Topic 5. Energy & Power Spectra, and Correlation. 5.1 Review of Discrete Parseval for the Complex Fourier Series

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1 Topic 5 Energy & Power Specra, and Correlaion In Lecure we reviewed he noion of average signal power in a periodic signal and relaed i o he A n and B n coeciens of a Fourier series, giving a mehod of calculaing power in he domain of discree frequencies. In his lecure we wan o revisi power for he coninuous ime domain, wih a view o expressing i in erms of he frequency specrum. Firs hough we should review he derivaion of average power using he complex Fourier series. 5. Review of Discree Parseval for he Complex Fourier Series You did his as a par of s ue shee Recall ha he average power in a periodic signal wih period T! is Ave sig pwr T +T T jf ()j d T Now replace f () wih is complex Fourier series f () C n e in! : I follows ha n Ave sig pwr T n n +T T C n (C n ) n C n e in! jc n j jc 0 j + +T m T f ()f () d : (C m ) e im! d (because of orhogonaliy) jc n j ; using C n (C n) : n

2 5/ 5.. A quick check I is worh checking his using he relaionships found in Lecure : (A m ib m ) for m > 0 C m A 0 for m 0 (A jmj + ib jmj ) for m < 0 For n 0 he quaniies are ( ) jc 0 j A 0 jc n j (A m ib m ) (A m + ib m ) ( A n + ) B n in agreemen wih he expression in Lecure. 5. Energy signals vs Power signals When considering signals in he coninuous ime domain, i is necessary o disinguish beween \nie energy signals", or \energy signals" for shor, and \nie power signals". Firs le us be absoluely clear ha All signals f () are such ha jf ()j is a power. An energy signal is one where he oal energy is nie: E To jf ()j d 0 < E To < : I is said ha f () is \square inegrable". As E To is nie, dividing by he innie duraion indicaes ha energy signals have zero average power. To summarize before knowing wha all hese erms mean: An Energy signal f () always has a Fourier ransform F (!) always has an energy specral densiy (ESD) given by E f f (!) jf (!)j always has an auocorrelaion R f f () f ()f ( + )d always has an ESD which is he FT of he auocorrelaion R f f (), E f f (!) always has oal energy E To R f f (0) E f f (!)d! always has an ESD which ransfers as E gg (!) jh(!)j E f f (!)

3 A power signal is one where he oal energy is innie, and we consider average power 5/3 P Ave lim T! T A Power signal f () T T jf ()j d 0 < P Ave < : may have a Fourier ransform F (!) may have an power specral densiy (PSD) given S f f (!) jf (!)j always has an auocorrelaion R f f () lim T! T T T f ()f ( + )d always has a PSD which is he FT of he auocorrelaion R f f (), S f f (!) always has inegraed average power P Ave R f f (0) always has a PSD which ransfers hrough a sysem as S gg (!) jh(!)j S f f (!) The disincion is all o do wih avoiding inniies, bu i resuls in he auocorrelaion having dieren dimensions. Insinc ells you his is going o be a bi messy. We discuss nie energy signals rs. 5.3 Parseval's heorem revisied Le us assume an energy signal, and recall a general resul from Lecure 3: f ()g(), F (!) G(!) ; where F (!) and G(!) are he Fourier ransforms of f () and g(). Wriing he Fourier ransform and he convoluion inegral ou fully gives f ()g()e i! d F (p)g(! p) dp ; where p is a dummy variable used for inegraion. Noe ha! is no involved in he inegraions above i jus a free variable on boh he lef and righ of he above equaion and we can give i any value we wish o. Choosing! 0, i mus be he case ha f ()g()d F (p)g( p) dp :

4 5/4 Now suppose g() f (). We know ha ) f ()e i! d F (!) f ()e +i! d F (!) ) f ()e i! d F (!) This is, of course, a quie general resul which could have been suck in Lecure, and which is worh highlighing: The Fourier Transform of a complex conjugae is f ()e i! d F (!) Take care wih he!. Back o he argumen. In he earlier expression we had ) f ()g()d f ()f ()d F (p)g( p)dp F (p)f (p)dp Now p is jus any parameer, so i is possible o idy he expression by replacing i wih!. Then we arrive a he following imporan resul Parseval's Theorem: The oal energy in a signal is E To jf ()j d jf (!)j d! jf (!)j df NB! The df d!, and is nohing o do wih he signal being called f (). 5.4 The Energy Specral Densiy If he inegral gives he oal energy, i mus be ha jf (!)j is he energy per Hz. Tha is: The ENERGY Specral Densiy of a signal f (), F (!) is dened as E f f (!) jf (!)j

5 5.5 Example [Q] Deermine he energy in he signal f () u()e (i) in he ime domain, and (ii) by deermining he energy specral densiy and inegraing over frequency. [A] Par (i): To nd he oal energy in he ime domain 5/5 ) f () u() exp( ) jf ()j d exp( )d 0 [ exp( ) d 0 0 Par (ii): In he frequency domain F (!) u() exp( ) exp( i!)d exp( ( + i!))d ( + i!) 0 [ exp( ( + i!)) Hence he energy specral densiy is jf (!)j +! 0 ( + i!) Inegraion over all frequency f (no! remember!!) gives he oal energy of jf (!)j df Subsiue an! jf (!)j df +! d! + an sec d d w hich is nice

6 5/6 5.6 Correlaion Correlaion is a ool for analysing wheher processes considered random a priori are in fac relaed. In signal processing, cross-correlaion R f g is used o assess how similar wo dieren signals f () and g() are. R f g is found by muliplying one signal, f () say, wih ime-shifed values of he oher g( + ), hen summing up he producs. In he example in Figure 5. he cross-correlaion will low if he shif 0, and high if or 5. f() Low g() High High Figure 5.: similar. The signal f () would have a higher cross-correlaion wih pars of g() ha look One can also ask how similar a signal is o iself. Self-similariy is described by he auo-correlaion R f f, again a sum of producs of he signal f () and a copy of he signal a a shifed ime f ( + ). An auo-correlaion wih a high magniude means ha he value of he signal f () a one insan signal has a srong bearing on he value a he nex insan. Correlaion can be used for boh deerminisic and random signals. We will explore random processes his in Lecure 6. The cross- and auo-correlaions can be derived for boh nie energy and nie power signals, bu hey have dieren dimensions (energy and power respecively) and dier in oher more suble ways. We coninue by looking a he auo- and cross-correlaions of nie energy signals. 5.7 The Auo-correlaion of a nie energy signal The auo-correlaion of a nie energy signal is dened as follows. We shall deal wih real signals f, so ha he conjugae can be omied.

7 5/7 The auo-correlaion of a signal f () of nie energy is dened R f f () The resul is an energy. f ()f ( + )d (for real signals) f ()f ( + )d There are wo ways of envisaging he process, as shown in Figure 5.. One is o shif a copy of he signal and muliply verically (so o speak). For posiive his is a shif o he \lef". This is mos useful when calculaing analyically. f() hen sum f() f( +τ ) f() Figure 5.: g() and g( + ) for a posiive shif Basic properies of auo-correlaion. Symmery. The auo-correlaion funcion is an even funcion of : R f f () R f f ( ) : Proof: Subsiue p + ino he deniion, and you will ge R f f () f (p )f (p)dp : Bu p is jus a dummy variable. Replace i by and you recover he expression for R f f ( ). (In fac, in some exs you will see he auocorrelaion dened wih a minus sign in fron of he.)

8 5/8. For a non-zero signal, R f f (0) > 0. Proof: For any non-zero signal here is a leas one insan for which f ( ) 6 0, and f ( )f ( ) > 0. Hence f ()f ()d > The value a 0 is larges: R f f (0) R f f (). Proof: Consider any pair of real numbers a and a. As (a a ) 0, we know ha a + a a a + a a. Now ake he pairs of numbers a random from he funcion f (). Our resul shows ha here is no rearrangemen, random or ordered, of he funcion values ino () ha would make f ()()d > f () d. Using () f ( + ) is an ordered rearrangemen, and so for any f () d 5.8 Applicaions f ()f ( + )d 5.8. Synchronising o hearbeas in an ECG (DIY search and read) 5.8. The search for Exra Terresrial Inelligence For several decades, he SETI organizaion have been looking for exra erresrial inelligence by examining he auocorrelaion of signals from radio elescopes. One projec scans he sky around nearby (00 ligh years) sun-like sars chopping up he bandwidh beween -3 GHz ino billion channels each Hz wide. (I is assumed ha an aemp o communicae would use a single frequency, highly uned, signal.) They deermine he auocorrelaion each channel's signal. If he channel is noise, one would observe a very low auocorrelaion for all non-zero. (See whie Figure 5.3: Chay aliens noise in Lecure 6.) Bu if here is, say, a repeaed message, one would observe a periodic rise in he auocorrelaion. τ increasing Figure 5.4: R f f a 0 is always large, bu will drop o zero if he signal is noise. If he messages align he auocorrelaion wih rise.

9 5.9 The Wiener-Khinchin Theorem Le us ake he Fourier ransform of he cross-correlaion f ()g( + )d, hen swich he order of inegraion, [ ] FT f ()g( + )d f ()g( + ) d e i! d f () g( + ) e i! dd Noice ha is a consan for he inegraion wr (ha's how f () oaed hrough he inegral sign). Subsiue p + ino i, and he inegrals become separable [ ] FT f ()g( + )d f () g(p) e i!p e +i! dpd p f ()e i! d F (!)G(!): g(p) e i!p dp If we specialize his o he auo-correlaion, G(!) ges replaced by F (!). Then For a nie energy signal The Wiener-Khinchin Theorem a says ha The FT of he Auo-Correlaion is he Energy Specral Densiy FT [R f f ()] jf (!)j E f f (!) a Norber Wiener ( ) and Aleksandr Khinchin ( ) (This mehod of proof is valid only for nie energy signals, and raher rivializes he Wiener-Khinchin heorem. The fundamenal derivaion lies in he heory of sochasic processes.) 5.0 Corollary of Wiener-Khinchin This corollary jus conrms a resul obained earlier. We have jus shown ha R f f (), E f f (!). Tha is R f f () E f f!)e i! d! where is used by convenion. Now se 0 5/9

10 5/0 Auo-correlaion a 0 is R f f (0) E f f!)d! E To Bu his is exacly as expeced! Earlier we dened he energy specral densiy as E To E f f!)d! ; and we know ha for a nie energy signal R f f (0) jf ()j d E To : 5. How is he ESD aeced by passing hrough a sysem? If f () and g() are in he inpu and oupu of a sysem wih ransfer funcion H(!), hen G(!) H(!)F (!) : Bu E f f (!) jf (!)j, and so E gg (!) jh(!)j jf (!)j jh(!)j E f f (!) 5. Cross-correlaion The cross-correlaion describes he dependence beween wo dieren signals. Cross-correlaion R f g () 5.. Basic properies f ()g( + )d. Symmeries The cross-correlaion does no in general have a denie reecion symmery. However, R f g () R gf ().. Independen signals The auo-correlaion of even whie noise has a non-zero value a 0. This is no he case for he cross-correlaion. If R f g () 0, he signal f () and g() have no dependence on one anoher.

11 5/ 5.3 Example and Applicaion [Q] Deermine he cross-correlaion of he signals f () and g() shown. f() g() a a 3a 4a [A] Sar by skeching g( + ) as funcion of. a+ τ 4a+ τ 0 a a+ τ 4a+ τ a+ τ 4a+ τ a+ τ 4a+ τ 0 a 0 a 0 a f () is made of secions wih f 0, f, hen f ( 0. a g( + ) is made of g 0, g a + a), g, hen g 0. The lef-mos non-zero conguraion is valid for 0 4a + a, so ha For 4a 3a: R f g () For 3a a: R f g () For a a: R f g () For a 0: R f g () f ()g( + )d f ()g( +)d f ()g( +)d 3a+ 0 3a+ a+ f ()g( + )d 4a+ 0 ( a a ( a a a a+ a d (4a + ) 4a ( + ) ) 4a+ d + a 3a+ ( + ) ) a d + a 3a+ ( a a ( + a ) ) d d a d a Working ou he inegrals and nding he maximum is lef as a DIY exercise.

12 5/ Figure 5.5: 5.3. Applicaion I is obvious enough ha cross-correlaion is useful for deecing occurences of a \model" signal f () in anoher signal g(). This is a D example where he model signal f (x; y) is he back view of a fooballer, and he es signals g(x; y) are images from a mach. The cross correlaion is shown in he middle. 5.4 Cross-Energy Specral Densiy The Wiener-Khinchin Theorem was acually derived for he cross-correlaion. I said ha The Wiener-Khinchin Theorem shows ha, for a nie energy signal, he FT of he Cross-Correlaion is he Cross-Energy Specral Densiy FT [R f g ()] F (!)G(!) E f g (!) 5.5 Finie Power Signals Le us use f () sin! 0 o moivae discussion abou nie power signals. All periodic signals are nie power, innie energy, signals. One canno evaluae j sin! 0j d. However, by skeching he curve and using he noion of self-similariy, one would wish ha he auo-correlaion is posiive, bu decreasing, for small bu increasing ; hen negaive as he he curves are in ani-phase and dissimilar in an \organized" way, hen reurn o being similar. The auocorrelaion should have he same period as is paren funcion, and large when 0 so R f f proporional o cos(! 0 ) would seem righ. We dene he auocorrelaion as an average power. Noe ha for a periodic

13 5/3 τ increasing Figure 5.6: funcion he limi over all ime is he same as he value over a period T 0 R f f () lim T!!! 0 T (T 0 )!0 T T T0 sin(! 0 ) sin(! 0 ( + ))d T 0 sin(! 0 ) sin(! 0 ( + ))d! 0 sin(! 0 ) sin(! 0 ( + ))d sin(p) sin(p +! 0 ))dp [ sin (p) cos(! 0 ) + sin(p) cos(p) sin(! 0 ) ] dp cos(! 0) For a nie energy signal, he Fourier Transform of he auocorrelaion was he energy specral densiy. Wha is he analogous resul now? In his example, FT [R f f ] [(! +! 0) + (!! 0 )] This is acually he power specral densiy of sin! 0, denoed S f f (!). The - funcions are obvious enough, bu o check he coecien le us inegrae over all frequency f : S f f (!)df [(! +! 0) + (!! 0 )] df [(! +! 0) + (!! 0 )] d! 4 [ + ] :

14 5/4 This does indeed reurn he average power in a sine wave. We can use Fourier Series o conclude ha his resuls mus also hold for any periodic funcion. I is also applicable o any innie energy \non square-inegrable" funcion. We will jusify his a lile more in Lecure 6. To nish o, we need only sae he analogies o he nie energy formulae, replacing Energy Specral Densiy wih Power Specral Densiy, and replacing Toal Energy wih Average Power. The auocorrelaion of a nie power signal is dened as R f f () lim T! T T T f ()f ( + )d : The auocorrelaion funcion and Power Specral Densiy are a Fourier Transform Pair R f f (), S f f (!) The average power is P Ave R f f (0) The power specrum ransfers across a sysem as S gg (!) jh(!)j S f f (!) This resul is proved in he nex lecure. 5.6 Cross-correlaion and power signals Two power signals can be cross-correlaed, using a similar deniion: R f g () lim T! T R f g (), S f g (!) T T f ()g( + )d 5.7 Inpu and Oupu from a sysem One very las hough. If one applies an nie power signal o a sysem, i canno be convered ino a nie energy signal or vice versa. To really nail i would require us o undersand Wiener-Khinchin in oo much deph.