Application of Digital Filters

Transcription

1 Applicatio of Digital Filters Geerally some filterig of a time series take place as a reslt of the iability of the recordig system to respod to high freqecies. I may cases systems are desiged specifically to respod oly to sigals withi a arrow freqecy bad. Recall a earlier discssio o the physical iterpretatio of the spectral eergy desity fctio. I sch cases the imm freqecy f preset i the recorded time series will be govered by the system respose. Filters of this type which limit the freqecy cotet of the time series i the recordig process are called prefilters. Frther filterig of the data after it is recorded may also be desirable. Optimm Filterig: Removal of o-statioaries The statistical aalysis techiqes discssed ths far are based o a assmptio of the "statioarity" of the radom process. There are several types of o-statioarity amely: that havig to do with the "short-term" mea - a time-domai o-statioarity o-statioarity of the ato-correlatio ad o-statioarity of the spectral eergy desity. Time-domai o-statioarity The discretely sampled versio of the series depicted i Figre IV.14 ( t ) wold have two ostatioarities of the first type becase of the tred ad the log period cycle. These are idetifiable o-statioarities becase as a geeral rle of thmb ay oscillatio with fewer tha 5 cycles over the observatio period T shold be cosidered to be as o-statioary. Ths it is desirable to remove both the tred ad oscillatio. Figre IV.14 A arbitrary oceaic time series (t) with a distict tred. 19 Febrary 2008 A2 Filterig Applicatios 8W. S. Brow 1

2 To remove the tred oe might cosider sig a liear regressio model or otherwise called the AVERAGE SLOPE METHOD or LEAST SQUARES METHOD. Cosider a times series of +1 vales ( = ); sampled at a iterval of t so that t = T - the Legth of Series or LOS Lets fit with a polyomial series û of degree K expressed as û k = û( t) = bk ( t ) K k=0 where e.g. K = 1 û = û( t) = b0 + b1( t) or e.g. K = 2 û t = t where...a liear tred = û( t) = b0 + b1 ( t) + b...a qadradic fit 2 ( t ) = ( t = T) For these polyomial fits the set of coefficiets b k are chose so that the followig expressio is miimized Q(b) = ( - û ) 2 = [ - K k=0 k bk ( t ) ] The coefficiets b k ca be fod by settig the K+1 partial derivatives Q(b) = 0 bk where k is the idex of the coefficiet of iterest eqal to zero. The geeral coditio for a Aproper@fit - the followig K + 1 eqatios 2. K k=0 b k ( t ) k+ l = where R = K. For K = 0 l ( t ) 0 b0 ( t ) = ( t ) where b o is the series sample mea 0 19 Febrary 2008 A2 Filterig Applicatios 8W. S. Brow 2

3 1 b = 0 For K = 1. l 1+ l l b0 ( t ) + b1 ( t ) = ( t ) ( l = 01) which leads to 2(2 +1) - 6 b0 = (-1) Ths ( ) b1 = t ( -1) ( +1) ca be calclated. (The OASP program REGRESS does this) With the first order fit (i.e. the tred) removed becomes accordig to = û( t) = b0 + b ( t) û 1 ad looks like Figre IV.15. = - û Figre IV.15. A time series from which the first order fit (liear tred pls mea) was removed. 19 Febrary 2008 A2 Filterig Applicatios 8W. S. Brow 3

4 Still there is a very low freqecy compoet i the series - like that i Figre IV.16 - that eeds to be removed. Oe approach is to solve for the free parameters (A f ad δ) of the best fit regressio model û^ (t) = A si (ft + δ ). Figre IV.16 A time series that has o-statioary statistics de to the cycle. Sbtractig the fitted sisoid " = - û ^ û ^ from reslts (see Figres IV.17) "(t) Figre IV.17 A statioary time series from which the o-statioarity has bee removed. 19 Febrary 2008 A2 Filterig Applicatios 8W. S. Brow 4

5 The resltig series "(t) appears to be "reasoably statioary". This process for removig ostatioarities is called optimm filterig. This type of filter (i.e. regressio) is called a "oe shot" filter becase it is applied to the etire time series at oe time. Ato-Correlatio o-statioarity Sometimes o-statioarity of the ato-covariace fctio might look like Figre IV.18 Figre IV.18 A sample cross covariace fctio of a o-statioary process. Geerally there is o way to correct for this type of o-statioarity so oe simply igores it. I other cases methods do exist for describig time varyig spectra of data with o-statioarities which are ot removable. (As we will ot be discssig them here see Bedat & Piersol). Sppose we have measred a statioary series t i which the eergy cotet was restricted to freqecies f # f ad the series was sampled at t 1 / 2f (i.e.1/(2 t) = f f ). The we ca compte the sample mea sample variace ad the sample ato-covariace as discssed i Chapter III. The ext estimator of iterest is spectral desity fctio. I priciple the ato-spectral desity fctio ca be compted by Forier trasformig the atocovariace fctio. However becase the fiite legth cotios ato-covariace fctio R ( τ ) (Eq. III.37) ad the discrete ato-covariace fctio ( τ ) (Eq. III.38) are ot defied 19 Febrary 2008 A2 Filterig Applicatios 8W. S. Brow 5 Rˆ

6 for τ > τ they mst be modified before their Forier Trasforms ca be calclated. The followig otlies some approaches. Dealig with Fiite Legth Series Effects Step 1. Costrct a Lag Widow The objective of this step is to costrct a lag widow h(τ) that is defied for all τ from -4 to 4 h(0) =1 ad h( τ ) = 0 for τ > τ (althogh mch of it is zero). Specifically a sitable eve fctio h(τ) is defied sch that (ote that for τ < τ h(τ) ca take o differet forms) Step 2. Form a Ifiite Cotios Ato-Covariace fctio "( τ ) by R "( ) = h( τ ). R τ ( τ ) R Step 3. Trasform "( τ ) to Prodce R S "(f) = H(f) *S i which H(f) is called a spectral widow. Ufortately it too is ot defied as f goes to ifiity. However we arge that by (f) (f) i (IV.29) is idetermiate becase S Step 4. Esemble Averagig which for ergodic processes is Eq. (IV.28) will become jst the weighted versio of the tre atocovariace fctio R (τ) accordig to ave[ ] = lim 1 T T/ 2 -T/ 2 [ ] dt Ths it follows that ave [S ave[r "(f)] = H(f)* S "( τ )] = h( τ ) R (f) tre atospectrm ( τ ) 19 Febrary 2008 A2 Filterig Applicatios 8W. S. Brow 6

7 or by ivokig the defiitio of covoltio is ave[s "(f)] = - S ( α) H(f-α) dα This form shows how H(f) has the effect of smearig eergy across freqecy bads. Ths the average calclated spectrm is eqal to the smoothed versio of the tre spectrm. Remember the previos discssio abot how g(t) [or h(t)] ca be covolved with a time series (t) i the time domai to prodce a smoothed (t) there. Here H(f) smooths the tre ato-spectrm throgh covoltio i the freqecy domai. Take-Home Message: With real fiite legth series we are limited i or ability to resolve freqecy cotet i the series. 19 Febrary 2008 A2 Filterig Applicatios 8W. S. Brow 7

8 Lets explore this poit a bit frther i terms of a lag widow that is defied like or rig mea filter (or boxcar) h( τ ) = 0 =1 for τ > τ for τ τ whose trasform is H(f) = 2τ si(2π f τ 2π f τ ) = 2τ sic(f τ ) The plot of H(f) (i.e. Eq IV.34) i Figre IV.19 shows how H(f) varies withi the fdametal freqecy bad of the spectral estimate betwee -1/2 < (f/ f = f τ ) <1/2 as well as otside of it at f τ > 1/2. ote the "side lobe" strctre icldig regios of positive ad egative vales. H(f) τ f τ (or f/ f) Figre IV.19 ormalized freqecy domai form (i.e. Forier trasform) of the "boxcar"or rectaglar lag widow f = 1/τ 19 Febrary 2008 A2 Filterig Applicatios 8W. S. Brow 8

9 Ths raw estimates of S "(f) are always smoothed versios of the tre spectral desities [i.e. covoltios of the tre spectrm S (f) ad the spectral widow H(f)]. This fact meas that there are two problems amely (1) Eergy Leakage There is leakage of the Atre eergy@ from the basic spectral bad to adjacet spectral bads throgh the effects of the Aside lobes@. (2) Itrodctio of egative Eergy. Oe way to sppress leakage is to modify the box car h(τ) i the time lag domai. However this actio ievitably broades the mai lobe of the lag widow trasform which i tr redces the freqecy resoltio of the process. Ths there are tradeoffs betwee freqecy resoltio ad leakage effects as demostrated i Figre IV.20 for the followig set of lag widows optios for which τ = M τ = M (with τ = 1); Bartlett Widow (triaglar) τ WB =1- for τ < τ (= M) τ = 0 τ > τ (= M) πτ WT ( τ ) =1/ 2 (1+ cos ) for τ < τ (= M) τ = 0 τ > τ (= M) Tkey Widow (cosie) SPECTRAL WIDOW LAG WIDOW 19 Febrary 2008 A2 Filterig Applicatios 8W. S. Brow 9

10 0 1/M 2/M 3/M... f 0 0.5M M Freqecy f ormalized Time (τ/τ ) Figre IV.20 (left) Forier trasforms of the rectaglar ( ) Bartlett ( ) Tkey ( _) ad Parze (----) lag widows (right) Time-domai distribtio of the lag widows with τ = M τ. ote that a decrease i freqecy resoltio is the cost for redcig side lobe eergy spreadig. 1. Power-Spectral Estimates from Ato-Covariace Fctio Estimates: Cosie Trasform For cotios series the tre spectrm is S (f) = - R ( τ ) cos 2π f τ dτ = 2 0 R ( τ ) cos 2π f τ dτ Ths for the fiite ato-covariace the spectrm is τ S "(f) = R ( τ ) -τ which ivolves a implicit lag widow ad problems cos 2π f τ dτ where τ = M τ Alterately. S "(f) = - h( τ ) R ( τ ) cos 2π f τ dτ where a lag widow is ivoked explicitly to deal with the fiite legth ato-covariace fctio. For examples see Figre IV Febrary 2008 A2 Filterig Applicatios 8W. S. Brow 10

11 Optioal Examples: DATA or LAG WIDOWS Optios Available I Specter Ipt Sigal Amplitde Spectrm Figre IV.21 A schematic of the (right) freqecy-domai Forier trasform of varios time-domai widow shapes. 19 Febrary 2008 A2 Filterig Applicatios 8W. S. Brow 11