Example Message bandwidth and the transmitted signal bandwidth

Transcription

1 4.6 Bandwidh-Eiien Modulaions We are now going o deine a quaniy alled he bandwidh o a signal. Unorunaely, in praie, here isn jus one deiniion o bandwidh. Deiniion The bandwidh (BW) o a signal is usually alulaed rom he dierenes beween wo requenies (alled he bandwidh limis). Le s onsider he ollowing deiniions o bandwidh or real-valued signals [3, p 73] (a) Absolue bandwidh: Use he highes requeny and he lowes requeny in he posiive- par o he signal s nonzero magniude sperum. This uses he requeny range where 00% o he energy is onined. We an speak o absolue bandwidh i we have ideal ilers and unlimied ime signals. (b) 3-dB bandwidh (hal-power bandwidh): Use he requenies where he signal power sars o derease by 3 db (/2). The magniude is redued by a aor o / 2. () Null-o-null bandwidh: Use he signal sperum s irs se o zero rossings. (d) Oupied bandwidh: Consider he requeny range in whih X% (or example, 99%) o he energy is onained in he signal s bandwidh. (e) Relaive power sperum bandwidh: he level o power ouside he bandwidh limis is redued o some value relaive o is maximum level. Usually speiied in negaive deibels (db). For example, onsider a 200-kHz-BW broadas signal wih a maximum arrier power o 000 was and relaive power sperum bandwidh o -40 db (i.e., /0,000). We would expe he saion s power emission o no exeed 0. W ouside o ± 00 khz. 77

2 Example Message bandwidh and he ransmied signal bandwidh (a) Baseband -B B (b) DSB-SC () - (d) Figure 29: SSB spera rom suppressing one DSB sideband BW Ineiieny in DSB-SC sysem: Reall ha or real-valued baseband signal m(), he onjugae symmery propery rom 2.30 says ha M( ) = (M()). The DSB sperum has wo sidebands: he upper sideband () and he lower sideband (), eah onaining omplee inormaion abou he baseband signal m(). As a resul, DSB signals oupy wie he bandwidh required or he baseband Rough Approximaion: I g () and g 2 () have bandwidhs B and B 2 Hz, respeively, he bandwidh o g ()g 2 () is B + B 2 Hz. This resul ollows rom he appliaion o he widh propery 8 o onvoluion 9 o he onvoluion-in-requeny propery. Consequenly, i he bandwidh o g() is B Hz, hen he bandwidh o g 2 () is 2B Hz, and he bandwidh o g n () is nb Hz. We menioned his propery in This propery saes ha he widh o x y is he sum o he widhs o x and y. 9 The widh propery o onvoluion does no hold in some pahologial ases. See [5, p 98]. 78

3 4.79. To improve he speral eiieny o ampliude modulaion, here exis wo basi shemes o eiher uilize or remove he speral redundany: (a) Single-sideband (SSB) modulaion, whih removes eiher he or he so ha or one message signal m(), here is only a bandwidh o B Hz. (b) Quadraure ampliude modulaion (QAM), whih uilizes speral redundany by sending wo messages over he same bandwidh o 2B Hz. 4.7 Single-Sideband Modulaion Transmiing boh upper and lower sidebands o DSB is redundan. Transmission bandwidh an be u in hal i one sideband is suppressed along wih he arrier. Deiniion 4.8. Conepually, in single-sideband (SSB) modulaion, a sideband iler suppresses one sideband beore ransmission. [3, p 85 86] (a) I he iler removes he lower sideband, he oupu sperum onsiss o he upper sideband () alone. Mahemaially, he ime domain represenaion o his SSB signal is x () = m() 2 os(2π ) m h () 2 sin(2π ). (55) where m h () is he Hilber ransorm o he message: m h () = H {m()} = π m(τ) τ dτ = m() π. (56) (b) I he iler removes he upper sideband, he oupu sperum onsiss o he lower sideband () alone. Mahemaially, he ime domain represenaion o his SSB signal is x () = m() 2 os(2π ) + m h () 2 sin(2π ). (57) Derivaion o he ime-domain represenaion is given in Seion 4.9. More disussion on SSB an be ound in [3, Se 4.4], [4, Seion 3..3] and [5, Seion 4.5]. 79

4 4.8 Quadraure Ampliude Modulaion (QAM) Deiniion In quadraure ampliude modulaion (QAM ) or quadraure muliplexing, wo baseband real-valued signals m () and m 2 () are ransmied simulaneously via he orresponding QAM signal: x QAM () = m () 2 os (2π ) + m 2 () 2 sin (2π ). Transmier (modulaor) Reeiver (demodulaor) m v H ˆm LP 2os 2 2os 2 2 2sin 2 xqam h Channel y 2 2sin 2 m2 v2 H ˆm2 LP Figure 30: QAM Sheme QAM operaes by ransmiing wo DSB signals via arriers o he same requeny bu in phase quadraure. Boh modulaed signals simulaneously oupy he same requeny band. The os (upper) hannel is also known as he in-phase (I ) hannel and he sin (lower) hannel is he quadraure (Q) hannel Demodulaion: Under he usual assumpion (B < ), he wo baseband signals an be separaed a he reeiver by synhronous deeion: { LPF x QAM () } 2 os (2π ) = m () (58) { LPF x QAM () } 2 sin (2π ) = m 2 () (59) 80

5 To see (58), noe ha v () = x QAM () 2 os (2π ) ( = m () 2 os (2π ) + m 2 () 2 sin (2π )) 2 os (2π ) = m () 2os 2 (2π ) + m 2 () 2 sin (2π ) os (2π ) = m () ( + os (2π (2 ) )) + m 2 () sin (2π (2 ) ) = m () + m () os (2π (2 ) ) + m 2 () os (2π (2 ) 90 ) Observe ha m () and m 2 () an be separaely demodulaed. Example () 2 os (2π ) + () 2 sin (2π ) Example os (2π ) sin (2π ) Suppose, during a ime inerval, he messages m () and m 2 () are onsan. Consider he signal m 2 os (2π ) + m 2 2 sin (2π ) Sinusoidal orm (envelope-and-phase desripion [3, p. 65]): x QAM () = 2E() os(2π + φ()), where envelope: E() = m () jm 2 () = phase: φ() = (m () jm 2 ()) m 2 () + m2 2 () 8

6 Example In a QAM sysem, he ransmied signal is o he orm x QAM () = m () 2 os (2π ) + m 2 () 2 sin (2π ). Here, we wan o express x QAM () in he orm x QAM () = 2E() os(2π + φ()), where E() 0 and φ() ( 80, 80 ]. Consider m () and m 2 () ploed in he igure below. Draw he orresponding E() and φ() m 2 os (2π ) + m 2 2 sin (2π ) 82

7 4.90. Complex orm: where 20 m() = m () jm 2 (). x QAM () = 2Re { (m()) e j2π } We reer o m() as he omplex envelope (or omplex baseband signal) and he signals m () and m 2 () are known as he in-phase and quadraure(-phase) omponens o x QAM (). The erm quadraure omponen reers o he a ha i is in phase quadraure (π/2 ou o phase) wih respe o he in-phase omponen. Key equaion: ( { LPF Re m () }) ( 2e j2π 2e j2π ) = m (). }{{} x QAM () 4.9. Three equivalen ways o saying exaly he same hing: (a) he omplex-valued envelope m() omplex-modulaes he omplex arrier e j2π, So, now you an undersand wha we mean when we say ha a omplex-valued signal is ransmied. (b) he real-valued ampliude E() and phase φ() real-modulae he ampliude and phase o he real arrier os(2π ), () he in-phase signal m () and quadraure signal m 2 () real-modulae he real in-phase arrier os(2π ) and he real quadraure arrier sin(2π ). 20 I we use sin(2π ) insead o sin(2π ) or m 2 () o modulae, x QAM () = m () 2 os (2π ) m 2 () 2 sin (2π ) = 2 Re { m () e j2π} where m() = m () + jm 2 (). 83