CODING & MODULATION Prof. Ing. Anton Čižmár, PhD. also from Digital Communications 4th Ed., J. G. Proakis, McGraw-Hill Int. Ed.

Transcription

1 CODING & MODULATION Pof. Ig. Ato Čižá, PhD. alo fo Digital Couicatio 4th Ed., J. G. Poai, McGaw-Hill It. Ed.

2 5 Optiu Receive fo the AWGN Chael I the peviou chapte, we decied vaiou type of odulatio ethod that ay e ued to tait digital ifoatio though a couicatio chael. A we have oeved, the odulato at the taitte pefo the fuctio of appig the digital equece ito igal wavefo. Thi chapte deal with the deig ad pefoace chaacteitic of optiu eceive fo the vaiou odulatio ethod, whe the chael coupt the taitted igal y the additio of Gauia oie. 5. OPTIMUM RECEIVERS FOR SIGNALS CORRUPTED BY AWGN Let u egi y developig a atheatical odel fo the igal at the iput to the eceive. We aue that the taitte ed digital ifoatio y ue of M igal wavefo { t,,,,m.}. Each wavefo i taitted withi the yol igalig iteval of duatio T. To e pecific, we coide the taiio of ifoatio ove the iteval t T. The chael i aued to coupt the igal y the additio of white Gauia oie a illutated i Fig.5.-. Fig.5.- Model fo eceived igal paed though a AWGN chael Thu, the e eceived igal i the iteval t T ay e expeed a t t + t t T 5.- with powe pectal deity of t Φ f N W/Hz Baed o the oevatio of t ove the igal iteval, we wih to deig a eceive that i optiu i the ee that it iiize the poaility of aig a eo. It i coveiet to udivide the eceive ito two pat the igal deodulato ad the detecto Fig.5.-.

3 The fuctio of the igal deodulato i to covet the eceived wavefo t ito N- dieioal vecto [ N ], whee N i the dieio of the taitted igal wavefo. The fuctio of the detecto i to decide which of M poile igal wavefo wa taitted aed of the vecto. Fig. 5.- Receive cofiguatio Received igal t Sigal deodulato Detecto Output deciio Two ealizatio of the igal deodulato ae decied i the ext two ectio. Oe i aed o the ued of igal coelato. The ecod i aed o the ue of atched filte. The optiu detecto that follow the igal deodulato i deiged to iiize the poaility of eo. 5.. Coelatio Deodulato We decie a coelatio deodulato that decopoe the eceived igal ad oie ito N-dieioal vecto. The igal ad the oie ae expaded ito a eie of liealy weighted othooal ai fuctio {f t}. It i aued that the N ai fuctio {f t} pa the igal pace, o that evey oe of the poile taitted igal of the et { t}, ca e epeeted a a liea coiatio of {f t}. I the cae of the oie, the fuctio {f t} do ot pa the oie pace. Howeve, the oie te that fall outide the igal pace ae ielevat to the detectio of the igal. Suppoe the eceived igal t i paed though a paallel a of N co coelato which aically copute the pojectio of t oto the N ai fuctio {f t}, a illutated i Fig Thu we have

4 T t f t dt + T [ t + t] f t dt,,,n 5.- whee T T t f t dt t f t dt,,,n 5.-3 The igal i ow epeeted y the vecto with copoet,,,,n. Thei value deped o which of the M igal wa taitted. We ca expe the eceived igal i the iteval t T a t N N f t + f t + t N f t + t 5.-4 whee N t t f t 5.-5 i a zeo ea Gauia oie poce that epeet the diffeece etwee the oigial oie poce t ad the pat coepodig to the pojectio of t oto the ai fuctio {f t} ad i ielevat to the deciio a to which igal wa taitted. The deciio ay e aed etiely o the coelato output igal ad the oie copoet +,,,,N. The coelato output { } coditioed o the th igal eig taitted ae Gauia ado vaiale with ea E E ad equal vaiace σ σ N 5.-9 Sice, the oie copoet { } ae ucoelated Gauia ado vaiale, they ae alo tatitically idepedet. A a coequece, the coelato output { } coditioed

5 o the th igal eig taitted ae tatitically idepedet Gauia vaiale. Hece the coditioal poaility deity fuctio of the ado vaiale [ N ] ae iply N p p,,,m 5.- whee p πn [ exp N,,,N 5.- By utitutig 5.- ito 5.-, we otai the joit coditioal PDF N p exp N [,,,M 5.- N πn The coelato output ae ufficiet tatitic fo eachig a deciio o which of the M igal wa taitted. All the elevat ifoatio i cotaied i the coelato output { }. Hece t ay e igoed. Exaple 5.- Coide a M-ay aead PAM igal et i which the aic pule hape gt i ectagula i the peiod,t with the aplitude a Fig The additive oie i zeo ea white Gauia oie poce. Let u deteie the ai fuctio ft ad the output of the coelatio-type deodulato. The eegy i the ectagula pule i E g T g t dt a dt a T Figue 5--4 Sigal pule fo Exaple 5-- gt a [ T [ T t Sice the PAM igal et ha dieio N, thee i oly oe ai fuctio ft. Thi i give a f t { / T t T g t a T othewie The output of the coelatio-type deodulato i T t f t dt T T t dt

6 It i iteetig to ote that the coelato ecoe a iple itegato whe ft i ectagula. If we utitute fo t, we otai T t t dt T [ + ] + whee the oie te E ad σ N. The poaility deity fuctio fo the apled output i 5.. Matched-Filte Deodulato p πn [ ] exp N Itead of uig a a of N coelato to geeate the vaiale { }, we ay ue a a of N liea filte. Suppoe that the ipule epoe of the N filte ae h t f T t, t T 5.-4 whee {f t} ae the N ai fuctio ad h t outide of the iteval output of thee filte ae y t t t τ f τ h t τ dτ T t + τ dτ Now, if we aple the output of the filte at tt, we otai T y T τ f τ dτ { } t T. The,,,N 5.-5,,,N 5.-6 Hece, the apled output of the filte at tie t T ae exactly the et of value { } otaied fo the N liea coelato. A filte whoe ipule epoe ht T-t i called atched filte to the igal t. Figue 5--5 Sigal t ad filte atched to t t ht T-t A A T t T t a igal t Ipule epoe The epoe of ht T-t to the igal t i

7 t y t τ T t + τ dτ 5.-7 which i aically the tie-autocoelatio fuctio of the igal t. Fig.5.-6 illutate yt fo the tiagula igal pule how i Fig Note thet the autocoelatio fuctio yt i a eve fuctio of t, which attai a pea at tt. Figue 5--6 The atched filte output i autocoelatio fuctio of t t y tfig.5.-6 τ T t + τ dτ yt T T I the cae of the deodulato decied aove, the N atched filte ae atched to the ai fuctio {f t}. Fig.5.-7 illutate the atched filte deodulato that geeate the oeved vaiale { }. Popetie of the atched filte a If a igal t i coupted y AWGN, the filte with a ipule epoe atched to t axiize the output igal-to-oie atio SNR, which i a follow SNR N T E t dt N 5.-4 Note that the output SNR fo the atched filte deped o the eegy of the wavefo t ut ot o the detailed chaacteitic of t. Thi i aothe iteetig popety of the atched filte The Optiu Detecto We have deotated that, fo a igal taitted ove a AWGN chael, eithe a coelatio deodulato o a atched filte deodulato poduce the vecto [ N ], which cotai all the elevat ifoatio i the eceived igal wavefo. I thi ectio we decie the optiu deciio ule aed o the oevatio vecto. We aue that thee i o eoy i igal taitted i ucceive igal iteval.

8 We wih to deig a igal detecto that ae a deciio o the taitted igal i each igal iteval aed o the oevatio of the vecto i each iteval uch that the poaility of a coect deciio i axiized. We coide a deciio ule aed o the coputatio of the poteioi poailitie defied a Pigal wa taitted,,,m which we aeviate a P. The deciio citeio i aed o electig the igal coepodig to the axiu of the et of poteio poailitie {P } iiize the poaility of eo ad axiize the poaility of coect deciio. Thi deciio citeio i called the axiu a poteioi poaility MAP citeio. Uig Baye ule, the poteio poailitie ay e expeed a p P P p whee p i the coditioal PDF of the oeved vecto give ad P i the a pioi poaility of the th igal eig taitted. The deoiato of ay e expeed a M p p P Soe iplificatio occu i the MAP citeio whe the M igal ae equally poale apioi, i.e. P /M fo all M. Futheoe, we ote that the deoiato i i idepedet of which igal i taitted. Coequetly, the deciio ule aed o fidig the igal that axiize P i equivalet to fidig the igal that axiize p. The coditioal PDF p o ay ootoic fuctio of it i uually called the lielihood fuctio. The deciio citeio aed o axiu of p ove the M igal i called the axiu-lielihood ML citeio. We oeve that a detecto aed o the MAP citeio ad oe that i aed o the ML citeio ae the ae deciio a log a the a pioi poailitie P ae all equal. i.e., the igal { ae equipoale. I the cae of a AWGN chael, the lielihood fuctio p i give y Equatio 5.-. To iplify the coputatio, we ay wo with the atual logaith of p, which i a ootoic fuctio. Thu l p N l π N 5.-4 N N

9 The axiu of l p ove i equivalet to fidig the igal that iiize the Euclidea ditace iiu ditace detectio N D, 5.-4 We call D,,,,,M, the ditace etic. Hece, fo the AWGN chael, the deciio ule aed o the ML citeio educe to fidig the igal that i cloet i ditace to the eceived igal vecto. We hall efe to thi deciio ule a iiu ditace detectio. Aothe itepetatio of the optiu deciio ule aed o the ML citeio i otaied y expadig the ditace etic i Eq.5.-4 a N N + D,. +,,,M 5.-4 N The te i coo to all ditace etic, ad, hece, it ay e igoed i the coputatio of the etic. The eult i a et of odified ditace etic D ', Note that electig the igal that iiize D, i equivalet to electig the igal that axiize the etic C, -D,, i.e., C, The te. epeet the pojectio of the eceived igal vecto oto each of the M poile taitted igal vecto. The value of each of thee pojectio i a eaue of the coelatio etwee the eceived vecto ad the th igal. Fo thi eao, we call C,,,,M, the coelatio etic fo decidig which of the M igal wa taitted. Fially, the te E,,,M, ay e viewed a ia te that eve a copeatio fo igal et that have uequal eegie, uch a PAM. If all igal have the ae eegy, ay alo e igoed i the coputatio of the coelatio etic C, ad the ditace etic D, o D,. It i eay to how Pole 5.5 that the coelatio etic C, ca alo e expeed a T C, t t dt E,,,M Theefoe, thee etic ca e geeated y a deodulato that co-coelate the eceived igal t with each of the M poile taitted igal ad adjut each coelato output fo the ia i the cae of eequal igal eegie. Equivaletly, the

10 eceived igal ay e paed though a a of M filte atched to the poile taitted igal{ t} ad apled at t T, the ed of the yolic iteval. Coequetly, the optiu eceive deodulato ad detecto ca e ipleeted i the alteative cofiguatio illutated i Fig I uay, we have deotated that the optiu ML detecto copute a et of M ditace D, o D, ad elect the igal coepodig to the allet ditace etic. Equivaletly, th optiu ML detecto copute a et of M coelato etic C, ad elect the igal coepodig to the laget coelatio etic. The aove developet fo the optiu detecto teated the ipotat cae i which all igal ae equally poale. I thi cae, the MAP citeio i equivalet to the ML citeio. Howeve, whe the igal ae ot equally poale, the optiu MAP detecto ae it deciio o the poailitie P,,,M, give y Eq o, equivaletly, o the etic PM, p P The followig exaple illutate thi coputatio fo iay PAM igal. Exaple Coide the cae of iay PAM igal i which the two poile igal poit ae E, whee E i the eegy pe it. The pio poailitie ae P p ad P - p. Let u deteie the etic fo the optiu MAP detecto whe the taitted igal i coupted with AWGN. The eceived igal vecto oe-dieioal fo iay PAM i ± E y T

11 whee y T i a zeo-ea Gauia ado vaiale with vaiace N σ. Coequetly, the coditioal PDF p fo the two igal ae exp E p σ πσ exp E p σ πσ The the etic PM, ad PM, ae exp, E p pp PM σ πσ exp, E p pp PM σ πσ 5.-5 If PM, > PM,, we elect a the taitted igal; othewie, we elect. Thi deciio ule ay e expeed a,, PM PM < > 5.-5 But + exp,, E E p p PM PM σ 5.-5 So that Eq.5.-5 ay e expeed a p p E E < > + l σ o equivaletly, p p N p p E < > l 4 l σ 5.-54

12 Thi i the fial fo fo the optiu detecto. It copute the coelatio etic p C, E ad copae it with thehold N l. Fig 5.- illutate the 4 p two igal poit ad. The thehold, deoted y τ h, divide the eal lie ito two egio, ay R ad R, whee R coit of the et of poit that ae geate tha τ h, ad R coit of the et of poit that ae le tha τ h. If E > τ h, the deciio i ade that wa taitted, ad if E < τ h, the deciio i ade that wa taitted. The thehold τ h deped o N ad p. If p/, τ h. If p>/, the igal poit i oe poale, ad, hece, τ h <. I thi cae, the egio R i lage tha R, o that i oe liely to e elected tha. If p</, the oppoite i the cae. Thu, the aveage poaility of eo i iiized. Figue 5-- Sigal pace epeetatio illutatig the opeatio of the optiu detecto fo iay PAM odulatio S - E τ h S E Regio R Regio R It i iteetig to ote that i the cae of uequal pio poailitie, it i eceay to ow ot oly the value of the pio poailitie ut alo the value of the powe pectal deity N, o, equivaletly, the oie-to-igal atio N /E, i ode to copute the thehold. Whe p/, the thehold i zeo, ad the owledge of N i ot equied y the detecto The Maxiu-Lielihood Sequece Detecto Whe the igal ha o eoy, the yol-y-yol detecto decied i the pecedig ectio i optiu i the ee of iiizig the poaility of a yol eo. O the othe had, whe the taitted igal ha eoy, i.e., the igal taitted i ucceive yol iteval ae itedepedet, the optiu detecto i a detecto that ae it deciio o oevatio of a equece of eceived igal ove ucceive igal iteval. I thi ectio, we decie a axiu-lielihood equece detectio algoith that eache fo the iiu Euclidea ditace path though the telli that chaacteize the eoy i the taitted igal. Let u coide, a a exaple, the NRZI igal. It eoy i chaacteized y the telli how i Fig

13 The igal taitted i each igal iteval i iay PAM. Hece, thee ae two poile taitted igal coepodig to the igal poit E, whee E i the eegy pe it. The output of the atched-filte o coelatio deodulato fo iay PAM i the th igal iteval ay e expeed a E + ± whee i a zeo-ea Gauia ado vaiale with vaiace N σ. Coequetly, the coditioal PDF fo the two poile taitted igal ae exp E p σ πσ exp E p σ πσ Fo ay give taitted equece, the joit PDF of,,, K ay e expeed a a poduct of K agial PDF, i.e., K K K K K p p exp exp,...,, σ πσ σ πσ whee eithe E o E. The, give the eceived equece,,, K at the output of the atched filte o coelatio deodulato, the detecto deteie the

14 K that axiize the coditioal PDF p,,..., K. Such detecto i called the axiu-lielihood ML equece-detecto. By taig the logaith of Eq ad eglectig the te that ae idepedet of,,, K, we fid that a equivalet ML equece detecto elect the equece that iiize the Euclidea ditace etic equece {,,..., } K D, 5.-6 I eachig though the telli fo the equece that iiize the Euclidea ditace D,, it ay appea that we ut copute the ditace D, fo evey poile equece. Fo the NRZI exaple, which eploy ojay odulatio, the total ue of equece i K, whee K i the ue of output otaied fo the deodulato. Howeve, thi i ot the cae. We ay educe the ue of equece i the telli each y uáy the Vitei algoith to eliiace equece a ew data i eceived fo the deodulato. The Vitei algoith i a equetial telli each algoith fo pefoig ML equece detectio. We aue that the each poce egi iitially at tate S. The coepodig telli i how i Fig.5.-. At tie tt, we eceive + fo the deodulato, ad at tt, we eceive +. Sice the igal eoy i oe it, which we deote y L, we oeve that the telli eache it egula teady tate fo afte two taitio. Thu upo eceipt of at tt ad theeafte, we oeve that thee ae two igal path eteig each of the ode ad two igal path leavig each ode. The two path eteig ode S at tt coepod to the ifoatio it, ad, o, equivaletly, to the igal poit E, E ad E, E, epectively. The two path eteig ode S at

15 tt coepod to the ifoatio it, ad, o, equivaletly, to the igal poit E, E ad E, E, epectively. Fo the two path eteig ode S, we copute the two Euclidea ditace etic D +, + E + E D +, E + E 5.-6 y uig the output ad fo the deodulato. The Vitei algoith copae thee two etic ad dicad the path havig the lage geate-ditace etic. The othe path with the lowe etic i aved ad i called the uvivo at tt. Siilaly, fo the two path eteig ode S at tt, we copute the two Euclidea ditace etic D D, + E + E, E + E 5.-6 y uig the output ad fo the deodulato. The two etic ae copaed ad the igal path with the lage etic i eliiated. Thu, at tt, we ae left with two uvivo path, oe at ode S ad the othe at ode S, ad thei coepodig etic. The igal path at ode S ad S ae the exteded alog the two uvivo path. Upo eceipt of 3 at t3t, we copute the etic of the two path eteig tate S. Suppoe the uvivo at tt ae the path, at S ad, at S. The, the two etic fo the path eteig S at t3t ae D +,, D, + 3 E D +,, D, + 3 E Thee two etic ae copaed ad the path with the lage geate-ditace etic i eliiated. Siilaly, the etic fo the two path eteig S at t3t ae D D,, D, + 3 E,, D, + 3 E Thee two etic ae copaed ad the path with the lage geate-ditace etic i eliiated. Thi poce i cotiued a each ew igal aple i eceived fo the deodulato. Thu, the Vitei algoith copute two etic fo the two igal path eteig a

16 ode at each tage of the telli each ad eliiate oe of the two path at each ode. The two uvivo path ae the exteded fowad to the ext tate. Theefoe, the ue of path eached i the telli i educed y a facto of at each tage. 5. PERFORMANCE OF THE OPTIMUM RECEIVER FOR MEMORYLESS MODULATION 5.. Poaility of Eo fo Biay Modulatio PAM atipodal - whee the two igal wavefo ae tgt ad t-gt, ad gt i a aitay pule that i ozeo i the iteval t T ad zeo elewhee. The eegy i the pule gt i E g. Fig.5.- Let u aue that the two igal ae equally liely ad that igal t wa taitted. The the eceived igal fo the atched filte o coelatio deodulato i + E whee epeet the additive Gauia oie copoet, which ha zeo ea ad vaiace σ N. I thi cae, the deciio ule aed o the coelatio etic give y Eq copae with the thehold zeo. If >, the deciio i ade favo of t, ad if <, the deciio i ade that t wa taitted. The two coditioal PDF of ae E / N p e 5.- π N p + E / N e 5.-3 π N

17 Thee two coditioal PDF ae how i Fig.5.-. Figue 5.- Coditioal pdf of two igal p/ p/ E E Give that t wa taitted, the poaility of eo i iply the poaility that <, i.e., E P e p d Q 5.-4 N Whee Qx i the Q-fuctio. Siilaly, if we aue that t wa taitted, E E + ad the poaility that > i alo P e Q. N Sice the igal t ad t ae equally liely to e taitted, the aveage poaility of eo i E P P e + P e Q 5.-5 N We hould oeve two ipotat chaacteitic of thi pefoace eaue. Fit, the poaility of eo deped oly o the atio E /N ad ot o ay othe detailed chaacteitic of the igal ad the oie. Secodly, we ote that E /N i alo the output SNR fo the atched-filte ad coelatio deodulato. The atio E /N i uually called the igal-to-oie atio pe it. The poaility ay e alo expeed i te of the ditace etwee the two igal ad. Fo Fig.5.-, we oeve that the two igal ae epaated y the ditace d E. By utitutig E d ito Eq.5.-5, we otai 4 d P Q 5.-6 N Biay othogoal igal Sigal vecto ad ae two-dieioal, a how i Fig.5.-3, ad ay e expeed a [ [ E ] E ] 5.-7

18 Fig.5.-3 The aveage eo poaility fo iay othogoal igal i Q N E Q P γ 5.- whee, y defiitio, γ i the SNR pe it. The eo poaility veu loge /N fo thee two type of igal i how i Fig A oeved fo thi figue, at ay give eo poaility, the E /N equied fo othogoal igal i 3 db oe tha fo atipodal igal. 5.. Poaility of Eo fo M-ay Othogoal Sigal The poaility of a -it yol eo i dy N E y dx e P M y x / exp π π 5.-

19 The ae expeio fo the poaility of eo i otaied whe ay oe of the othe M- igal i taitted. Sice all the M igal ae equally liely, the expeio fo P M give i Eq.5.- i the aveage poaility of a yol eo. I copaig the pefoace of vaiou digital odulatio ethod, it i deiale to have the poaility of eo expeed i te of the SNR pe it, E /N, itead of the SNR pe yol, E /N. With M. each yol covey it of ifoatio, ad hece E E. Thu, Eq.5.- ay e expeed i te of E /N y utitutig fo E. Soetie, it i alo deiale to covet the poaility of a yol eo ito a equal poaility of a iay digit eo. Fo equipoale othogoal igal, all yol eo ae equipoale ad occu with poaility PM P M M 5.- Futheoe, thee ae way i which it out of ay e i eo. Hece, the aveage ue of it eo pe -it yol i PM P M 5.-3 ad the aveage it eo poaility i jut the eult i Eq.5.-3 divided y, the ue of it pe yol. Thu, P P P M M, >> 5.-4 The gaph of the poaility of a iay digit eo a a fuctio of the SNR pe it, E /N, ae how i Fig.5.-5 fo M,4,6,6,3, ad 64. Thi figue illutate that, y iceaig the ue M of wavefo, oe ca educe the SNR pe it equied to achieve a give poaility of a it eo. Fo exaple, to achieve a P -5, the equied SNR pe it i a little oe tha db fo M, ut if M i iceaed to 64 igal wavefo 6 it/yol, the equied SNR pe it i appoxiately 6 db avig of ove 6 db!!!. What i the iiu equied E /N to achieve a aitaily all poaility of eo a M? [E/N > l db]. Thi iiu SNR E /N i called the Shao liit fo a AWGN chael.

20 5..3 Poaility of Eo fo M-ay Biothogoal Sigal 5..4 Poaility of Eo fo Siplex Sigal The poaility of eo fo iplex igal i idetical to the poaility of eo fo othogoal igal, ut thi pefoace i achieved with a avig of M log ρ log db M i SNR. Fo M, the avig i 3 db. Howeve, a M i iceaed, the avig i SNR appoache db.

21 5..5 Poaility of Eo fo M-ay Biay-Coded Sigal e If di i the iiu Euclidea ditace of the M igal wavefo, the the poaility of a yol eo i uppe-ouded a e e d i < < d i PM M P M Q exp N 4N The value of the iiu Euclidea ditace will deped o the electio of the code wod, i.e., deig of the code Poaility of Eo fo M-ay PAM Poaility of a yol eo M 6log M Eav P M Q M M N

22 5..7 Poaility of Eo fo M-ay PSK π π PM Q γ i Q γ i M M Poaility of Eo fo QAM Recall, that QAM igal wavefo ay e expeed a t A g tcoπf t A g ti πf t 5.-7 c c c whee A c ad A ae the ifoatio-eaig igal aplitude of the quadatue caie ad gt i the igal pule. The vecto epeetatio of thee wavefo i [ ] Ac Eg A Eg To deteie the poaility of eo fo QAM, we ut pecify the igal poit cotellatio. We egi with QAM igal et that have M4 poit. Fig.5.-4 illutate two fou-poit igal et. The fit i a fou-phae odulated igal ad the ecod i a QAM igal with two aplitude level, laeled A ad A, ad fou phae. Becaue the poaility of eo i doiated y the iiu ditace etwee pai of igal poit, let u ipoe the coditio that d e i A fo oth igal cotellatio ad let u evaluate the aveage taitte powe, aed o the peie that all igal poit ae equally poale. Figue 5.-4 Two fou-poit igal cotellatio d + A A A da

23 Fo the fou-phae igal, we have P av A A Fo the two-aplitude, fou-phae QAM, we place the poit o cicle of adii A ad 3 A. Thu, d e A, ad i P av A A 4 [3 + ] A which i the ae aveage powe a the M4 phae igal cotellatio. Hece, fo all pactical pupoe, the eo ate pefoace of the two igal et i the ae. I othe wod, thee i o advatage of the two-aplitude QAM igal et ove M4-phae odulatio. Next, let u coide M8 QAM, a how i Fig d e i A. Auig that the igal poit ae equally poale, the aveage taitted igal powe i M M A P av Ac + A ac + a M M whee a c,a ae the coodiate of the igal poit, oalized y A. a c P av 6A P av 6.83A d P av 4.73A

24 Theefoe, the fouth igal et equie appoxiately db le powe tha the fit two ad.6 db le powe tha the thid to achieve the ae poaility of eo. Fo M 6, thee ae ay oe poiilitie fo electig the QAM igal poit i the two-dieioal pace. Fo exaple, we ay chooe a cicula ultiaplitude cotellatio fo M6. I thi cae, the igal poit at a give aplitude level ae phae-otated y π elative to the igal poit at adjacet aplitude level. Howeve, 4 the cicula 6-QAM i ot the et 6-poit QAM igal cotellatio fo the AWGN chael. Rectagula QAM igal cotellatio have the ditict advatage of eig eaily geeated a two PAM igal ipeed o phae-quadatue caie. I additio, they ae eaily deodulated. Although they ae ot the et M-ay QAM igal cotellatio fo M 6, the aveage taitted powe equied to achieve a give iiu ditace i oly lightly geate tha the aveage powe equied fo the et M-ay QAM igal cotellatio. Fo thee eao, ectagula M-ay QAM igal ae ot fequetly ued i pactice. If we eploy the optiu detecto that ae it deciio o the optiu ditace etic, it i elatively taightfowad to how that the yol eo poaility i tightly uppe-ouded a 3Eav P M 4Q 5.-8 M N fo ay, whee Eav/N i the aveage SNR pe it. The poaility of a yol eo i plotted i Fig.5.-6.

25 5.. Copaio of Digital Modulatio Method The digital odulatio ethod ca e copaed i a ue way. Fo exaple, oe ca copae the o the ai of the SNR equied to achieve a pecific poaility of eo. Howeve, uch a copaio would ot e vey eaigful, ule it wee ade o the ai of oe cotait, uch a a fixed data ate of taiio o, equivaletly, o the ai of a fixed adwidth. With thi goal i id, let u coide the adwidth equieet fo eveal odulatio ethod. Fo ultiphae igal the chael adwidth equied i iply the adwidth of the equivalet low-pa igal pule gt, which deped o it detailed chaacteitic. Fo ou pupoe, we aue that gt i a pule of duatio T ad that it adwidth W i appoxiately equal to the ecipocal of T. Thu, W/T ad, ice T/Rlog M/R, it follow that R W log M Theefoe, a M i iceaed, the chael adwidth equied, whe the it ate R i fixed, deceae. The adwidth efficiecy i eaued y the it ate to awidth atio, which i R W log M The adwidth-efficiet ethod fo taittig PAM i igle-idead. The, the chael adwidth equied to tait the igal i appoxiately equal to /T ad, ice T/Rlog M/R, it follow that R W log M Thi i a facto of ette tha PSK. I the cae of QAM, we have two othogoal caie, with each caie havig a PAM igal. Thu, we doule the ate elative to PAM. Howeve, the QAM igal ut e taitted via doule-idead. Coequetly, QAM ad PAM have the ae adwidth efficiecy whe the adwidth i efeeced to the ad-pa igal. Othogoal igal have totally diffeet adwidth equieet. If the M othogoal igal ae cotucted y ea of othogoal caie with iiu fequecy epaatio of /T fo othogoality, the adwidth equied fo taiio of log M ifoatio it i

26 W M M M R T / log M I thi cae, the adwidth iceae a M iceae. Siila elatiohip otai fo iplex ad iothogoal igal. I the cae of iothogoal igal, the equied adwidth i oe-half of that fo othogoal igal. I the cae of PAM, QAM ad PSK, iceaig M eult i a highe it ate-toadwidth atio R/W. The cot of achievig the highe data ate i a iceae i SNR pe it Thee odulatio ethod ae appopiate fo couicatio chael that ae adwidth-liited R/W> ad whee thee i ufficietly high SNR to uppot iceae i M telephoe chael, icowave adio chael M-ay othogoal igal yield a it ate-to-adwidth atio of R/W<. A M iceae, R/W deceae due to a iceae i the equied chael adwidth. SNR pe it equied to achieve a give eo poaility deceae a M iceae. Thee odulatio ae appopiate fo powe-liited chael that have ufficietly lage adwidth to accoodate a lage ue of igal

27 A M, the eo poaility ca e ade a all a deied, povided that E / N > db. Thi i the iiu SNR pe it equied to achieve eliale taiio i the liit a the chael adwidth W ad the coepodig it ate-yo-adwidth atio R / W. The atio C/W, whee CR i the capacity i it/, epeet the highet achievale it ate-to-adwidth atio o thi chael. Hece, it eve a the uppe oud o the adwidth efficiecy of ay type of odulatio.

28 SELECTED PROBLEMS 5. A atched filte ha the fequecy epoe j e H f jπf a Deteie the ipule epoe ht coepodig to Hf. Deteie the igal wavefo to which the filte chaacteitic i atched. πft 5. Coide the igal A/ T t co πf ct t t T othewie a Deteie the ipule epoe of the atched filte fo the igal. Deteie the output of the atched filte at tt. c Suppoe the igal t i paed though a coelato that coelate the iput t with t. Deteie the value of the coelato output at tt. Copae you eult with tha i.

29 5.4 A iay digital couicatio yte eploy the igal t, ta, t T t T fo taittig the ifoatio. Thi i called o-off igalig. The deodulato cocoelate the eceived igal t with t ad aple the output of the coelato at tt. a Deteie the optiu detecto fo a AWGN chael ad the optiu thehold, auig that the igal ae equally poale. Deteie the poaility of eo a a fuctio of the SNR. How doe o-off igalig copae with atipodal igalig?

30

31 5.5 The coelatio etic give y Equatio ae N N C,,,,, M whee T t f T t f t dt t dt

32 Show that the coelatio etic ae equivalet to the etic T T dt t dt t t C, 5.6 Coide the equivalet low-pa coplex-valued igal l t, t T, with eegy T l dt t ε Suppoe that thi igal i coupted y AWGN, which i epeeted y it equivalet low-pa fo zt. Hece, the oeved igal i T t t z t t l l +, The eceived igal i paed a filte that ha a equivalet low-pa ipule epoe h l t. Deteie h l t o that the filte axiize the SNR at it output at tt.

33 5. I Sectio 4.3. it wa how that the iiu fequecy epaatio fo othogoality of iay FSK igal with coheet detectio i f/t. Howeve, a lowe eo poaility i poile with coheet detectio of FSK if f i iceaed eyod /T. Show that the optiu value of f i,75t ad deteie the poaility of eo fo thi value of f.

34 5.6 Coide that octal igal poit cotellatio i Figue P5.6 a The eaet-eigho igal poit i the 8-QAM igal cotellatio ae epaated i ditace y A uit. Deteie the adii a ad of the ie ad oute cicle. The adjacet igal poit i the 8-PSK ae epaated y a ditace of A uit. Deteie the adiu of the cicle. c Deteie the aveage taitte powe fo two igal cotellatio ad copae the two powe. What i the elative powe advatage of oe cotellatio ove the othe? Aue that all igal poit ae equally poale. Figue P5.6

35 a 45 8-PSK 8-QAM 5.7 Coide the 8-poit QAM igal cotellatio how i Figue P5.6 a I it poile to aig thee data it to each poit of the igal cotellatio uch that eaet adjacet poit diffe i oly oe it poitio?

36 Deteie the yol ate if the deied it ate i 9Mit/. 5.8 Suppoed that iay PSK i ued fo taittig ifoatio ove a AWGN with a powe pectal deity of N W/Hz. The taitted igal eegy i ε A T, whee T i the it iteval ad A i the igal aplitude. Deteie the igal aplitude equied to achieve a eo poaility of -6 whe the data ate i c it/. d it/ e Mit/ 5.9 Coide a igal detecto with a iput

37 ± A + whee +A ad A occu with equal poaility ad the oie vaiale i chaacteized y the Laplacia PDF how i Figue P5.9. a Deteie the poaility of eo a a fuctio of the paaete A ad σ. Deteie the SNR equied to achieve a eo poaility of -5. How doe the SNR copae with the eult fo a Gauia PDF? Figue P5.9 p e σ / σ

38 5. Coide the two 8-poit QAM igal cotellatio how i Figue P5.. The iiu ditace etwee adjacet poit i A. Deteie the aveage taitted powe fo each cotellatio, auig that the igal poit ae equally poale. Which cotellatio i oe powe-efficiet? Figue P5. a

39 5. Fo the QAM igal cotellatio how Figue P5., deteie the optiu deciio oudaie fo the detecto, auig that the SNR i ufficietly high o that eo oly occu etwee adjacet poit. Figue P

40 5.3 Two quadatue caie co πf c t ad i πf c t ae ued to tait digital ifoatio though a AWGN chael at two diffeet data ate, it/ ad it/. Deteie the elative aplitude of the igal fo the two caie o that the ε / N fo the two chael i idetical.

41 5.6 Coide a digital couicatio yte that tait ifoatio via QAM ove a voice-ad telephoe chael at a ate of 4 yol/. The additive oie i aued to e white ad Gauia. a Deteie the ε / N equied to achieve a eo poaility of -5 at 48 it/. Repeat a fo a ate of 96 it/ c Repeat a fo a ate of 9, it/ d What cocluio do you each fo thee eult?

42

43 5.7 Coide that fou-phae ad eight-phae igal cotellatio how i Figue P5.7. Deteie the adii ad of the cicle uch that the ditace etwee two adjacet poit i the two cotellatio i d. Fo thee eult, deteie the additioal taitted eegy equied i the i the 8-PSK igal to achieve the ae eo poaility a the fou-phae igal at the high SNR, whee the poaility of eo i deteied y eo i electig adjacet poit. Figue P5. d d

44 5.8 Digital ifoatio i to e taitted y caie odulatio though a additive Gauia oie chael with a adwidth of Hz ad N W/Hz. Deteie the axiu ate that ca e taitted though the chael fo fou-phae PSK, iay FSK, ad fou-fequecy othogoal FSK, which i detected ocoheetly.

45 5.36 A peech igal i apled at a ate of 8 Hz, logaithically copeed ad ecoded ito a PCM foat uig 8 it pe aple. The PCM data i taitted though a AWGN aead chael via M-level PAM. Deteie the adwidth equied fo taiio whe a M4. M8. c M6.