor paths are updatcd, only the path with the sinallest cumulative inctric is kept at cach state. Ultimatcly, the survivor path table (SFT) is traced back and the decoding table yields the decoded output scqucnce.

To implcincnt the demodulator, ii number of tables arc dcfincd and built with the aid of the decomposed CPM modulator. To do this, the CI'E is initialised to a known state, and for cach initial slate an input is taken horn the set of possible inputs 0 to A4 ~ I , with the corresponding outputs being the next CPE state and the waveform gencr- ated by the MM. This is done in a systciniitic and iiuto- mated way, and for each diffcrcnt ring Zw it needs lo be done only once. All the input and output data generated by the modulator arc stored in individual data files that arc suhsec~uently combined to build thrce diffcrcnt tables. These tables are tlicii used by Llic dcinodulator to decode tlic tr;insmitted symbol sequcncc. The tables that arc required to he coiistructcd are tlic statc transition table (STT) (which takes a s its inputs the currelit s k t c and input symbol and outputs tlie next slate), the branch waveform table (BWT) (which lakcs iis inputs the current statc aiid symbol input and gciierates tlie output waveform), and the decoding table (Dl) (which takes as its inputs the current state and previous state and outputs tlic transmitted symbol). It is assumed tlicrc arc no pal-allcl transitions bctwccn slates. The xlwiiitqe or using these Lihlcs is that for the same value of L, tlic Viterhi soft-decision demodula- tor docs not need to be redesigned Tor each iicw schcine lo accommodate niorciless statcs and possible waveforms, or a different symbol alphabet s i x . A more detailed diagram or the detector is shown i n Fig. 4, which gives the format of the lablcs and how they are used for the i-cdisation of tlrc ring Viterbi detector.

The addresses of the STT aiid the BWT arc essentially the same, and are computed by niultiplying the state valuc by the order (M) ol'tlic input data alpliabct and adding the input symbol that Caiiscs the transition to the next state (tlic value in the STT). Simultancously, the BWT outputs a stored waveform Because this is a discrete time system, each waveform is represcntcd by a number of equally spaced samples, eight in this cxample. Consequently, Cacli BWT address reI'crrcd to iibovc lids eight sub-addresses,

When a sigilal vk is received representing one symbol ol' data (cight samples in this case), the increincntal squared Euclidean disbince (SED) bctwcen it and all of the possible waveforms held in the BWT can be computed. These val- ues arc known a s the branch nietrics (EM). Under the con- trol of' thc STT, the coinpare select and update (CSU) [unction adds the appropriate EM values lo tlie cumulalivc nietrics held in state mctrics tahle I (SMTI). SO, at each new state a number of candidate braiichcs will arrive. Tlic function of the CSU is to select Llic branch arriving in each new state which caiises tlic lcast increment in the new cumulative metric value. All othcr hranches arriving at a new stale arc discarded. l h e new cumulative inctric valncs arc stored in SMT2 ready Cor the next iteration of the algo- rithm. Of course, SMTI and SMT2 SWAP roles for the next iteration. The sclcctcd branchcs determine thc survivor paths through the trellis (one for cach state) so simultane- ously with metric update, the surviving branches arc storcd in the SPT in the Comm of their previous slkitcs. The SPT is arranged so that the rows (memory addrcsses) rcprcscnt the current states, inrd tlie columns (memory contents) reprc-

o ~ i e Tor tach sample value.

I40

sent the previous statcs for ii given statc at ii given time instant. This incans that the most proliable transmitted synibol is contained in one or the selected branches. In addition, at each iteration (i.e. after the reception of a new symbol of' data), the SFT is processed by an algorithm that searches back L, (the truncation path length) symbols, starting Trom the state that currently holds the least cuinu- lalive iiietric as shown in Fig. 4. The DT is used to yield the most likcly triinsmittcd symbol with a delay of L,,, . T seconds, where T is the symbol pcriod. Computer siinula- lions have shown tliat using a LTp value of fives times tlie constraint length (number of delay cells) of the overall cncodcr results in only a slight degradation compared to the infinite length path case.

An cxample of how thcsc tables arc used hy the modula- tor and the Viterbi detector is shown i n Fig. 5. In Fig. 511 we can sec the trellis structure produced by the modulator (CPE + MM), where branches rcpreseiit both state triinsi- lions and the output or a waveform. A transition bctwccn svates and a waveform output occurs when a new data symhol arrives at the input to the nioduliitoi-. For example, if we are currently in state 0 and the wiving input symbol is a binary 'l', tlic ncxt state will bc I and the wavcl'orm wl is output from l l ic dcniotlnlator. In thc decomposed CPM model, cach waveform (channel symbol) corresponds to a codcword output by tlic CPO and iiipul to the MM. Fig. 5h shows how this trellis structure is related to the STT and the HWT utilised in the Viterbi demodubator/ detector. Fig. 5c shows the structure of the DT which simply reverses the S'M operation, i.e. given a current and picvious state held in the SPl' it oulputs tlrc clccodcd syinbol.

Note that our purpose for thc adaptable Viterbi detector is to permit the rapid assessnicnt of varioiis coded CPM schemes, so flexibility aiid the minimisation of change arc primary considerations. Its general purpose structure will also permil aspects such as sol1 input and soft output to be accoinmodated with case. Consequently, the structure adopted will not he optimum f'or any particular scheme.

4 Error rate performance for the AWGN and Rayleigh flat fading channels

Using the 2RC CPM modulator and the Vitcrhi deniodda- toridctcctor described previously, several coinniunication scheincs were designed and simulated in order to assess their pcrl'orinance in terms or bit error rate (BER) against SNR. Results arc presentcd Cor uncoded binary and uncoded quaternary systems with both additive white

111 noise (AWGN) and Rayleigh flat I'ading (RFF) channcls. I n wjireless tninsmissions at UHF and microwave frequencies, tlic arrival of niultiple/dekiyed signals at the receiver gives rise to envelope fading which can he mod- cllcd by introducing a nmltiplicativc fading channcl. Fur- therinorc, the movement oI' the mobile station (MS) CAUSCS Doppler frequency shifts in the received signals. The maxi- mum Doppler shift (f;,,) is given by.f,;, = v/d, where vis the speed ol' the MS and d is the carrier wavelength. In this work, the inaxitnuni Doppler l'rcqoency is norinaliscd wing ,fiI = /,',, . T, whcrc T is the symbol pcriod. For the simula- tions results presented hcrc, ,f;) = 0.1.

I n gencral, it is B straightrorward process to cstimate tlie RFF channel phase response. Thus, the channel induccd phase shirts can be eliminated to permit 'coherent' demodi- Iwlion which improves the decoding perI'orinance. In 0111'

simulations wc eliminate the phase shift introduced by the RFF channel, thereby emulating idcal coherent dcinodula- tion.

I K h l'~,,~.-C~,,,,,,,~,,., Vu/. 147, NU 3. J u m 2001l

current state next state

. , , ,

........... ~, '.. decoded symbol

To validale the simulations the results tire compared with at1 estimate [7] of tlic prohahility 01' tlic hit error in AWGN given by

where is the normalised tiiiiiiiiiiim squared Euclidean dislance (SED), E/,/iJN(, is the ratio between llic energy per inlbrnuition bit (E,]) and the noise power spectral density (No), and (I(.) is Ihc cotnplctiicntary error rmxion. I'urllicrmorc, we have (&,J2 = (Dt>,iJ2/2E,s, where (D,,,iJ2 is tlic tiiinimiiin SED and E.s i s the cncrgy per synihol. Note tlu11

whcl-c M is the size of the input alphabet and F is llic cod- ing wtc, which is I in our simiilations. It is also possible to rclalc E.s/N(, to the signal to noise d o (SIN) or SNR a s shown, &/No = (SlN)HT, where the producl BT is the nor-

//!/< /'lo< ~C.',~!m,w,t,, Viii 147, No. i, .hsw 2Ullil

maliscd bimdwidlh, H is llic transmittcd signal bandwidth conhining a spccilicd percentage ol' the totid transmitted power and Tis the sytnhol period.

Bounds on tlic mitiinitirn squared Euclidcan for various LRC CPM sclicnics liavc hccn determined [XI. For 2RC sclienies with h = UP, the upper bound for and for the following schemes is,

M = 2, P = 2 d:",,,, = I .07 h J = 4, P = 4 + d;,,,,, = 1.38

In both of the simulatcd sclicnies (M = 2 and h.l = 4) 21 normalised biindwitllh of BT = 1.2 was assumcd. lior hinary 2RC this handwidth cont;iins ;ipproxiinatcly 99.72'X) 0 1 the total Iransniilted power, while Ihr quatcriiary 2RC it conliiins approximatcly 99.97% of tlic tolill power.

As sliown i n Fig. 6, the RER pcrlbrnianccs l'or Ihe iiiicodcd binary and qualcrtiary syslcins for the AWGN cliatiiiel ;ipproach the tlicorcticiil lower bounds (I,,,,,,, and

whcn tlic SNR i~icraiscs. In both systcms, the S a m norinaliscd bandwidth, 137' = 1.2, and the Saiiic haud ralc

141

were used. To achieve a BER of 10~5, the quaternary sys- tem iiccds a SNK 4.5dB higher than that Tor the binary system, but it should be noted that the overall bit rate of the quaternary system is twicc that oT the binary system. If the symbol period in the quaternary system is doubled (i.c. the symbol rate is halved) we can halve the required SNR, corresponding to a gain of 3d13, and consequcntly the ticlvmtagc of the binary system will be reduced to only 1.5dB. Clearly, the quaternary system ( / I =1/4) is spcclrally more eficicnt than the binary system (/I = ID) with only 81

small SNR penalty.

. . , : :

# : : ~ e i : i / / : . . . ! , . , . : . . . . , 1 0 0 " 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4

SNR, dB

1

0.1

0.01

-3 10

-4 10

10

I o 0 2 4 6 8 10 12 14 16 18 20 2224 26 26 30 32 34 36 36

-5

-6

SNR. dB

It can be seen in Fig. 7 that in the Rayleigh chiinnel, to acliicvc a BEK or 10F5 the uncodcd quaternary system iiccds an SNR approximately 6dB higher (or 3dB in tcrrns oT E,IN0) than that required Tor the uncodcd binary system. Ncvcrtheless, il should be noted that the quaternary system ( / I = 1/4) is spectrally more efficient than the binary system (h = 112).

Comparing the BEII pcrformancc 01' these scheincs with AWGN and Raylcigh channcls, it can be observed that while Tor the AWGN channel the HER decreases rapidly with increasing SNR, for the Raylcigh channcl tlie BER decreases much more slowly. Clearly, [or the Raylcigh channel it is necessary to use much more signal energy to achieve the Sainc RER compared with that required for the

142

AWGN alone. For example, the uncoded binary system requires a 21dB incrcasc in the SNR to achieve a BER of 10-5.

5 Conclusions

A complete digilal connnunication system was imple- mented and novel bit error rale (BER) results obtained for uncodcd biliary and quaternary CPM schemes, operating over AWGN and RFF chiiiinels. The transmitter utilised ii decomposed model of CPM such that a simplified maxi- iinini likelihood receiver could be used. To achieve this, considerable effort was directed towards tlie desigii of a Vitcrbi detector which could be easily adapted to the ring of integers used in the CPM transmitter. This resulted in a novel implementation of a ring Vilcrbi detector which could he quickly niodilicd to decode the chosen ring CPM scheme, leading the way to a feasible iinplerncntatioii in hiirdwirc. This enablcd perbnnance enhancing trellis coded modulation schemcs to be implemented and evalu- ated both easily aiid quickly. The proposed schemes were validated via Monte-Carlo based simulations.

6

I

2

3

4

5

6

7

R

7

References

UALDINI FILIIO, I<., PESSOA, A.C.P., and ARANTES, D.S.: 'Svstematic linear C O ~ C S over ti iine Cor cncadcd nllase madulntion'.

Appendix

Using cqn. 6 in eqn. 2 allows eqn. 4 to be written as

7>(T + Tl,T, U )

d I ( M - 1)(7+ nT) T +

T h ( M - l ) T + Th(lV1 - I)n + T

- 2nh /,-I

( M - 1)q( T + i T ) - R h ( lV1 - 1) ( n -1, + 1 ) %=n

noting that the value of q(z + i7) is for i 2 L 171. Finally wc obtain

n- I , 1 , ~ I

'l/>(T+nT,U) = 271hCUi + 4 7 1 h ~ u n - i d T +iT) i=D i=ll

aiid it can bc wrillcn that

Note: inod,J4 = 0 - l012zl 2n (1.1 dcnotcs tllc lai'gcsl intcgccr 1101 cscccdiiig the enclosed number).

The ininiiniini nuinbcr 01' syinbol intervals n in which the mnximuni valiic or the ring XI, , i.e. P - I , can he reached, is obtained iTin thc sum above Ul is rcplaced by its mi~siiiiun~ valiic, i.e. M ~ I , giving

+ rh(1l.I - 1)(L - 1) 0 < 7 < T (10)

It can be seen that the first term is data-dependent only, the second term is data and tinie-dependent, the third and lburth terms are time-dcpcndcnt only, and tlic fir111 is nci- ther data nor time-dependent. Considering the d&ta dependelit only term of 114.) taken modulo 2ngives

This implics that

r - i w 7i,,Li,L ? L - 1 + M - 1 This implies that all possible values of Z,, can be reachctl

in symbol intervals (using different combinations or Ui). For cxamplc, iI" P = M , all possible values will be reached for n = I,, i.e. on the ( L + l)th syinbol interval. From then on the phise trcllis will s(art to repa t itselC

Eqn. IO taken modulo Zngivcs the physical tilted plmc a s

q ( T + n ~ , U ) = inorla,,

Abbreviating, we cim wrile