4 Recursive Linear Predictor

Transcription

1 4 Recusve Lnea Pedcto The man objectve of ths chapte s to desgn a lnea pedcto wthout havng a po knowledge about the coelaton popetes of the nput sgnal. In the conventonal lnea pedcto the known coelaton sequence of the nput data s used to detemne the pedcto coeffcents. Howeve, n ths case the coelaton sequence of the sgnal, whch needs to be estmated, wll be fomed n a ecusve way by usng wndow-based technques. Snce the pedcto updates ts coeffcents based on the ecusvely-tacked changes n the auto-coelaton sequence, the pedcto wll be named a Recusve Lnea Pedcto (RLP). As the pedcto coeffcents ae updated fo evey nput sample, the ecusve lnea pedcto s expected to pefom bette than the conventonal lnea pedcto when the coelaton popetes of the nput samples change wth tme. To make the RLP obust n the tme-vayng case, a fogettng facto - as n the WRLS [9] algothm wll be used whle updatng the coelaton sequence. At the end of ths chapte the desgned pedcto wll be used to pedct the envelope of a complex flat fadng channel. 4.1 One Step Ahead Pedcto The block dagam of a one step ahead pedcto s shown n Fgue 4.1 whee the output of the pedcto, x ˆ( n) s the estmate fo the pesent nput sample x(n). The pedcto output x ˆ( n) can be wtten n tems of the past values of the nput sequence as shown below: xˆ( n) = a = 1 = a T x x( n ) (4.1) Hee, [ ] T a = a a a s the pedcto coeffcent vecto and 1... T x = [ x ( n 1) x( n ) L x( n )] s the nput vecto. s the ode of the pedcto. 51

2 The negatve sgns wth all of the pedcto coeffcents ae used fo mathematcal convenence and confom to cuent pactce n the techncal lteatue. x(n) x(n-1) z -1 z -1 z -1 z -1 x(n-) a 1 a a 3 a 1 a x ˆ ( n ) Fgue 4.1: Block Dagam of a One Step Ahead Pedcto. The geneal equatons fo a D step ahead lnea pedcto ae gven n Appendx A. The equatons fo the one step ahead pedcto can be deved fom these equatons by smply settng D equal to 1 (one). The nomal equaton fo the one step ahead pedcto s: ( l) + a k ( k l) = 0, l = 1,,,L (4.) k = 1 In matx fom, (0) (1) ( L 1) (1) (0) ( L ) L L L L ( 1) a1 ( ) a ( L ) a = (1) () ( L) (4.3) o, Ra = b (4.4) In (4.) and (4.3), s the ode of the pedcto and L detemnes whethe the pedcto system nomal equatons ae ove-detemned o not. If L=, the system of equatons epesented by (4.3) s called the Yule-Walke equatons [7]. In ths case the coelaton matx R s not only Toepltz but also Hemtan symmetc. If L>, the system of nomal equatons becomes ove-detemned. ow the man objectve of desgnng a lnea 5

3 pedcto s to detemne a, the negatve of the coeffcent vecto of the pedcto, fo a gven auto-coelaton functon (ACF). If L =, the Yule-Walke equatons can be solved fo a by nvetng the coelaton matx R, and then multplyng t by b. The Yule-Walke equatons can also be solved by usng the Levnson algothm, whch explots the symmety popetes of R, to solve (4.4) fo a, and eques only 0.5 multplcatons and addtons. So thee s a sgnfcant educton n the computatonal complexty fom the usual matx nveson usng Gaussan elmnaton, whch eques on the ode of 3 multplcatons. If the system s ove-detemned (L>), nstead of the usual matx nvese, the pseudo-nvese of R s used to get a least squaes soluton to (4.4). The ove-detemned system geneally yelds bette esults than the symmetc system, when the nput sgnal s nosy. 4. Recusve ethod fo Changng the Pedcto Coeffcents In the absence of a po knowledge, the auto-coelaton functon (ACF) of the nput sequence to the pedcto needs to be estmated to detemne the coeffcents of the lnea pedcto by usng (4.3) o (4.4). In ths secton a ecusve appoach to estmatng the ACF wll be descbed whee the auto-coelaton matx R and the auto-coelaton vecto b, needed fo solvng (4.4), wll be updated fo evey new nput sample. Ths ecusve algothm poduces an estmate of the equed a po knowledge of the ACF. Agan, fo a sgnal wth tme vayng statstcal popetes, the auto-coelaton sequence needs to be estmated fo the pesent condton. At the same tme, the olde past values of the estmated auto-coelaton need to be dscaded fom the pesent statstcs. Fo ths pupose a wndowbased technque wll be used n the ecusve method of estmaton of the ACF. In the ecusve method, the ACF at any nstant s estmated by usng the pesent data sequence and the past estmate fo the ACF. Snce we want the system to foget the effect of the past nput samples, a wndow of length s used to consde only the most ecent samples of nput data. Let ths sequence be x (n). These nput samples ae then used to estmate the pesent value of the auto-coelaton sequence. An length auto-coelaton 53

4 (sngle-sded) sequence s then calculated by usng x (n). Let ths auto-coelaton sequence be (n). Θ( m) = DFT ( p) = IDFT ( n) = { x ( n) }, { Θ( m) Θ ( m) } m = 0,1, L, 1, [ (0) (1) L ( 1) ] p = 0,1, L, 1 (4.5) The pesent value of the auto-coelaton sequence s then estmated by usng the pevously estmated ACF and (n) n the followng way. λ ( n 1) + ( n) ( n) =, 0 λ 1 (4.6) λ + 1 In (4.6) λ s the fogettng facto that has the same effect as the fogettng facto n the well-known exponentally weghted RLS algothm. Snce λ s less than o equal to 1, t wll povde an exponentally decayng wndow effect, as shown n Fgue 4.. Howeve the RLS knd of fogettng facto has a lmtaton wokng n the non-statonay envonment. Ths s because t can only apply an exponental wndow to the tme sees and the exponental wndow not only has an nfnte tal but also penalzes nea pesent data too much (Fgue 4.). 1 n - λ umbe of teaton n Fgue 4.: RLS Wndow Functon. 54

5 To mpove the fogettng wndow functon, a ectangula wndow of length W wll be used n addton to the exponental wndow to foget data n the fa past completely. The algothm equvalent to the RLS algothm but usng ths knd of fogettng facto s called the Wndowed Recusve Least Squaes (WRLS) algothm [9]. The wndow functon of WRLS s shown n Fgue 4.3. Due to the addtonal ectangula wndow, the WRLS algothm pefoms bette than RLS fo non-statonay sgnal estmaton [9]. ow f the WRLS based fogettng facto s used to estmate the pesent coelaton sequence, update equaton (4.6) needs to be changed n the followng way: λ ( n 1) + ( ) λ W n ( n W ) ( n) =, 0 λ 1 (4.7) λ + 1 λ W In (4.7), W s the length of the ectangula wndow as shown n Fgue 4.3. Snce the wndow used to collect the nput data s of length, at the begnnng of the estmaton pocess t s mplctly assumed that the coelaton popety of the nput sequence wll not be the same afte nput samples. Theefoe, whle updatng the coelaton sequence (n), the length of the ectangula wndow W should easonably be less than o equal to the length of the wndowed nput sequence x (n),.e., W. (4.8) 1 λ 1 n - W + 1 n n - λ W n W + 1 umbe of teaton n Fgue 4.3: WRLS Wndow Functon. 55

6 Snce the ectangula wndow takes nto account only the most ecent W samples and completely fogets the past nfomaton, a hghe value of the fogettng facto λ can be used to gve moe weght to the mmedate past samples n WRLS. The estmated auto-coelaton sequence (n) ((4.6) o (4.7)) s then used to fom the coelaton matx R(n) and the coelaton vecto b(n) n (4.5) fo the tme nstant n. The pedcto coeffcents fo that tme nstant ae then calculated by solvng (4.5). The steps of the Recusve Lnea Pedcto algothm, whch updates the pedcto coeffcents fo evey tme nstant, can be summazed as follows: Step 1 Table 4.1: Algothm fo Recusve Lnea Pedcto. Intalzaton x 1) = [ x(0) 0 L 0] T (, s the length of x Step Step 3 Instantaneous Coelaton Sequence Estmated Coelaton Sequence (0) = L zeo matx Θ( m) = DFT ( p) = IDFT ( n) = { x ( n) }, { Θ( m) Θ ( m) } m = 0,1, L, 1, [ (0) (1) L ( 1) ] p = 0,1, L, 1 λ ( n 1) + ( n) RLS Wndow: ( n) =, 0 λ 1, o 1+ λ λ ( n 1) + ( n) λ WRLS Wndow: ( n) = 1+ λ W ( n W ) Step 4 Detemnaton of the pedcto coeffcents R = a = R 1 b, (0) (1) ( L 1) f L =, (1) (0) ( L ) o, a = pseudonvese( R) b, L L L L f L > ( 1) ( ), ( L ) b = (1) () ( L) xˆ( n + 1) = a x ( n) Step 5 Update x ( n + 1) = [ x( n) fst 1elements of x ( n) ] Step 6 Recuson Repeat fom Step. T 56

7 4.3 Pedcton of the Fadng Envelope In ths secton, the fadng envelope of a flat fadng channel wll be pedcted by usng the desgned ecusve lnea pedcto. Dffeent ways of pedctng the values of the fadng envelope fo a flat fadng channel have been dscussed [10]. If the fadng envelope s sampled at a ate of moe than two tmes f m, the maxmum Dopple spead, t s possble to ntepolate the value of the fadng envelope at any pont n between two samples [10]. Ths knd of pedcton s possble because of the nheent coelaton among the samples of the fadng envelope. It has been shown [11] that the powe spectum of a flat fadng envelope can be expessed by the followng equaton: SE C 1.5, f fc fm f f = c ( f ) π fm 1 (4.9) fm 0, f fc > fm In (4.9), f c and f m ae the cae fequency and the maxmum Dopple spead of the fadng channel espectvely. E ( f ) s the powe spectum of the fadng envelope. The S C base-band equaton can be deved fom (4.9) by settng f c to zeo, whch yelds: 1.5, f fm SE B ( f ) = π fm f (4.10) 0, f > f m Fgue 4.4 shows the base-band powe spectum of a fadng envelope fo a maxmum Dopple spead of 100 Hz. Fom (4.10) and fom Fgue 4.4, t s clea that the maxmum fequency component n the fadng envelope s f m. Accodng to the yqust samplng theoem, the fadng envelope can be epesented completely by ts samples as long as the samplng ate s hghe than f m. 57

8 Fgue 4.4: Base-Band Powe Spectum of the Fadng Envelope wth Dopple Spead of 100 Hz. As the powe spectal densty (PSD) of the fadng envelope s not whte, thee exsts coelaton between the tme samples of the fadng envelope. The coelaton s stonge fo a smalle f m, o hghe f s. Theefoe, f the fadng envelope s sampled at a ate much hghe than f m, the futue value of the fadng envelope can be pedcted easonably well fom ts pesent and past samples. A smple lnea pedcto (specfcally a one step ahead pedcto) can be used to pedct the mmedate next value of the fadng envelope. To vefy ths statement to nvestgate the pefomance of the ecusve lnea pedcto, we wll geneate the Raylegh fadng envelope fo the non-statonay stuaton. The statonay fadng envelope s a specal case of the non-statonay envelope when all of the tme-vayng quanttes ae constants on-statonay Raylegh Fadng Envelope Smulato The geneal block dagam of the Raylegh fadng smulato [1] s shown n Fgue 4.5. Whle smulatng ths system, the man dffculty n the block dagam s the mplementaton of the base-band Dopple flte whose PSD s gven by (4.10). The magntude esponse of the flte can be deved by takng the squae oot of the ght hand sde of (4.10). The equaton fo the magntude esponse of the Dopple flte s gven below. 58

9 Independent H f ) = π D ( 4 fm f, f, f f > f m m (4.11) Baseband Gaussan ose Souce Baseband Gaussan ose Souce Baseband Dopple Flte Baseband Dopple Flte Absolute Value Opeato Fadng Envelope j Fgue 4.5: Geneal Block Dagam of Raylegh Fadng Envelope Geneato. Snce (4.11) esults n nfnty at f = f m and zeo fo f geate than f m, t s mpossble to fnd a dgtal flte that povdes a magntude esponse equal to that gven n (4.11). A fequency doman appoach fo mplementng the fadng geneato (Clak s model [1]) avods the dffcultes of fndng the appopate mpulse esponse of the Dopple flte. Clak s model also uses FFTs fo effcently calculatng the fadng envelope. The poblem wth Clak s model s that ths model s desgned fo pocessng a block of data wth a constant Dopple spead f m. Theefoe, Clak s model cannot be used to smulate the fadng envelope when the moble velocty changes contnuously wth tme. The elatonshp between the moble velocty and the maxmum Dopple spead of the fadng envelope s gven by the followng equaton. f m v v = fc = (4.1) c λ c In (4.1), v s the moble velocty, 8 c = 3 10 m/ sec s the velocty of lght, and λ c s the wavelength of the lght fo the cae fequency f c. Any change n the moble velocty dung the obsevaton tme wll esult n a contnuous change n the Dopple spead. Snce the maxmum Dopple spead detemnes the coeffcents of the Dopple fltes n Fgue 4.5, 59

10 these fltes wll be tme-vayng f the moble velocty changes wth tme. In the smulaton, the Dopple flte was mplemented by usng a lnea phase FIR flte desgned usng the fequency doman samplng method. The desgn pocedue s descbed below. 1. Snce the Dopple flte s beng desgned n the fequency doman, the total fequency band s defned fom -f s / to f s / Hz, whee f s s the samplng fequency of the flte. The magntude esponse of the flte, defned ove the total fequency band, s then sampled at K equally spaced fequences. Theefoe, the spacng between two adjacent fequency samples s: fs f = (4.13) K To make the calculaton effcent, K s chosen so that log K s an ntege.. Snce ou objectve s to desgn a eal FIR flte, we fst defne K/ samples fo the postve fequences only. The magntude samples fo the postve fequency band ae defned as: K H ( k) = H D ( k f ), k = 0,1, L 1 (4.14) 3. To make the flte ealzable, the magntude esponse of the flte should not be nfnty at f = f m. The magntude esponse at f = f m was detemned by lnealy ntepolatng the pevous two fequency samples. Let, d be the fequency ndex coespondng to the Dopple fequency,.e., expesson d fm ± δ = and δ f < f. In the ± δ s used to compensate fo the fact that f m may not coespond to any fequency doman sample pont. H ( ) H ( 1) H ( ) (4.15) d = d d 4. Even though the magntude esponse of the Dopple flte goes to zeo mmedately afte f = f m, to educe the pples n the magntude esponse of the desgned flte, two 60

11 sample ponts s used to defne the tanston band of the flte. The values of the two tanston band samples s defned as (expementally detemned fo the lowest value of the pck pple): H ( H ( d d + 1) = 0.6H ( + ) = 0.1H ( d d ) ) (4.16) 5. Agan, to educe the pple n the pass-band of the flte [13], the sampled values of the magntude esponse wee changed n the followng way: k K H altenate ( k) = ( 1) H ( k), k = 0,1, L 1 (4.17) 6. A lnealy vayng phase esponse was added to the magntude samples. The phase esponse s zeo at f = 0 (.e., fo k = 0) and s equal to -π at f = f s / (fo, k = K/). The fequency esponse of the flte fo the fst K/ samples s gven below: H dopp ( k) = H altenate ( k) e π j k K, k K = 0,1, L 1 (4.18) 7. To make the phase esponse of the flte lnea, the next half of the fequency sample aay was flled up by takng the complex conjugate of the fst half n evese ode,.e., H H dopp dopp K K ( + k) = H dopp( k), K ( ) = 0, fo ths case. K k = 1, L 1 (4.19) 8. The mpulse esponse of the desed flte, h dopp (n), s then calculated by takng the nvese DFT of H dopp (k). Ideally the mpulse esponse of the flte s a eal sequence. Due to the fnte wod length effect of the computng devce, h(n) can be a complex sequence wth a vey small magnay pat. Ths magnay pat s then neglected. 61

12 Fo the non-statonay case, to get an output (fadng envelope) wth a smooth vaaton of ts statstcal popetes a tme-vayng flte of length K s mplemented. The coeffcents of the tme vayng flte ae the same as the coeffcents of the Dopple flte h dopp (n) calculated fo evey nstant of tme wth the nstantaneous velocty. At nstant m, let the flte coeffcent vecto be h(m).e., h(m) = [h(0) h(1) h(k-1)] T calculated at nstant m. At that nstant, let the flte state vecto, o the nput vecto be w(m), whee [ w( m) w( m 1) w( m K 1 ] T w ( m ) = L + ) (4.0) In (4.0), {w(m)} s the complex..d. zeo mean Gaussan andom sequence of vaance (Fgue 4.5). The output of the tme-vayng flte at nstant m s then calculated as follows: comp T ( m) = h ( m) w( m) (4.1) Fo the equed length of fadng envelope, the complete sequence of { comp (m)} s fst geneated. The geneated sequence s then nomalzed fo an ms value of one. The absolute value of the esultant sgnal s the desed fadng envelope. Fo the statonay case, the coeffcents of the Dopple flte ae calculated only once. All othe pocedues ae smla to the non-statonay case Smulaton Results In ths secton smulaton esults wll be pesented to vefy the pefomance of the ecusve pedcto unde statonay and non-statonay condtons. Befoe gong to those esults, some ntemedate esults wll be pesented to show that the mplemented Dopple flte s a good appoxmaton fo the desed non-ealzable flte. The magntude esponses of the dealzed (lnea appoxmaton nea maxmum Dopple fequency) Dopple flte and the mplemented Dopple flte ae shown n Fgue 4.6. The paametes used to geneate these two magntude esponses ae: Samplng fequency of the flte, f s = 8000 Hz; 6

13 oble velocty, v = 50 mph; Cae Fequency, f c = 900 Hz; axmum Dopple spead, v fm = fc = = Hz c Fgue 4.6 shows that, except n the vcnty of f m, the maxmum Dopple spead, the mplemented flte has almost the same magntude esponse as the dealzed Dopple flte. The dscepances nea f m ae due to changes (Secton 4.3.1) made to make the flte ealzable. Fgue 4.6: agntude Response of the Idealzed and Smulated Dopple Flte Fadng Envelope wth Constant Velocty Snce the maxmum Dopple spead f m of the fadng envelope s constant fo a constant velocty, f m wll be used as the ndependent vaable fo all of the esults. A Raylegh fadng envelope wth f m = 100 Hz and f s = 8 khz was geneated by usng the pocedue explaned n Secton The fadng envelope s shown n Fgue 4.7. Ths geneated envelope sequence s then used as the nput to the ecusve lnea pedcto. The output of the pedcto when an RLS type wndow functon s used to update the auto-coelaton 63

14 sequence, along wth the ognal pedcted envelope, s shown n Fgue 4.7. An ovedetemned system was used to detemne the pedcto coeffcents. The sze of the R matx used n the smulaton was 0x7. The paametes used n the smulaton ae gven below: axmum Dopple spead, f m = 100 Hz; Samplng fequency, f s = 8 khz; Length of the Pedcto, = 7; Fogettng facto, λ = 0.9. Fgue 4.7: Actual and Pedcted Fadng Envelope (RLS Type Wndow Used to Update the Coelaton Sequence). Fgue 4.8 shows the actual and the pedcted fadng envelope when the WRLS type wndow functon s used to update the auto-coelaton sequence. The wndow sze W used n the smulaton was 50. All othe paametes used to geneate Fgue 4.8 wee the same as those used to geneate Fgue 4.7. Fom Fgue 4.7 and Fgue 4.8, t s clea that both ecusve lnea pedcto appoaches can follow the nput sequence. The pefomances deved fom both appoaches ae compaed on the bass of the pedcton eo, whch s the dffeence between the ognal and the estmated envelope. To get a constant quantty fo measung the pefomance of the dffeent appoaches, the tme 64

15 aveage of the squaed pedcton eo was measued and the esultant quantty was named the ean Squaed Pedcton Eo (SPE). Theefoe, 1 n= SPE = e ( n) = ( x( n) xˆ( n) ) (4.) n= 0 Fgue 4.8: Actual and Pedcted Fadng Envelope (WRLS Type Wndow Used to Update the Coelaton Sequence). The SPE of the ecusve lnea pedcto wth two dffeent updatng wndows ae shown n Fgue 4.9. Fgue 4.9 shows that fo the same Dopple spead, the SPEs ae dffeent fo dffeent ealzatons of the fadng envelope. Theefoe, to get an estmate of SPE, the smulaton was un fo 50 tmes wth the same ealzatons fo dffeently wndowed estmate updates. Ffty dffeent SPEs wee then used to calculate the ensemble aveaged SPE fo both appoaches. The esultant ensemble aveage wll gve us an estmate of the expected value of the SPE and ths esult wll be used to compae the pefomance of the dffeent appoaches. Fgue 4.10 shows the estmated SPE fo both appoaches fo dffeent values of maxmum Dopple spead, f m. Due to the tansent effect, the fst few samples of the pedcted sequence ae not the same as the ognal nput samples. Theefoe, the fst 100 samples wee not used n the calculaton of the SPE. 65

16 Fgue 4.9: ean Squaed Pedcton Eo wth RLS and WRLS Wndows fo Dffeent Realzatons (f m = 100 Hz). Fom Fgue 4.10 we obseve that wth the ecusve lnea pedcto, the pedcton eo nceases wth an ncease n f m, the maxmum Dopple spead. Ths s because fo hghe f m, o n othe wods, fo hghe moble velocty the andomness of the fadng envelope nceases. Ths nceased andomness esults n educed coelaton among the samples of the fadng envelope and thus yelds an ncease n the pedcton eo. Fgue 4.10 also shows that, fo any f m, the ecusve lnea pedcto pefoms bette f the WRLS type wndow functon s used fo estmatng the coelaton sequence of the nput. Fgue 4.10: Oveall ean Squaed Pedcton Eo fo Dffeent f m. 66

17 4.3.. Fadng Envelope wth Constant Acceleaton The same block dagam as shown n Fgue 4.5 was used to geneate the Raylegh fadng envelope fo constant acceleaton. Snce the moble velocty s changng at evey nstant n tme, the Dopple flte of Fgue 4.5 was changed fo evey nput sample. Whle geneatng the fadng envelope fo constant acceleaton, the fst 400 samples of the envelope wee geneated wth constant velocty of 30 mles/hou. The acceleaton was mposed afte the fst 400 samples. To educe the numbe of samples, and consequently educe the pocessng tme, a lowe samplng fequency (f s = 000 Hz) was used n the smulaton. The velocty pofles fo dffeent values of acceleaton ae shown n Fgue A poton of the fadng envelope geneated fo an acceleaton of 10 mles/sec s shown n Fgue 4.1. Fgue 4.11: Velocty Pofle of the Fadng Envelope fo Dffeent Acceleatons. Fgue 4.1 also shows the output of the ecusve lnea pedcto. A WRLS type wndow functon was used n the smulaton to geneate the pedcted envelope shown n Fgue 4.1. The smulaton was pefomed wth a wndow sze W = 15 and a fogettng facto λ = The same wndow functon was used to geneate the mean squaed pedcton eo (SPE) fo dffeent values of acceleaton. Smla to the case of constant velocty, the SPE s dffeent fo dffeent ealzatons of the fadng envelope fo the same velocty pofle (.e., fo a constant acceleaton). Theefoe, ten dffeent ealzatons wee 67

18 used to estmate the SPE value fo a patcula acceleaton. The values of the SPE fo dffeent acceleatons ae shown n Fgue Fgue 4.13 also shows the SPE vs. acceleaton cuve when the RLS type wndow s used to update the auto-coelaton sequence of the nput sgnal to the pedcto. The value of the fogettng facto used n the smulaton fo the RLS type wndow was 0.9. Fgue 4.1: Raylegh Fadng Envelope fo Constant Acceleaton of 10 mles/sec and the Pedcted Envelope. WRLS Type Wndow Used n the Pedcto. Samplng Fequency, f s = 000 Hz. Fgue 4.13: ean Squaed Pedcton Eo fo Dffeent Acceleatons. 68

19 Fom Fgue 4.13 t can be sad that, unlke n the constant velocty case, the pefomance of the ecusve lnea pedcto (measued by SPE), ethe wth the RLS wndow o the WRLS wndow, does not degade wth an ncease n the acceleaton. Fgue 4.13 also shows that the pedcto wth the WRLS wndow pefoms appoxmately 1 db bette than the pedcto wth the RLS wndow when the moble velocty changes wth a constant acceleaton. Ths mpovement n the pefomance comes n a tade-off wth the nceased memoy equements of the WRLS wndow ove the RLS wndow. 69