EGR 544 Communication Theory
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- Ευθύμιος Κοντολέων
- 7 χρόνια πριν
- Προβολές:
Transcript
1 EGR 544 Commucato hory 8. Spctral charactrstcs of Dgtally Modulats Sgals Z. Alyazcoglu Elctrcal ad Computr Egrg Dpartmt Cal Poly Pomoa Spctral charactrstcs of Dgtally Modulats Sgals Spctral charactrstcs Chal badwh lmtato for ral chal Radom procss Powr spctral dsty PAM (Larly modulatd dgtal sgal) CPM (CPFSK) (Nolar Modulatd dgtal sgal) Cal Poly Pomoa Elctrcal & Computr Egrg Dpt. EGR 544-8
2 Powr Spctra of Larly Modulats Sgal h bad-pass sgal s gv as form of j f t { vt π } c st () = R () whr v(t) s quvalt low-pass sgal h autocorrlato fucto of s(t) j f { } c φ ( τ) = R φ ( τ) ss π τ h Fourr trasform of φ ss (τ) spctrum ss( f) = ( f fc) +( f fc) [ ] gvs us th th powr dsty whr (f) s th powr dsty spctrum of v(t). So, w d to fd th th powr dsty spctrum of v(t) frst. Cal Poly Pomoa Elctrcal & Computr Egrg Dpt. EGR Powr Spctra of Larly Modulats Sgal For th lar dgtal modulato mthods, v(t) ca b gv as puls shap fucto vt () = Igt ( ) = whr trasmsso rat s /=R/k symbol/s, I rprstato of k-bt blocks dgtal formato, g( ) s a rmstc puls fucto Not that {I } s a dscrt tm radom procss. I s ral ad corrspods to ampltud valus of trasmsso sgal for PAM I s complx sc sgal s two-dmsoal for PSK, QAM, ad PAM-PSK. h autocorrlato fucto of v(t) s prodc wth prod * φ( t+ τ; t) = E v ( t) v( t τ) + * * = E ( ) ( ) IIm g t g t+ τ m = m= Cal Poly Pomoa Elctrcal & Computr Egrg Dpt. EGR
3 Powr Spctra of Larly Modulats Sgal Assum that {I } s wd-ss statoary wth ma ad autocorrlato fucto s * φ ( m) = E II + m h autocorrlato fucto of v(t) * ( t+ ; t) = ( m ) g ( t ) g( t+ m) = m= * = φ( kg ) ( t gt ) ( + τ k) k= = * = φ( k) g ( t ) g( t+ τ k) k= = φ τ φ τ = k= φ ( k) K( t, τ, k) whr Lt k=m- * (, τ, ) = ( ) ( + τ ) K t k g t g t k = Cal Poly Pomoa Elctrcal & Computr Egrg Dpt. EGR Powr Spctra of Larly Modulats Sgal Sc K( t+, τ, k) = K( t, τ, k) s prodc ad prod φ ( t+ + τ, t+ ) = φ ( t+ τ, t) Ad th ma valu of v(t) [ ] E v() t = E[ I ] g( t ) = µ g( t ) = = PAM, v(t) has prodc ma ad autocorrlato fucto wth prod. hrfor, t s calld a cyclostatoary procss or a prodcally statoary procss th wd ss Powr spctrum dsty of a cyclostatoary φ (, radom t τ procss s ) wo dmsoal Fourr trasform of Avrag powr spctrum dsty of a cyclostatoary radom procss s / / * ( ) = ( + ; ) = [ ( ) ( )] / + / φ τ t t E v t v t φ τ τ Cal Poly Pomoa Elctrcal & Computr Egrg Dpt. EGR
4 Powr Spctra of Larly Modulats Sgal m-avrag autocorrlato fucto for cyclostatoary procss Whr / φ( τ) = ( t ; t) φ + τ / / * = φ( k) g ( t ) g( t k) / k + τ = = / * = φ( k) g ( t) g( t k) / k + τ = = * = φ( k) g ( t) g( t τ k) + k= = φ( k) φgg ( τ k) φ ( ) gg τ k= df tm-autocorrlato fucto of g(t) as * gg ( ) = g ( tgt ) ( + ) φ τ τ t' = t t for t = / t' = / W trprt ths tm autocorrlato fucto Cal Poly Pomoa Elctrcal & Computr Egrg Dpt. EGR Powr Spctra of Larly Modulats Sgal jπ fτ ( f) = φ ( τ) dτ φ( τ) = φ ( m) φgg ( τ k) k= jπ fτ = φ( k) φgg ( τ k) dτ k= jπ fτ = φ( k) φgg ( τ k) dτ k= jπ fk j π fτ' = φ( k ) φgg ( τ ') dτ ' k= jπ fk = φ( k ) j π fτ ' gtgt ( ) ( + τ' ) dτ ' k= = ( f) G( f) Avrag powr spctrum dsty h avrag powr spctrum dsty of PAM sgal s rmd by th puls shap, as wll as th dgtal put formato Cal Poly Pomoa Elctrcal & Computr Egrg Dpt. EGR
5 Powr Spctra of Larly Modulats Sgal h Fourr trasform gvs us th avrag powr spctrum dsty of v(t) ( f) = G( f) ( f) G(f) Ergy spctrum dsty whr G(f) s Fourr trasform of g(t) ad (f) th powr dsty of th formato squc ( f) = φ( m) m= jπ fm h autocorrlato fucto φ (m) ca b obtad form (f) as ( (f) prodc ad prod ) / π fm m = f df / φ ( ) ( ) Cal Poly Pomoa Elctrcal & Computr Egrg Dpt. EGR Powr Spctra of Larly Modulats Sgal Exampl: Iput formato s ral ad mutually ucorrlatd Whr σ σ + µ ( m = ) φ( m) = µ ( m ) dots th varac of a formato squc h powr dsty spctrum of formato squc ( f) = σ + µ jπ fm m= hs s prodc ad prod s, thrfor, µ m = + ( f) σ δ( f ) m= Rmmbr that F{}=δ(f) If Samplg wth prod m F{} = δ ( f ) m= Cal Poly Pomoa Elctrcal & Computr Egrg Dpt. EGR 544-8
6 Powr Spctra of Larly Modulats Sgal h powr dsty spctrum of v(t) wll b σ µ m ( f) = G( f) + G( f) δ ( f ) m= h powr dsty spctrum of v(t) σ µ m ( f) = G( f) + G( f) δ ( f ) m= Cotuous spctrum Dscrt spctrum Wth zro ma put formato,th dscrt spctrum s zro, th avrag powr dsty spctrum s rmd by G(f) Cal Poly Pomoa Elctrcal & Computr Egrg Dpt. EGR Exampl: Powr Spctra of Larly Modulats Sgal A jπ ft G( f) = A hrfor g(t) s( π f ) = A π f = A s c( f ) jπ f jπ f G( f) = ( A) sc ( f) σ µ m ( f) = G( f) + ( ) ( ) G f δ f m= = σ As c ( f) + µ Aδ( f) t h avrag powr spctrum dsty for rctagular pulss f <.5 F{s c( t)} = ( f) = f >.5 f F{( x at)} = X( ) a a jπt f F{( x t t)} = X( f) F{ x( t)} = X( f) F{ X( t)} = x( f) Dscrt spctra compots Cal Poly Pomoa Elctrcal & Computr Egrg vash Dpt. for m EGR 544-8
7 Exampl: Powr Spctra of Larly Modulats Sgal A g(t) / jπ ft G( f) = cos ( t ) A π t A s c( f ) j f π = f t A π t g() t = cos ( t ), t h avrag powr spctrum dsty of rasd cos sgal g(t)? σ µ m ( f) = G( f) + G( f) δ ( f ) m= σ As c ( f) µ A µ A µ A = + δ( f) + δ( f ) + δ( f + ) 4( f ) Cal Poly Pomoa Elctrcal & Computr Egrg Dpt. EGR Powr Spctra of Larly Modulats Sgal Exampl: What th avrag powr spctrum dsty of PAM sgal for th put formato squc I? {I } assum to b mutually ucorrlatd,ach havg zro ma ad ut varac ( m = ) j φ( m) = ( f) = φ( m) o.w. m= = ( f) = G( f) ( f) = G( f) π fm Cal Poly Pomoa Elctrcal & Computr Egrg Dpt. EGR
8 Powr Spctra of Larly Modulats Sgal Exampl: If th put formato squc I gv as followg, what th avrag powr spctrum dsty of I? I = b + b {b } assum to b mutually ucorrlatd,ach havg zro ma ad ut varac ( m = ) φ( m) = ( m= ± ) ( ow..) ( f) = φ ( m) m= jπ fm jπ f jπ f = + + = + = [ π f ] cos( ) 4cos( f ) h corrspodg powr dsty spctrum of th low-pass modulatd sgal v(t) s Cal Poly Pomoa Elctrcal & Computr Egrg Dpt. EGR ( f) = G( f) cos ( π f) π Powr Spctra of CPFSK ad CPM sgal Costat Ampltud CPM sgal: Assum that I ca tak {±, ±3, ±(M-) ad thy ar statstcally dp ad cally dstrbutd wth pror probablts P = P( I = ) = ±, ± 3,..., ± ( M ) k CPM sgal: whr s() t = Acos[ π f t+ φ(; t I)] φ(; t I) = πh I q( t k) k= k c ad /, t L qt () =, t, h quvalt low-pass sgal vt () = j (;) t φ I Cal Poly Pomoa Elctrcal & Computr Egrg Dpt. EGR
9 Powr Spctra of CPFSK ad CPM sgal h autocorrlato fucto of quvalt low-pass sgal * φ( t+ τ; t) = E ( ) ( ) v t+ τ v t jφ( t+ τ; I) jφ( t; I) = E j[ φ( t+ τ; I) φ( t; I)] = E = E xp j πh Ik[ qt ( + τ k) qt ( k)] k= φ( t+ τ; t) = E xp( [ ( ) ( )]) j πhik qt+ τ k qt k k= = E xp( j πhik[ q( t + τ k ) q( t k )]) k= j h[ q( t k) q( t k)] P π + τ = k= { Ik } Cal Poly Pomoa Elctrcal & Computr Egrg Dpt. EGR { I } d... k Powr Spctra of CPFSK ad CPM sgal h avrag autocorrlato fucto φ( τ) = ( t ; t) φ + τ Lt β τ< (β+) j πh[ q( t+ τ k) q( t k)] = P k= { Ik } ad t < qt ( ( L ) ) t ( L ) [( L ), L ) qt ( k) qt () t [, ) k -L qt ( + τ ( β + L ) ) t + τ ( β + L ) [( L ),( L + ) ) qt ( + τ k) qt ( + τ ( β + ) ) t + τ ( β + ) [(, ) ½-/= k (β+)-l (β+) max(,) (M-) β + j πh[ q( t+ τ k) q( t k)] φ( τ) = P k= m( β + L, L) = ( M ) odd Cal Poly Pomoa Elctrcal & Computr Egrg Dpt. EGR
10 Powr Spctra of CPFSK ad CPM sgal φ τ β ( β + ) j πh[ q( t+ τ k) q( t k)] ( ) = P, f k= ( L) { Ik } Cosdr th cas of (β+) L > (β L) φ β L j πh[ q( t k)] j πh[ ] ( τ) = P P k= ( L) { Ik} k= { Ik} β + j πh[ q( t+ τ k)] P k= β + L { Ik } Cal Poly Pomoa Elctrcal & Computr Egrg Dpt. EGR Powr Spctra of CPFSK ad CPM sgal M- j π h[ q( t k)] φ( τ) = P [ ( jh)] Ψ k= ( L) = ( M ) odd M- m= L = ( M ) odd P β L = [ Ψ ( jh)] λτ ( β ) whr M - ψ ( jh) = P π = ( M ) odd j h j πhqt [ ( + τ ( m+ β) )] β L λξ ( ) = (a) Charactrstc fucto of radom squc {I } M- j π h[ q( t k)] M- j h[ q( t k)] P π + ξ P k= ( L) = ( M ) k= L = ( M ) odd odd Cal Poly Pomoa Elctrcal & Computr Egrg Dpt. EGR 544-8
11 Powr Spctra of CPFSK ad CPM sgal Avrag Powr spctral dsty of CPFSK jπ fτ f d ( ) = φ ( τ) τ jπ fτ jπ fτ = φ ( τ) dτ + φ ( τ) dτ jπ fτ jπ fτ = φ( τ) + φ( τ) dτ = R ( ) jπ fτ φ τ dτ φ ( τ) = φ ( τ) sc L jπ fτ jπ fτ jπ fτ ( ) = ( ) + ( ) L φ τ dτ φ τ dτ φ τ dτ L ( m+ ) jπ fτ jπ fτ ( ) ( ) m m= L = φ τ dτ + φ τ dτ Cal Poly Pomoa Elctrcal & Computr Egrg Dpt. EGR Powr Spctra of CPFSK ad CPM sgal L jπ fτ jπ fτ jπ fτ φ( τ) dτ = φ( τ) dτ + φ( τ) dτ L L ( m+ ) jπ fτ m L jπ fτ = ( ) + [ ( )] ( ) m m= L Lt ξ = τ m φ τ dτ ψ jh λ τ m dτ L jπ fτ jπ fτ m L j π f ( ξ+ m) ( ) = ( ) + [ ( )] ( ) m= L φ τ dτ φ τ dτ ψ jh λ ξ dξ Lt = m L L π τ ( ) jπ fτ j f φ( τ) dτ = φ ( τ) dτ jπ f j π f ( ξ+ L) + [ ( )] ( ) = ψ jh λξ dξ Cal Poly Pomoa Elctrcal & Computr Egrg Dpt. EGR 544-8
12 f ψ ( jh) < = Powr Spctra of CPFSK ad CPM sgal [ ψ ( jh)] jπ f = ψ ( jh) jπ f h avrag Powr spctral dsty of CPM sgal s L j π fτ j π f ( ξ+ L) ( f ) = R φ( τ) dτ + λ( ξ) dξ jπ f ψ ( jh) Lt s look at th cas of Ψ ( jh) = M- jπ hi ψ ( jh) = E = P = ( M ) odd jπ h Cal Poly Pomoa Elctrcal & Computr Egrg Dpt. EGR Powr Spctra of CPFSK ad CPM sgal Dscuss th cas of ψ ( jh) = ( ) j π ψ jh = P h =, f h = tgr { ( M )...,,..( M )} Wthout loss of gralty, w ca st ψ jπvh jπv ( jh) = = v <, sc ψ ( jh) = j f π j π v j π f [ ψ ( jh)] = = = = = j π( f v/ ) h Fourr rasform of samplg ut stp fucto v v π( f ) jπ f [ ψ( jh)] = + δ( f ) j = = = Cal Poly Pomoa Elctrcal & Computr Egrg Dpt. EGR
13 Powr Spctra of CPFSK ad CPM sgal jπ f [ ψ( jh)] = + δ( f ) j cot π ( f ) = = v v Fourr trasform of u(t) s U( f) = δ ( f ) j π f Fourr trasform of u(t) samplg wth samplg prod j U( f ) = δ ( f ) = = = π( f ) j = δ ( f ) cot π ( f ) cotπ t = = = π ( t ) Cal Poly Pomoa Elctrcal & Computr Egrg Dpt. EGR Powr Spctra of CPFSK ad CPM sgal hus, th avrag powr spctrum dsty of CPM sgal cotas mpulss locato at f + v =, =,,... h tr avrag powr spctrum dsty cluds cotus compot ad a dscrt spctrum compot. Cal Poly Pomoa Elctrcal & Computr Egrg Dpt. EGR
14 Powr Spctra of CPFSK ad CPM sgal Spcal cas o th avrag powr spctral dsty of CPFSK sgals. Assum that I s qual probabl P =,forall. M ( M ) jπ h ψ ( jh) = M = ( M ) odd j( M ) πh jπhm ( ) = jπ h M jmπh jπhm = jπh jπh M s( Mπ h) = It s ral umbr. M s( π h) Cal Poly Pomoa Elctrcal & Computr Egrg Dpt. EGR Powr Spctra of CPFSK ad CPM sgal Also, φ τ β ( β + ) j πh[ q( t+ τ k) q( t k)] ( ) = P, f k= ( L) { Ik } ( πhmqt+ τ k qt k ) ( π + τ ) ( ) β + s [ ( ) ( )] = k= ( L) M s h[ q( t k) q( t k)] It s also ral umbr.. L j π fτ j π f ( ξ+ L) ( f ) = R φ( τ) dτ + λ( ξ) dξ jπ f ψ ( jh) L ( L+ ) jπ fτ = φ( τ)cos( π fτ) dτ + R λ( τ L) dτ jπ f ψ ( jh) L L Not: φ( τ) = [ ψ( jh)] λ( τ β) for β τ < ( β + ) mpls that φ( τ) = λ( τ β) for L τ < ( L+ ), (applyg β = L) Cal Poly Pomoa Elctrcal & Computr Egrg Dpt. EGR
15 Powr Spctra of CPFSK ad CPM sgal L ( L+ ) jπ fτ ( f ) = φ( τ)cos( π fτ) dτ + R φ ( ) j f τ dτ π ψ ( jh) L jπ fτ L ( L+ ) = φ( τ)cos( π fτ) dτ + φ( τ)r dτ L ( ) jπ f ψ jh L ( L+ ) cos( π fτ) js( π fτ) ( f) = φ( τ)cos( π fτ) dτ + φ( τ)r dτ L ψ ( jh )cos( π f ) + jψ ( jh )s( π f ) L ( f) = φ( τ)cos( π fτ) dτ ( L+ ) cos( π fτ) js( π fτ) ψ( jh)cos( π f) jψ( jh)s( π f) + φ( τ )R d L τ ψ ( jh )cos( π f ) + jψ ( jh )s( π f ) ψ ( jh )cos( π f ) jψ ( jh )s( π f ) L ( f) = φ( τ)cos( π fτ) dτ ( L+ ) cos( π fτ)[ ψ( jh)cos( π f)] ψ( jh)s( π f)s( π fτ) + φ( τ) d L τ ψ ( jh) ψ( jh)cos( π f ) Cal Poly Pomoa Elctrcal & Computr Egrg Dpt. EGR Powr Spctra of CPFSK ad CPM sgal Fally, + [ ψ( jh) cos( π f )] ( L+ ) φ( τ)cos( π fτ) dτ ψ jh ψ jh π f L L ( f) = φ( τ)cos( π fτ) dτ ( ) ( )cos( ) Ψ( jh)s( π f ) ( L+ ) φ ( )s( ) τ π fτ dτ ψ ( jh) ψ( jh)cos( π f ) L W oly d to drv tgrato of th avrag autocorrlato fucto ovr [,L) ad [L,(L+)). Cal Poly Pomoa Elctrcal & Computr Egrg Dpt. EGR
16 Powr Spctra of CPFSK ad CPM sgal Assum that th puls shap s rctagular puls g(t) full rspos mployd (L=) β + j πhqt [ ( + τ k) qt ( k)] φ( τ) =, f ( ) β τ < β + k= LM { Ik } β + j πh[ q( t+ τ k) q( t k)] = k= M { Ik } β + M j πh( M)[ qt ( + τ k) qt ( k)] = k= M m= Cal Poly Pomoa Elctrcal & Computr Egrg Dpt. EGR Cas: β = ( τ < ) φ ( τ) φ ( τ) Powr Spctra of CPFSK ad CPM sgal M j πh(m M )[ q( t+ τ k ) q( t k )] = k= M m= ( πhm q t + τ k q t k ) ( π + τ ) s [ ( ) ( )] = k= M s h[ q( t k) q( t k)] Cas: β = ( τ < ) M j πh(m M)[ qt ( + τ k) qt ( k)] = k= M m= ( πhm q t + τ k q t k ) ( π + τ ) s [ ( ) ( )] = k= M s h[ q( t k) q( t k)] Cal Poly Pomoa Elctrcal & Computr Egrg Dpt. EGR
17 Powr Spctra of CPFSK ad CPM sgal Fally,w obta th avrag Powr Spctrum dsty of CPFSK as = + M M M ( f ) A ( f) ( ) ( ) ( ) Bm f A f Am f M = M = m= whr h A ( f ) = sc f - (-- M) B m cos( π f αm) ψ cos( αm) ( f ) = + ψ ψ cos( π f ) αm = πhm ( + M) Modulato dx s Mπ h ψ = ψ ( jh) = M sπ h h = f d Cal Poly Pomoa Elctrcal & Computr Egrg Dpt. EGR Powr Spctra of CPFSK ad CPM sgal Avrag powr spctrum dsty of th quvalt low-pass CPFSK sgal s plottr Oly Uppr sd bad plottd bcaus of avrag powr spctrum dsty s symmtry Spctrum dsty Spctral dsty for two lvl CPFSK M= =.5 H=f d Normalz frq f Cal Poly Pomoa Elctrcal & Computr Egrg Dpt. EGR
18 Powr Spctra of CPFSK ad CPM sgal Spctrum dsty Spctral dsty for two lvl CPFSK M= =.5 H=f d Normalz frq f Cal Poly Pomoa Elctrcal & Computr Egrg Dpt. EGR Powr Spctra of CPFSK ad CPM sgal Spctrum dsty Spctral dsty for two lvl CPFSK M= =.5 H=f d Normalz frq f Cal Poly Pomoa Elctrcal & Computr Egrg Dpt. EGR
19 Powr Spctra of CPFSK ad CPM sgal Spctrum dsty Powr Spctrum dsty for two lvl CPFSK M= =.5 h=f d Normalz frq f Avrag spctrum rlatvly smooth for h< Avrag spctrum bcoms broadr. hat s th raso CPFSK s usd h< Cal Poly Pomoa Elctrcal & Computr Egrg Dpt. EGR Powr Spctra of CPFSK ad CPM sgal M=4 =.5 h=f d Impuls appars wh h s clos to, h badwh bcoms almost twc of M=, 4 lvl CPFSK carrrs twc dgtal formato tha -lvl CPFSK Powr Spctrum dsty for quatrary CPFSK Cal Poly Pomoa Elctrcal & Computr Egrg Dpt. EGR
20 Powr Spctra of CPFSK ad CPM sgal M=8 =.5 h=f d Powr Spctrum dsty for octal CPFSK Impulss appars wh h clos to, h badwh bcoms almost twc of M=4, 8 lvl CPFSK carrrs twc dgtal formato tha 4-lvl CPFSK Cal Poly Pomoa Elctrcal & Computr Egrg Dpt. EGR Powr Spctra of CPFSK ad CPM sgal Spcal cas of MSK sgal h=/ ad M= ad ψ= s Mπ h sπ ψ = ψ ( jh) = M sπh = s π / = A ( f ) = sc f - (-3) 4 cos( π f α ) ψ cos( α ) B ( f ) = = cos( π f α ) m m m + ψ ψ cos( π f ) π αm = πhm ( + M) = ( m+ 3) hrfor, th avrag powr spctrum of MSK sgals 6 cos ( f) ( f ) A ( f) ( ) ( ) ( ) Bm f A f Am f M M = = m= π ( 6 f ) = + = m π Normalzd avrag powr spctrum of MSK sgals ( f ) cos ( π f ) = () ( 6 f ) Cal Poly Pomoa Elctrcal & Computr Egrg Dpt. EGR
21 Powr Spctra of CPFSK ad CPM sgal Offst QPSK wth rctagular puls shap v() t = [ I g( t ) ji g( t )] = + t < OQPSK wth rctagular puls shap gt () = o.w. φ ( t+ τ; t) = E[[ I I ] g( t ) g( t+ τ m) m = m= + E[[ I+ Im+ ] g( t ) g( t+ τ m ) = m= j E[[ IIm+ ] gt ( ) gt ( + τ m ) = m= j E[[ I+ Im] g( t ) g( t+ τ m) = m= Cal Poly Pomoa Elctrcal & Computr Egrg Dpt. EGR Powr Spctra of CPFSK ad CPM sgal If {I }ar statstcally dp ad cally dstrbutd wth uform margal dstrbuto φ( t+ τ; t) = gt ( ) gt ( + τ ) + gt ( ) gt ( + τ ) = = = gt ( ) gt ( + τ ) = τ τ ( t ; t), t φ + τ = = < τ τ φ( t + τ; t) = + =, t < φ ( ) τ = + τ, τ <, o.w. th avrag powr spctrum of Offst QPSK sgals f s ( π f) ( f ) = F /\ sc ( f) = = ( π f) Cal Poly Pomoa Elctrcal & Computr Egrg Dpt. EGR
22 Powr Spctra of CPFSK ad CPM sgal MSK fall off cosdrably fastr tha Offst QPSK log ( f ) () Cal Poly Pomoa Elctrcal & Computr Egrg Dpt. EGR Powr Spctra of CPFSK ad CPM sgal MSK s mor badwh-ffct tha Offst QPSK Cal Poly Pomoa Elctrcal & Computr Egrg Dpt. EGR
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