Модулированные колебания, спектральный анализ, линейная фильтрация, страница 4

15): ñïåêòð èìïóëüñàC(ω) (ñ ìíîæèòåëåì 1/T ) ïðåäñòàâëÿåò ñîáîé ¾îãèáàþùóþ¿ ÷àñòîêîëà ñïåêòðàëüíûõ êîìïîíåíò cn ïåðèîäè÷åñêîé ïîñëåäîâàòåëüíîñòè èìïóëüñîâ. Âìåñòî äèñêðåòíîãî ñïåêòðà {cn } ïîëó÷àåì íåïðåðûâíûé ñïåêòð C(ω).Ìîäóëü óíêöèè C(ω) îïðåäåëÿåò àìïëèòóäû ãàðìîíè÷åñêèõ êîëåáàíèéðàçíûõ ÷àñòîò, ñóììà êîòîðûõ îáðàçóåò èìïóëüñ f (t). Êàê âèäíî èç ãðàèêà,îñíîâíîé âêëàä äàþò ãàðìîíè÷åñêèå êîëåáàíèÿ, ÷àñòîòû êîòîðûõ çàïîëíÿþò èíòåðâàë |∆ω| < 2πτ . Ýòî ïîëóøèðèíà ãëàâíîãî ìàêñèìóìà óíêöèèsin ωτ /2.Äèàïàçîí÷àñòîò∆ω ìîæíî íàçâàòü øèðèíîé ñïåêòðà C(ω).ωτ /2Ìû ïîëó÷èëè çàìå÷àòåëüíîå ñîîòíîøåíèå, ñâÿçûâàþùåå ìåæäó ñîáîéäëèòåëüíîñòü ñèãíàëà ñ øèðèíîé åãî ñïåêòðà:ò.

å. ñïåêòð ïåðåíîñèòñÿ ïî îñè ÷àñòîò íà âåëè÷èíó ω0 (ðèñ. 17).Èç îðìóëû (27) ÿñíî, ÷òî åñëè ñïåêòð óíêöèè f0 (t) ëîêàëèçîâàí â îáëàñòè ÷àñòîò |ω| 6 Ω (ðèñ. 17à), òî ñïåêòð êîìïëåêñíîé óíêöèè z(t) íåñîäåðæèò îòðèöàòåëüíûõ ÷àñòîò (ðèñ. 17á) Z(ω) ≡ 0 ïðè ω < 0, åñëè ω0 > Ω.(26)τ · ∆ω ≈ 2π.Ýòî ñîîòíîøåíèå èìååò óíèâåðñàëüíûé õàðàêòåð. Îíî îêàçûâàåòñÿ ñïðàâåäëèâûì ïî ïîðÿäêó âåëè÷èíû äëÿ ïðîèçâîëüíîãî ñèãíàëà f (t). ×åì áîëüøåäëèòåëüíîñòü ñèãíàëà (ëèáî áîëüøå èíòåðâàë âðåìåíè, â òå÷åíèå êîòîðîãîïðîèñõîäèò åãî çàìåòíîå èçìåíåíèå), òåì óæå ñïåêòð ñèãíàëà ∆ω , è, íàîáîðîò, ÷åì êîðî÷å ñèãíàë (èëè áûñòðåå ïðîèñõîäèò èçìåíåíèå ñèãíàëà), òåìøèðå åãî ñïåêòð, ò. å. òðåáóåòñÿ áîëåå øèðîêèé èíòåðâàë ÷àñòîò ãàðìîíè÷åñêèõ êîëåáàíèé, îáðàçóþùèõ â ñóììå äàííûé ñèãíàë.

 ýòîì ñîñòîèò ñìûñëçàìå÷àòåëüíîãî ñîîòíîøåíèÿ (26), êîòîðîå íàçûâàåòñÿ ñîîòíîøåíèåì íåîïðåäåë¼ííîñòåé.Ñïåêòð óíêöèè f0 (t) åñòü C0 (ω): f0 (t) ↔ C0 (ω). Íàéòè ñïåêòðZ(ω) êîìïëåêñíîé óíêöèè z(t) = f0 (t)eiω0 t .Ñîãëàñíî (25) çàïèøåìÇàäà÷à 8.Íàéòè ñâÿçü ìåæäó ñïåêòðîì ìîäóëèðîâàííîãî êîëåáàíèÿf (t) = a(t) cos[ω0 t + ϕ(t)]è ñïåêòðîì êîìïëåêñíîé óíêöèèãäå f0 (t) = a(t)eiϕ(t) êîìïëåêñíàÿ óíêöèÿ, ñîäåðæàùàÿ âñþ èíîðìàöèþî ìåäëåííî ìåíÿþùèõñÿ çàêîíàõ ìîäóëÿöèè τ > 2π/ω0 . Ïîëó÷àåì, ÷òî ñïåêòðC0 (ω) óíêöèè f0 (t) îòëè÷åí îò íóëÿ ëèøü â îáëàñòè ÷àñòîò |ω| 6 Ω ≃ 2π/τ ,ïðè÷¼ì ω0 > Ω.Îòìåòèì, ÷òî ñàìè óíêöèè f (t) è z(t), ñîãëàñíî îðìóëå Ýéëåðà, ñâÿçàíû î÷åâèäíûì ðàâåíñòâîì:f (t) = Re z(t) =Z(ω) =−∞f0 (t)eiω0 t·e−iωtdt =∞Zf0 (t)e−∞20−i(ω−ω0 )tdt = C0 (ω − ω0 ).(27)1[z(t) + z ∗ (t)].2Ïîñêîëüêó Z(ω) ≡ 0 ïðè ω < 0, òî z(t) ìîæíî çàïèñàòü â âèäå1z(t) =2πÇàäà÷à 7.∞Zf (t) = Re z(t),z(t) = a(t)ei[ω0 t+ϕ(t)] = f0 (t)eiω0 t ,∞ZZ(ω)eiωt1dω =2π∞ZZ(ω)eiωt dω.0−∞Òîãäà äëÿ êîìïëåêñíî-ñîïðÿæ¼ííîé óíêöèè z ∗ (t) èìååì1z (t) =2π∗∞Z∗Z (ω)e−iωt1dω =2π0Z0−∞21Z ∗ (−ω)eiωt dω,îòêóäà ÿñíî, ÷òî óíêöèÿ z ∗ (t) èìååò ñïåêòð Z ∗ (−ω), îòëè÷íûé îò íóëÿ ëèøüâ îáëàñòè ÷àñòîò ω < 0.

Ñêëàäûâàÿ ëåâûå è ïðàâûå ÷àñòè äâóõ ïîñëåäíèõðàâåíñòâ, íàõîäèì∞Z12f (t) =2πplaements[Z(ω) + Z ∗ (−ω)]eiωt dω.−∞Òàêèì îáðàçîì, ñïåêòð C(ω) ìîäóëèðîâàííîãî êîëåáàíèÿ f (t) èìååò âèäC(ω)12 Z(ω)1 ∗2 Z (−ω)Ìíîæèòåëü e−iωτ (íå çàâèñÿùèé îò ïåðåìåííîé èíòåãðèðîâàíèÿ t′ ) âûíîñèòñÿèç-ïîä çíàêà èíòåãðàëà:∞Z′−iωτC(ω) = e(28)f0 (t′ )e−iωt dt′ = C0 (ω) · e−iωτ ,−∞èëè ñèìâîëè÷åñêè f0 (t − τ ) ↔ C0 (ω)e−iωτ , ò.å. ñìåùåíèå ñèãíàëà âî âðåìåíèíà τ (çàïàçäûâàíèå) ïðèâîäèò ê óìíîæåíèþ åãî ñïåêòðà íà e−iωτ (òåîðåìàñìåùåíèÿ).11PSfrag replaementsZ(ω) + Z ∗ (−ω),22ò. å.

ðàâåí ñóììå äâóõ íåïåðåêðûâàþùèõñÿñïåêòðîâ (ðèñ. 18).ÑîîòâåòñòâåííîC0 (ω)C(ω) =−ω00ωω0èñ. 18Z(ω) =2C(ω) ïðè0ïðèω > 0,ω < 0.Âî ìíîãèõ ñëó÷àÿõ öåëåñîîáðàçíî ðåøàòü çàäà÷ó ðàçëîæåíèÿ â ñïåêòððåàëüíîãî ìîäóëèðîâàííîãî êîëåáàíèÿC(ω)ωz(t) = a(t)ei[ω0 t+ϕ(t)] ,êîòîðûé íå ñîäåðæèò îòðèöàòåëüíûõ ÷àñòîò, åñëè ðå÷ü èä¼ò î ìåäëåííî ìåíÿþùèõñÿ âî âðåìåíè çàêîíàõ ìîäóëÿöèè a(t) è ϕ(t). Òîãäà ñïåêòð C(ω) ðåàëüíîãî êîëåáàòåëüíîãî ïðîöåññà ñîâïàäàåò (ñ òî÷íîñòüþ äî ìíîæèòåëÿ 1/2) ñîñïåêòðîì êîìïëåêñíîãî ñèãíàëà z(t) ïðè ω > 0.

Ñïåêòð æå C(ω) ïðè ω < 0 âñèëó ñâîéñòâà ýðìèòîâîñòè óðüå-îáðàçà äåéñòâèòåëüíîé óíêöèè íàõîäèòñÿñ ïîìîùüþ ðàâåíñòâà C(−ω) = C ∗ (ω).Çàäà÷à 9. Ñïåêòð óíêöèè f0 (t) åñòü C0 (ω): f0 (t) ↔ C0 (ω). Íàéòè ñïåêòðóíêöèè f (t) = f0 (t − τ ).Èìååì∞ZC(ω) =f0 (t − τ )e−iωt dt.−∞Ïîñëå çàìåíû ïåðåìåííûõ t = t − τ (dt = dt′ ) ïîëó÷àåì′C(ω) =∞Z′f0 (t′ )e−iω(t +τ ) dt′ .−∞22ω0−ω0èñ. 19PSfrag replaementsà)f0 (t)á)f (t) = a(t) cos[ω0 t + ϕ(t)],íàéäÿ ñïåêòð êîìïëåêñíîé óíêöèèωf (t)ttτ2− τ2− τ2τ2èñ. 20Ïóñòü f0 (t) ↔ C0 (ω).

Íàéòè ñïåêòð C(ω) óíêöèè f (t) == f0 (t) cos ω0 t.Èñïîëüçóÿ îðìóëó Ýéëåðà, çàïèøåìÇàäà÷à 10.11f0 (t)eiω0 t + f0 (t)e−iω0 t .22Ñîãëàñíî ðåøåíèþ çàäà÷è 7, èìååìf (t) =11(29)C0 (ω − ω0 ) + C0 (ω + ω0 ),22ò. å. ñïåêòð C0 (ω) (óìíîæåííûé íà 1/2) ïåðåíîñèòñÿ ïî îñè ÷àñòîò âëåâî èâïðàâî íà íåñóùóþ ÷àñòîòó ω0 (ðèñ. 19).  ÷àñòíîñòè, ïóñòü f0 (t) ïðÿìîóãîëüíûé èìïóëüñ äëèòåëüíîñòè τ (ðèñ.

20à):(1 ïðè |t| 6 τ /2,f0 (t) = Pτ (t) =0 ïðè |t| > τ /2.C(ω) =23Òîãäà f (t) = f0 (t) cos ω0 t îáðûâîê êîñèíóñîèäû (öóã) äëèòåëüíîñòè τPSfrag replaements(ðèñ. 20á). Ñîãëàñíî (29), ïîëó÷àåìτ sin(ω + ω0 )τ /2τ sin(ω − ω0 )τ /2C(ω) =+,2(ω − ω0 )τ /22(ω + ω0 )τ /2ihSfrag replaementsãäå C0 (ω) = τ sinωτωτ/2/2 ñïåêòð èìïóëüñà f0 (t) (ñì. çàäà÷ó 6).Ñïåêòðû C0 (ω) è C(ω) ïðåäñòàâëåíû íà ðèñ. 21.τà)C0 (ω)|S(ω)|C(ω)á)ω−3ω0 −2ω0τ /2ωωω0−ω0−ω002∆ωω02ω03ω0èñ. 23PSfrag replaementsC(ω)èñ.

21C0 (ω)Íàéòè ñïåêòð C(ω) ñèãíàëà f (t), ÿâëÿþùåãîñÿ ïåðèîäè÷åñêèìïîâòîðåíèåì ñèãíàëà f0 (t) ↔ C0 (ω):Çàäà÷à 11.f (t) =N−1Xn=0(ïðè êîíå÷íîì N ).f0 (t − nT )2∆ΩÈñïîëüçóÿ òåîðåìó ñìåùåíèÿ (28), ïîëó÷àåìθC(ω) = C0 (ω)plaementsNθθR θθÂûðàæåíèåS=N−1Xe−inθ(θ = ωT )ìîæíî èíòåðïðåòèðîâàòü êàê ñóììó êîëåáàíèé,àçû êîòîðûõ ñîñòàâëÿþò àðèìåòè÷åñêóþ ïðîãðåññèþ. Íà âåêòîðíîé äèàãðàììå êàæäûé ïîñëåäóþùèé âåêòîð åäèíè÷íîé äëèíû ïîâ¼ðíóò íà óãîë θ ïî îòíîøåíèþ ê ïðåäûäóùåìó. Ïîëó÷àåì ÷àñòü ìíîãîóãîëüíèêà, âïèñàííîãî â îêðóæíîñòü, ðàäèóñR êîòîðîé ðàâåí (ðèñ. 22)1.R=2 sin 2θ242ω03ω0 4ω0Ñóììàðíûé âåêòîð èìååò äëèíóe−inωT .|S| = 2R sinsin N2θ2π − N θ=.2sin θ2Óãîë ïîâîðîòà âåêòîðà ϕ = (N − 1) 2θ .

Ïîëó÷àåìN−1Xn=0èñ. 22ωω0èñ. 24n=0S1N−1X0n=0Îêîí÷àòåëüíî èìååìeinωT =sin N 2ωT i N −1 ωT.e 2sin ωT2C(ω) = C0 (ω)sin N 2ωT i N −1 ωTe 2.sin ωT2Ïîëó÷àåì íàáîð óçêèõ (ïðè N ≫ 1) ñïåêòðàëüíûõ ìàêñèìóìîâ âûñîòîé,sin N ωT2ðàâíîé N (ò. ê. sin ωT= N íà ÷àñòîòàõ ωn = nω0 , êðàòíûõ ω0 = 2πT ),225ñ ïîëóøèðèíîé ∆ω = N2πT , êîòîðóþ ìîæíî íàéòè èç óñëîâèÿ sin N ωT= 02(ðèñ.

23). Ìàêñèìóìû ïðîìîäóëèðîâàíû ¾îãèáàþùåé¿ óíêöèåé C0 (ω),ñïåêòðàëüíàÿ øèðèíà êîòîðîé îïðåäåëÿåòñÿ äëèòåëüíîñòüþ ñèãíàëà f0 (t):∆ω ≃ 2πτ . Ôóíêöèÿ C(ω) èìååò âèä, ïîêàçàííûé íà ðèñ. 24 (óíêöèÿ C0 (ω)èçîáðàæåíà íà ðèñóíêå ïóíêòèðîì).Ïðè N → ∞ âûñîòà ñïåêòðàëüíûõ ìàêñèìóìîâ ñòðåìèòñÿ ê áåñêîíå÷íîñòè, à èõ øèðèíà ∆ω ê íóëþ: ìû ïîëó÷àåì ¾÷àñòîêîë¿ δ -óíêöèé.Ñóììèðóÿ îòêëèêè, ïîëó÷àåìg(t) =12πZC(ω)H(ω)eiωt dω.Âûõîäíîé ñèãíàë èëüòðà g(t) (êàê è ëþáîé ñèãíàë) ìîæíî ïðåäñòàâèòü ëèáîâ âèäå ðÿäà Ôóðüå:Xg(t) =bn eiωn t ,(32)ëèáî â âèäå èíòåãðàëà Ôóðüå:5. Ëèíåéíàÿ èëüòðàöèÿ (ñïåêòðàëüíûé ìåòîä)Åù¼ ðàç îïèøåì àëãîðèòì ðåøåíèÿ çàäà÷è ëèíåéíîé èëüòðàöèè (ñïåêòðàëüíûé ìåòîä).1. Ïåðâûé øàã ïðåäñòàâëåíèå âõîäíîãî ñèãíàëà èëüòðà â âèäå ñóïåðïîçèöèè ãàðìîíè÷åñêèõ êîëåáàíèé: ëèáî â âèäå ðÿäà Ôóðüå (13), ëèáî èíòåãðàëàÔóðüå (24).

Ñïåêòð C(ω) âõîäíîãî ñèãíàëà f (t) íàõîäèòñÿ ñ ïîìîùüþ ñîîòíîøåíèé (22) èëè (25). ñëó÷àå äèñêðåòíîãî ñïåêòðà, íàïðèìåð ïåðèîäè÷åñêîé óíêöèè f (t),íàõîäèì íàáîð êîýèöèåíòîâ cn ðÿäà Ôóðüå ñ ïîìîùüþ îðìóëû (22).2. Âòîðîé øàã íàõîæäåíèå ÷àñòîòíîé õàðàêòåðèñòèêè èëüòðà H(ω), ò. å.îòêëèêà èëüòðà íà ãàðìîíè÷åñêîå âíåøíåå âîçäåéñòâèå åäèíè÷íîé àìïëèòóäû:eiωt → L → H(ω)eiωt .Ñóììèðóÿ îòêëèêè íà êàæäîå ãàðìîíè÷åñêîå ñëàãàåìîå âõîäíîãî ñèãíàëàcn eiωn tcn eiωn t → L → cn H(ωn )eiωn t ,3.íàõîäèì ðåçóëüòèðóþùèé âûõîäíîé ñèãíàë èëüòðà g(t) (îòêëèê íà çàäàííîåâõîäíîå âîçäåéñòâèå f (t)):(31)g(t) =12πZB(ω)eiωt dω.(33)Èç ñðàâíåíèÿ (30) è (32), (31) è (33) ñëåäóåò, ÷òî ñïåêòð âûõîäíîãî ñèãíàëà(íàáîð êîýèöèåíòîâ ðàçëîæåíèÿ bn â ñëó÷àå äèñêðåòíîãî ñïåêòðà) ëèáîóíêöèÿ B(ω) (â ñëó÷àå íåïðåðûâíîãî ñïåêòðà) íàõîäÿòñÿ ñ ïîìîùüþ ðàâåíñòâbn = cn H(ωn ),B(ω) = C(ω) · H(ω),(34)êîòîðûå ëåæàò â îñíîâå ñïåêòðàëüíîãî ìåòîäà ðåøåíèÿ çàäà÷è ëèíåéíîéèëüòðàöèè. ÷àñòíîñòè, ðàâåíñòâà (34) ïîäñêàçûâàþò ïóòü ðåøåíèÿ çàäà÷è ñåëåêöèè,êîòîðàÿ âîçíèêàåò ïðè ïðè¼ìå ðàäèîñèãíàëîâ.

Ïóñòü íà âõîä êîëåáàòåëüíîãîêîíòóðà ïðè¼ìíèêà ïîñòóïàþò ñèãíàëû äâóõ ðàäèîñòàíöèé, âåäóùèõ ïåðåäà÷è íà íåñóùèõ ÷àñòîòàõ ω0 è ω1 . Ýòî ìîäóëèðîâàííûå êîëåáàíèÿfs (t) = a(t) cos(ω0 t + ϕ1 (t))èfn (t) = a(t) cos(ω1 t + ϕ2 (t)).1C(ωn )eiωn t dωîòêëèê íà êàæäîå ãàðìîíè÷åñêîå ñëàãàåìîå âõîäíîãî ñèãíàëà 2πåñòü11C(ωn )eiωn t dω → L →C(ωn )H(ωn )eiωn t dω.2π2πÈõ ñïåêòðû Cs (ω) è Cn (ω). Òðåáóåòñÿ âûäåëèòü ïîëåçíûé ñèãíàë fs (t) èîòñåÿòü ïîìåõè (ñèãíàë fn (t)).Òðåáîâàíèÿ, ïðåäúÿâëÿåìûå ê ÷àñòîòíîé õàðàêòåðèñòèêå êîíòóðà, âûòåêàþò èç ñîîòíîøåíèé (34).

Âî-ïåðâûõ, êîíòóð íåîáõîäèìî íàñòðîèòü íà íåñóùóþ ÷àñòîòó ñèãíàëà fs (t), ò. å. ðåçîíàíñíàÿ ÷àñòîòà êîíòóðà ωð äîëæíàñîâïàäàòü ñ ω0 : ωð ≃ ω0 . Ïðè ýòîì äîáðîòíîñòü êîíòóðà Q äîëæíà áûòü äîñòàòî÷íî áîëüøîé, ÷òîáû â ïðåäåëû ïîëîñû ïðîïóñêàíèÿ êîíòóðà ∆ω = ωQ0íå ïîïàëè ñïåêòðàëüíûå êîìïîíåíòû ïîìåõ ω1 , |ω1 −ω0 | > ∆ωê . Ñ äðóãîé ñòîðîíû, ÷òîáû ïîëåçíûé ñèãíàë áûë ïðèíÿò áåç èñêàæåíèé, íåîáõîäèìî, ÷òîáûïîëîñà ÷àñòîò ïîëåçíîãî ñèãíàëà ∆Ω, îïðåäåëÿåìàÿ õàðàêòåðíûì âðåìåíåì τèçìåíåíèÿ óíêöèé a(t) è ϕ(t), îïèñûâàþùèõ çàêîí ìîäóëÿöèè (∆Ω·τ ≈ 2π ñîîòíîøåíèå íåîïðåäåë¼ííîñòåé), áûëà ìåíüøå ïîëîñû ïðîïóñêàíèÿ êîíòóðà∆Ω ≪ ∆ωê .