arped, Chirp Z-Trasform: Radar Sigal Processig by Garimella Ramamurthy Report o: IIIT/TR// Cetre for Commuicatios Iteratioal Istitute of Iformatio Techology Hyderabad - 5 3, IDIA Jauary
ARPED, CHIRP Z TRASFORM : RADAR SIGAL PROCESSIG Garimella Rama Murthy, Associate Professor, Iteratioal Istitute of Iformatio Techology, Gachibowli, Hyderabad 3, AP, Idia ABSTRACT It is well realied that arped Discrete Fourier Trasform has bee utilied i may sigal processig applicatios. Also, Chirp Z Trasform algorithm eables samplig the trasform of a fiite legth sequece o a spiral cotour. Combiig the advatages of these trasforms, the author iovated the cocept of arped, Chirp Z trasform. Applicatio to radar sigal processig is briefly discussed. Itroductio: Fourier aalysis has bee successfully utilied i may braches of egieerig ad sciece. Particularly, i the research area of digital sigal processig, Fourier aalysis of discrete time sigals was etesively studied ad may iterestig results / algorithms were developed. Specifically Z trasform cocept is proposed as a geeraliatio of discrete time Fourier trasform. It is realied that Fast Fourier Trasform algorithm ca be utilied for efficiet computatio of samples of the trasform of a fiite legth sequece tae at equally spaced poits aroud the uit circle [OpS]. e may be iterested i samplig the trasform o a cotour other tha the uit circle or we may ot require samples of the trasform over the etire uit circle. ith this motivatio, researchers proposed the Chirp Z Tasform CZT algorithm [OpS]. Also, researchers proposed the cocept of arped Discrete Fourier Trasform DFT which eables samplig the trasform at o uiformly spaced poits o the uit circle [Mitra]. After carefully uderstadig the logical basis of CZT ad DFT, the author proposed the cocept of arped Chirp Z trasform CZT i this paper. This trasform eables
samplig the trasform at o uiformly spaced poits o may iterestig cotours i the plae. Based o well ow ideas, this ovel trasform is proposed for applicatio i radar sigal processig. This research paper is orgaied as follows. I Sectio, the cocept of arped Discrete Fourier Trasform is discussed. I Sectio 3, the Chirp Z Trasform is briefly summaried. I Sectio 4, the ovel cocept of arped Chirp Z trasform is proposed. Applicatio of CZT to radar sigal processig is briefly discussed i Sectio 5. The research paper cocludes i Sectio 6.. arped, Discrete Fourier Trasform: Let us first review the Discrete Fourier Trasform DFT. The followig discussio is etracted from [Mitra]. It is well ow that the poit DFT X [ ],, of a sequece of legth, [ ],, is defied as X[ ] X j π,... where X is the Z trasform of the sequece []. It is give by e X [ ].. The descriptio of the iovative idea of warped discrete Fourier trasform requires the followig cocept of all pass fuctio.. All Pass Fuctios: ω The frequecy respose A e j of a all pass filter A ehibits uit magitude respose at all frequecies i.e. ω A e j for all ω...3 The trasfer fuctio A of such a filter has poles ad eros occurrig i cojugate reciprocal pairs: M * jθ λ A e..4 λ To esure stability, it is assumed that all the poles of A lie iside the uit circle i.e. λ < for all.5
If A is a real coefficiet all pass fuctio, the θ or θ π, ad ay comple pole at λ must be * accompaied by a comple cojugate pole at λ. Thus, i this case A ca be epressed as A D D D M..6 ow we relate the above discussio to that of Discrete Fourier Trasform. The result will be a iterestig trasform called the arped Discrete Fourier Trasform DFT: arped Discrete Fourier Trasform DFT: The poit DFT X [] of a legth [ is give by the equally spaced frequecy sequece ] samples of a modified Z trasform X by applyig the trasformatio X obtaied from C.7 where C is a M th order real coefficiet all pass fuctio as discussed above. It should be oted that the all pass fuctio warps the frequecy scale. Thus, uiformly spaced samples o the uit circle i the plae are mapped oto ouiformly spaced poits o the uit circle i the plae. e readily have that Hece we have X [ ] C P X [ ] Q j π e j π e P Q.8 P[ ],.9 Q[ ] where P [ ] ad Q[ ] are poit Discrete Fourier Trasforms. Also Q [] does ot deped o [] ad ca be pre computed. Applicatios: arped Discrete Fourier Trasform DFT was successfully utilied i may sigal processig applicatios. Some of them iclude i Spectral Aalysis: Usig a short legth DFT, higher frequecy resolutio is achieved. ii Tuable FIR filter desig
iii ouiform perfect recostructio filter ba desig. 3. Chirp Z Trasform : The chirp trasform algorithm is directed toward computatio of samples of the trasform o a spiral cotour equally spaced i agle over some portio of the spiral. The followig discussio is tae from [OpS]. To be precise, let desigate a poit sequece ad X desigate its trasform. Usig the chirp trasform algorithm, X ca be computed at the poits give by A,,, M where.3. jφ e ad.3. jθ A A e 3.3 I the above equatios, ad A are positive real umbers. Cosequetly, the cotour alog which samples are obtaied is a spiral i the plae. The parameter cotrols the rate at which the cotour spirals: if is greater tha uity, the cotour spirals toward the origi as icreases, ad if is less tha uity, the cotour spirals outward as icreases. The parameters A ad θ are the locatio i radius ad agle, respectively, of the first sample, i.e. for. The remaiig samples are located alog the spiral cotour with a agular spacig of φ. Cosequetly, if, the spiral is, ifact, a circular arc ad also if A, this circular arc is part of the uit circle. 4. arped, Chirp Z Trasform: The mai cotributio of this research paper is to propose the iovative idea of combiig the trasforms proposed i sectios ad 3. e first cosider oe possible way of combiig warpig the frequecy scale ad chirpig. The other possibility is briefly discussed at the ed of this sectio. Thus, we are aturally led to the followig sequece of ideas. For the purpose of clarity, we utilie the same otatio as that proposed i Sectio. Specifically, we have that
] [ C X Q P.4. ow usig the idea of sectio 3, the warped chirp trasform is computed i the followig maer: A A A D P X Y.4. Thus, i this case, the frequecy scale is first warped ad the usig the idea of chirp trasform algorithm, X is computed at fiitely may poits o a spiral cotour. The other approach is to evaluate the trasform o a spiral cotour first ad the warp the frequecy scale. 5. Applicatio to Radar Sigal Processig: To uderstad the utility of warped chirp trasform, we eed to tae a closer loo at the chirp trasform algorithm. The followig iterpretatio is well documeted i [OpS]. ith the values of give by a equatio of the form i 3., we readily have that X.,,,...,, M A 5. Usig the followig idetity ] [ +,..5. Equatio 5. ca be rewritte as. A A X 5.3 ow lettig A g,. 5.4 e readily have that
X g,,,,..., M... 5.5 I the above epressio, we recogie that the summatio term as correspodig to the covolutio of the sequece g with the sequece. Thus the computatio of equatio 5.5 is as depicted i the followig Figure, where h. Figure : Spectrum Aalysis i Radar usig CZT he A ad are uity, the sequece h ca be thought of as a comple epoetial sequece with liearly icreasig frequecy. I radar systems, such sigals are called chirp sigals. A system similar to the oe i the above figure is commoly used for spectrum aalysis i radar problems. Usig the ovel arped Chirp Z trasform, we propose to develop high resolutio spectral aalysis for applicatio i radar problems. 6. Coclusios: I this research paper, the well uderstood cocepts of arped Discrete Fourier Trasform DFT as well as Chirp Z Trasform CZT are summaried. Combiig the advatages of these trasforms, a ovel trasform called arped
Chirp Z Trasform is iovated. It is epected that this ovel trasform will fid may applicatios i sigal processig. Refereces: [OpS] A.V. Oppeheim ad R..Schafer, Digital Sigal Processig, Pretice Hall of Idia Private Limited, ew Delhi, [Mitra] S.K.Mitra, Digital Allpass Filter : A Versatile Sigal Processig Buildig Bloc, Ivited Tal delivered at the Iteratioal Coferece o Recet Advaces i Commuicatio Egieerig RACE 8, Osmaia Uiversity December 3, 8.