Spectral Characteristics of Digitally Modulated Signals
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1 Strl Chrtrt of Dgtlly odultd Sgl 6:33:56 Wrl Couto holog Srg 5 Ltur7&8 Drtt of Eltrl Egrg Rutgr Uvrty Ptwy J 89 ught y Dr. ry dy ry@wl.rutgr.du Doutd y Bozh Yu ozh@d.rutgr.du trt: h ltur frt trodu th tdrd rrtto of ol d gl of dgtl odultd gl d t owr trl dty PSD. h t how how th PSD of d gl lwy otd y t ol d volo. Flly two ortt odulto h OFD d PSK r trodud d th trl hrtrt of th two odultd gl r dud.. PSD of odultd Bd Sgl w ow tht dwdth ffy vry ortt ftor for hoog odulto h. o lr th dwdth ffy of o odulto h w d to gt th owr trl dty of th orrodg odultd gl. Grlly dgtlly odultd gl wrtt t R{ t j f whr t th d quvlt ol volo d th rdo h. o lult th PSD of t w rwrt t : t { t + v t j f * jf h th PSD of t drvd y: τ t + τ t t + τ t j + v * t + τ t + t + τ v * t + v * t + τ v * t jf jf f τ + f jf f τ + Orv tht W got ± f jf τ + jfτ τ * τ + τ * τ w got τ jf τ f [ f f + * f f rl d v w got f [ f f + f + f So th PSD of d gl t oltly dtrd y th PSD of t ol d vlo t. oqu th rult of rfor of vrou odulto d dodulto thqu r ddt of rrr frqu d hl frquy d... Stdrd Fort: Stdrd Rrtto of Col Bd Sgl W ow ov o to tudy th tdrd rrtto of ol d gl of dgtlly odultd gl. Grlly th d gl t wrtt : v t t whr th rrr ltud... th our K yol qu K th ory lgth whh dd o odulto h th yol durto d t quvlt hg futo of durto.
2 h th tdrd rrtto of th ol d gl.. El of ltud-shft Kyg SK odulto For SK odulto w hv whr yol qu v t h t { { + j h t th our th ltud hg ul qur wv or othg l Hr whh o ory. d th quvlt hg futo: t h t h d gl t wrtt : t { h t o[f t + rg whr + d rg t. PSD of Bd Sgl w hv gv th tdrd rrtto of th d ol vlo of th odultd d gl. W u t to gt th PSD of th d gl. Rll: v t t h t utoorrlto gv y: t + τ t t + τ v * t t + τ * t Cl: t ylottory ro.. t + τ t rod t wth rod Proof: t + + τ t + t + + τ v * t + t + + τ * t + Lt - d - t + + τ t + t + τ + * t u th our yol ttory.. th w hv + t + + τ t + t + τ * t t + τ t So t ylottory. hu τ otd y tg th t vrg of t + τ t.. τ + t * t dt + τ Chg vrl: t- w got τ + z * dz + τ Lt - τ + + z + τ z + τ So t PSD gv y f S [ * z + τ + * dz z + τ jfz dz * dz * dz jf j f τ jf z+ τ
3 whr [ * t τ jfz B f B f dz jfτ jf B * f jf h for of th quvlt ul hg futo t. { vl t l v t v t. v v t t dt t j t dt δj th Fourr trfor of Orvto: h PSD of t dd o: h orrlto rort of our qu. Whl th ov gv frquy do rrtto or hrtrt for f ddt t. Wht out th rrtto of whol l of fl wvfor? Grlly t durg yol trvl log to rrtd tr t of orthoorl futo { t <... whr d t E t + E t whr E th yol rgy th wvfor. O oulr SK - wth our yol : + j whr { ± ± 3 ± 5... ± d. For th of 6- th gl otllto how Fg t E t Fg Sgl Cotllto of 6- h PSD of 6- gl : S f H f Whr th vr of yol H f th Fourr trfor of h t f w hoo h t rd o wth rolloff ftor β. 5 th Fg how th PSD of th 6 gl. j f For th SK t R{ t t. t of for th gl : t h tof t t h tf t t t Fg PSD of 6- wth h t tr of d w hv rd o futo
4 V. OFD whr t. Wht OFD Orthogol Frquy Dvo odulto OFD lo odulto h dgd to ot th fft of ultth frquy ltv fdg. OFD lo of gl our yol h of durto od to lo of rlll odultd yol h of wth durto. ho uh tht: >> δ τ th RS dly rd. So th hl wll loo l flt fdg hl d th d for qulzto vod. lo h our yol th lo of lgth trttd rlll y loyg orthogol urrr th yol rt o h urrr uh l th th rl our rt. rult th fft of dly rd rdud. h r th dvtg of OFD.. Rrtto of OFD gl h ol vlo of OFD odultd gl : v t t whr { t r orthogol wvfor t ho h t{ j t -. d h t u t whh rtgl ul. th frquy rto { t r orthogol. W tht t t tt our yol r trttd ug th dtt urrr. Uully r ho fro otllto. h tdrd fort th gv y v t t h t j t { { Why OFD ttrtv? OFD odulto ttrtv u t hvd y ug thr vr drt Fourr trfor DF or vr ft Fourr trfor FF. Codr d gor th frquy fft j t tr {. lo hoo h t t w hv t u t j { t f w l t t t w got { j { j Dot { - h { { r jut th FF of th lo whr... So th trttr y to lt. Fg 3 how th h of OFD trttr. Fg 3 OFD trttr. PSD of OFD Sgl u th our yol r zro d th ltud hg ul h t th th PSD of t gv y
5 S f δ H f whr δ f h t u t th H f f. Fg how th PSD of t th : hg ul th z of lht { ± ± 3... ± Fg 5 how th gl otllto of th 8PSK Oft w hoo h t u t d Fg 5 Cotllto of 8PSK h t u t or rd o ul. Fg PSD of OFD gl W fro th fgur tht th lo rr or rt of th rgy r. h l tht w gt ttr trl ffy r. V. Ph Shftg Kyg PSK Wht PSK PSK rh th gr for of odulto ot wdly utlzd otorry rt rgg fro vo-d od to hgh-d tllt tro. th uggt th gl t grtd y h odulto of uodl rrr to o of qud h oto. For -ry PSK gl th tdrd fort of t ol d gl gv y v t t whr t h t{ j h t whr o ory h t th ltud hg ul h t th h. PSD of PSK gl Lt u uorrltd our yol d h t u t lo w u our yol r qul rol d dfd y t: { :... h t h t{ j h t d t h t f h t whr f h rult followd fro th followg: { j α t f α t Rll: f S [ * τ jfz dz jfτ jf For uorrltd our yol w hv
6 S f jf τ z τ * dz j [ h τ h z h τ h z dz { [ h τ h z h τ h z dz { [ h τ h z t l dt t > t Fg 6 PSD of PSK wth dffrt W hv S f h H f Choo τ h h t u t th f S f [ f f E [ f whr E log z jf τ z dz f [ f th yol rgy d. For fr oro of dwdth ffy wth dffrt w uttut wth log d gt: S f log f log [ f log Fg 6 how th PSD of th PSK gl wth dffrt. Hr th dwdth ffy R dfd η B d th dwdth B B hr dfd th ull-to-ull dwdth. Fro th fgur w gt tl whh how th dwdth ffy of PSK wth dffrt. l Bdwdth ffy of PSK η B Fro th tl w tht th dwdth ffy η B r r for -ry PSK. Howvr t th owr ffy dr r du to th lor dtt tw dffrt gl th otllto. V. Coluo th ltur th PSD of odultd gl r dud. Frt w howd tht th PSD of d gl t oltly dtrd y th PSD of t ol d vlo t. h two ortt odulto h OFD d PSK r trodud. OFD odulto ot oly good to fght ult-th fdg ut lo y to lt g t vry ttrtv for hgh t-rt wrl lto ultth rdo vrot. -ry PSK lo wdly ud otorry rt rgg fro vo-d od to hgh-d tllt tro. Rfr: [. Rort Wrl Couto.
7 Prl d Prt. d Edto Prt-Hll Eglwood Clff J: 996. [ J. Pro Dgtl Couto th Edto Grw-Hll Y:. [3 S. G. Wlo Dgtl odulto d Codg. Prt-Hll 996 [ ry dy Ovrvw of OFD htt:// y/cour/wsd/wd-lf.pp.
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