ose ad Varably Homewor # (8), aswers Q: Power spera of some smple oses A Posso ose A Posso ose () s a sequee of dela-fuo pulses, eah ourrg depedely, a some rae r (More formally, s a sum of pulses of wdh Δ τ ad hegh / Δ τ, ad he probably of a pulse bewee me ad +Δ s rδ, ad we osder he lm of Δτ ad Δ ) Calulae he power sperum P ( ω ) of hs ose Soluo he sraghforward approah s surprsgly dfful beause oe has o be areful abou he lms Sar wh he defo P = lm ( ) We osder frs ω ; for ω = we have o remember he fe pr ha we eed o subra a osa value (here, r) from () so ha s mea s ae a small Δ, ad approxmae he egral by a sum over a large umber = / Δ of / Δ ωδ ervals of legh Δ, e, e () d e, where = f here s a eve he = h erval, ad = f here s o eve Eah s depede, ad he probably of = s rδ ωδ ωδ ωδ ω( m) Δ e () d e e e e m = = =, m= = = he las quay s bes aalyzed aordg o wheher m= or o For m=, m = = (se a oly be or ), so he expeed value of eah erm s rδ here are = / Δ values of, so her orbuo o P ( ω ) s ( rδ ) = r he harder par s o show, rgorously, ha he orbuo of he erms ω ( m) e Δ m for whh m do o orbue Say = m here are m, = pars of m ad he sum, for whh = m Se ad m are depede ad eah has a probably of rδ of beg equal o, he her produ has a probably of ( rδ ) of beg equal o So he poro of he sum for whh = m has a expeed value of
ω( m) Δ ( )( ) ωδ ( ) ωδ e m r e r e = Δ = Δ Summg hs over m= he ere rage of (from o ) ad leg Δ (wh = / Δ ad = / Δ ) yelds ωδ lm r Δ( ) e = r ( ) Δ = he ragle-wave (see ear Sysems heory homewor Q), so s( ω / ) r ( ) = r fragle ( ) = r f ragle r = Provded ω, ω/ hs goes o (as / ) for suffely large hs las egral s he Fourer rasform of For ω =, he above expresso does o go o ; fa, dverges (as ) he problem s ha a ω =, maers ha he mea value of he sgal () was ozero s r We a oly expe a overge value for P = lm ( ) f we sar wh a sgal of mea he ω = -ase for P = lm ( ( ) ( ) ) < > s bes hadled by a speal argume P r d We wa o fd () = lm ( ( ) ) ( ) () r d= d () r he frs erm ous he umber of Posso eves a segme of legh ; r s he expeed mea of hs quay So ( () ) r d s he varae of a Posso dsrbuo wh mea r, amely, r r (he varae of a Posso dsrbuo s equal o s mea) So P () = = r ω ( m) Summarzg: For ω, he self-erms of e Δ m orbue r o P ( ω ), m, = ad he ross-erms have a lm of For ω =, a argume based o Posso oug sass shows ha P () = r So for all ω, P ( ω ) = r Alerave paral soluo wh a useful sgh
Reogze ha () a faser Posso proess s also a Posso proess, ad () speedg up he proess s a hage of sale of he power sperum herefore, ω u ωu u ωu P ( ω) = lm ( ) = lm ( ) e du = lm ( ) e du where we ve used u =, ad M u ωu u ωu lm ( ) e du = lm ( ) e du M M where we ve used M = he Posso proess represeed by u ( / ) has a rae r (wh me measured by u), ad hus a be regarded as a sum of depede Posso proesses of rae r So he power sperum of he Posso proess u ( / ) s mes larger ha he power sperum () So P( ω ) = P( u/ ) ( ) P( ) ω = ω, e, P ( ω ) s depede of ω Bu sll aes some wor o fd hs value For example, le ω he eah ozero erm he Fourer esmae s dephased, ad has magude hs shows ha as ω, P ( ω ) s he average umber of eves per u me, e, r B Sho ose A sho ose u () s a proess whh opes of a sereoyped waveform x(), ourrg a radom mes, are supermposed ha s, u () = x ( ), where he mes are deermed by a Posso proess of rae r he shos x() are ypally osdered o be ausal, amely, x () = for < ve he Fourer rasform u = u( ), fd he power sperum Pu of u x() u() Soluo he sho ose proess s he resul of flerg a Posso proess (of rae r) by he lear fler wh mpulse respose x() By par A, he Posso proess has power sperum r Flerg a sgal by a lear fler wh rasfer fuo X mulples he power sperum by X So = r X Pu C Sho ose, varable sho sze hs s a proess v () whh he ampludes of he shos vary radomly ha s, v () = ax ( ), where he ampludes a are hose
depedely ve he Fourer rasform u = u( ) ad he momes of he dsrbuo of he a, fd he power sperum Pv of v Soluo We a use he reasog of par B, bu appled o a modfed Posso proess ha has mpulses of ampludes a We alulae he power sperum of hs proess by he mehod of par A Se hese ampludes are depede, every sep of par A s readly exeded, yeldg he resul ha he power sperum of he modfed Posso proess s r a So P = r a X Q: Ipu ad oupu ose v Reall he behavor of a lear sysem wh addve ose (pages 6-7 of AV oes), ossg of a lear fler (haraerzed by s rasfer fuo g : s() r() z() If he pu s s() = s e ω ad here s a addve ose z () wh power sperum Pz ( ω ), he he quay F(, r ω,,) r(), whe alulaed for daa leghs ha are a mulple of he perod π / ω, has a mea value s g ad a varae P z ( ω ) Aalyze he suao whe here s also some ose added pror o, dagrammed below: s() r() y() z() Soluo: Deoe eral sgals as follows: s() x() u() r() y() z()
Based o he smpler sysem osdered o pages 6-7, sgals x() are haraerzed by F( x, ω,,) x( ), whh has mea s ( ω ) ad varae P y ( ω ) (he sysem ossg of s, y, ad x s deal o he oe o page 6-7, bu wh = I ) Fourer ompoes of he sgal a u are equal o hose a x, mulpled by g herefore, sgals a u are haraerzed by F( u, ω,,) u( ) wh a mea s g ad varae Py g( ω ) Addg a depede ose erm z() does o hage he mea, bu adds o he varae aordg o s power sperum So F(, r ω,,) r() has mea s g ad varae ( Py g + Pz )