ECE542, Fall 2004 Homework 7 Solutions

Transcription

1 EECE54: Dgtal Cmmuncatns Thry Prf. M. Hayat ECE54, Fall 004 Hmwrk 7 Slutns Prblm. Unn Bund: Ρ Ρ annunc, 0 { snt} dy Q But Ρ{ annunc snt} f ( y ( (Assum ( If <, Ρ{ annunc snt} f ( y dy Q 4 6 Ρ 6Q 4Q Q 4 Nt: Fr xampl, 6Q cms frm sx cmbnatns fr whch. Ths ar (,, (,3, ( 3,4, ( 4,3, (,, ( 3,. Fr th spcfd, valus f and w hav: 3 Ρ Q( Q( 4 Q( Th unn bund s farly tght n ths xampl.. Bhattachanya Bund: Ρ [ ( y f ( y ] f, Us th fact that [ f y f ( y ] t dy dt ( 8 ( dy π and sm algbrac manpulatns t fnd: 9 Ρ Ths s largr than th unn bund bcaus Ρ ( s nw bng uppr bundd whr n th unn bund cas t was calculatd xactly.

2 EECE54: Dgtal Cmmuncatns Thry Prf. M. Hayat. Gallagr Bund: Frm class nts: [ ] ( ( Ε Ρ π π π π ( ( ( ( ( ( ( (, 0, ( ( Y y y y y y dy dy dy y f y f whr Y s a zr-man Gaussan r.v wth varanc. Nw [ ] N Y α α Ε fr any R α S, π Ε Y P ( ( (, By Jnsn s Inqualty (assumng 0 Hnc, Ε Y P π π ( ( ( ( ( ( ( ( ( ( (, S MATLAB pt t s th dpndnc f th bund n. Nt that th mnmum s at. Th Gallagr bund s quvalnt t th Bhattachanya bund n ths xampl. %Gallagr- bund plt

3 EECE54: Dgtal Cmmuncatns Thry Prf. M. Hayat clar r[0.6:.00:] ; %r0.5; Nrlngth(r; sg0.5; al./(*(sg^*(lr; %ba/(lr; aa(a./r.*{(i./(r.*(r - ; bba.*((l./r -; m [ 3 5 7]; Pzrs(4,Nr; fr l:4, pzrs(l,nr; fr l:4, f ( ~ ppxp(aa*(m(.^bb*(m(.^*m(*m{.*(a./(r.* (r; nd nd P(,:p.^r; nd P 0.5*sum(P,; subplt (,, smlgy(r,p,'k-' xlabl( {\rh} ' ylabl('gallager BOUND' subplt(,, plt(r(flr(nr/.:nr,p (flr(nr/.:nr,'k-' xlablc ( {\rh}' ylabl('gallager BOUND' 3

4 EECE54: Dgtal Cmmuncatns Thry Prf. M. Hayat Prblm 5.6 a By nspctn, a A ; b a b d r r r sn 45 r ( sn 45 r A A c 8-QAM: Ρ r r r A r A A 8-PSK: Ρ 8 8 av ( 0 5 av 0 Pwr advantag 6 (8-QAM s mr advantagus (.37 ( A 8 Prblm 5.7 By nspctn, 4

5 EECE54: Dgtal Cmmuncatns Thry Prf. M. Hayat r d 0 Nw, d r r r cs(45 s Pwr: Ρ 4 d 4 ; Ρ 8 d d d Addtnal pwr ( 5.3dB r d Prblm PSK: Max bt BFSK: rat Max symbl rat lg M 00 KHz 00 Kbts / s As abv wth M ; w gt 00 Kbts / s. Sparatn s btwn adacnt T frquncs. S max sparatn s. T S, W and # bts / symbl bts / sym. T 5 T 0 s wth bts / sym W S # bts / s 00 Kbts / s T 5

6 ECE54, Sprng 004 HW#8 Slutns Cnsdr an all-zr cdwrd cnsstng f k bts. Thr s a ttal k - rrr k k k pattrs. On th thr hand, th ttal numbr f bt rrrs s k. Thus, by assumng that all symbls ar qually prbabl, th avrag numbr f bt rrrs pr symbl rrr s k k- / k - (bt rrrs/pr symbl rrr. Nw lt P b b th prbablty f a bt rrr. Thn, th avrag numbr f bt rrrs pr symbl rrr s kp b. But, ths numbr s als qual t th prbablty f symbl rrr, P S, tms k k- / k -, whch stablshs th dsrd rlatnshp. In th cas f gray cdng, w nly nd t cnsdr n-bt symbl rrrs and gnr all thr symbl rrrs. Wth ths assumptn and by assumng as bfr that w cnsdr th all-zr cdwrd, thr wll b a ttal f k symbl rrrs. Nw nt that n ths cas th ttal numbr f bt rrrs s als k, snc ach symbl rrr rsults n nly n bt rrr. Hnc, th avrag numbr f bt rrrs pr symbl rrr s. Thus, by xamnng th avrag numbr f bt rrrs pr symbl rrr (as n Prblm w btan k P b P S. 3 Part I: Pck ε>0. Thr xsts δ >0 such that f(x- f(y < ε whnvr x- y < δ. Assum x n cnvrgs t x. Thn, w can chs an ntgr n 0, dpndng n δ, s that x n - x < δ whnvr n>n 0. Thus, f(x- f(x n < ε fr all n>n 0, r statd dffrntly: f(x n f(x. Part II: Lt A b a clsd and bundd st. Put B f(a and cnsdr a cnvrgnt squnc y n B. Lt y lm y n. W nd t shw that y B. Nt that y n f(x n fr sm x n A. Snc A s bundd, th squnc x n has a cnvrgnt subsqunc squnc x n,. Mrvr, snc A s clsd, lm x n, ls n A. Lt x lm x n,. Snc f s cntnuus, lm f(x n, f(x. But x A, whch mpls that f(x B (snc x A. But y lm n y n lm n f(x n lm f(x n, f(x B, whch prvs that B s clsd. Prf that B s bundd (by cntradctn: Assum that B s unbundd. Ths mans that w can fnd an ncrasng squnc y n B wth lm y n. Cnsdr th squnc x n A dfnd by y n f(x n. Snc A s bundd, th squnc x n has a cnvrgnt subsqunc x n,, and lm x n, x A snc A s clsd. By cntnuty, hwvr, lm y n, lm f(x n, f(x, and f(x< by dfntn f a functn vr A. Ths cntradcts th assumptn that y n, s dvrgnt. Hnc, B s bundd. 4 Pck a cnvrgnt squnc f vctrs x m frm th prscrbd st, whch w wll call E. Namly, x m x. W nd t shw that x s n th st E. Ths mans that x m - x 0 as m, whr. dnt Eucldan nrm. Ths m-dmnsnal

7 Prblm 4.3 : (a E [z(tz(t τ] E [{x(t τy(t t}{x(ty(t}] E [x(tx(t τ] E [y(ty(t τ] E [x(ty(t τ] E [y(tx(t τ] φ xx (τ φ yy (τ[φ yx (τφ xy (τ] But φ xx (τ φ yy (τand φ yx (τ φ xy (τ. Thrfr : E [z(tz(t τ] 0 (b E ( V frm th rsult n (a abv. Als : V T T 0 0 T 0 z(tdt E [z(az(b] dadb 0 T 0 E (VV T 0 E [z(az (b] dadb T T 0 0 N 0δ(a bdadb T 0 N 0da N 0 T Prblm 4.4 : E [x(t τx(t] A E [sn (πf c (t τθsn(πf c t θ] A cs πf cτ A E [cs (πf c(t τθ] whr th last qualty fllws frm th trgnmtrc dntty : sn A sn B [cs(a B cs(a B]. But : E [cs (πf c (t τθ] π 0 cs (πf c (t τθ p(θdθ π 0 cs (πf c (t τθ dθ 0 π Hnc : E [x(t τx(t] A cs πf cτ 46

8 Nt : Rlatnshp ( can als b btand by smpl dffrntatn f th rsdual rrr wth rspct t th cffcnts {s n }. Snc s n s, n gnral, cmplx-valud s n a n b n w hav t dffrntat wth rspct t bth ral and magnary parts : d da n E d [ da n s(t Kk s k f k (t ][ s(t K n s n f n (t ] dt 0 a nf n (t [ s(t K n s n f n (t ] a n f n (t [ s(t K n s n f n (t ] dt 0 a n R { f n(t [ s(t K n s n f n (t ]} dt 0 R { f n(t [ s(t K n s n f n (t ]} dt 0, n,,..., K whrwhavxpltdthdntty:(x x R{x}. Dffrntatn f E wth rspct t b n wll gv th crrspndng rlatnshp fr th magnary part; cmbnng th tw w gt (. Prblm 4.7 : Th prcdur s vry smlar t th n fr th ral-valud sgnals dscrbd n th bk (pags Th nly dffrnc s that th prctns shuld cnfrm t th cmplx-valud vctr spac : c s (tf (tdt and, n gnral fr th k-th functn : c k s k (tf (tdt,,,..., k Prblm 4.8 : Fr ral-valud sgnals th crrlatn cffcnts ar gvn by : km Ek E m s k(ts m (tdt and th Eucldan dstancs by : d ( km { E k E m } / E k E m km. Fr th sgnals n ths prblm : E, E, E 3 3, E

9 and: d ( d ( d ( d ( 4 5 d ( d ( Prblm 4.9 : Th nrgy f th sgnal wavfrm s m(t s: E s m (t dt s m(t M s k (t dt M k s m(tdt M M s M k (ts l (tdt k l M s m (ts k (tdt M s m (ts l (tdt M k M l E M M Eδ M kl k l M E E M E ( M M E E M Th crrlatn cffcnt s gvn by : mn s E m (ts n (tdt ( s E m (t ( M s k (t s n (t M k M ( s E m (ts n (tdt M M s M k (ts l (tdt k l ( M s E n (ts k (tdt M s m (ts l (tdt M k M l ME E E M M M M M M M s l (t dt l Prblm 4.0 : (a T shw that th wavfrms f n (t, n,...,3 ar rthgnal w hav t prv that: f m (tf n (tdt 0, m n 49

10 (c Th transtn matrx s : I n I n B n I n B n Th crrspndng Markv chan mdl s llustratd n th fllwng fgur : / / / / / 0 - /4 /4 æ Prblm 4. : (a I n a n a n, wth th squnc {a n } bng uncrrlatd randm varabls (. E (a nm a n δ(m. Hnc : φ (m E [I nm I n ]E[(a nm a nm (a n a n ] δ(m δ(m δ(m, m0, m± 0,.w. (b Φ uu (f T G(f Φ (f whr: Φ (f m φ (mxp(πfmt xp(4πft xp(4πft [ cs 4πfT] 4sn πft and G(f (AT ( sn πft πft 60

11 Thrfr : ( sn πft Φ uu (f 4A T sn πft πft (c { If {a n } taks } th valus (0, wth qual prbablty thn E(a n / ande(a nm a n /4, m 0 [ δ(m] /4. Thn : /, m0 φ (m E [I nm I n ]φ aa (0 φ aa ( φ aa ( [δ(m δ(m δ(m ] 4 and Φ (f m φ (mxp(πfmt sn πft Φ uu (f A T ( sn πft πft sn πft Thus, w btan th sam rsult as n (b, but th magntud f th varus quantts s rducd by a factr f 4. Prblm 4.3 : x(t R [u(txp(πf c t] whr u(t s(t ± ŝ(t. Hnc : S : U(f S(f ± Ŝ(f whr Ŝ(f { S(f, f > 0 S(f, f < 0 U(f { S(f ± S(f, f > 0 S(f S(f, f < 0 } { S(f r0, f > 0 0rS(f, f < 0 Snc th lwpass quvalnt f x(t s sngl-sdband, w cnclud that x(t s a sngl-sdband sgnal, t. Supps, fr xampl, that s(t has th fllwng spctrum. Thn, th spctra f th sgnals u(t (shwn n th fgur fr th cas u(t s(tŝ(t and x(t ar sngl-sdband } } 6

12 and : Φ vv (f a T U(f Fr th rctangular puls f QPSK, w hav : ( φ uu (τ T τ, 0 τ T T Fr th MSK puls : φ uu (τ u(t τu (tdt T τ T ( τ T cs π τ τ T π 0 sn πt T π τ sn T π(tτ sn dt T Prblm 4.8 : (a Fr smplcty w assum bnary CPM. Snc t s partal rspns : q(t T 0 u(tdt /4 q(t T 0 u(tdt /, q(t /, t > T s nly th last tw symbls wll hav an ffct n th phas : φ(t; I πh n k I k q(t kt, nt t nt T n k I k π (I n q(t (n T I n q(t nt, π It s asy t s that, aftr th frst symbl, th phas slp s : 0 f I n, I n hav dffrnt sgns, and sgn(i n π/(t fi n, I n hav th sam sgn. At th trmnal pnt t (n T th phas s : φ((n T ; I π n k Hnc th phas tr s as shwn n th fllwng fgur : I k π 4 I n 67

13 t0 tt tt t3t t4t 7π/4-5π/ π/ π/ π/ π/ π/ π/4 - æ (b Th stat trlls s btand frm th phas-tr mdul π: t0 tt tt t3t t4t - - φ 3 7π/4 - - φ 5π/ φ 3π/ φ 0 π/ æ 68

14 (c Th stat dagram s shwn n th fllwng fgur (wth th (I n, I n r(i n, I n thatcaus th rspctv transtns shwn n parnthss (-,. (-,- (-,. φ 5π/4 φ 3π/4 (, (-,- (, (, (-,- (, φ 3 7π/4 φ 0 π/4 (-,-.. (-,, (-, æ Prblm 4.9 : φ(t; I πh n k I k q(t kt (a Full rspns bnary CPFSK (q(t /: ( h /3. At th nd f ach bt ntrval th phas s : π nk I 3 k π nk I 3 k. Hnc th pssbl trmnal phas stats ar {0, π/3, 4π/3}. ( h 3/4. At th nd f ach bt ntrval th phas s : π 3 nk I 4 k 3π nk I 4 k. Hnc th pssbl trmnal phas stats ar {0, π/4, π/, 3π/4, π, 5π/4, 3π/, 7π/4} (b Partal rspns L 3, bnary CPFSK : q(t /6, q(t /3, q(3t /. Hnc, at th nd f ach bt ntrval th phas s : πh n k I k πh (I n /3I n /6 πh n k I k πh 3 (I n I n Th symbl lvls n th parnthss can tak th valus {3,,, 3}. S : ( h /3. Th pssbl trmnal phas stats ar : {0, π/9, 4π/9, π/3, 8π/9, 0π/9, 4π/3, 4π/9, 6π/9} 69

15 ( h 3/4. Th pssbl trmnal phas stats ar : {0,π/4,π/, 3π/4,π,5π/4, 3π/, 7π/4} Prblm 4.30 : Th 6-QAM sgnal s rprsntd as s(t I n cs πftq n sn πft, whr I n {±, ±3},Q n {±, ±3}. A suprpstn f tw 4-QAM (4-PSK sgnals s : s(t G [A n cs πft B n sn πft]c n cs πft C n sn πft whr A n,b n, C n,d n {±}. Clarly : I n GA n C n,q n GB n D n. Frm ths quatns t s asy t s that G gvs th rqurs quvalnc. Prblm 4.3 : W ar gvn by Equatn ( that th pulss c k (t ar dfnd as L c k (t s 0 (t s 0 [t (n La k,n ], 0 t T mn[l( a k,n n] n n Hnc, th tm supprt f th puls c k (t s 0 t T mn[l( a k,n n] n W nd t fnd th ndx ˆn whch mnmzs S L( a k,n n, r quvalntly maxmzs S La k,n n : ˆn argmax [La k,n n], n,..., L, a k,n 0, n It s asy t shw that ˆn L ( f all a k,n, n 0,,..., L ar zr (fr a spcfc k, and ˆn max{n : a k,n } ( thrws. Th frst cas ( s shwn mmdatly, snc f all a k,n, n 0,,..., L ar zr, thn max n S max n n, n 0,,..., L. Fr th scnd cas (, assum that thr ar n,n such that : n <n and a k,n, a k,n 0. ThnS (n L n >n ( S (n, snc n n <L du t th allwabl rang f n. S, fndng th bnary rprsntatn f k, k 0,,..., L, w fnd ˆn and th crrspndng S(n whch gvs th xtnt f th tm supprt f c k (t: k 0 a k,l 0,..., a k, 0, a k, 0 ˆn L S L k a k,l 0,..., a k, 0, a k, ˆn S L k /3 a k,l 0,..., a k,, a k, 0/ ˆn S L 70