Sgal,autocorrelato Phase ose p/.9.3.7. -.5 5 5 5 Tme Sgal,autocorrelato Phase ose p/.5..7.3 -. -.5 5 5 5 Tme Sgal,autocorrelato. Phase ose p/.9.3.7. -.5 5 5 5 Tme Sgal,autocorrelato. Phase ose p/.8..6. -.6 5 5 5 Tme
Sgal,autocorrelato 5 Addtve ose S/.5..7.3 -. -.5 3 6 9 5 Tme Sgal,autocorrelato 5 Addtve ose S/.5..7.3 -. -.5 3 6 9 5 Tme Eve o harmoc fuctos ca be etracted, but wth some dstorto 5.3 The Auto correlato of Uform, Sparse, Radom seres Let us start wth a radom occurrg square pulses (partcle detecto) Sgal Tme A pulse ampltude; T pulse wdth E A T Pulse Eergy. lm Average pulse desty T T R ( τ) ( t) ( t+ τ) R ( τ) for τ > T Whe τ > T Ether ( t) or ( t+ τ) R A T ( ) T τ ( τ ) lm E( ) T T T Detrmed by T of the sgle pulse!!! τ 3
For smplcty let us assume that T ; E - τ for τ < R ( τ ) for τ > ( ) ( ) ωτ ωτ S ( ω) R ( τ) e dτ τ e dτ ω s S ( ω) ω π Frst zero at ω! T S ( ω) τ cos( ωτ) dτ Sc(/)^..8.6.4.. 5 5 5 Tme. The wdth of the sparse seres s determed by that of a sgle pulse.. o correlated samples results f we sample at a rate of ω π T 4 Eamples:. A fast photomultpler wll geerate s pulse for each photo The ose spectrum wll reach GHz.. Electros coductors collde at femtosecod (average) The ose spectrum wll reach up to the IR rego. Usually we assume a dealzed Bad Lmted Whte ose S( ω) The power spectrum of the ose Ω Ω S( ω) for ω< Ω ad zero elsewhere ωt Ω s( Ωτ ) R( τ) S( ω) e dω π π ( Ωτ) π Frst zero at τ whch s the samplg rate for Ω o correlated ose samples 5 3
6. Geeralzed flter: Suppose we start from a sample of a measured sgal ad we wat to flter out some frequeces. We could geerate the FFT elemets: S( t) ( FFT ) G( ω) ( G( ω) H ( ω) )( flter) ( IFFT ) F( t ) ( Fltered ) ( ) { s } Istead of performg two fourer trasforms, wth ther Gbbs oscllato, we could fd the coeffcets drectly: ωt g( ω) s( t) e dt; F(t) deotes the fltered sgal. ωt Ft () g( ω) H( ω) e dω st ( τ) h( τ) dτ π Ft ( ) st ( t) ht ( ) Recursve flter the tme doma. "Memory" fucto. 6 A Eample- Low pass flter R C ωt ωt e H ( ω) ; h( τ) H( ω) e dω d + ωrc + ωrc ω for smplcty let us assume RC (tme uts) we have a smple pole at ω ; usg the resdue theorem h( τ ) e τ. for τ >. Usg causalty we kow that h( τ ) for τ < Ft ( ) st ( t) ht ( ) st ( ) ht ( t ) Memory fucto..8.6.4.. -4-4 Tme 7 4
A eample: Idealzed Bad Pass flter H ( ω) ω < Ω ω >Ω Ω Ω ωt ωt s( Ωt) ht () H( ω) e dω e dω t Ωt Ω Ω s( Ωt ) h( t ) Flter coeffcets. Ω We ca dgtally sythesze ay flter we eed. 8 6. Matched flter What s the best set of weghtg factors ( the frequecy doma) to mamze sgal to ose? (If we are ot terested the sgal shape!) S( ω) ws W s the weghtg vector ad S s the sgal vector. each sample pot S has a ose compoet assocated wth t. the total measured ose varace (from ucorrelated samples) w ( ws ) ( w ) ; The measured (power) sgal to ose: S P P ma; w 4 ( Ss) S ( w ) s s w S S 9 5
w If we remember that S ca be comple, ad S SS * tha we get s cost * ; For Whte ose all are equal ad the flter should trasmt all frequeces that the sgal carres. For o whte ose we should frst whte the ose! ad the tme doma, for Whte ose: ht ( ) cost s( t ); ( ) fltered( τ) s( t + τ) + ( t + τ) s( t) 6. Sources of ose. Thermal ose (also kow as Johso or yqust ose). Based o thermodyamcs: equpartto law, secod law of thermodyamcs ad eumeratg cavty modes coected to a resstor V 4 k TRΩ B for M Ω resstor, at room temperature, V 3V Hz 6. Matched flter What s the best set of weghtg factors ( the frequecy doma) to mamze sgal to ose? (If we are ot terested the sgal shape!) S( ω) ws W s the weghtg vector ad S s the sgal vector. each sample pot S has a ose compoet assocated wth t. the total measured ose varace (from ucorrelated samples) w ( ws ) ( w ) ; The measured (power) sgal to ose: S P P ma; w 4 ( Ss) S ( w ) s s w S S 6
. Shot ose. Resultg from the dscrete flow of electros, (or os or photos). f the average flow cossts of partcles, the fluctuato the umber s eiω For ao-ampere curret 8 fa Hz 3. Flcker or /f ose. Theoretcal bass s stll shaky, but s foud every case, ad domates at low frequeces (<Hz). 4. Pck up ose (Iterferece ose). Maly at 5 ad Hz ad hgher harmocs of le frequecy. at AM ad FM frequeces (-Mhz) at dmmer le swtchg frequeces ( KHz) TV, Radar ad Moble phoes (.3-3GHz) 6.3 Trasverse hardwred flters: Surface Acoustc Wave All our flterg procedures ca be dagramed as follows: Delay Delay Delay Delay Multply Multply Multply Multply Add 3 7
7. Cross correlato fuctos: If we ecte a lear system, we kow that ts respose s determed by the Trasfer fucto, ad ectato fucto, eve whe t s hdde ose. 7. Let us defe a Cross correlato fucto : S ( ) ( ) ( ) lm y τ t y t + τ T ( t ) y ( t + τ) dt Some mmedate propertes:. Sy ( τ) Sy ( τ) From statoary propertes of the system.. Sy ( τ ) T T R () R yy () Mootoc decay tme. 3. If ether t ( ) or yt ( ) are perodc tha S y ( τ ) s perodc too. Sy ( τ + T) t ( ) yt ( + τ + T) t ( T) yt ( + τ) Sy ( τ) 4. If both t ( ) ad yt ( ) are perodc wth the same perod tha: 4 ω t ( ) ae ; yt ( ) be ωt mω( t+ τ) ( + m) ωt m S ( ) abe e abe e S τ y m m m, m, ( τ ), m a y t mωt m m b e mωτ m + m ab e δ ωτ S wll have large compoets whe both ad y have large compoets at the same! 6. Whe or y are ot perodc we ca epad them Fourer tegral: ωt ω( t+ τ) t ( ) ( ω) e dω; yt ( τ) y( ω) e dω π + π ωτ 5 8
( τ ) ( ω ) ( ω) ω ω ( π ) ( ω+ ω) t ωτ S y e e d d y ( ω ) ( ω) ω ω ( π ) ( ω+ ω) t ωτ y e e d d d ωτ ( ω) y( ω) e δ( ω ω) dωdω π + ωτ Sy( τ) ( ω) y( ω) e dω π * S ( ω) ( ω) y( ω) ( ω) y( ω) y t Both ad y cotrbute to the cross correlato To mamze the cross correlato fucto t s advsable correlate the system respose wth a replca of the ectato fucto. 6 7. Matched flter may be realzed by cross correlato (whte ose) The output of a matched flter at tme t f f ( ) ( τ ) S( T) st ( ) + ht ( ) s( ) + ht ( τ) dτ for a matched flter h s kow: st ( t),... t T ht () elswhere T ( τ ) S( T) s( ) + s( T + τ) dτ T t s: Cross correlate the sgal wth addtve ose wth a replca of the ectato sgal T 7 9
Sgal,autocorrelato Adtve ose S/..8.6.4.. 5 5 5 Tme Sgal,crosscorrelato Addtve ose S/..8.6.4.. 5 5 5 Tme Auto ad Cross correlato of Addtve ose Sgal() t s( ω t) + radom S ( t) (s( ω t) + radom) s( ω t) y ( ω ) S ( t ) Sgal( t )s ( t t ) y 8 Multplcatve ose S/ Sgal,autocorrelato..8.6.4.. 5 5 5 Tme Multplcatve ose S/ Sgal,crosscorrelato..8.6.4.. 5 5 5 Tme Auto ad Cross correlato of Multplcatve ose Sgal() t s( ω t)* radom y S ( t) (s( ω t) * radom) s( ω t) 9
7. Uses of cross correlato fuctos 7. Iput-Output cross correlato () t H( ω) y() t S t y t S y * y( τ ) ( ) ( + τ ; y( ω) ( ω) ( ω) [ ] S H * y( ω) ( ω) ( ω) ( ω) f ( ω) cost (Whte ose) S ( ω) cost* H( ω) y We ca measure the trasfer fucto of a commucato etwork o le whe there are o sgals, oly ose, by computg the put output cross correlato