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1 D-A APPLICATION OF LATTICE FILTERING TECHNIQUES TO CM ITI MEASURED TINE DOMAIN DATA(U) LAWRENCE LIVERMORE EIIhhEEEIIIhIIhEIIIE NATIONAL LAB CA S GILES ET AL JAN 87 N881t4-83-D-9689 UNCLASSIFIED F/G 9/1 i L Eh/hhE h hh he I

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3 YrW MAPPLICATION OF LATTICE FILTERING TECHNIOUES 00 TO CW MEASURED TIME DOMAIN DATA S. Giles H. G. Hudson S. R. Parker R. J. King January 1987 DTIC SELECTE SEP 02 j 87 1 ~E tm NSTATE-V ' A Ap1,qoved kr 1publi' : k1 Dbbdtrij~on Unjjroile This work was supported by the Office of Naval Technology/American Society of Engineering Education. The work vas performed at the Lawrence Livermore National Labs '.

4 / / I INTRODUCTION This paper concerns the application of lattice filtering, 3] " techniques to discrete frequency domain data. Time domain models for two different antennas are found, the monopole and cavity-back spiral. Lattice models and a layered medium model 45] are obtained. The following is a brief list of important notation used in the work, basically that of Parker [1]. j N Yi emn emn T s Y1 Number time domain sampled points Discrete time domain signal Forward prediction error Backward prediction error Time domain sampling period Discrete Fourier transform of Yi R i k i m Autocorrelation coefficient of lag i T s Reflection (partial correlation) coefficient O 0 Fundamental angular frequency Order of difference equation for yi s bim WR WL Ith coefficient in difference equation for yi s Right going planar wave Left going planar wave u I Wave propagation speed in boundary I 1 4 Tit Characteristic impedance in boundary I 0 0 z Single sampling period delay -1-2 ~A'-6,J:'b'N'.y Codes i Av&Ixid I cr... LTED T'.i t Ca

5 PRODUCTION OF AUTOCORRELATION COEFFICIE,'4TS We assume all Y 1 (Lw 0 ) 1-O,...,m are known. complex quantities are obtained is immaterial. The method by which these However, for this work they were obtained through experiments using the EMP Engineering Research Omnidirectional Radiator (EMPEROR), the heart of which is an HP8510 Network Analyzer. The calculation of R i i-o,...,m is straight forward. R i cos Wi T s i-o,...,m. (1) W 0 is related to the sampling period in equation (1) by W " [2m + 22pTS) where [2m P is the nearest power 2 operator on 2m+1. The number (N) of time samples is taken to be 12m+112 P. R i is useful in determining an estimate of Yi i-o,...,n-1. PRODUCTION OF LATTICE STRUCTURES For a linear causal stationary real autoregressive (AR) process of order m, the forward prediction error is given by m emn a Yn -1 b m Yn- n = 0,..., N-i. (3) Lai I -SNL= -mis LaZ h

6 The backward prediction error (sometimes called postdiction error) is the following expression: - m emn = Yn-m - i-i bmi Yn-m+1 n a 0,..., N-i. (4) The bmis in equations (3) and (4) are the difference equation coefficient for an AR process of order m. It is assumed that emn and emn have the same statistical properties, that eon - eon - Yn, and that m S 250. Using the Levinson-Durbin algorithm and equations (3) and (4) we can write the following important relations: em+ln " emn -km+l em,n-1 (5) em+in - -km+i emn + em n-i (6) where the quantity km+ 1 (reflection coefficient) is given by either the expectation E(emn im,n-i) ratio or equivalently km+ 1 " b m+1,m+ I (8) Another way of calculating the k-values is to use the Shur algorithm [6], since by now the Ris are known. This algorithm involves the following steps: -3-

7 1. Define a generator matrix I T [R 0 R 1.. R m1 0 L R 1... RmJ (9) 2. Shift the first column of the generator matrix and define T [0 R 0 R 1 1 L 0 RI"'" Rm I 1 (10) 3. Calculate k1 k1 m 0 m) (i) 4. Define a matrix 8(k 1 ) 1 -k e [ 1 (12) 5. Produce new generator matrix T T G - e(kj) G (13) -4-

8 6. Go to step 2 and repeat until all ks are found. We have used the above algorithm to calculate k-values. Equations (5) and (6) can be represented by the basic lattice diagram (Fig. 1). We shall call this Oa -m 1 Figure 1. Basic Lattice Structure (k ) structure the transmission lattice as will become obviously justified from the following lattice, which we shall call the scattering lattice (Fig. 2). From the scattering lattice it is clear that the backward error and forward error can be viewed as right going A MU am * l.n Figure 2. Scattering Lattice Structure ( km+) -5- I II l~ libili llr 'r? N" - ' "

9 and left going signals, respectively; and the transmission lattice relates quantities at one port with those at the other port. If we drop the second subscript on the error and represent single sampling period delays by z - 1, the transmission lattice looks like this (Fig. 3). rki + Figure 3. Transmission Lattice Z -I Delay By cascading transmission lattice we can produce a "whitening" filter as shown in Fig. 4. That is, yoe e, " -J e1+ Figure 4. Whitening Filter -6- I I " ' r r-,j r" cm

10 from Fig. 4 we have a relationship between the outputs em+1 and em+ 1 and the single input yi of the following form: em+1 eo T em+i T[ (14) where the matrix T is the overall transmission matrix: T 1 -k 1 l Z-10 1 ri**km* I i Z_ k1 Z10 [-i Lkm+ 1j L0 1i Lkm 1] 1 1 (15) ) Now, one can examine the T 22 entry of T for the characteristic equation or use Levinson-Durbin to calculate the AR parameters. Before leaving the transmission matrix we consider the case when error signals are delayed according to a factor z - 0, where B is a constant. It follows from equations (5) and (6) that the lattice structure will be as shown in Fig

11 z.lf +I + Z'6ek Figure 5. Transmission Lattice Delay Z - 0 in Each Branch Cascading scattering lattices without the z - 0 factor, we generate the following filter (Fig. 6), which was used to predict yn n-o,1,...,n-1. By driving the filter at the em+1 terminal with a shifted pulse, we Z " kk k. + Figure 6. Cascaded Scattering Lattices obtain a response which estimates the original discrete time signal, yi. If we include the z - 0 factors in each basic lattice, we get the diagram shown in Fig. 6 but with z - 0 delays with the e and e 0 signals treated appropriately. The z - 0 factor is important in considerations of EM waves in layered media. -8-

12 WAVES IN A LAYERED MEDIUM We assume from an EM field theory point of view that the source free medium of interest is linear and stationary, but that it is divided into sections which are anisotropic and homogeneous bounded by short sections of materials which are isotropic and nonhomogeneous. We assume that right going and left going planar waves exist in the different media which make up the overall medium and variations in y and z directions are negligible. Figure 7 depicts this situation [5]. 1 2 m+i W 3 o WRI W.m WN.. I..,g Ww WL WU WU4 WL*. - x dx dx d"x Figure 7. Layered Medium. The sections numbered 1 to m+1 and shaded are the isotropic -- but not necessarily homogeneous -- boundaries. Between the boundaries are sections of anisotropic homogeneous materials. To the right of boundary m+1 and left of 1 are infinite isotropic homogeneous regions. Considering the boundaries first, from transmission line theory, we can write after a few manipulations the relationship between right and left going waves as the following [4]: -9-

13 WR t -a (x) WR _R(x) 1 W (16) LLJ L! ult LLJ For the Ith section a,(x) in operator equation (16) is given by at CX ) - 1 ( C17) where n, is a function of position. We can solve equation (17) for n,, i.e., ) -(xa exp [2 fx a(x) da) (18) where A is a constant. Now, it can be shown using forward discretization with step sizes Ax and At and for small 1 Axf factors that the scattering lattice structure for equation (16) is the following: 1 - k + Wnl 10. V"w,. WL, WLI..I Figure 8. Transmission line lattice for boundary I (I - 1,...,m+1) -10- mmmmilliii

14 k in the above Fig. 8 is 6x, where -a is the average value of a in the boundary 1. For a very small magnitude value k is approximately the reflection coefficient of EM wave theory. From Fig. 8 we recognize that WR m+1 is analogous to em+ 1 and that W, m+i is like em+1. The only problem m+ is the representation of the delay in the basic scattering lattice of Fig. 2. This problem can be rectified by considering waves in regions between boundaries. We simply use the propagation times to establish the delays. Since the medium is homogenous between boundaries, corresponding reflection coefficients are zeros and the scattering lattices are sections of straight lines. Then it follows that the z - $ factor (8 * 0) of Fig. 5 represents the time it takes a left going wave to be emitted by one boundary and propagate to the next boundary. Since from the basic lattice structure analogous right going "waves" must take longer than left going ones (the z - factor), the material between boundaries must be anisotropic with propagation speed in one direction different form that in the opposite direction. So, not only do we have the lattice models of Figs. 4 and 6, we also have a layered medium model which will have the structure of Fig. 9. kmj k.., Figure 9. Basic Layered Medium Lattice for Errors -11-

15 RESULTS The above modeling theory was applied to data obtained for experimental use of EMPEROR where a 22.5 cm monopole was used as a test object. The response of the monopole for 250 k-values and for 15 non-zero k-values modeling was obtained. The reduced order (15 non-zero k-values) cases result from setting statistically insignificant k-values equal to zeros, and correspond to elimination of certain exponentially damped sinusoidal components. Discrete Fourier transforms (DFT) and time estimate signals are shown in Figs The only problem in producing these results was selecting the proper shift for the input pulse. This was done by trying different shifts until the DFT of the estimate time signal matched that of the measured frequency domain data. In Figs is shown how well the scheme performs for a more complicated antenna test object, the cavity-backed spiral antenna. These values were reproduced for 250 k-values, only. As in the monopole case the lattice model does yield agreeable results for frequency domain. For time domain, the results for CB spiral are not as good as those for the monopole. CONCLUSIONS It has been shown that autocorrelation coefficients and reflection coefficients are key quantities in structuring lattice models. It has been * shown that frequency domain data can be used to 6uild time domain lattice models. It has been demonstrated that the method works for monopole and cavity-backed spiral antenna data. -12-

16 RE FERENCES 1. Parker, S. R., "Lattice Modeling of Stochastic Signals: Some Fundamental Concepts." LLNL/UCRL 15802, SIC , Makhoul, John, "Linear Prediction: A Tutorial Review," Proceedings of IEEE, Vol. 63, April 1975, pp Kay, S. M. and S. L. Marble, "Spectrum Analysis--A Modern Perspective," Proceedings of IEEE, Vol. 69, November 1981, pp Bruckstein, A. M., B. C. Levy and T. Kailath, "Differential Methods in Inverse Scattering," SIAM Journal of Applied Mathematics, Vol. 45, April 1985, pp Orfanidis, S. J., Optimum Signal Processing: An Introduction. New York: Macmillan Publishing Company, Dewilde, P., A. C. Vieira and T. Kailath, "On a Generalized Szego - Levinson Realization Algorithm for Optimal Linear Predictors Based on a Network Synthesis Approach," IEEE Transactions on Circuits and Systems, Vol. CAS-25, September 1978, pp

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