AD-AL14 996 ROME AIR DEVELOPMENT.CENTER GRIFFISS AFB NY F/B 20/AT HE FEE ,A NEW TUNABLE HIGH RESOLUTION SPECTRUM ESTIMATOR. (U)FEB B2 H E WEBB, 0 C YOULA. 0 M GUPTA

UNCLASSIFIED RADC-TR-81-9 NL

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APPROVED:

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Coummad and Control Division

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REPORT DOCUMENTATION PAGE SEFO!EFR1REPORT 4UMU ft 12._GOVT ACCESSION4 O 3. RECIPIECATLO NMMER

RADC-TR-81-397 t- /' ____________

4. TITLE (and Subtitle) S. TYPE OF REPORT & PERIOD COVEREDIn-House Report

THE FEE, A NEW TUNABLE HIGH RESOLUTION 1978 - 1981SPECTRUM ESTIMATOR 6. PERFORMING 0*4G. REPORT NMER

N/A7. ATHORs) . CONTRACT OR GRANT NUMMER(s)

Haywood E. Webb*N/A

9. PERFORMING ORGANIZATION NAME AND ADDRESS IC. PROGRAM ELEMENT. PROJECT. TASKAREA A WORK UNIT NUMBERS

Rome Air Development Center (COTD)Griffiss AFB NY 13441 J.O. 5581PROJ

11. CONTROLLING OFFICE NAME ANO ADDRESS 12. REPORT DATE

Rome Air Development Center (COTD) February 1982Griffiss AFB NY 13441 I11. NUMBER OF PAGES

___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ 110Ilk. MONITORING AGENCY NAME & AOORESS(iI different ko Control~ing Office) 15. SECURITY CLASS. (of this report)

Same UNCLASSIFIEDIS& OECLASSIVICATION/00OOUNGRAOING

____________________________________________ N/A SCEULE

16. DISTRIBUTION STATEMENT (of tis Report)

Approved for public release; distribution unlimited.

17. DISTRIBUTION STATEMENT (of the abstract entered in Block 20. It different from Repo")

Same

Ill. SUPPLEMENTARY NOTES

*Dante C. Youla - Polytechnic Institute of N.Y. Farmingdale, NY

*OM Gupta - Digicomp Corporation, Ithaca, NY

It. KEY WORDS (Continue on ewei .51 side it necessay end Identify by block nusbow)

Spectrum Estimation

Circuit TheoryPositive Real Functions

20. ABSTRACT (Continue on everse1. side if noesemy onsd Identify by black nube)

This report consists of three parts:

1l. A circuit theory model of the spectral estimation problem isgiven. The flat echo estimator is defined. Some of its properties areexplored.

1 2. Part two addresses a software implementation in an interactive Imode. Examples are given.

DD I IJAN 73 1473 SOTIono OF 1 NOV6 Gsis OBSOLETE UNCLASSIFIED

SECURITY CLASSIFICATION OF TIS PAGE (owY. Do's . s~ud

UNCLASSIFIED

SECURITY CLASSIFICATION OF THIS PAGE(U91 Data E-e4tre)

3. Part three generalizes the scaler problem of Part 1 to themulti-dimensional vector cases.

UNCMASSIFI ED

SECURITY CL ASSIICATION Or 0 'w* PAG6r1'la DudW Enter")

.. ..-

PREFACE

In problems of the sort treated here, one is given a projection (segment)

of a non-negative definite function. The problem is to extend (continue) the

function over all its domain in such a way that it is even and non-negative

definite. Subsequently, the Fourier transform of an even and non-negative

definite function is even and non-negative, we.have generated a spectrum. In

the problem of spectral estimation, the given segment of data is from a

covariance function, so that the Fourier transform of it together with its

extension, is a power spectrum. All solutions considered here have the

property of consistency, that is, the inverse Fourier transform of the

estimated spectrum reproduces itself over the given data interval.

This report represents work accomplished under the technical direction

of the undersigned with the support of the RADC Laboratory Director's fund.

HAYWOOD E. WEBB, Jr.

Acoession For

NTIS GRA&IDTIC TABUnannounced ElJustificatio Ot.C

Distribution/

Availability CodesAvail and/or

Dist Special

- .4.I

r

PART 1

BACKGROUND THEORY AND DERIVATION

by

DANTE C. YOULAPolytechnic Institute of N.Y. Farmingdale N.Y.

PART II

A DIGITAL IMPLEMENTATION

by

OM GuptaDigicomp Corporation Ithaca N.Y.

PART III

MULTI CHANNEL CASE

by

DANTE C. YOULA

Polytechnic Institute of N.Y. Farmingdale N.Y.

PART IV

SOME OBSERVATIONS AND

SUGGESTIONS FOR NEW WORK

by

HAYWOOD E. WEBB Jr.

iii V,

PART I -BACKGROUND THEORYAND DERIVATION4

by

D.C. YOULA

TABLE OF CONTENTS

Pagje

ABSTRACT

INTRODUCTION AND PRELIMINARY RESULTS

II. "TUNED" SPECTRAL ESTIMATORS 16

III. THE DESIGN OF TUNED FEE'S 20

APPENDIX A - NETWORK FOUNDATIONS 22

APPENDIX B - PHASE INTERPOLATION WITH A 26REGULAR ALL-PASS

REFERENCES 35

FIGURE 1 36

!~I,

I

I--I

ABSTRACT

The FEE, or Flat-Echo Estimator, represents an entirely new solu-

tion to the problem of spectrum recovery from a given finite set of error-

free covariance samples. In concept it is based squarely on frequency-

domain network-theoretic ideas and its principal features are the following.

1) Numerical robustness, as defined in this report, can be assigned

in advance by the adjustment of a single parameter.

2) It is possible to "tune" the estimator to produce selective ampli-

fication of a finite number of desired frequencies without impairing either

its interpolatory character or its numerical robustness. Enhanced reso-

lution is to be expected.

3) For the parameter setting which yields maximum robustness, the

FEE coincides with the MEE, the Maximum-Entropy Estimator.

In addition to providing a rather leisurely account of some of the

network ideas behind the FEE, this research report also contains two

appendices, A and B. The first supplies an explicit and compact formula

for the class of all interpolatory spectral densities, while the second is

devoted to the design of minimum-degree regular all-passes with prescribed

phase at a given finite set of frequencies.

k-j. .. . . . .. . .. .... .. . ... . . . ... . . i ... . . . lm .. .

I. INTRODUCTION AND PRELIMINARY RESULTS

Suppose that we have a discrete-time random process x where MtM canttraverse all positive and negative integers. Let us also suppose that the

process is zero-mean and second-order stationary. Then, (

E(xt~t+k) = C(k) , Ik = O-+co , (1)

is the associated covariance function. Clearly, for all integers k,

C(-k) = E(xtxt k = E(xt+kxt) = Z(k) . (2)

As is well known [1],

C(k) fT- -eJke dF(e) (3)2 r

-T.

where F(e) is monotone-nondecreasing, continuous from the left and of

bounded-variation:

iTVarF- f dF(B)=ZTTC(O) < . (4)

-T

Further, the derivative

K(e) = dF(_) > 0 (5)

exists for almost all 8, is L over -1T < @ < 17 and defines the spectral density

of the process. Evidently, if F(e) is absolutely continuous,

C(k)= fejke K(O)d (6)

-T

Let us assume first that F(8) is absolutely continuous and that K(e)

satisfies the Paley-Wiener criterion,

f InK(e)de > -o . (7)

-IM

a complex conjugate of a.

I-I

Then, there exists [ I] a unique function

00B(z) = Zbrz , b0 >0 (8)

r=O

which is analytic and devoid of zeros in z < 1 and is such that(2 )

Z 0b r < (9)

r=O

and43 )

K(e) IB(ee) Iz a.e. (10)

Moreover, x t can be realized as the output of a causal digital filter with a

square-surnmable impulse response( 4 )

h =b" r=O-o,(Ir r

driven by an appropriate white noise source sequence ut:

00x t ht r u (-cc t < 0o0 (2

t r -r=O

E(Ur O, E(UrUm rm < r, m< oc) (13)

00

E(xjk C C(k) + Ek h, r oc < k < 00) * (14)

r=O

42 1B(z) is of class H .

B(eiO) limit B(reie) exists for almost all e.r-* 1-0

(4) 00

Z Ihrlz - T lb < cc

r=O r=0

1-2

We can now pose and solve several problems in spectral estimation.

Problem 1. It is known that F(e) is absolutely continuous and that the

spectral density K(8) = dF(8)/de satisfies the Paley-Wiener condition (7).

Under the assumption that the n+l covariance samples C(O), C(l), ... ,C(n)

are known absolutely and without error, determine, if it exists, a compatible

K(8) which maximizes h in the input-output model (12).

Solution. Let the numbers C(k), k=n+I - oo, constitute any admissible

completion of the given data C(O), C(1), ... ,C(n) and let K(e) be the corres-

ponding spectral density.

Let

TT

dk = f eijkelnK(6)de , (15)-TT

k=O-oo, let a 0 = 1 and let the coefficients ak, k= 1 -oo, be determined from

the equation

00

exp( £dr)= yr akz k (16)r= k=

Then [I

k,

hr 7 r exp(d0 /2) , r = 0-o . (17)

Theorem 1. To construct an admissible completion C(k), k=n+I - co,

which maximizes

h = exp(d0 /Z) (18)

is equivalent to choosing K(O) to maximize

d = ' f InK(e)de (19)-rr

But [9 ] this leads immediately to the MEE, the unique maximum-entropy

estimator, KME(e), Eqs. (73) and (74).

Comment 1. What does it mean to maximize ho and is it also possible

to select spectral density estimators which maximize other meaningful

1-3

functionals defined on the impulse response sequence [hr ]? These questions

deserve to be studied and one possible approach is to combine (17) with the

following explicit formulas for the akIs and dkIS [2]:

a,= dl, a 2 = d + , a= d3+d -- (20)

and in general, (5)

d -1 0 ... 0

2d2 d1 -2 ... 012

ar =r. det ............ ......................... r> 1,

(r-l)d r. (r-Z)dr 2 (r-3)dr 3 .•. .d1 , -(r-1) (21)

rd r (r-1)dr l (r-Z)d ... Zd , d 1

= dr+ d1drl +..., r>2 * (22)

Problem 2. Let F(e) and K(9) be subject to the same constraints as in

problem I and assume once again that the covariance samples C(0), C(1), .

C(n) are known absolutely and without error. Provide, if feasible, a geo-

metric time-domain interpretation of the process whereby every admissible

completion C(k), k=n+l - 00, is generated.

Solution. As is well known [ 2 ], under the given hypotheses, (6)

C(0) C(1) . . . C(k)

Ak = C(-1) C(0) . . . C(k-1) >o, k=0 - o , (23)........... .. .. .. .. ..

C(-k) C(-k+l) . . . C(0)

for any possible admissible extension C(k), k=n+l -oo, where

(5) detA= IAI = determinant of square matrix A.

(6) The Paley-Wiener condition (7) precludes the possibility Ak= 0.

J-4

C(-k) S C(k). (24)

Thus, at the first step, C(n+l) must be chosen from the set of values t which

satisfy the inequality

C(o) C(l) .. C(n)

C(-1) C(o) ... C(n-1) C(n)

a M ............ . . . . ... > 0 . (25)C(-n) C(-n+l) ... C(o) c(l)

C(-n) ... C(-1) C(o)

However, it is not difficult to show [ 2 ] that this set is the interior of a

circle 0 with radiusnA

rn n (Z6)n-A

and center

C(1) C(2) C(n) 0

n=- C(O) C(t) . . . C(n-1) C(n) (27)n- 1 & . . . .. . . .. . * . . . °

C(-n+l) C(-n+2) . . . C(O) C(1)

In fact, if AkI denotes the minor of Ak that is obtained by striking out

its first column and last row, it follows from a fundamental determinantal

result applied to An+l( ) that

IA+() 2 n n n+l (28)

But as is clear from (Z5) and (26),

nAn + I' (__ _ A n -l_ I__ _ _ _ _(2

9 )

(7 )If D is any determinant, and S is the second minor obtained from itby striking out its ith and kth rows and its jth and Ith columns, and if wedenote by Aij the cofactor of the element which stands in the ith row and jthcolumn of D, then C 3 J,

Aij Ait: (-l)i+ j +k+l DS (29a)

Akj Ak

I-5

and (28) yields,

= < r (30)n-I n

since An+i(I) and An. 1 are both positive.

Once having chosen C(n+l) = consistent with (30), C(n+Z) is then

selected from the interior of a circle of radius r+ 1 = A+I/A and center

tn+l given by (27) with n replaced by n+l, etcetera. Evidently, except for

r n and tn which are uniquely determined by the prescribed data C(O), G(1),

C(n), all successive rk and k depend on the particular rule used to pick

C(k+l) from the interior of (k' k=n+1 -,oo.

Theorem 2. Under the conditions prevailing in problem 1, the following

is true.

1) Let C(k), k=n+l -00, denote any admissible completion of the initial

covariance data C(O), C(1), ., C(n) and let K(e) be the corresponding spectral

density estimator. Let K(8) = IB(eJe)1 2 where B(z), Eq. (8), is the associate

Wiener-Hopf factor. Then, the sequence of radii rk, k=n- oo, generated by

this completion is always monotone-nonincreasing and

Ak 2 2limit rk= limit a = h >0 (31)k-koo Ak- o

In addition, for the sequence of centers, k' k=n-oo,

oAnimsup glb 2< n (3 Z)

k-*oo n o- Ain-

2) The maximum-entropy estimator KME( 8 ) is the (unique) spectral

density that is obtained by identifying each C(k+l) in the completion with

the center of circle 0 k

C(k+l) = k =n-*0 . (33)

Under this ME rule,

limit _ k 0 (34)

k-6

1-6

and all circles 0 k possess the same maximum possible radius rn:

An A k

r = 1- -1 k (35)

k = n-oo

Proof (in outline). Given any admissible completion and its corres-

ponding spectral density estimator K(O)= IB(eJ)1 2 where B(z) is the associate

Wiener -Hopf factor, it is well known [4] that

A~k 2limit- = b > 0 (36)k- oo Ak-I o

Furthermore, from (28) with n replaced by k,

r k+=~IzAk -IAk+il, Z\ Akr k +I = k .1 k I I J - k .

r- = _( I )< -= r k , (37)k1 A k a kI A k A k-i

and it is clear that the sequence rK, for k=n-)co, is monotone-nonincreasing.

Thus,

2 Ak '/f

b limit-- <- = r (38)Sk k-I n-i n

with equality iff

A k+ = 0, k=n-oo (39)

Since K(9) is integrable,

limit C(k+l) = 0 (40)k- oo

which implies that all circles (k contain the origin as an interior point for

k sufficiently large. Hence, for large enough k,

I 'kl < rkS r (41)

so that2 A

imit supIk limit rk= b 2 < r n (42)k-oo k- oo n-i

1-7

To establish 2), observe that (39) and (Z9) lead to the conclusion that

b0 assumes its maximum possible value

(bo)M.AX (43)n-I

iff for every k= n -oo,

= C(k+l) - k ' (44)

the center of circle @k. But according to (11) and (18), b 0 = exp(d0 /Z) so

that (bo)MAX is necessarily paired with the estimator K(e) KME(e) which

maximizes the entropy

rd =-f nK(e)dG . (45)

-rr

Lastly, since the ME completion rule identifies k with C(k+l) for

every k>n, it is evident that

limit gk = limit C(k+l) = 0 , (46)k-*oo k-*co

Q.E.D.

Comment 2. Theorem Z suggests that the ME estimator KME(B) achieves

good performance because its completion rule possesses a kind of optimal

robustness against loss of positive-definiteness. Or, stated somewhat

differently, since each additional covariance sample is chosen at the center

of its circle of admissibility, it appears plausible to conjecture, on the

grounds of symmetry, that the ME rule possesses a fair degree of self-

correction in the face of (unsystematic) sample error. Admittedly, these

considerations are heuristic in nature but they seem to be reinforced even

more by the realization that the initial data C(0), C(l), ... , C(n) must actually

be estimated from samples of the random process xt itself.

Problem 3. Under the assumption tha'&t C(k) is real, place "the gerieral

problem of spectral estimation from a prescribed set of error-free covariance

samples C(O), C(I), ... , C(n), in a network-theoretic setting involving positive-

real functions.

1-8

Solution. First, it is easily seen that C(k) real implies that the

function00

Z(z) = C(O) + ZC(k)zk (47)

k= 1

is positive-real (pr) in Izk 1, i.e., Z(z) is real for z real and

Re Z(z) - 4~z)>0 Iz I < . (48)

The proof is quite sinple.

In fact, it follows readily from the reality of C(k) and (3) that(8 )

Z z(z) I e je +z dFIe) , zI z < (49)

Hence,

Re Z(z) = f z 1e- z dF(8)>0, IzI < 1 (50)

since dF(e)>0, Q.E.D.

It can be shown C5 ], that for almost all e, the "interior" radial limit

Z(e e) a limit Z(re ) (51)r-*1-0

exists and that

K(G) = ReZ(e j6 ) . (5Z)

However, even more is true. Let F (6) denote the singular part 9 ) f

F(8) and let the countable set of discontinuities of F(e) occur at e= i with

jumps Pi>O, i= 1-00o. Then, (49) may be rewritten as

f eje+zdFl1+1i_ e 1 +z 1 f eJl+zZ (z) = r / - dF IPi J' + -I f J z K e d

Tr e -z (53)

Zs(z) + ZF(z) + Z(z) ( 54)

(8)The interchange of summation and integration is justified by dominatedconvergence.

(9)Fs(e) is monotone-nondecreasing and its derivative equals zero almosteverywhere [6].

I-9

Clearly,

= limit (1 -ze ) Z(Z) , 1-0(r z-Oe j ei

j e.and the PI's emerge as residues of Z(z) at its poles zi=e (all simple) on IzI =I.

These poles are revealed by the presence of the Foster reactance

z F(Z) PejZ (56)

i=1 e -z

Of course,

Re ZF(e J) 0 (57)

and since F(6) is of bounded-variation,

00

Z i < co (58)i=l

so that the infinite sum in (56) is well-defined.

Thus, given C(k), k= 0-*n, the problem of spectral estimation is equiva-

lent to that of generating all pr functions \1 0'

Z(z)=C(0)+C(1)z+ ... + C(n)zn+O(z n + l (59)

We shall now present an elementary solution based on the synthesis of ideal

* TEM transmission-line cascades [7, 8].

The Interpolatory Cascade. With the given data of real numbers C(k),

* k= 0-n, construct the real symmetric (n+l) x (n+l) matrix

C(O) C(l) . C(n)

c( ) C(0) C(n-)T = ....... ................... . (60)

C(n) C(n-1) . . . C(0)

(10)Phrased differently, find all pr function Z(z) such that d Z(0)/dzk= 0-*n, are prescribed real numbers.

1-10

An admissible completion C(k), k=n+ -oo, exists, if and only if Tn> 0n 0 1)

Two cases must be distinguished when this criterion is satisfied.

Case 1: An = detT n> 0. Then, rank T = n+1 and all interpolatory prnnn-

solutions Z(z) are constructed in the following manner.

Step 1. Let Tk denote the (k+1) x(k+1) Toeplitz matrix obtained by sub-

stituting k for n in (60), let

ak =det Tk , k>0 , (61)

and let Ak denote the minor of Ak obtained by striking out its first column

and last row. Let N denote the 2-port reactance-cascade (Fig. 1) composed

of n+l ideal TEM lines all of the same I-way delay T>0 and positive charac-

teristic impedances,

R. :C(O) (62)

and

k-i AVkRk-I .:Rk 1 , k= 1-n . (63)

Rk - k-i AV:' -(-I) .'1

k-l

Then, under the identification ( 12)

z e (64)

every admissible interpolatory solution Z(z) is obtained as the impedance

seen looking into the input port of N with its output port closed on an airbitrary

pr function W(z). (In Fig. 1, terminals cd are connected to ab.)

(lIk and Ok denote the kxk identity and zero matrices, respectively.For any matrix A, ., A' and A (-A) denote its complex conjugate, transposeand adjoint. If A(=A ) is kxk and hermitean, A>Ok(>0k) means that it ispositive -definite (nonnegative-definite). As usual x- column-vector.

S(12) p= +jW is the standard complex-frequency variable used in network

theory.

I-ii

The quantities

Rk'Rk-i k-l -knk= Rk+Rk i (-) .Ak i , k=i-*n , (65)

are the n junction reflection coefficients. Their physical significance is not

difficult to grasp. Indeed, let the terminated structure depicted in Fig. 1

be excited by an input current source i(t) = 6(t). This impulse immediately

launches a voltage impulse C(0)6(t) of strength a 0 t = C(O) into line 0 which

then travels towards line 1.

After T secs., a t reaches the first junction separating line 0 and line 1

and part of it is reflected and part of it is transmitted.'(13) If alr and ait de-

note the strengths of these reflected and transmitted impulses, respectively,

conventional transmission line theory yields,

(66)

aOt

and

(\ _)= 1-s 1 (671Ot

In the same way, after T more secs., ait reaches the second junction

and is partly reflected and partly transmitted to generate a Zr and a t where

2 r = s (6 8 )

it

and azt .z S z (9a t 2

In general therefore, upon reaching line k, a(k- l)t is divided into an impulse

of strength a kr reflected back into line k-I and one of strength a kt transmitted

into line k. Clearly,

,a 2=akr " 2 ~kt 2Z

a- ) = kk akt ) -s , k=l-n (70)

(13)It is convenient to label an impulse by its strength.

1-12

Moreover, since interaction with the passive load W(z) commences only

after the lapse of (n+i)T secs., the input voltage response v(t) over the time

interval O<t< 2(n+l)T is determined solel, by the (n+l)-line cascade. We can2now relate the radii rk to the squared amplitudes akt.

For according to Eqs. (26), (28) and (65),

A 2 = k k-ZAk r kI-n (71)-i Ak- k-I

Thus, since r 0 = C(O) a0 t, a comparison of (71) with (70) immediately gives

the important physical connection formula,

2aktrk = C(O) , k =0-n (72)

Obviously then, since successive transmitted impulses can never in-

crease in strength, the corresponding sequence of successive radii of the circles

of admissibility must be monotone -nonincreasing.

It follows from this interpretation, that to avoid any decrease in circle

radius for k>n, it is necessary that the last transmitted impulse in line n suffer

no reflection upon reaching the load W(z). But this is only possible if W(z) =

Rn, the characteristic impedance of the last line in the (n+l)-line cascade.

Consequently, KME(e) is the real part of the driving-point impedance of

the n+l -line structure N closed on W(z) =Rn . Explicitly [9, 10, Appendix A],n

KME(e) = iPn(e~6 ) (73)

where

C(O) C(l) . . . C(n)

1 C(i) C(o) . . . C(n-1)P(Z) : . ...................................... (74)nan-IC(n-1) C(n-2) . .. C(l)

n n-lz z,,

1-13

is a degree n strict-Hurwitz polynomial.( 1 4 )

Case 2: A =detTn =0. Now rankT n<n+l andthere exists a p <n such

that A = det T > 0 but A = det T = 0. The interpolatory pr function

Z(z) is unique, is Foster and is of degree p+l = rankTn. Thus, according to

(53) and (54), Zs(z) = Za(Z) a 0 and

P+l jei

Z(z) = ZF(z) 2 e e+Z (75)i=l e -z

The poles z. e i , and the residues pis i= 1-# 4+1, can be found in three steps

(1 5)

I) Find a real nontrivial column-vector solution x= (x o x 1 , . . • P. 1)

of the homogeneous equation

T +x= . (76)

(There exists only one such linearly independent x.)

2) The p+l zeros of the polynomial

x(z)=x 0 + x 1Z+... +x (77)

j6.are all of unit modulus and are precisely the desired poles z = e , i 1-p+l.

je.3) With the e s known from 2), the quantities pI, P2" Pp +l are

determined as the unique solutions of the linear system of the Vandermonde type,P+l -jke.

C(k) = V pie 0-' (77a)

i=l

A much more enlightening physical interpretation of case 2 is available.

Namely, since A >0 implies that T > 0 then

Ak > 0 , k =0 -. (78)

(14)P n(z)0, Iz < 1. Incidentally, observe that P (z) is the determinantof the matrix that results by replacing the last row in n by zn, zn-, . . ., z, 1.

(15) This three-step rule goes back to F. Riesz [2] but its use in the context

of spectral estimation is of recent vintage [1 1 ].

1-14

Hence, from (71),

sk<I , k=l-*P. (79)k

However, since A =0, s +l1; i.e., s I and consequently, in view

of (65), R 0= or 0. In other words, the interpolatory Z(z) is the input

impedance of a cascade of p+l TEM lines of characteristic impedances

Rot R i p. R Pclosed on either an open or short circuit (but not both!). This

explains why the process terminates and why Z(z) is unique. Alternatively,

we can say that the choice of covariance sample C(p+l) has been made from

the boundary of the circle of admissibility ( and as a result the transient

voltage impulse transmitted into line p+1 must be zero. This situation is

realized only because the load that terminates line p is either an open or a

short. (We conclude that the two associated real sample values C(P+1) are

the two intersections of 0 with the real axis.)(1 6 )

In the next section we shall undertake a much deeper study of the

properties of the spectral estimator, especially with regard to its dependence

on the choice of pr load W(z). Nevertheless, for now we can assert that

any selection of covariance sample made from the boundary of its circle of

admissibility immediately forces Z(z) to be a unique Foster function of

finite degree and C(k) to be a finite sum of pure tones (F (e) = K(e) 0):5

UI -jkeC(k) pe (80)

i=(

( 1 6 )Since the data C(k), k=0-pi, are real, the center of 0 lies on thereal axis.

I-is

II. "TUNED N SPECTRAL ESTIMATORS

From this point on it is assumed that the given data C(k), k= O-n, are

real and error-free, that T is positive-definite and that F(B) is absolutely-ncontinuous. Thus, the objective is to estimate the spectral density K(6) =

dF(e)/de and we restrict our attention exclusively to estimators defined by

rational pr functions W(z).

Let

W(z) -Rp_ W(= z)+n (81)

n

denote the reflection coefficient of the load normalized to R, the charac-

teristic impedance of the last line. Since W(z) is pr, p(z) is bounded-real

(br); i.e.,

Ip(z)I <I , IzI<L . (82)

Let

ZP (z)Z MEWZ -_ 1 ~z83)

denote the input impedance Z(z) with W(z) = R . Clearly, this correspondsn

to p(z) = 0 and Pn(z) is given by (74). It can be shown (Appendix A) that

Qn(z) and Pn(z) are two degree-n polynomials that satisfy the even-part

condition, (17)

(84)

It is also a consequence of the same analysis that the even part of the

impedance Z(z) seen looking into the n+I-line cascade with the output closed

on a general pr termination W(z) is given by

Z(z) + Z(z) 1 - p(z)p*(z)n P

(85)-D (z)D

where

Dn(z) = Pn(z)-Zp(z)pn(z) (86)

(17) A*(z) a A(I/z) for A(z) real and rational.

1-16

and ( 1 8 )

13n(z)z nPn* (z) (87)

Consequently, Z(z) generates the spectral estimator1 -1 p(e J e) 12

K(8) = (88)IDn (e J ) 1

Observe that if p(z) =0, we recover KME(B), Eq. (73). Clearly, under the

assumption that F(6) is absolutely continuous, I p(e j) { 1 and W(z) is non-

Foster. In general, an estimator is said to be tuned if the associated re-

flection coefficient p(z) is nonconstant.

Let

i-P(z) *(z) = r(z) r,(z) (89)

where r(z) is a real rational function analytic together with its inverse in

Izj < 1. (This Wiener-Hopf factor is uniquely determined by the normaliza-

tion condition r(0)> 0. Furthermore, 2(0) < I unless p(z) = 0.) Hence,

from (85),

K(G)= IB(e j3 2z (90)

where

B(z) = (z) (91)* I n

Since D (z) is free of zeros in IzI< l,(09) B(z) is the Wiener factor for K(e)'i n* Theorem 3. Let the pr function W(z) generate the spectral estimator

K(8), given the error-free covariance data C(k), k= 0-n. Let (rk) denote

the sequence of radii of the corresponding circles of admissibility. As we

(18) fn(z) is the polynomial reciprocal to P n(z).

(19)P (z) is strict Hurwitz and

nz Pn(z)

is positive-real.

1-17

know, rk r.= b 0 where b =B(0). In addition, from (11) and (17),

do= f InK()d8= 2nb0 (92)-1tf

Thus, (91) yields( 2 0 )

A

1) r (K) = r j o)--- rj0). r (KME) < r w(KME) (93)n-i

with equality iff K(e) = KME(6).

2) Entropy(K) = entropy (K ) < Entropy (K.E) (94)ME r (0)

with equality iff K(G) = KME(8). Or, as an alternative to (94) we may also

write,_L TT

Entropy(K) = entropy(KME) - 1 f In 1 de (95)

l- 7 p(eJ )I

We are now in a position to describe an entirely new type of spectral

estimator with the capability of controlling both resolution and robustness

(as measured by r00) in a parametric fashion. To this end, observe that

Eq. (95) permits us to define

1-- . In 1 ) (96)

to be the entropy echo-loss density at angle 6.(21)

A flat-echo estimator (an FEE) is one for which the entropy echo-loss

density is independent of 8. According to (96), this requires that

I le j = constant. (97)

But this is possible iff

p(z) =ud(z) (98)

Bear in mind that Dn(O) =Pn(0)

The phrase "echo-loss" is taken from filter theory [7 .

1-18

where .±is a constant constrained to lie in the range

0 < <1 ,(99)

and d(z) is a regular all-pass; i.e., d(z) is a real-rational function analytic

in z <1 which satisfies the paraunitary, condition

d(z)d*(z) =1 *(100)

1- 19

III. THE DESIGN OF TUNED FEE'S

Let us consider the properties of an FEE defined by the reflection

coefficient p(z) = p d(z), d(z) a regular all-pass and 0 < pi < 1. Then,

r(z) = 1 - , (101)

r,,(K FE) 4 (1 2 roo(KME) (102)

and K FE (e) = IB(e ie) 12 where

B (z) = (103)P (z) - kizd(z)P (z)

is the associate Wiener-Hopf factor. Thus,

KFE(e) =KME(e). 11 -4ae ed(e j )U n(eil 2 (14

where

U (z) = n~z (105)n

is also a regular all-pass (because P n(z) is s tric t-Hurwitz). Clearly, U n(Z)

is uniquely determined by the prescribed data C(k), k =0-n.

We can now list some of the anticipated advantages of an FEE.

A) Numerical robustness, as measured by r'.. is assignable in ad-

vance by fixing the constant 4 and is unaffected by any subsequent tuning.

A2) On Izi 1 1, d(z) and U n(z) are both of unit modulus so that

d(eie = exp [j *(8)~ (106)

and

U n(e ) exp Cj (e)~ 1 (107)

1-20

where *(e) and n (8) are real functions of e. Thus, if

cp( e) = *(e) +1ne ) + e (108)nn

the factor that multiplies KME(G) in Eq. (104) reduces to

I-2

2 " (109)1-2iacos cp(e) +

Consequently,

I KFE(e) <+(110)

1+P' - KM(9) I-P

and any such ratio can be realized at one or several prescribed values of eby an appropriate choice of all-pass d(z). The FEE is therefore capable of

providing selective controlled magnification of different parts of the spectrum.

In our opinion, if used properly, this estimator could prove enormously

valuable in situations requiring exceptional resolution. (The design of d(z)

is one of phase equalization with a regular all-pass and is explained in detail

in Appendix B.)

A3) The poles of d(z) are zeros of B(z). Hence, the FEE not only in-

cludes the MEE as a limiting case (p = 0 = maximum robustness), but also

yields exact model matching for a broader class of random processes than

those generated by pure finite-order autoregressive schemes.( 2 2 )

A 4 ) KFE(9) > 0, -n7< 9< 1r, a property that makes good physical sense

whenever an exact spectral null is highly unlikely. (Such is usually the case

if x t is the sum of two or more independent random processes.)

Appendix B contains design formulas for a singly and doubly-tuned FEE.

These realize maximum gain (1+)/(l-p), at one and two prescribed values of

e, respectively.

(22) The only zero of

BME(Z) = 1n

is Z=00.

1-21

APPENDIX A

NETWORK FOUNDATIONS

Let the reactance 2-port N' shown in Fig. 1 be described by its ZxZ

scattering matrix S(p) normalized to R on the right and R on the left. Thenn 0

17 ]

1 F) h(z) e flpT S 1 1(p) s 12 (p) 1(Al)= gT -e_npT - znh(z) s 1 2 (P) s 2 2 (P)

where z = e and h(z), g(z) are two real polynomials of degree n in z, the

latter being strict-Hurwitz. Moreover,

g(z)g,(z) -h(z) h(z) = 1 (A2)

Let a load with pr driving-point impedance W(z) close terminals ab.

Then, the load seen from terminals ef has a reflection coefficient normalized

to Rn given by

W(z) -Rs (z) =zp(z) = z.W()+ n (A3)

n

Hence 112],

s 1 S21sIs = s + 1s 22 s (A4)1 1 1 -2 s 1s

is the resultant input reflection coefficient normalized to R 0 . Making use of

(Al) and (A2) we obtain,

s(z) = h(z) + z(z)Z(z) (A5)gz +z (5g (z) +zp(z) 9(z)

where gz g= and =zn h.

Consequently, the input impedance is given by

1-22

(0+zPd)Z(z) =R. ~1+s n nl-I (A6)

where

0 n(z) =g(z) +h(z) (A7)

JR-0 Z

p (z) = g(z) -h(z) (A8)

Note, that with W(z) R n, p(z) 0 and

Z~~~~2 (z) ME(z MZ~) P n (A9)

Lastly, by a direct calculation,

Qn Pn* +Qn* Pnl1 (Ala)

so that (A6) yields (details omitted),

z~z* -pP ~(All1)

(P -ZPP )(Pn-ZPP )*

Thus,

)+*( D 1 ej8

in which

Naturally, if p(z) =-0,

K( e) = KmE(e) =(A 14)

and for the sake of completeness we shall verify the deterrninantal expression(74) for P (z).

nl

1-23

From (A9) and (A10),

ZME (Z) + ZME*(z) Qn(z) Qn*(z) 1

2 P (Z) z) 15n P(z)P *(z) (l5

Let

Pn(z)=a0+a z+.+an zn 0>0 (Al6)

Then, sincen

Z ME(z)=C()+ T ZC(k)zk+ O(zn+ l ) (A17)

k=l

(A15) and (A16) yield

n

k= I k=nz

It follows easily from (A10) that Pn(Z)/0, IzI< 1, so that 1/Pn(Z)

admits a power series expansion

2b 0 +b 1Z+b2z +... (A19)

whose radius of convergence is greater than unity. Of course,

b i >L 0 , (AZ0)0 a0b= n(0 ) = 0

2 n

Consequently, we may compare the coefficients of 1, z, z ,...,z on

both sides of (A18) to obtain

T a = b e (AZ1)!n-. oa-n(Ai

where the Toeplitz matrix T is given by (60),n

a =(an ,ani, ... a0 )' (A22)

and

e2 (0, 0, ... , 1)' (A23).-n

1-24

is composed of n O's and a single I in the last row. Thus,

n n-i n n-i - IPn(z) =(z , z ,...,)a =b 0 (z , z ,.,1)T e 2 (AZ 4)

C(O) C(i) .* C(n)

C(1) C (0) .* . C(n-i)

I . . .. .. .. .. .. .. .. .. ....... (AZ 5)7- C(n-l1) C(n-2) ... C(i)

n n-iz z.. 1

However, if we now use Cramer's rule to solve Eq. (A21) for a0we

obtain

a= n~ (A26)0 an

and (AZ5) immediately reduces to (74), Q. E. D.

1-25

APPENDIX B

PHASE INTERPOLATION WITH A REGULAR ALL-PASS

The phase of a regular all-pass can be made to assume any prescribed

set of real values, mod Z1, at given values of 8= i I 6e..., e, . However,

the degree of this all-pass can far exceed the integer I unless the phases

and the angles 6i. i= 1-41, are interrelated in a special way. In this appendix

we investigate the possibility of designing a degree-I regular all-pass to

interpolate at precisely I points. This of course represents the most economic

general situation.

Any real rational regular all-pass dI (z) of degree 1>0 has the structure

f (z) zi Ej , (z )

dI(z) =E .E I = (. 7z (B1)

where C= tI and

EI (z)= I+fIz+..+f I z 1 +fIzI (BZ)

is a real strict-Hurwitz polynomial with lowest-order coefficient normalized

to unity and f, / 0.

Since d1 (1) = E, the problem of most economic phase interpolation may

be posed as follows: Let d I(eie)=exp j*I (e)]. Then, I (0) and

(ej e)S )- (0) + arg e * (B3)

E I (e3 )Given the two sequences

0=60<81<...<6I <n (B4)

and

O= C0,*i, ... , 0 (B5)

where e= ( " $ (0), find E1 (z) such that

1-26

= e k= 0+l (B 6)E l{ j 8 ki

E1ek

The original recursive p-plane solution to this problem is due to

Rhodes [7, 13] and will repay careful study. Our own z-plane development

of his approach places less emphasis on synthesis-theoretic ideas and has

a somewhat different motivation.

For every r = 0-+1, let

r

E (g) = 1 + f(r)k (B7)

k= I

denote the real strict Hurwitz polynomial of degree r which interpolatescorrectly at e= e0, el ... , er Then, according to (B6) the difference

r+l rr E" r = 1 (dr+ l - d r) (B8)

r+I r r

-j ei .must vanish at z=1 and z=e , i= 1- r. Thus, invoking reality,

r(z) () T (1-z) . H (z Z-z cos e.+i)r+ . r =r i=_I (B 9)

Ei(z) E(z) - (z) E (z) (9r+ rr+l r

rr a constant. Specifically, by setting z=0 it is immediately verified that

f(r+ 1) _ f (r ) SlOr = r+I r "

Clearing fractions,r

Itr+I (Z) Er (Z) -Er+I(Z)f r(Z)=T'r (Iz). 1 (zZzzCos a i+1) (B 11)

i=l

so that

r+2 Erl r+2 r+I _r~ l 2

Er+ 1 E r E r + rr = -- C r+ l -A r+ I

1-27

A simple rearrangement now yields

Er+l(f r+2 +A r+l r =r+ (Er+2 A r+IEr ) (B13)

Since E r+l(z) is strict Hurwitz, it is relatively prime to 2 r+l (z) and musttherefore divide E r+(z) + A r+l(z)Er (z) without remainder. Clearly, the

quotient q(z) is at most of degree one and self-inversive; i.e., q(z) = (z).

Consequently,

q(z) = %r 1 (1+z) , (B 14)

gr+l a real constant, and

E+z(z) + -Ir+-- (z - Zz cos e +ll)Er(Z) = r (l+z)Er~ (z) (BIS)

Obviously,

n-f(r+Z) f(r)1 +r+I r+2 r ( 6

r= 1 +~-7 Dr f(r+l)f(r) (B16)r+l r

and (B 15) takes the final convenient form,

E+Z(z)= gr+i(+z)Er+2(z) )(z -zcos er+I+)Er(z) , (B17)

1sin -( 1 -a 1 )

E 0 (z)=1, E1 (z)=l + 2 1 1- z (B18)sin -(8+a1)

r=0-1£ -2+.

Clearly,

sin T 1 +a- 11 (B 19)f~)=sin 1(8 Ol+

and El(z) is strict Hurwitz iff

If 1) . (BZO)

1-28

This granted, we now prove that the remaining polynomials are also strict-

Hurwitz provided

0< r < I , r=1-1 -1 . (B21)

The proof is inductive in character and divides naturally into several parts.

1) From (B17), by direct calculation and iteration,

Sr+z E Er+l Er+2 r+l = (gr+1-1)(z 2 cosr+l +1)(2r+l Er Er+lr)

(B22)

r+l=' 7r+l0-z). II (z 2 - zcos e.+l) (B23)

1=1

where

r+l= (f(l) ) i1) o . (B24)r+ 1 i=l

Thus, Ek(z) has exact degree k, k= 0-*1.

Z) Let us assume that all Ek(z), k= 1-*r+l, are strict-Hurwitz and let

us write E (z) = E (z, r ) to stress its dependence on the choice ofr+2 r+2 r+1parameter 9r+l.

Clearly, as is seen from (B17), Er+?(z,0) has two distinct zeros

Zl,2 = exp[+je r+l (B25)

on IzJ = l, while the remaining r lie in IzJ > 1. As r+l is increased, z1 and

z either slide together on I z 1, or move as a pair into I zI< 1, or penetrate

jointly the region IzI > 1. In the first case, there exists r+l' 0< tr+l < 1,

such that Er+Z(z, r+I) has a zero z = e of unit modulus.

Since Er+ (z, 1) has the zero z=-1 and r+l zeros in Izi > 1, it is obvious

that in case two, at least one root locus must leave the interior of the unit

circle. Hence, again there exists a g r+ 0< tr+l < 1, such that Er+2 (z, gr+1)

has a zero z = ej @ of unit modulus.

Consider case three and suppose that for some §, O< 9 < 1, Er+2 (z, g)

is not strict-Hurwitz. By continuity, this can only be realized if some root

1-29

i.

i

locus located in z > 1 crosses the boundary Jzj-- 1 for some

0< tr+l<1 .(1) In summary, the assumption that Er+2 (z, 9) is not strict-

Hurwitz for some t in 0< < I, necessarily implies the existence of a0< g < 1, for which E (z, r ) has a zero z=e e of unit modulus.

r~' ~lr+Z r+1This we now show is impossible.

3) If z = eje is a zero of Er+Z(z, r+I), it is also a zero of fr+Z(Z, tr+1l

and according to (BZ3) and (BZ4), either 8 = 0 or it coincides with some Biti1i = l-4r+l.

If e=0, the zero is at z=I and (B17) yields

O= Fr+Er+l()+(l-1r+l)(l-Cos ar+l)E rl . (B6)

However, E r(0 ) = Er+ I(0) = I so that Er(1) > 0 and Er+1 (1) > 0. (If, say,

E r(1)< 0, then Er (z) must vanish for some z in 0< z< 1, a contradiction.)

Consequently, the right-hand side of (BZ6) is a sum of positive terms and

we have an inconsistency.

If 6= E (z )has the zeror+ 1' r+z' r+1

z=expj or+ I1

and (B 17) gives

erj 6er+1=os-r+1 . E (e (BZ7)-)r+1 2 r+1

e 1 TTBut cos(e +i/2) /0 because 0< < - and (B27) is therefore invalid.

Lastly, suppose that for some i, i= I- r, e=e.. Then, from (B17),t i'(e)+(L )co e-cse ~ i.Jei

0= r(1+e re l (e j)+2(1- )or+(s el-Cos er+ )Er(e ). (B28)

By construction and the fact that i< r,J el e.)

arg -y e-i) arg - J~i)- (B29)Er+l(e E(e ')

( 1)The coefficient of the highest power of z in Er+2(z, fr+l) is unequalto zero for all %r+l in 0< r+l < 1. This means that no root locus can passfrom the exterior to theTnteri3r of the unit circle, or vice-versa, by escapingthrough the point at infinity.

1-30

and since for any k

arg k(ej e} = ke-argEk(eJe) , (B30)

we deduce from (BZ9) that

jei e. j o.argE (e ) L + argEr(e 1) . (B31)

With the aid of (B31), Eq. (B28) is now reduced to the purely real form,° .I ir , i il(e.- cos e0 r+lC° "T r+le j+(1- rdcsr+l)(C s r

(B32)

But a moments thought reveals that we are once again faced with a

contradiction because 0< e < 8r+ 1 < TT implies that,

Cos 9i'Cos Or+1 > 0 (B33)

Thus, it has finally been established that every polynomial Er(z), r 0-1 , is

strict-Hurwitz, Q. E. D.

The restrictions

0< Fr < 1 , r 1-1 -1 , (B34)

impose constraints on the admissible interpolatory pairs (8,. ), i= 0-1.

These constraints reflect the well-known fact that the frequency variation

of a regular all-pass is not entirely arbitrary.

For example, as a consequence of the strict-Hurwitz character of

Ek(z) we can easily prove thatd k (e ) d* k(8 )-ar g =(B 35)

and

) k k(0) = kTT .(B36)

Thus, the a's must at least form a monotone-increasing sequence:

1-31

0 a 0< CL1< . .. < aI< IT .(B37)

Unfortunately, (B37) does not always guarantee (B34) and the general

situation is rather complicated. Nevertheless, because of their great

practical importance, we shall carry out the designs for a single and doubly-

tuned FEE in detail. Recall that according to Eqs. (104), (108) and (109),

aKFE(e) = KME(). 1-4 2 (B38)

l-Zcos Cp(e) + P

where

CP = O) )+ On(8)+8 , (B39)

and *(0), n() are the phase angles of d(e i ) and Un(e j), respectively.

Design of a Singly-Tuned FEE. Suppose that maximum gain

1+__ (B40)1 -4

is desired at some single suspected peak observed in KME( 6 ), say, at location

e=e,. Clearly, we must have c481) =0, mod Z1, and this we attempt to

achieve with a regular all-pass dI(z) of degree one.

Let

Y = n(e 1) + al, mod 27 , (B41)

so that 0< Y < Zr and let(Z)

e=sign(Y1 -r T) . (B4Z)

If r=0 we pick d 1 (z) -1. Otherwise, if E=+I we choose arg i(eL ZTT-?Y

and if e=-1 we set 7T(+1)= r+a 1 = -y i . Thus, from (B18),

(2)( 1, x>0

sign x O, x=O

-I, x< 0

T-32

I. (61+,y1+ .- ii)sin +Y+ I

(Z) I + -Iz (B43)sin -- (1+yl + - - Tr)

and

dlI(Z) (Z. ) (B44)

It is easily confirmed that

sin -(1 + Y -+ -- c T)f(l) = s 1 y (B45)1 sin 1(81 Y1+ - T)

has magnitude less than one, as required.

Design of a Doubly-Tuned FEE. Let us now attempt to achieve maxi-

mum gain at two locations 61 and ezo 0< 61< 62< 17, with a regular all-pass

dZ(z) of degree two. Let

Yi= On(0i) + ei . mod 2r, (B46)

i=1,2

Since * z (I) - *z(0) = a1 and *2 (6) - 2(0) = aL2 and since the phase angle

ye) - 42(0) of the regular all-pass l 2 (z)/E?(z) must be monotone-increasing,

it is necessary that a1 < C2"

This latter observation leads to the following assignment.

1) If Y ; c = , CL i =2-Yi, i=1,Z.

2) If y < Y2 , y < IT, Y 17; = -1, C=L 7-Yl , CL = 317-y?.

3) Any other case is impossible with a degree-two regular all-pass. (3)

In cases 1) and 2), E 1 (z) is given by Eq. (B43).

To calculate E 2 (z) we can use (B 17) with r = 0:

E (z) = ( l(+Z)El(Z) +(1 - 1 )(z 2- Zz cos e1+1) (B47)

(3)The maximum phase increment attainable with dZ(z) over 0< 6< r equals 217and occurs at e= T.

1-33

Of course, l is unknown and must be determined from the interpolatory

condition, 2 ( ez(

arg j = Z(B48)

E?(e z)

Clearly,

je2 )El(e 2 + e 2(cos ecos e 1)

and we find that

e :c (B49)

y+x

where

jee2 -je2/2 1 (cos e- (B0)y=E(e )e , (B5co0-

COS

Finally, solving for ' 1 with the help of (B43) we get (algebra omitted),

82-1S co-T sin- ( 2 +y 2+-7- 1r -fl1 sin. 7( 2 -y2 -=-- IT)

cos 1 -cos in 1 -Esin VY z+ - TT)

(B51)

Let us check to see if 0< I < 1 when E=1. For this it suffices that

I sin Ile2+ Y3 sin-L(el+Y)I>sin e -Y2 sin -L(e-y < 0 , (B5Z)

depending on whether

ez-' 2 0 (B53)

respectively. These inequalities are not automatic and we shall attempt

to clarify this matter in Part II.

1-34

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Z. N. I. Ahiezer and M. Krein, "Some Questions in the Theory ofMoments," American Math. Society, Math. Monograph, Vol. Z, 1962.

3. M. Bocher, "Introduction to Higher Algebra, " MacMillan Company,New York, 1907.

4. D. C. Youla and N. N. Kazanjian, "Bauer-Type Factorization ofPositive Matrices and the Theory of Matrix Polynomials Orthogonalon the Unit Circle," IEEE Trans. on Circuits and Systems, Vol. CAS-Z5,No. 2, February 1978.

5. G. M. Goluzin, "Geometric Theory of Functions, " American Math.Society, Math. Monograph, Vol. 26, 1969.

6. H. L. Royden, "Real Analysis, " MacMillan Company, New York, 1963.

7. J. D. Rhodes, "Theory of Electrical Filters, " John Wiley and Sons,New York, 1976.

8. D. C. Youla and G. Gnavi, "A Network Solution of the 1-DimensionalInverse Problem by Time-Domain Reflectometry", Part I; TechnicalNote. No. 2, PINY, Farmingdale, New York.

9. J. Burg, "Maximum Entropy Analysis," presented at the 37th AnnualMeeting, Soc. Explor. Geophysics, Oklahoma City, Oklahoma, 1967.

10. Ya. L. Geronimus, "Polynomials Orthogonal on a Circle," AmericanMath. Society Translation, No. 104, 1954.

11. V. F. Pisarenko, "The Retrieval of Harmonics from a CovarianceFunction," Geophys. J. R. Astr. Soc., 33, pp. 347-366, 1973.

12. H. J. Carlin and A. B. Giordano, "Network Theory: An Introductionto Reciprocal and Nonreciprocal Networks, " Prentice-Hall, EnglewoodCliffs, New Jersey, 1964.

13. J. D. Rhodes, "Filters Approximating Ideal Amplitude and PhaseCharacteristics," IEEE Trans. on Circuit Theory, Vol. CT-20, No. 2,March 1973.

1-35

w>0

I~c0

I)

-J-00

0.U.w- 0

zr 0

0.0

1-36

PART I I

A DIGITAL IMPLFIBENTATION

by

%I GuptaDigicomp Corporation Ithaca NY

Chapter 3

1. Introduction

The solution to the problem of the spectral estimation as proposedby Professor Youla of Brooklyn Polytechnic offered many attractivefeatures. One of them is the ability to tune the spectrum at desiredfrequencies. The tednique, called Flat Echo Estimator (FEE) is describedin a paper by Professor Youla. 1 A computer program, described in thefollowing, was developed to implement this technique in FORTRAN IV on?MLTICS. The brief description of the algorithm is followed by the blockdiagram, description of various subroutines and a listing of the FORTRANprogram itself. A user documentation to rum this program is also providedin this report.

2. Algorithm

Youla has shown that the maximum entropy spectral estimates K (e),for the finite sequence of error-free covariance samples C(1), C(2),...,C(n+l) given by

1ME-e..-21(1)1%(e] )12

can be viewed as the real part of the driving-point impedence of a cascadeof (n+l) comensurate lines teriinated on a load W(z) (Rn). The resistance

is the same as the characteristic impedance of the last line in thecascade. He showed that any other positive-real load termination will alsoprovide an admissible spectral estimate for the same oovariance samplesequence. If the load network is an all-p:; net or u:ith reflectionfactor

P(e j S) = md(ejo)= uexp[jP(e)J , ,(2)the spectral estimates provided will be KFE(6) and could be derived tobe

1-u2

FE(e) KM l-2u cos p(e) + 2 (3)

Vherei (e) = Vi(6) + 8n(6) + 8

The phase factor an(8) is given by the expression

exp [j an(e)] = P = / , z -- ej

P"(W P(z) (4)

The polynomial Pn(z) here is the same as that in the equation (1)and is given by

( C() C42) ... C (n+)-

C(2) C(1) ... C(n)Pn (Z) I"AnAn-1 . . . . . . .

C ~ n ) ( n -l . . . C ( 2 )

Izn zn-1 . 1 _ (5)

The deterninant A is given by

C(1) C(2) ... C(k+l) 1

c (2) C(1) ... c(k) IAk= I . . . . . . . . . . .. . . .

I (6)_ c(k+l) C(k) ... C(1) J

The factor G(O)1-p 2

G(8) =

1-2ucos *(e) + p2 (7)

in the expression for KFE() is similar to a tuned amplifier gain factor.The maximum antropy thus can be tuned to provide for peaks at anydesired frequency by a proper choice of the all-pass load network's phasecharacteristics 4)(e). The choice of (O<lj<I) provides for adjusting theamplitude of the peak. It should by noted that u=0 reduces KFE to FK.Professor Youla has demonstrated a link between the variable u and thenumerical robustness of the spectral estimator. The conditicn u=Oreduces KFE to KME and represents the maximum numerically robust choice.

3. Computer Program

The flow chart to ccmpute KFE(O) for a given sequence C(M), C(2),...,C(n+l) is shown in Figure 1. The first step to obtain KFE(6) is to camqxteKMF(), the spectral estimates by the maximum entropy method, then designan all-pass network and finally oarpute G(e). The multiplication ofG(e) and KE(8) will provide the value of KF(e). It should be noted thatwe don't have to find the actual element values of the all-pass network.

IH-2

- f

COMPUTEKMs(e)

DESIGN ANALL PASS

BOUNDED REALFUNCTION d(z)

COMPUTEGl(e)

KFE(e) = G(e)KME(e) IFIG. 1: FLOW CHART TO ESTIMATE SPECTRUM

BY FLAT ECHO ESTIMATOR METHOD

11-3

-

Rather, the aumputation of an appropriate all-pass bounded-real function

d(z) (see equation 2) will be enough.

3.1 Caputation of KME(o)

TKhe ( (), the spectral estimate by maximum entropy method is

omputed as follows. First a matrix tn is formed from the ovarianoecoefficients C(l), C(2), ... C(npts) as

C(l) C(2) ... C(npts)

C(2) C(1) ... C(npts-l)

tn = . .. ... .. (8)

C(npts-1) ... ... C(2)

Then the coefficients of a polynomial Qn(z) (which is the same as Pn(z)except for the constant multiplying factor ( = i/ AnAn l ) in eqn. (5))are obtained as the determinant of a matrix t,, formed by deleting anappropriate column i of matrix t n . -

The coefficient of Zn for exaple, is found as the determinant of amatrix tn obtained by deleting the first column of t n . The reason fordoing so can be seen by noting that Qn(z) is the determinant of thematrix formed by t n appended by a vector [zn, zn - l, .....,.1 as its last

F tn tn2 0'" tnnpts-/6nan-I Pn(z) = Qn(Z) = I znptsl znpts- 2 ... 1 _

nptsE E (-l)m ItJznrts ' l (9)

MTl

The amputation of det [tr]is performed by calling the subroutine

pine oof.

A constant multiplying factor COtult is obtained as the product ofdet n (=An) and An-, It can be seen that A-, is same as the constantterm in on(z). The determinant an is obtaind in the do loop toompute coefficients of Qn(z).

11-4

Finally, the coefficients "coef" of polynomial Pn(z) are obtained fromthe coefficients of Qn(z) by dividing each one of them by 1/v'Cmult.

The spectrum KE(O) may be obtained at this point as e 12

if so desired. The flow chart to compute N (E) is sh Mn in Figure 2.

3.2 Computation of FEE Spectral Estimates

Our goal is to obtain KFE(). Therefore, instead of simply obtainingKj(e) at the above step, a subroutine spectral estimator is called.This subroutine first queries if only Y is desired or KFE is desired.For KFE, it asks further questions as to singly tuned or doubly tuned,

the value of these frequencies (in radians) and the maximm robustnessfactor AM. It then calls an appropriate subroutine design-all-passor design-all-pass-2 to determine a first order or second order all-passreflection factor. The subroutine "response" is used to cupute thespectrum nres points for e between 0.0 and "thmax". Upon return fromspectral-estimator, the results can be written in an output file to providefor readouts and/or plotted on Tek screen by calling plot routinesappropriately. The flow chart of the computer programs is shown inFigure 3.

4. Subroutine Description

The computer program has been broken down into very many smallerDroqramns called subroutines. A list of subroutines used herein is shownwith their interconnection in Figure 4. A brief description of varioussubroutines is provided in the followinq:

4.1 Subroutine "Spectral-Estimator"

4.1.1 Purpose: The purpose of this subroutine is to query the userif FEE or MEM spectral estimates are required. In case FEE is desired,it queries the user about the robustness factor AM and the tunedfrequencies. It then designs an appropriate all-pass reflection coefficient(first order for sinqly tuned by calling "desin-all-pass" or second order

for doubly tuned by calling "design-all-pass-2". The queries for FEEparameters and the design of all-pass network is skipped if only MEN isdesired. It then calls "response" to cxmImte the spectral estimatesusing equation (3). The flow diagram of this st)routine is shown inFigure 5.

4.1.2 List of Variables: Following is the list of variables usedin "Spectral Estimator"nfoe - integer variable = 0 for W4

= I for singly tuned FEE

TI-5

FORM THEMATRIX

[tn]FROM SAMPLES

COMPUTE COEFF.

OF Qn(Z)'IyDIVIDE COEFF.OF Qo(Z) BYIAnAn-, TO

FORM Pn(Z)

IKME()1,= Pn(e j #)12

FIG. 2: FLOW CHART TO COMPUTE KME()

11-6

COMPUTE [-4 c77Pn (Z)

COMPUTE SPECTRALSPECTRALESTIMATES ESTIMATOR

I DESIGN SINGLYPLOT THE TUNED ORSPECTRUM AND DOUBLY TUNEDWRITE THE ALL PASS FUNCTIONVALUES INOUTPUT FILE

RESPONSE

FIG. 3: OVERALL PROGRAM-FLOW DIAGRAM

HI-7

CD SPECTRAL- ESTIMATORi=-

D ESIGN-ALLPASS DSN-ALLPASS-2

RESNSZI AMP. ARG

POLYVAL i

ARGMVNT

2' PHASE

FIG. 4: INTERCONNECTION OF VARIOUS SUBROUTINES

11-8

FEE? NO

YES

GET P ANDTHE TUNEDFREQUENCIES

YS DESIGN

SIGYTUNED FIRST ORDER

J ALLPASS

DESIGN I

SECOND- ORDERI,ALL PASS ,

COMPUTE _

SPECTRALESTIATES

FIG. 5: FLOW CHART -SUBROUTINE SPECTRAL ESTIMATOR

11-9

= 2 for doubly tuned FEEaiu - the numerical i'bustness factorpeak - a vector of frequencies (in radians) at which tuning of the spectrzum

is desired. The frequencies are arranged in the increasing order.The smallest desired frequency must be peak (1).

nerror - a flag to indicate that all-pass network function cannot beobtained

npts-d - the order of the polynmnal Et in the all-pass reflection factorE (z)

d(z) =E (z)

npts-d = 2 for singly tuned case= 3 for doubly tuned case

cf-d - the oefficients of the polynomial

4.2 Subroutine Design all pass4.2.1 Purpose: This subroutine is to desiqn a first order all-

pass network function d(z) whose phase is prescribed at one frequency(peak (1)). This computer program is based on the algorithm provided byYoula in Reference 1. First the phase GAHM 1 = [8(ej)+ e1]mod 2wat 81 = peak (1) is computed. The all-pass phase function is to bechosen in such a way that *(OI) + GMM 1 = [0) mod2w where ,(el) isthe phase of the all-pass d(z) at peak (1).

4.3 Subroutine amp-arq4.3.1 Purpose: Given a polynomial R(z)

R(z) = a0 + a (z) + a 2 z +...+ zn

This subroutine computes the ampltude and phase ofR(a) for any given "th"

z = e j (th)The subroutine polyval is used to compute the complex value of polynomialR(z) and argmt is used to determine the phase of R(z) between 0 and 2.

It is frequently desired in this program to evaluate the phase of theall-pass function.

S-E(z) E (l/z)d(z) = ej Y - e JYz n I

E (z) E (z)

arq rd(z) y + (n-l)e + arq [E (z*)1 - arq [E(z)]= V + (n-1)8 - 2a(0)

and y is sane constant phase (0 or 2n)

ir-io

4.4 Subroutine Polyval

This subroutine computes the value (ocmlex) of a polynomial.R(z) = a, + a2 z + a3z

2 +...+ anzn-lfor ocmplex z. The coefficients ai are real.

4.5 Subroutine Argument

This subroutine calculates the arquzent of a complex nuxber. Theresult is between 0 and 2w dependinq upon the siqns of the real andIinary parts.-

4.6 Subroutine "response"

This subroutine calculates the spectral estimates for K or K.,using appropriate expressions at nres points for 6 ranging from 0to "thmax". For maximum entropy, KM is qiven by

fx(i) = KC(0i) =

v Pn(ejui)12

Fbr K_, K, is multiplied by a factor Amult qiven by equation (7).Mbst of the variables used in this routine are self-explanatory.

4.7 Subroutine design-all-pass.2:

4.7.1 Design of a doubly tuned FEEThe purpose of this routine is to desiqn an all-pass network

function whose phase has been prescribed at two frequencies. Exceptfor same minor changes, the algorithm used is the same as providedby Youla in Reference(l). The first step in the process is to ccmputethe desired phases Gamma 1 and Gamma 2. (eq. B(46) of Reference (1))at peak (1) and peak (2). Then the phases alpha 1 and alpha 2 arecrnputed depending upon the relative values of Gamma 1 and Ganma 2.

We know that (see eq. B48 of Reference (1))

E2 (z)a g - I .. ...0 ......... (4.7.1)E 2 (Z) I zeJO

2=z2

Z2 (E2 (l/z)J jze 2 Ja2= e -2arg[E2(z)J e

E2 (Z)

II-II

arg[E2(] W 1 - c2a/2 . . ... (4.2.2)J q 2

But also (see eq. (B-47) of Reference (1))

E2 e = e 2 " l e (I ) (2j8 e 38 c s 1

982/2 9e 2 )(2e 2 2)2 1 1

= 2(COS-2/2 • e )El(e ) + (1-C,)e (2coe - 2cose)

j82 e j e

2 Je/2 l-c coxse 2 - cose 1

=2C cos 2 /2e (E1 (C e +-1 cose 2/2

jo

arg E2 (z)I = arg[e (y+x) ............... (4.7.3)

z:=e J02

where y = El(e 2 )e - 2 *4.7.4)

i-s cs2 2 -ose 1El cose2/2 ...... (4.7.5)

Fran eq. (4.7.2)

82 - c2/2 = 2 + arg (y+x)

or arg(y+x) = a2/2 = a

i.e., Re(y+x) = R cosAIm(y+x) = R sinA

9e2

but E(z) = 1 + f z = 1 + fe

-j 2/2 -Je/2 *Je2y = El(Z)e = e + fe

11-12

Y= cos82 /2 - j sin 2 /2 * fl x°s82 /2 + f1j sine2/2

y= (1 + fl)cos0 2 /2 + j(fl-l)sinse2 12 (4.7.6)

Therefore fran eq. (4.7.6)

x + (1+f 1)oos82 2 = RcosA . . . . . . . . . . .(4.7.7)

(fl-l)sine2 /2 = RqinA . . . . .. . . . . . .. (4.7.8)

Since f1 is known (beinq the coefficient of z in E1 (z) of sinqly tunedcase), R can be determined fran eq. (4.7.8). This R is then substitutedin eq. (4.7.7) to obtain x .

once x is know, EI can be onmipted as, from eq. (4.7.5)

1-E x cos82/2=B

&i cos8 2 -°os6I

& I +B .... . .. ..... ... . (4.7.9)

Then the coefficients of polyncmial E 2(z) are determined by eq. B(47),proof in Pef (1), as

E2 (z) = Fl(+z) (l+fl(z)) +U- (-l) (2 2 +2)

= 1[14l+fl)z + flz 2I + (1- l) (1-2z cose 1 + z2 )

= 1 + z[&l(l+f I ) - 2 cos 1 (l-El)] + z 2 (&if I + (1-&1))

4.4.2 List of Variables:

The variables used in this subroutine are fairly self-explanatory.

Vector cf-d = coefficients values of polymial E2 (Z)npts-d = nunber of coefficients of L2 (Z)

(=3 for doubly tuned case)ar = A of eq. (4.7.6)br = B of equ. (4.7.9

TI-13

I1

5. User Instructions

The cmputer program is implemented in FORTRAN IV and the listingis shown in Appendix A. The program is named as FEE2 on Multics.Before running the progrm, the user mist make sure that the graphicsmode for his system is turned on, A typical user session is describedbelow. The system's response or questions are marked as "Sw whereas theuser response is marked as U. (see figure 7)

S: yU: fee2S: enter two digit file number (note 1)

enter number of parts to be usedS: enter 0 for me, 1 for singly tuned and 2 for doubly tuned fee

(note 2)U: 0,1, or 2The following questions are not asked it the user enters 0 in the* aboveresponse.S: enter robustness factor - amuU: any real number v, O<u<l (note 3)S: enter desired resonant peaks (in radians)U: any real number between 0 and 1 (note 4)S- The estimated spectral plots are plotted, the u and resonant peakvalues printed and then, the following statement typed: If you wishto make a hard copy, do it now.

Then type 1, if you wish to plot the phase of the all pass network.Else, type in 0.

U: 0 or 1S: depending upon the user's response to the above question, the phase

of the all pass FFZ network load is plottedS: StopS: y

5.1 Notes:

5.1.1 Note 1: The program at this stage is asking you to inputbdo digits XX (=25 in test case) of the file XX on the terminal. Theprogram assuznes that the following input data is pre-stored in file XX(file 25,for example).

Input data in file XX:(i) nres (numzber of points at which estimated spectral response

is to be evaluated in format 14)

11-14

(ii) tinax (the maximum value of 8 (in radians) up to whichspectral estimates are to be cxrouted in format E18,10)

(iii) npts (in format 12, the number of error-free covariance sampledata points)

(iv) The covariance values at npts points (in the format E18.10)

Fbr the test case, the data stored in file 25 is shown in fig.6. TheMM and FEE spectral estimates are shown in figures 8 through 11.

5.1.2 Note 2: The program at this point is askinq if the maximumentropy spectral estimates or FEE estimates are desired. If the usertypes in 0 (for MEM), the program does not ask any further questions,and simply plots the ME4 spectral estimates on the terminal screen. Inthe case of FEE (user response of 1 for singly tuned and 2 for doublytuned), the user has to type in the robustness factor v and the desiredresonant peaks.

5.1.3 Note 3: The robustness factor u should be a real number, between0.0 and 1.0. If you type 0.0, the vector will be the same as MINestimates. The program would bcmb if u = 1,0.

5.1.4 Note 4: The values of resonant peaks in radians that you are askedto input (real numbers) must be 1 for singly tuned and 2 for doublytuned case. Furthermore, the peaks for doubly tuned case must be inincreasinq order, i.e. the first input value must be less than the secondinput value.

5.1.5 Note 5: At this point, the system has computed and plottedthe desired spectral estimates. It has also printed the other usefulinformation, such as u and peak values for the hard copy records. Theprogram can also plot the phase of the all pass function d(Z). It asksthe user if he wishes to plot this phase. Since the program shallblank out the screen before plottinq this all pass phase function, theuser must make a hard copy of spectral estimates before responding tothis question.

1I-15

IP

10246.2831842.0200001.6222271.0767960.5164770.1171780.0121310. 0959290.3461920. 5909480.7289070.7431160. 5967890.3961380.1900080.0431110.0044630.0352870.1273530.2173940 .268148

Fig. 6: Data for Test Example (omtained in file 25)

11-16

Setup graphics-table tek 4014segment Setup-graphics no~t found.r 1217 0.055 1.750 50

Setup graphics-table tekc 4014r 1213 0.351 3.906 62

fee2enter two digit file numbter25enter 0 for mn, 1 for singly and 2 for doubly tuned fee2enter robustness factor :amu

0.5enter desired resonant peak (in radians)0 .6035enter desired resonant peak (in radians)2.502

Fig. 7: A typical user session

IT-17

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LUU

r- -- T- cv LU I

LU5.

LLI

LL.'4

I- It

J -w -- -v -- In -' -- -CD

C40. CL4 CS -L

C4 C-18

uJLUL

II j%

LL.

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I I I I I I I I I I I s

L I I I '1 I I I I I I I I II I I a I I I I I I I I I UI l - I I I I O I I I - I

I I1 I I I I I I I I T

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L ~ ~ I --- 1.. . , . . .' - ----. . J.... -- -. I. ... I.. . I.. ---- 1 -....-

L.L

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IT-19

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I~~A I.- I I I

L. AJ

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r -- - I --- -r - I' Lai

I act

! * 1 ~C.3I I I I I

II

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----- ---- I ------ ---- r --- 4- ---- cn ~I I ~ II I cc

I -I

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--- - - -- -- -- C14I

;1U7 -- -- IA -- - - L -- - - -L - - -- --L -- - -- J C

LAo cm cc co I I-C%-----------4---------- JN C,, C

W IM. IA C. I- 1I MC J > c i= L

I- I IIT-21

4=;;

-- 4----r--- --- , cc-

I I

LLU

-~~U W -cc . ~ J

ULU

LJ- - ---- .-- --- l O.LA

t ~C0

co C4Acc

C4 cc cc. cc

11-22

REFERENCES

1. D. C. Youla, "The FEE: A New Tunable High-Resolution Spectral Estimator. Part I -

Background Theory and Derivation," Technical Note No. 3. PINY, Contract No.F30602-78-C-0048, RADC.

11-23

1.60pdizmension c(20), tn(20,20), peak(20dimension cxief (20), fx(1024), thi(1024)character*16 header (3)ow~plex z, fz j cexp, cthcuvm/respostallpass (1024)

external plotsetup (descriptors)external plot _(descriptors)external ioa (descriptors)external read list (descriptors)external write list_(descriptors)header (1) - "Maximni Entropy"header (2) - "Singly tuned FEE"header (3) - "Doub~ly tuned FEE"call ioa (enter two digit file numb~er)call read list (nfil)read (nfil ,113) nresread (nfil,l111) thnMx

113 format(14)read(nfil,ll0) npts;do 10 i=l, npts

10 read(nfil,lll) c(i)110 format(I2)111 format (E18. 10)

nptsi--npts-ldo 40 jr--l,nptsido 20 i=l,jr

20 tn(jr,i) = c(jr+l-i)kd = npts-jr+ldo 30 i=2,kdjc--jr+i-l

30 tn(jr,jc)=c(i)40 aritinue

nfilot=18nsiqn=ldeln-0 .0do 60 i=l,np:tscall pmcof (nptsi,i,tn,cf)coef (npts-i+l) -nsign*cf

* d&,in =(nsiqn*c (npts-i+l) *cf) +delnnsign =-nsiqn

Appendix -Fortran Prcrrani Listing

A-1

(Appendix, continued)

60 continuewrite (nfilot, 114)

114 fonat(Sx,"Spectral Estimates follow")ault = deln * coef(1)

amult = abs(amwlt)amult = 1.0 / cmultcmult = sqrt(i ult)do 75 i71, npts

75 coef(i) = cmult * coef(i)call spectral,_ estimator (thi,fx,ooef,thmax,amu,peak,npts,nres,nfes)do 80 i=l,nreswrite (nfilot,l115) i, thi (i) , fx M), ph allpassiW

115 format(14, 5x, E12.4, 4(2x,E14.5))80 continue

call plotSsetup(header (nfee+l) ,"freq" ,"spectralvalue" ,3,10.0,1,0)call plot_(thi,fx,1024,l,)if(nfee.eq.0)go to 95call ioa ("robustness factor and peak freq are")call write_list_(amu)do 90 i=l,nfee

90 call write list (peak(i))call ioa("Input data is in file No.")call write list_(nfil)

95 continuecall ioa ("If you wish to make a hard copy, do it rxw")call io-(" Then type i if you wish to plot")call ioa ("the phase of all pass fee network")call ioa ("else type in 0")call read list_(ntemp)if(ntemp .ne. 1) go to 96call plotSsetup("phase of tmed fee network","freq","ise",l,l0.0,l,0 /c)call plot_(thi,ph allpass,1024,1," ")

96 continueclose (nfil)close (nfilot)stopend

A-2

(Apendix, continued)

subroutine spectral estimator (thi ,fx ,cf ,tlvmx,amipeak ,npts ,nres ,nfee)dirension thi (1024) , fx (1024) ,peak (20) , cf (20) ,cf-d (20)external ioa- (descriptors)external read-list- (descriptors)call ioa- ("enter 0 for mem 1 for singly and 2 fordoubly tuned fee")call read-list-(nfee)if(nfee.eq.0) qo to 20call ioa- ("enter robustness factor --: ama##)call read-list- (anti)do 10 i=l,nfeecall ioa-("enter desired resonant, peak(inxwiig)")call read-list- (peak i))

10 continuecall design-allpass (npts, npts-d,cf ,peak, cphase-d,cf-d)if(nfee.eq.l) go to 20if(peak(2) .gt.peak(lt) cp to 15call ioa- ("error: 2nd res freq less than 1st res; f req.Treated as singl/cy tuned")go to 20

15 nerror = 0call desiqni-allpass-2 (npts ,npt-- d,nerrordf,peak,cphase-d,cf -d)if(nerror .eq. 0) go to 20call ioa- ("error: All pass can not be desiqned. Treatedas singly tuned/c")

20 continuecall responise (nres ,nfee,npts ,npts-d ,cf ,thmax,cf-d,cphase-d,amu,fx,thi)returnendsubroutine response(nres ,nfee,npts ,npts-d,cf,tm,cf-d,cphase-d,mu, fx/c,thi)

am /re sps-allpass (1024)nfee = 0 if mew~ is desired, - niuiter of resonantpeaks for fee)dinmnsicn fx(1024), thi(1024) ,cf(20) ,cf-d(20)delttrqnesdelth-thmax/delth

amlt-num = 1.0e+00 - amisq

A- 3

(rypendix, continued)

aimlt-den =1.0e+00 +anusgqamu2 = auado 80 i=1,nres

th = i*delththi(i) = thcphase = O.Oe+00

20 continuea = (peak(l)-alphal)/2.e+0Ob = (peak(l)+alphal)/2.0e-00cf-d(2) =sin(a)/sin(b)cf-d(l) = 1.e+00npts-d =2do 30 i=3,20

30 cf-d(i) - 0.e+00call axp-arq (npts-d ,cf-d,peak (i) ,couase-d,amip,oiise-d,all-ikase-d)nfil=40

101 foniiat(2x,"Design" ,4x5El3.6)th =0.e+00close (nfil)returnendfunction tvopi-nrd (a)twopi = 6.283184e+00na = a/twopiboopi-nrx3=-ra*tw.opi +-aif (AWoi-mod. it. 0.e+00) twopi-moe-Wpi+thwvpi-wodreturnendsubroutine desiqn-all-pass-2 (npts ,npts-d,rnenror,cf ,peak,cphase-d,cf-d)dimiension cf (20) ,cf-d (20) ,peak (20)

c.... This subroutine assumes that singly tuned been calledonce

pi =3.141592e+00

twopi =6.283184e+00

threepi =9.424696e+00

crkiase = .e+00call aaap-arq (npts ,cf ,peak (1) ,cpase ,aip1,ioaael,all-riiasel)call anmp-arg (npts ,cf ,peak (2) cuvasvp2 ,phie2,all-oiase2)gammal =all-jIasuel + peak(l)

A- 4

(Appendix, continued)

gamwna2= all-phase2 + peak (2)gauiwul = twapi-md(ganmad)gamna2 t-WPi-Md (gammna2)nfil=40call ip.-arg (npts-d ,cf-d ,peak (2) ,cjise-d,aii,

call at-arg (rpts-dcf-d,peak (1),cphase-d,al ,1 il ,aphl)aljlial = twpi - qaialalpia2 = twopi - qauuua2eps = 0.Oe + 00if (gamna1 .ge. qwina2) go to 10if(qammal .gt. pi) go to 100if (gamma2 . lt. pi) go to 100eps = 1.0e+00

alphal = pi - qamma 1alpiia2 = threepi - qanuua2

10 criftinuenfil=40

ar -0. 5e+00 * arar =(cf-d (2) -1.e+O0) *sin (peak(2) /2.Oe+00) /sin (ar)x =(cf--C&(?)+1.0e+0O)*cos(peakc(2)/2.0e+00)x - -1, (rcs abr ^v xo(peak (2) /2.0e+00) /(oos (peak (2)) )-cos(peak (M)epsi = 1.e+00/(l.e+00 + br)epend. = 1.0e+00 - epsicf-d(3) = epsi. + (epsi*cf-.d(2)cf-d(2) = (ePsi*(l.e+00+cf.-d(2))) - 2.0e+00*os (peak (1))

*epsnd.cphase-d = eps * pi

npts-d = 3(;all anpq-arg (npts-d,cf-d,peak (2) ,cphase-d,aip,ph2d,aph2d)

A-5

PART II I

MULTI CHANNEL CASE

by

D.C. YoulaPolytechnic Institute of NY Farmingdale NY

TABLE OF CONTENTS

Page

ABSTRACT

I. INTRODUCTION 1

II. FORMULATION 3

IllI. THE Zm-PORT INTERPOLATORY CASCADE 5

IV. THE CLASS OF ALL SPECTRAL ESTIMATORS 30

V. THE MULTICHANNEL FLAT-ECHO ESTIMATOR 37

REFERENCES 40

APPENDIX A 41

APPENDIX B 44

III-i

ABSTRACT

Let x t denote a discrete-time, zero-mean, second-order stationary,

full-rank m-dimensional random vector process. Let us assume that the

spectrum of x is absolutely continuous and let K(G) and C(k) denote the

associated mxm spectral density and covariance matrices, respectively.

Suppose that the partial information C(O), C(1), .. , C(n) is known in

advance without error. The first major contribution of this research report

is contained in theorem 2 which supplies, via Eqs. (173) and (174), a general

explicit parametric formula for the class of all spectral densities K(O) con-

sistent with the given n+l (matrix) pieces of covariance data.

The free parameter is an mxm matrix function p(z) which is analytic

in I z I < 1 and of euclidean norm less than unity almot everywhere on Iz = .

Thus, p(z) is essentially an arbitrary passive scattering matrix .

Most of the numerical calculations can be carried out efficiently and

recursively. Indeed, our algorithm can be constructed around the generalized

Levinson recursions for the left-right matrix orthogonal polynomials generated

by the sequence C(O), C(l), ... , C(n). Consequently, all the highly developed

and sophisticated software that has come into existence during the last decade

because of the intense effort devoted to Burg's maximum-entropy estimator,

can be used intact.

The second major contribution is the identification of the FEE or Flat-

Echo Estimator. The FEE is based squarely on frequency-domain concepts

and represents an entirely new particular solution to the problem of spectrum

recovery. It emerges by an appropriate restriction of the functional form of

the parameter p(z). Some of its principal advantages are the following.

1) "Numerical robustness" can be assigned in advance by the adjustment2

of m numerical parameters.

2) It is possible to "tune" the estimator to produce selective spectral

amplification at a finite number of desired frequencies without impairing

either its interpolatory character or its numerical robustness. Enhanced

resolution is to be expected.

I- --i.i

3) For the setting of the numerical parameters which yields maximum

robustness, namely p(z) s" Om p the FEE coincides with the MEE.

In conclusion, we should like to mention that our development of the

general parametric formula for K(e) alluded to above, constitutes the solu-

tion to a long open problem which some researchers, Burg included, have

considered to be prohibitively difficult.

III-iV

I. INTRODUCTION

In a previous publication [1 ] the author developed an entirely new

solution to the problem of spectrum recovery from a given set of covariance

samples of a single random process.

The flat-echo estimator (the FEE) incorporates the following important

features:

1) Numerical robustness is assignable in advance by the adjustment

of a scalar parameter and can be traded off against resolution.

Z) It is possible to "tune" the estimator to provide selective amplifi-

cation at any set of prescribed frequencies without impairing either its

interpolatory character or its numerical robustness.

3) For the parameter setting which yields maximum robustness, the

FEE coincides with the MEE, the maximum-entropy estimator.

Our principal objective in this continuing study is to generalize the

FEE to the multichannel case and we therefore assume that the reader is

familiar with the geometric and analytic points of view adopted in Ref. 1.

Moreover, since this is a research report, our demands on his knowledge

will vary widely but we have nonetheless attempted to be tutorial whenever

possible.

With regard to the matter of notation, most of it is standard and self-

explanatory. However, let us point out that A, A', A (- A ) and det A denote,

in the same order, the complex conjugate, transpose, adjoint and determinant

of the matrix A.

Column-vectors are written a, x, etc. , or as

x= (x1,x2,...,x) (0.0)

whenever it is desirable to exhibit the components. Further, Im is the mxmn

identity, 0m k the mxk zero matrix and 0 is the m-dimensional column-

vector.

As usual, if A=A is hermitean, A>0(>0) means that x*Ax>0(>0) for

every vector x /0. We then say that A is nonnegative or positive-definite,

as the case may be.

IrT-1

In this paper, we distinguish carefully between the right-half p-plane

and the interior of the unit circle since some of our arguments are couched

in both analogue and digital terms.

Thus, if W(p, z) denotes a matrix whose entries are meromorphicje

functions of both p= C+jW and z= r , it is convenient, for typographical

reasons, to set

W, (p, Z) - *-p , 1/z) (0. Z)

Clearly, for p=jw and z=e , w and e real,

W (jW, e j ) = W*(jW, eje) (0.4)

and the sub-star coincides with the top-star.

Let x stand for either p or z and let / 2 denote either the open right-half

p-plane, Rep> 0, or the open unit-circle, IzI <1.

An mxm matrix Z(x) is said to be positive if 1) it is analytic in R2and

2),

Z(x) + Z*(x) > 0 , x E (0.6)

Similarly, an mxm matrix W(x) is said to be bounded if 1) it is analytic in/2

and 2),

1 -W(x)W*(x) >0 , xE/2 (0.8)m

If in addition, Z(x) or W(x) is real for all real xEI2, it is said to be

positive-real or bounded-real, respectively.

A positive matrix Z(x) which is a. e. skew-hermitean on the boundary

of 2is said to be lossless while a bounded matrix W(x) which is a.e. unitary

on the boundary of/Ris said to be regular paraconjugate unitary. If Z(x) is

also real, it is Foster and if W(x) is real it is regular paraunitary.

111-2

II. FORMULATION

Let x t denote a discrete-time m-dimensional random vector process

where "t" can traverse all positive and negative integers. Let us also sup-

pose that the process is zero-mean and second-order stationary. Then,

E(xtxt+k) = C(k) 1k! = o - (1)

is the associated mxm covariance function. Clearly, for all integers k,

C(-k) =E(xtxk) E(xtxk') = C*(k) . (2)E( txl~~ -k= (t+k- t

As is well-known [2], if the power spectrum F(e) of the process is

absolutely continuous, there exists an mxm hermitean nonnegative-definite

spectral density matrix K(9), whose entries are absolutely integrable 2 over

-T< e< r and is such that,

C(k) = 1" (3)e

Ck f ejkK(e)de (3)

Hence,

Qo

K(6) - T C(k)e j k e (4)

and the mxm covariance samples C(k), 1kI =0-x, emerge as the Fourier

coefficient of K(O).

Moreover, if the process is free of essentially determined components,

K(O) satisfies the Paley-Wiener criterion,

f f ndetK()de> - o (5)

- r

IVery often, we do not distinguish between the function C(k) and theparticular sample C(k). Hopefully, when taken in context this should nevercause confusion.

2For short, K(e)EL I .

111-3

Clearly, such a process automatically obeys the full-rank condition,

detK(e) / 0 , a.e., (5a)

but not conversely.

From the inequality K(8)> 0 it is easily shown [6 ] that

c(0) Cl) . . . C(n)

C(-l) C(0) . . . C(n-l)Tn =. . . . . . . . . . . . . . . . . >-- =0-oo . (b

C(-n) C(-n+l) . . . C(0)

Furthermore, if the process is full-rank (and, a fortiori, if it satisfies

Paley-Wiener), only the inequality sign prevails.

Unless stated explicitly otherwise, we shall adhere to two main working

assumptions.

A1 ) The power spectrum F( e) of the process is absolutely continuous,

i.e.,

dF(@) = K(e)de (6)

where K(e)£L1 and

K(9) = K(,(0) > 0 (7)

A2 ) Either the process is full-rank or its satisfies the Paley-Wiener

condition (5) or both. The precise situation can always be ascertained from

the c ontext.

Our primary task is to develop a physically insightful solution to the

problem of estimating K(6) from a knowledge of n+l given error-free mxm

covariance samples C(0), C(1), .... C(n).

II1-4

III. THE Zm-PORT INTERPOLATORY CASCADE

By way of motivation, let us recall the properties of a cascade N of

n ideal TEM lines which have the same 1 -way delay T> 0 but arbitrary positive

characteristic impedances R 0 , RI, .... Rnl1 [I .

Let SN(p) denote the scattering matrix of N normalized to R0 at the in-

put and any positive port number at the output. Then, SN(p) is completely

parametrized by the specification of a single arbitrary real polynomial h(z)

of the form3

h(z) = ZhI+z 2h +...+Zn h. (8)

Explicitly,

CI) with h(z) so prescribed, choose the real polynomial g(z) (of degree

< n-i) to be the strict-Hurwitz solution of the equation

g(z)g- (z) = 1+h(z)h..:(z) . (9)

This Wiener-Hopf factor is analytic together with its inverse in Izi <I1

and is uniquely determined under the restriction g(0) >0.

C2 ) Let-ZpT

z = e . (9a)

Then,

-n[ep (10)gN( ) 9 z) z n n h (z)

It is readily seen that SN(p) is a real meromorphic matrix function of

p that is analytic in Rep>0 and satisfies the identity

SN(P)SN.(p) = 12 . (11)

Thus, SN(p) is regular paraunitary.

3 The property h(0)= 0 reflects the fact that the input-side normalizationnumber has been chosen equal to R 0 , the characteristic impedance of thefirst line.

TII-S

The immediate goal is to generalize the above construction to the 2m-

port case, m> 1. However, due to the lack of commutativity of matrix

multiplication, several subtleties arise for m > 1 which car. only be dealt with

by modifying the procedure in a substantial way.

Lemma 1. Let H(z) denote an arbitrary (and not necessarily real) mxm

polynomial matrix of degree < n of the form

H(z) = zH +z H 2+.. .+z n H . (12)

There exist two mxm polynomial matrices GI(z) and G2 (z) of degrees <n-I

such that

1) det GI(z)" detG2 (z) /0, 1zj< I . (13)

2) The matrix

L(z) = G1 (z)H(z)G 2 (z) (14)

is polynomial.

3) '*I(rer the identification z = e T > 0, the ZmxZm matrix

GlI(z)H(z) e-nP TGI(z)

enS n(-1) (15)N e-rPG2 l(z) -zn G I (z)H. (z)

is meromorphic in p, analytic in Rep>0 and paraconjugate unitary:

SN(P)SN*c(p) = Zm . (16)

Proof. Let G (z) and G,(z) be chosen as the mxm Wiener-Hopf factors

of the two matrix equations

G (z)G,*(z) = lm+H(z)H,(z) , (17)

G 2 (z)Gz2 (z) = Im+H,(z)H(z) . (18)

Clearly, because of the form (12) of H(z), both H(z)H .,(z) and FH,(z)H(z) are

paraconjugate hermitean matrices with powers of z restricted to lie between

-(n-1) and n-i.

111-6

From a classical result [3*, Gl(z) and GZ(z) are polynomial and uniquely

determined up to multiplication on the right by arbitrary constant mxrn unitary

matrices. In addition, their degrees do not exceed n-I and they are minimum-

phase, i.e.,

detGI(z) detG 2 (z) / 0 , Izi< 1 . (19)

But, as it is obvious that the right-hand sides of (17) and (18) are also

nonsingular for all z =ej, 8 real, we actually have the stronger inequality (13)

which includes the boundary of the unit-circle, Izi = 1.

If we now observe that

S()=G 1 1 (Z) [ H(z) e~nT1m (0N 0 GZ1(z e-npTlm -znH(z) (

it is straightforward to verify (16 ) with the aid of (17) and (18). Thus, SN(P) is

meromorphic and regular paraconjugate unitary. 4

Since a right-inverse is also a left-inverse, (16) implies that

SN,(P)SN(P) = l2m . (21)

In turn, (21) yields5

(G G,) "I + H- (G 1G I, )11 H = 1m (22)

H(GzGz*) - (G 1GI*)H 0 m (23)

(G1 G,1 *) + H(G2 GZ*)-H* = 1 m (24)

From (Z3),

G I(z)H(z)G2 (z) L(z) = Gl*(z)H(z)Gz*(z) (25)

4It follows from (13) that G, (z) and G 2 (z) are analytic in zII so thatthe domain of analyticity of SN(p) includes the jW-axis in the p-plane.

5Arguments are omitted wherever convenient.

111-7

But the right-hand side of (25) is analytic in Iz I whereas the left-hand

side is analytic in I z I 1 and this means that L(z) is analytic in the entire

finite z-plane. By Liouville's theorem, L(z) is a polynomial matrix and the

proof of lemma 1 is complete, 0. E. D.

Comment 1. If H(z) is chosen real, G 1 (z) and G 2 (z) can also be con-

structed real and SN(p) is a real regular paraunitary matrix.

Let us now suppose that a Zm-port N of generic type defined in lemma 1

is closed on its output side on a passive m-port load with normalized scattering

matrix description r(z). Of course, r(z) is mxm and bounded and therefore

analytic in I z I< 1.

Let the four mxmn partitions in (15) be labeled S 1 1 (p), S 1 2 (p), SZ 1 (p) and

S 2 2 (p). Then, from first principles [4],

S=S 1 1 + Slzr(1m-S2 2 7)' S2 1 (26)

is the normalized scattering matrix of the resultant m-port NM

Thus, making the appropriate substitutions and using (17),

1~ n-i-I -S(z)- GH+z G Ir(i +znG ir-I G (27)

_1 1 m 2 11' 2

=G 1 1(HG 2 (lm+ zn G 2 1H ) r)+zn)(G znH, r) (28)

=G 11 (HG2 + zn(Im+ Hi,)F)(G + znH 1) -1 (29)

=- 1 (HG + znG iGlr )(G 2 + zn-, F r (30)

G1 2 z 1 -1

(L+znGl.3)(G 2+z H,)I-r (31)

= (L(z) + d1(z) r(z))(Gz(z) + i4(z) F(z)) 1 (32)

66

where L(z) =G[ (z)H(z)G 2(z) and 6

l(Z)-a z nGl*:(z) ,(33)

6 (i(z ) and A(z) are the mxm matrix polynomials reciprocal to Gl(z) and

H(z), respectively.

111-8

-

fl(z) -z n H, (z) (34)

Consequently, since it has been shown in lemma 1 that L(z) is polynomial,

S = (L+ G 1 ")(G2 + 14 r)' (35)

appears as a bilinear matrix transform of r with polynomial matrix coef-

ficients that depend on z alone.

Lemma Z. Let

r(z) = zp(z) (36)7

where p(z) is an arbitrary mxm bounded matrix in z. Then, the first n+l co-

efficients in the power-series expansion of

S = (L+ zd1 p)(G 2 + zflp) - (37)

about z =0 are independent of the choice of p(z).

Proof. Let S (z) denote S(z) with P(z) =0 . It suffices to prove that0 m

S(z) - S (Z ) = 0(z n + l ) (38)

in the neighborhood of z = 0.

Evidently,

S(z)= L(z)GI(z)=G 1 (z)H(z) (39

since G 1 L=HG 2. Thus,

S-S o =(L+ z1P )(G 2 + zAp) -G IH (40)

so that

G 1(S-So)(G 2+ z1p)= G I (L+ zGI ) - H(G2+ z P) (41)

-- z(G 1 j 1 - Hft)p (42)

= zn+l P (43)

Such a r(z) is automatically bounded. Moreover, every S(z) generatedby (37) is also bounded if c(z) is bounded.

II-9

from (17). Or,

S-S = z +lG_ 1 P(G +zflp) -1 (44)

However,because G1 (0) and G,(0) are nonsingular and p(z) is analytic in

Iz I:l, (44) implies the desired result (38), Q. E.D.

Comment 2. Since S(z) is bounded,

Z(z) = (1 +S(Z))(1 M-S(z)) 1 (45)

is positive whenever it exists. Clearly, as is seen from (37) and (12),

S(0) = L(0)G2 (0) =G1 1 0 ()= 0 m (46)

so that neither lm+ S(z) nor 1m -S(z) are identically singular. Thus, Z(z) and

Z (z) are both well-defined.

Let us write

n k +Z(Z) = 1 m+ E ZC(k)z + O(Zn'l (47)

k= 1

According to lemma 2, the mxan coefficients C(k), k= 1-*n, are independent of

the particular choice of bounded p(z) and are therefore calculable from the

power-series expansion of

Z (z)=( +S (z))(1 S S(z))1. (48)

Since the factors in (48) commute we can also write

Z0 =(l -s) -(1+S ) (49)

= ( m-G I H) (1 M+G 1H) (50)

= (G-H) '(G +H)=2P- Qn (51)

where

P(z) G I(z) -H(z) (52)

and

111-10

G (z) + H(z)Qn (Z) (53)

By direct calculation and Eq. (17),

P(Z)Qn--(Z) + Qn(z)Pn*(z) = 'm (54)

and we are now ready for the remarkable lemma 3.

Lemma 3. 1) The polynomial matrices P n(z) and Qn(z) are nonsingular

in the closed unit circle, i.e.,

detPn(z), detQn(Z)#O , IzI < i (55)

Z) Z z (P,(Z)Pz W 1 (56)2 n(*()Pn)

3) The coefficients C(k), k= l-*n, determine the interpolatory

Zm-port N completely up to within a frequency insensitive all-pass cascaded at

the output side.

Proof. 1) Suppose first that for someto, zo < 1, det Pn(zo) = 0. Then,0nothere exists a nontrivial m-vector a such that a*P (zo) = 0 .

Thus, from (51) and the analyticity of Zo(z) in Izl < 1,

a*Pn (z )Z(z )=0' =Za*Q (z) (57)n o 0 m n n

Therefore, invoking (5Z) and (53),

a*(P (z ) + n(zo))= a'G (z (58)n o n o 1 o -m

which is impossible because Gl(Zo) is nonsingular in I zI <1.

On the other hand, if Izol =-1, Pn*(zo)= Pn(zo), Qn*(zo)-Q*(Zo) and

(54) evaluated at z = z reads

Pn(zo)Q*(z) + Qn(z)P (z= 1 (59)

n0 0-11

Thus, by multiplying (59) on the left and right by a* and a, respectively,

we obtain a*a = 0, another contradiction.

2) According to (51) and (54),

Z + Z o p - (.0 n1 - -( n Qn* n * (60)

3) Let

Pn(Z)=A 0 +AlZ+...+Azn (61)

where the A's remain to be determined in terms of the C's.

From (60),

Z 0 (z) + p ( z ) = z n p-Iz) 62

n n

where

p (z)=zp Pn(z)=A*+A- lZ+...+Azn " (63)n Wei n n-i 0

Part 1) permits us to conclude that Zo(z) and P'l(z) are both analyticn

in IzI <1 and it is therefore legitimate to set z = eje, e real, and to equate

coefficients of e on both sides of (62), all k.

Carrying this out for k= 0 - n we obtain the linear system,

A*XT = [1 Om... 0 (64)0 n mm m

where

X-d [AA0 A1 .. An (65)

and 8

rI m C(1) ...G(n)

n C(-n) 1 n" "+ ... ] n (66)

C(-n) C(-n+1) ... I j)[ m8 For any k, C(-k) -C*(k). Matrices which possess the structure in (66)

are said to be normalized block Toeplitz.

I-12

Since

Z (eje) + Z*(eJe)Ko 0 0 ( n (eJO)Pn(eJ) 1 (67)

is obviously positive-definite and L over -17"< e< T, and since it also satisfies

the Paley-Wiener criterion (5), it follows [6] that

T=T* > 0 (68)

n

The solution of (64) is now routine.

Let A 0 be determined from the Gauss factorization

A*Ao = (1 -EnT_ iVn ) - 1 (69)

-o 0-lnn-

and let T ) denote the matrix formed with the first m rows of Tn. Then,n 1; n

X =(AP) (T n)l; (70)

Equation (69) fixes A0 uniquely up to multiplication on the left by a

constant mxm unitary matrix U. In view of (70), P (z) is then also unique upn

to within the same pre-factor. The next step is to produce a formula for the

coefficients of Q (z).

Let

Qn(Z) =F0 + FZI z + Fnz (70a)

and let

Y =F 0 F ... Fn (71)

From (51),

2 Qn (z) = pn(z)Zo(z) (72)

and a comparison of coefficients of like power of z , k= 0 -*n, yields

ZY =X (73)

n

where

111-13

i

I m ZC(1) ... ZC(n)

1

X =(74)n

2C(1)

1M

Thus, Qn (z) is also uniquely determined up to within the unitary premultiplier U.

1 +Consequently, H(z)=P (z) - (z) and Gl(Z) = P(z)+Qn(z) are both

determined up to multiplication on the left by U, and then G 2 (z) can be found

as the minimum-phase polynomial solution of (18). As mentioned previously,

G (z) is unique up to multiplication on the right by an arbitrary constant mxM

unitary matrix V.

If we now return to (15), it is immediately seen that S1 1 (p) is unique,

S1 2 (p) is unique up to the right multiplier U*, S2 1 (p) up to the left multiplier

V* and S 2 2 (p) up to the left-right multipliers V* and U, respectively. But

this can be interpreted to mean [4] that every compatible interpolatory Zm -port

N is obtained from a particular one, Np, say, by cascading N on the right in a

Zm-port all-pass with normalized scattering description

! m(75)v* I l m

This completes the proof, Q. E.D.

The preceding lemma suggests a very natural definition.

Def. I. A set of mxmn matrices C(k), k= 1- n, is said to admit the

strong interpolation propert? (the SIP), if there exist two mxm polynomial

matrices P (z) and Qn(Z) of degrees less or equal to n such that,

1I1" det P (z) detQn(z) / 0, 1zI < I ;

1 . Pn (Z)Q n*(z)+Qn(z)Pn*(z) = 1m

111-14

1 3' For IzI < I,

nZ(z) E ZP n(Z)Q n(z) 1 ZC(k)z k + nZ+ I

k= I

1 4* Z 0 (z) is positive.

Lemma 4. A set of mxm matrices C(k), k=I 1-n, admits the SIP iff

I 1 CMi C (n)

Tn C(-I1) 1 * C(n-1) >0 . (76).... .I. *LC(-n) C(-n+ 1)

Proof. Let the data admit the SIP. Then, from Ii. 12V and the definition

of Z (z),

K((8es)Z(e) 0 (P*(e36je)P (eije) (77)

is L1 and positive -definite over - T< < TT.

Furthermore, in view of 1 31

fe _jK (e)de= C(k) ,-n<k<n ,(78)

and since it is obvious that K0 (e) is nonsingular for all e, it follows from thediscussion at the end of Section II that T > 0.

Sufficiency. A review of the proof of lemma 3 shows that the poly-

nomials Pn(z) and Qn (z) must be given by the formulas,

Pn(Z) = X (Z) , Qn(Z) = Y4 (Z) (79)

where X and Y are determined by means of Eqs. (69)-(74) and

9(z) 11 i m . m ) (80)

III-is

Of course, the positive -definiteness of Tn guarantees that P (z) and

n(z) are well-defined and it remains to be shown that they possess all the

requisite properties.

Note first that I is equivalent to

. ,Z* + n g*)X = * 1 (81)

But, as is easily seen,

(Z) ,(Z)= 1+1Q(z) + 1 (z) (82)

where

0m

z1 0m m

zzImz 1 zi

zn n-1 *.zz 1 zn'1 . . . Z1l

rn m m M

Hence, since

T - 1 + ) (84)n Z n n

9and 06 commutes with M n we find that

n n1 Tn++1 (P) (85)

T T+ rl* n n +""2- n (86)

=T +SIT +T 0 87n n . (87)

9 To a great extent, it is this commutativity that is responsible for so

many of the remarkable properties exhibited by Toe litz matrices.

111-16

Now, by using the explicit formula X (A 0 ) - 1 (T n we obtain,

in succession,

XT = (A*)) (A*') 1 m (88)n 0A) 0(8

XTnn = (A*) [ 0 . . . r =0 (89)

and

X(T +,T +T )X*=i . (90)n n n m

Thus, 12 is validated.

The remaining three properties can be disposed of almost simultaneously

by proving that Z o(Z)=2P1(z)Qn(z) is positive in IzI< I.

First, P n(0)=A But, in view of (69) and the premise T n > 0, A 0 is

nonsingular so that detP n(z)k 0. Thus, Z 0 (z) is well-defined and analytic in

some neighborhood of z = 0.

By construction, its power series expansion about z = 0 must be of the

form

cc

Z (Z) = 1 r+ E ZC(k)zk (91)

k= 1

where C(1), C(Z), .... C(n) constitute the prescribed data. Let r> 0 denote

the radius of convergence of (91). Of course, since we have not yet estab-

lished that Pn(z) is minimum-phase, i.e., that detP (z)/0, Izi < 1., it can-nn

not be asserted that r>1.

Nevertheless, by equating coefficients on both sides of the identity

n 00 n

7A1 Z' ).(m+ C (k)zk2~ F1 z (92)

I =0 k=1 1 =0

we obtain the formulas,

111-17

where a , I =O0 - (93)

wher Yis given by (74) with n replaced by 1,

C = (94)

and

F 0 F IFI

(95)

F 0

(The matrices OLIand 0,are upper -triangular block Toeplitz.)

Our aim is to prove that

TI LJ z =(-1 (a + 6 B * C* > 0 (96)

for all I > 0. Or, equivalently, that

Y,~ + C*> 0 0 1=- (97)I II

By hypothesis, T n> 0 and (97) is correct for I 0 -n. Clearly then,

the desired result V> 0, 1 > n, follows trivially if it can be shown that for

all such I

, m (I -n+ 1) Yn-I (8

111-18

THE FEE , A NEW TUNABLE HIGH RESOLUTION SPECTRUM ESTIMATOR.IW)

FEB 82 H E WEBB, D C YOULA, 0 M GUPTA

UNCLASSIFIED RADCTRBIL397 NL

2 2-A 1 99 f l f IR D L OP ENl f lT R Rf l f l f l flBNYf f l f l f l

M fEMELEhhh

IB .1 IIIJII 111 .

MiCROCOPV RE IS[Ii ION T t T (-HAW

To accomplish this we shall exploit the already-derived "even-part"

condition,

Pn(Z)Qn(Z) + Q(z)Pn (z) = 1 * (99)

We leave it to the reader to show that a direct comparison of like

powers of z on both sides of (99) leads to the 1-step update,

Yn = Im + Yn-I (100)

Since (100) is (98) with I =n, the natural way to proceed is by induction,

Evidently,

10and ,+ (101)

where10

A1 z= [AIAZ ... A + (102)

and

F1 2 = IF FF...F1 " (103)

By direct calculation,

Y+ + (104)

in which,

1 +1

Y1 1 = (A F*+ FRA) (105)

k= 0

and

Y1 2 1--AA"A 1 + 1 1+ F 1 F 2''F,+ 1 ) (106)

1 0 For I>n, A I =F 1 - OM .

111-19

-I --

If we now take stock of the previous footnote, it is quickly observed

that (105) and (106) reduce to

n

Yll = (AkF + FkA) (107)

k=0

and

1 2 [AA...A]n + [F F2."" Fn - (108)

respectively.

However, by equating the (1, 1) and (1, 2) blocks on the two sides of

(100) it is found that Y = I and yi1= 0 whence,

Y1 +1 = lrn1 m •. (109)

But from the induction hypothesis,

m( = In(I -n+1) Yn-I (110)

so that (109) expands into

+1 + V = -) -1 +l,2 n -)(I _n+l) +Yn-l = I -n) + r n-l ti l

and the induction step is completed. Thus,

T I > 0 , I = 0- 0. (112)

According to a generalization of a theorem of Schur (Appendix A),

(112) is a sufficient condition for the power-series0

I + E ZC(k)zk (113)

k= 1

to define a positive function of z in IzI< 1.

However, since this series and Z0 (z) coincide in jzj < r, they must,

by analytic continuation, coincide in all of IzI < 1. We have therefore shown

that Z(p) =2Pn1(z)Qn(z) is positive and it remains to check property II.

111-20

rI

The positivity of Zo(z) automatically guarantees its analyticity in

I z < 1. Hence, as shown in lemma 3, if for some zoo IZ I < 1, there

exists a nontrivial vector a such that a*P = ' , then necessarily,Zn n 0 0ma*Q (z )OZ n o =m

Consequently, from (99),

naPn(zo)d n(Zo) + *Qn(zo)P(n(Zo) = zn 0~ = m (114)

which implies that z = 0. But this is a contradiction because P n(0) = A° is

nonsingular.

Similarly, since it follows immediately from Eq. (59) that Pn (z) and

Qn (z) are both nonsingular on the boundary of the unit circle, the assumption

I Zo = I is equally untenable. Finally, by observing that

Zol11(z) l(z)Pn(z) (115)

is also positive, we can apply exactly the same reasoning to establish the

nonsingularity of Qn(z) in Izi < 1, Q.E.D.

Corollary. Let the mxm matrices C(k), k= 1-*n, admit the SIP. Then,

there exists an interpolatory Zm-port N such that

nto(z) = lm+ 7 2C(k)z k+0(z n + 1 (116)

k= 1

Furthermore, N is unique up to a frequency-insensitive Zm-port all-pass

cascaded on its output-side.

Proof. Let us employ the procedure de. ' - '-4 in lemma 4 to construct

P (z) and Q (z) from the prescribed data. The i know,n n

detPn(z), detQn(z) 0 , IzI< 1 (117)

and Z (z) ZP (Q (z) is positive.0 n n

111-21

As before, let

G

(z) T Pn(z) + Qn(z) (18)

and

H(z) = P(z) - Q(Z) (119)

Since property 12 is satisfied,

G 1 (z)G 1,*(z) 1 +H(z)H*(z) (120)

and it remains to prove that GI(z) is minimum-phase.

Obviously, the positivity of Z 0 (z) implies that of 1 m + Z 0 (z) and this

entails that the latter be nonsingular in Iz I < 1. But in view of (118),

det (I+ Z omdet G I(z)det( m +0 (z)) = 2 m detPn(Z) (121)

and it follows that detG (z) is free of zeros in IzI < 1.11

The construction of N is now completed by determining G 2 (z) as the

minimum-phase solution of (18). Of course, the arbitrary constant unitary

matrix right-multiplier V can again be absorbed in an output Zm-port all-

pass, Q.E.D.

Comment 3. The recursion in (98) was discovered (in the scalar

case) by the author and published for the first time in Reference 7. It is

another manifestation of the incredibly rich substructure exhibited by block-

Toeplitz matrices.

It is also important to understand that the integer n is det, -mined

by the number of data and serves only as an upper bound on the degrees of

Pn(z) and Q n(z). For example, the set C(k) 0, k= 1-n, admits the SIP

sinceT n = 1> 0 . (122)

Actually, detGI (z) 0 in Iz< 1.

111-22

Clearly, in this case

(T) l( = [ 1 0 . . . 0 (123)

and

En =[0 m 0 m . . . 0 ] (124)

so that

A0=1 (125)0 m

and

Pn(z)1 =m = (z) (126)

are compatible choices. Thus, P (z) and Qn(z) are constants and total de-

gree reduction has occurred.

At this stage we are ready to state and prove our first major theorem

which serves as the foundation stone for the multichannel spectral estimation

technique developed in the next section. However, we must first insert

another definition.

Def. 2. A matrix Z(z) is said to be full-passive if it is positive

and

Z( ej e ) + Z*(eJ)K(e) ? ~( (127)

is nonsingular for almost all e in -iT< e< rr; i.e., if

detK(8) 0 , a.e. (128)

Similarly, an mxm matrix r(z) is full-bounded if it is bounded and

det(l1 - r*(eJ )r (e e ) 0 , a.e. (129)m

Theorem 1. Let C(k), k= 0 -n, be any given set of n+1 constant

mxm matrices such that C(0) = I m and T > 0, and let N denote tlie (unique)

associated interpolatory Zrn-port defined in the corollary to lemma 4.

111-23

Let n stand for the collection of all mxm full-passive matricesn1

nZ(z) = 1 + Z ZC(k)zk + O(z ) . 1130)

mk= I

(Any such Z(z) is said to interpolate to the prescribed data.)

Then, the class 9 coincides with the set of all mxm positive matricesnZ(z) generated by formulas (45) and (37) in which p(z) is an arbitrary mxm

full-bounded matrix.

Proof. For n=O, Q0 is the set of all full-passive mxm matrices

Z(z) such that Z(O) = Im. Hence, in particular,

S(z)= (Z(z) - lm)(Z(z) + m' (131)

is bounded and equal to 0m for z = 0. Thus, by SchwartzIs lemma, 12

S(z) = zp(z) , (13?)

p(z) bounded.

In other words, all positive Z's that satisfy the interpolation con-

straint Z(0) = 1 are obtained by letting P(z) range over all bounded manmmatrices in the formula,

-1Z (I + z)(lM- z) . (133)

But Eq. (132) is (37) with H(z) = 0 and Gl( G2(z ) = Im so that them I1 z G2 z) s

corresponding interpolatory Zm-port N has the normalized scattering description,

F0 1iS (p) Jni m (134)

[im mI

Algebraic manipulation of Eq. (133) yields,

1 0IM-je P(eJe ))*K( m)(m- eje P(eje)) = I- p*(eJe)¢(e j ) (135)

1 1Let r (z) be an mxm bounded matrix such that r(0) 0m . Then [9r(z)/z is also bounded.

111-24

Consequently, Z(z) is fuli-passive iff p(z) is full-bounded and the case

n0O is verified. For n>I1 we proceed by induction.

12'First, Z(z) e 0 obviously implies that Z(z) CO Let the Zm-portn n-V*

interpolatory cascade N 1 constructed with the data G(k), k= 0 -1n- 1, be 1parametrized as described in lemma 1 by the mxm polynomial matrix H 1 ) 13

Of course, degree H I(z) <n-i and H 1 (0) = 0 m Let G II(z) and G 2 (z) denote

the companion left-right mxm polynomial Wiener-Hopf factors:

G 1 1 G 1 Im+ HIHI* ,(136)

Also, let

L1()=G I I(z)H I(z)G 1 2 (z) .(138)

The induction hypothesis permits us to assert that

Z = (1 + S)(1m - S)(139)

where (Eq. (37)),

S = L 1+ za11 1)(G1?' ZA -n(140)

and -n(z) is some full-bounded mxmn matrix. Thus, making the appropriate

substitutions in Eq. (44) and replacing n by n- I we obtain,

5(z) -S I (z)10 = G I r(G 1 2 + z 1 1)' (141)

in which

S (z) -1,IzHlz (142)

According to lemma 2, the power-series expansions of S(z) and S I (z)

about z = 0 agree up to z nland this means that the quotient on the left-hand

1Z n>0 implies that Tn..1i> 0. Incidentally, we already know fromlemma 4 that in is not empty.

13 This Hl(z) must not be confused with the constant matrix H, appearing

in (8).

ITI-25

side of (141) is analytic in Izi < 1. However, (139) determines the power-n nseries expansion of S(z) up to z because that of Z(z) is known up to z

If we couple this with the observation that S, (z) is completely fixed

by the data C(k), k= 0 -n-1, it is concluded from (141) that

S(z) - Slo(Z)

S(0) = G 1 1 (0) n 0 G12(0) (143)z=,zz= 0 .

is a known quantity. Hence, rj(z) must be selected from the class of mxm

bounded matrices which at z = 0 interpolate to the prescribed value given

in (143).

Clearly, Vn(z) bounded implies that fl r(0) 11 < 1 but actually, f1 '(0) 1 < 1.

Suppose otherwise. Then, Im - r#(0)7i(0) is singular and there exists a

vector a of unit norm such that

a = *(0) p(0)a . (144)

Or,

a = '*(0)b (145)

where

= 0)a . (146)

Evidently,

a*a = b*b = 1 (147)

Construct two constant mxm unitary matrices V and U which incorporate

a and b, respectively, into their first columns and let

6(z) = U*r)(z)V . (148)

The matrix 6(z) is mxm and bounded and 65, i(0) = 1.

Since all elements of 6(z) are analytic and bounded by unity in I zI < 1,

we conciude by maximum-modulus that 61 1 (z) 1 1. In turn, the requirement

i m - 6*(z)6(z)> , jzj< 1 , (149)

111-26

then forces all the other entries in the first row and columnn of 6(z) to be

identically zero so that

6(z) =1 16 1(z) ,(150)

61(z) an (m-l)x(m-1) bounded matrix. Thus,

1 V(1~)=V*( 1 (l -6*6))V (1)M_ TIm-6in=V( 1( - 11

is clearly singular for all I z <1 which contradicts the full-bounded character

of n~(z). Consequently,

1m - n*(0)7)(0) >0 (152)

and the renorrnalization scheme described in Appendix B becomes available.

Let Ra and Rb be any pair of hermitean matrix solutions of the Eqs.,

2Rba = 1 rn-'nl() *(0) .(154)

Then, from Schwartz's lemma and the renormalization theorem in Appendix B,

R (1 TI TI * (0))l 1 1~~O)R =zpz ,(15

p(z) an mxm bounded matrix. In addition, it is seen from Eq. (B 10) that 1(z)

is full-bounded iff p(z) if full-bounded. The last step is to salve (155) for TI(z),

substitute into (140) and then rearrange properly.

The end result is

S5 = (L+zd IP)(G 2 + zIRP) 1(156)

where

a R (H + zn(00)12 ) ,G1Ra (G 1 1 + zn(0)L(157H aR 1G a 1, 17

111-27

L = (L +zG151(0))R~1 , GZ=(GlZ+zHlTI(O))pRI o (158)

The proof that G (z) and G (z) are minimum -phase is easy. For ex-

ample, since

G= Ra1 (I+ 1(0) Gll) 1 1 (159)

it is clear that G I is minimum-phase iff the middle factor in (159) is minimum-

phase. But L =G II HI G1 so that =G- 1 G and

1 1* 1 * 12*1

LIG 1 1 =-G . (160)

However, from (137) and maximum-modulus it is seen that G 1 2 (z)H 1 (z)

is bounded. Thus, since 11zn (0)I1 < I, for Izj < 1,

Im + zr(O)Ll(z)G8 (z) (161)

is a positive matrix and its determinant is necessarily devoid of zeros in

Iz[< 1. A similar proof applies of course to G(z).

Equations (157) and (158) can be combined into the single matrix version,

L. Hi, [ 1, H,1=T(z) • [ I I j (162)Li.

where

T(z)= (R- I 1 R ) m z n j0) (163)a z'n 71(O) 1m

Let

1 0]S'p 0 0m i a .

Then, by direct verification,

111-28

T pT, IFo ( 164)

and

-1 = -

L1,L 1 Glz Gz= Gz HIGIIGIIH1Gz -G zGz

=G1(HH1-G G - )G , -1 m 164a)

Henc e,

GZ P ap P (z • O165)

i. e.,

*GGI= I m +HH* (166)

*G Z*z = 1 m + H*H (167)

and

L = Gxo I HG I, HGZ* ( 168)1 2 G1 *HZ*(6)

In short, it has been shown that every member of 1 can be synthesizedn

by an appropriate full-bounded termination of the output-side of the 2m-port14

N defined by H(z). Since part 3) of lemma 3 assures us that this 2m-port is

I an essentially unique construct from the full set of coefficients C(O), C(l), C(n),

the proof of theorem 1 is complete, Q.E.D.

In the next section the procedure developed in theorem I will be given a

condensed algorithmic formulation and then applied to the problem of spectral

estimation.

1 4 Note that degree H(z) < n and H(O) 0m

111-29

IV. THE CLASS OF ALL SPECTRAL ESTIMATORS

Let us now suppose that we have available n+l error-free mxm co-

variance samples C(O), C(l), ... , C(n) of an m-dimensional full-rank

random vector process xt. We shall produce a parametric formula for the

class of all spectral densities K(s) consistent with the given data.

Theorem 2 (The Master Result). Let xt denote a full-rank m-dimensional

random vector-process and suppose that its first n+l matrix mxin covariance

samples C(0), C(1), ... , C(n) are known without error.

Let T n and T n(-I) denote the Toeplitz matrices constructed on C(0),

C(l), ... , C(n) and on the "reversed" data C(0), C(-I), ... , C(-n), respectively.

(Clearly, Tn(-I) is obtained by replacing each block C(k) in Tn by C(-k), k= 0-tn.)

Let

"(z) = 1 z n 1 (169)m in m

Then, the following 3-step algorithm generates all spectral density matrices

K(O) compatible with the given data.

1) Affect the (unique) Gauss factorizations,

T = A* A , T (-1) = (-1(-1)an(' ) (170)n n n n n n

where An and A n(-I) are both lower-triangular with positive diagonal scalar

entries. Let M and M (-I) represent the two submatrices formed with then -1n .first m columns of A and A n (-I), respectively.

2) Define two mxm polynomial matrices

Pn(z;l) = M n (z) (171)

and

P (z;-1)= 1Zn~;l '(z)M (-l) ,(172)

of degrees < n and let 15

15Pn(z;-1) = znP (z;-I).

111-30

D (z) P (z;1)- zp(z)P (z;-]) (173)n n n

3) Then, 16

K(O) D Dn(e Je)(, - p(e je ) P*e Je ID (eJe) (174)

where p(z) is an arbitrary full-bounded mxm function.

Proof. Assume that C(0) = I m and let us show that Pn (z;1) Pn(z).

As we known, Pn (z) = X (z) where

X-- (A-) (Tn )1; (175)

and

AA0 = (I -EnTn_ )* . (176)

But clearly, from T'_n = A(A- ) and the lower-triangular charactern n nof An, it follows immediately that

(T )=A -1. M4 17n 1; 00 n (177)

where A0 0 is the upper left-ha lid corner mxm piece in A . Furthermore,0 thn

since the matrix on the right-hand side of (176) is the upper left-hand corner

mxmn piece in T-1 we can also make the identification,n

A* = A00" (178)

Thus,

-X A.0* MS M (179)X = A00 100A00 n n

and Pn(Z) = M*n (z) Pn(z;1). To arrive at an understanding of the meaning of

Pn(Z;-1) we must derive a general expression for K(O).

Let C(k) denote the covariance function of the full-rank process x t . Then

necessarily,

16 D jej@) = D(e J), e real.

111-31

T k > 0 and T k(-1 ) > 0 (180)

k = 0-oo

Hence, invoking the multivariable Schur theorem,

00

Z(z) = Im + Z ZC(k)zk (181)

k= 1

is a positive mxm matrix.

Clearly,

00- ke = Z(eJe) + Z*(eJ0O (182)K(e) - F C(k)e (12

k=-oc

where

Z(e ) -- limit Z(re i ) . (183)r- 1-0

To describe all spectral densities K(8) consistent with the data it is

sufficient to exhibit a formula for the hermitean parts of all full-passive

mxm matrices,00

Z(z) = I + ZC(k)zk + 0(zn) . (184)

k= I

As a preliminary step we shall use lemma 2 to generate all corresponding mxm

bounded functions,-I

S(z) = (Z(z) - 1 M)(Z(z) + 1 M ) (185)m m

Indeed, every such S(z) is given by Eq. (37) with P(z) an arbitrary full-

bounded mxm matrix. Nevertheless, it turns out to be more convenient to

work with the equally valid expression

S(z) = (G + zPL) '(H + zP , (186)

that is obtained by reducing

111-32

S S + Sl1 2 (- rS 2 2 'IrS 1 (187)

instead of (26).

A straightforward calculation with the aid of (186) yields, for iz 1,

(G 1 + zpt)(l- SS*)(G 1 +zp)* = 1 -O (188)

Or, since

-S) (l -SS'M1~* 19m m - S- ) (189)

and

Im -S(G 1 +zPL) - 1(G I -H-zp(G 2 - L)) , (190)

we obtain

Z + Z -1 * -D Om- PPIDn*l Iz = I (191)

where

Dn(z) = Gl(z) - H(z) - zp(z)(G (z) - L(z)) (19z)

Hence,

Dn(Z) =Pn (Z;) - zp(z)P (Z;- ) (193)

if it can be demonstrated that

Pn(z;- 1) =G 2 (z) - L(z) (194)

agrees with (172).

From Eqs. (25) and (39),

LG 2 G1 H = ( (195)

so that

II I- 33

%-

Zo-- (1+ S)( -( ) - S = (Gz+L)(Gz-L) - . (196)

Thus,

Zo+Zo10 0*_2 ((G 2 -L)(Gz -L)) ; (197)

i.e.,

Zo + Zo -1(G -L) z = • (Gz -L) . (198)

Since

GGZ (z)G(z) = Im + L(z)L(z , (199)

it follows that uSo(z)II < 1, 1z < 1. Hence,

(I m - (z))G,(z) = Gz) - L(z) (zoo)

is nonsingular in Iz < I and the exact same reasoning used in lemma 3 to derive

(171) also yields (172) when applied to (198).

To complete the proof we must remove the restriction C(0) = 1 andmthis is accomplished by exploiting the following series of observations.

O . There exists a lower-triangular matrix R with positive diagonal

entries such that

(R*)-C(O)R-l = 1 m • (201)

O. The use of (174) in conjunction with the normalized data

(Ro)c(k)Ro, k=0-n , (202)

is tantamount to reconstructing all admissible normalized spectral densities,

(R'" K(e)Ro - (203)0 0

111-34

0 3 The lower-triangular factorizations associated with the "forward"

and ureverse-direction" Toeplitz matrices constructed on these normalized

samples equal, respectively, An(l) and A n(-I) multiplied on the right by

A'1 =(R R . 2 Z04)

Thus,

M (l) -A M (1) , (205)n o n

M n(-l1) -~ A oM(- 1) ,(206)

Pn(z;-l) -1 '(z) AoMl-) n(208)

and

Dnl) - M* (1) A*WzzP(z)M*(- 1)A*1(z) - z (209)

0 Since

(R= KR 1(I m'P D (210)

we see that

K = (1- P )Dn* (211)

where

Dn(z)=D .(R)" =M (z)-ZPIz)M*(- )i(z)=: (z;l)-Z0(z)P (z;-1) =D (z )

(212)

Q.E.D.

Comment 4. The simplicity of the general formula (174) is truly re-

markable, especially when it is realized that Pn(z;l) and Pn (z;-1) are inti-

mately related to the right and left matrix orthogonal polynomials defined bythe sequence C(k), k= 0 -*n. This means [6, 103 that both can be generated

recursively (and simultaneously) by readily available and highly efficient

algorithms of the Levinson type, thereby obviating any need to compute the

inverses An I and An1(-1).n n 1-35

Although we intend to resume our studies of these matters in depth

we shall nevertheless qive the explicit connection formulas for

the benefit of the reader who wishes to proceed directly to the numerical

implementation of (174).

Namely, if 00 and A(-l) denote the upper left-hand corner mxrnNaely repetvey (n (n)A0,

blocks in An and A(-l), respectively, and if Q(n)(z) and R(n)(z) represent,

in the same order, the monic left and right degree-n matrix orthogonal

polynomials alluded to above, then [63,

Q(n) (z) = P~n(Z;l 00 (ZZa)

and

R (z) = A00 (-I)P n (Z; - l) •(212b)

Consequently, Eqs. (136)-(142) in Reference 6 become immediately available

after a suitable 1-i map of some of the symbols.

111-36

V. THE MULTICHANNEL FLAT-ECHO ESTIMATOR

Under the assumption that x t is a full-rank random vector-process with

absolutely continuous spectral function, a knowledge of its covariance samples

C(O), C(l), ... , C(n) permits us to assert that all admissible interpolatory

K(G) are given by (174) where p(z) is any full-bounded mxm function.

Evidently, as is seen from this expression, K(8) satisfies Paley-Wiener

iff the same is true of

I - (eje ) P*(e je)

This granted, there exists [61 an mxm bounded matrix function B (z) which is

analytic together with its inverse in IzI < 1, such that

m - Pie j ) p*(ejel = B (e j )B(e j e (213)

for almost all 8 in -TT < G< t.

From (213) and (174) it follows that

K(e) = B(eJe )B*(e Je (214)

where

B(z) = DI (z)B (z) (215)n P

is also analytic together with its inverse in Iz I< 1. Hence, formula (215)

constitutes a complete parametrization of the set of all admissible spectral

density Wiener-Hopf factors consistent with the given data, in terms of the

free bounded matrix-function parameter B P(z).

Since 1 7

B(0) = D (O)B (0) = P((0;1)B (0) = Pn(0)B (0) (216)

n p n p n p

and

1 -B (0)B*(0) > 0 , (217)m p P1 7 Remember that Pn (z) and Pn (z;l) are identical.

111-37

'4 .- 'n -

we see that

P(O)Pn(0) -B(0)B*(0) = Pn(0)(I m - B (0)B*(0))I*(O) > 0 (218)

In particular, 18

IdetP n(0)> IdetB(0)l (219)

Furthermore, because Pn(0) is nonsingular, equality in (219) is possible

iff B (0) is a unitary matrix. But then, it follows routinely from maximum-

modulus that B (z) = B P(0) for all z and this in turn implies that

p(z) O . (220)m

According to (173) and (174), p(z) = 0 m corresponds to the choice of

estimator

KME(O) = (Pn(eJe)Pn(eJe))- 1 (221)

and invoking the result19

iT

21n~det]Pn(0) 1 + Z2njdetB (0)1 = Z1nIdetB()I J fIn detK(e)d -Entropy (K)

-TT

(222)

it is immediately concluded that KME(G) is the maximum-entropy estimator

[I I ]. In other words, KME(e) is obtained by the termination of the basic

interpolatory Zm-port N in a matched m-port load, r(z) o m

Alternatively, it is seen from (174) that the entropy of any admissible

K(e) associated with the choice of bounded function p(z) is given by, 2 0

Entropy(K) = gntropylKE - ft e"11 - 0e @ *(eJ )lde <Entropy libEl.~M)fin det (1m -P(e p(~)e1 - -1

-T E(223)

1 8 1f A and B are both hermitean nonnegative-definite, det(A+ B) >detA+ detB.

19 Apply Cauchy's theorem to the analytic function IndetB(z) and observe

that

detK(@)= IdetB(e )I

z0_ je * jSince Im - P ( e ) p(e )> 0, a. e., the reciprocal of its determinant is a

number which is greater than unity for almost all e.

111-38

- ' . .. " ..... .. .. .... ... |i == nm .. . ... ... .. . .. ...... ... .•

Clearly, equality is possible iff p(e J) 0 a. e. in -T<e<TT; i.e., iffm -

K(e) = KME(G).

In view of (2Z3), it is legitimate to define

I In det" ( p(e0 3)p*(eJe)) (224)

to be the entropy echo-loss density at angle e. A flat-echo estimator (an

FEE) is one for which this density is a constant independent of e.

One particularly simple (and numerically effective way) to generate

FEE's is to choose

p(z) = pd(z) (225)

where IL is an arbitrary constant mxrn matrix such that

l - P *P> 0 , (226)

and d(z) is an arbitrary rational mxm regular paraconjugate unitrary matrix. 21

Thus, for this subset of FEE's,

KFE(e) = D'l(ej9)(Il-,UPP)D: (e j o (227)

where

Dn(Z) = Pn(z)- zid()1(z;-I) , (228)

and

Entropy(K FE Entropy(KE) - In det-l m -( m PP (229)

Note, that the entropy of KFE(e) can be fixed in advance by an appropriate

assignment of p and d(z) can then be designed to accomplish the necessary

tuning. The latter emerges as a problem of interpolation with regular mxm

matrix all-passes.

Although a complete (recursive) solution has already been obtained for

the case m I 1 2, much work remains to be done for general m.

d(z)d*(z) = d*(z)d(z) Im and d(z) is analytic in Izi _ 1.

111-39

REFERENCES

1. D. C. Youla, "The FEE: A New Tunable High-Resolution SpectralEstimator. Part I - Background Theory and Derivation, " TechnicalNote No. 3, PINY, Contract No. F3060Z-78-C-0048, RADC.

2. N. Wiener and P. Masani, "The Prediction Theory of MultivariateProcesses, " Part I, Acta Math., Vol. 79, 1958.

3. D. C. Youla, "On the Factorization of Rational Matrices, " IRE Trans.Information Theory, Vol. IT-7, pp. 172-189, July 1961.

4. H. J. Carlin and A. B. Giordano, "Network Theory: An Introductionto Reciprocal and Nonreciprocal Networks, " Prentice-Hall, EnglewoodCliffs, New Jersey, 1964.

5. N. I. Ahiezer, "The Classical Moment Problem," Hafner PublishingCo., New York, 1965.

6. D. C. Youla and N. N. Kazanjian, "Bauer-Type Factorization ofPositive Matrices and the Theory of Matrix Polynomials Orthogonalon the Unit Circle, " IEEE Trans. on Circuits and Systems, Vol. CAS-25,No. 2, February 1978.

7. D. C. Youla, J. D. Rhodes and P. C. Marston, "Recent Developmentsin the Synthesis of a Class of Lumped-Distributed Filters, " CircuitTheory and Applications, Vol. 1, 59-70, 1973.

8. V. Belevitch, "Classical Network Theory, " Holden-Day, San Francisco,1968.

9. V. P. Potapov, "The Multiplicative Structure of J-Contractive MatrixFunctions, " Trans. of the Amer. Math. Soc., Series Z, Vol. 15, 1960.

10. A. C. Vieira, "Matrix Orthogonal Polynomials with Application toAutoregressive Modeling and Ladder Forms," Ph.D. Dissertation,Stanford University, Stanford, CA, August 1977.

11. J. Burg, "Maximum Entropy Analysis, " presented at the 37th AnnualMeeting, Soc. Explor. Geophysics, Oklahoma City, Oklahoma, 1967.

111-40

APPENDIX A

The Multivariable Schur Theorem. The power series

00

Z(Z) = I + Z ZG(k)z k (Al)

k= 1

defines an mxrn positive matrix in I zI< 1, iff

Tk >0, k=I-c (AZ)

Proof. Suppose that Z(z) is positive. Then, by definition, Z(z) is

analytic in I z < 1 and

Z()+Z()>0,IZt< 1 .(A3)

Hence, for every fixed r, 0 < r < 1, and for every mxni trigonometric

polynomial matrix

f(e) f 0 + f Ieie+.+f ke-k (M4)

e real, k> 1,

1if T*(eZ(re + Z*(reje) f(e)d > 0 .(AS)

-TT

Or, carrying out the integration,

&I~IfC(q-1 )f > 0 (A6)I ,q=0

where C(0) aIM

In the limit as r -+1 1-0 with k held fixed, (AM) goes into

k

I ,q=

111-41

and because of the arbitrary character of the rnxm matrices fop f 1 " fk

we conclude that Tk> 0.

Sufficiency. From Tk>0 we deduce, in particular, that

[c* lZ> 0 (A8)

i.e., for every k>0,

I C(k) 11 <I • (A9)

Thus, if Z(z) is defined by (Al) we obtain the estimate,

00

IIZ(z) 1 I + Z 2 lC(k)j• IzIk < (AlO)k= I

oo l+l

< I + E , (Al1)

k=l 1-1zi

I z < 1. Consequently, Z(z) is analytic in Iz < I and it only remains to

demonstrate its positivity.

This can be achieved by observing [53 that

00 goZ (Z) +Z*(Z) 1 ( kk 7- -k I

zz ~C Ikz+ C(-k)z ) (AIZ)2(l -zz) I=0 k=0 k=l

00 00 00 0

S (k) z z C(-k)zI=0 k=0 I =0 k=l (A13)

00 00 00 0:2 2c(a-s)ZI7O+ T c((x- pz-O0-O a=p M--O O=m+l

(Al4)

111-42

- . P.I i

00 C kE C(X-O)z z lim~it AcaOf>

(A15)

where

(In reaching (A15) we have used (A7) and have also noted that the first

sum in (A14) contains all the elements on and above the main diagonal of the

doubly-infinite array

MCL O~ M 0 0 0 00(AP7)

while the second contains all those strictly below.)

Since 1 - Izi? > 0 for IzI < 1, (A15) yields

Z (z) + Z* (z)>0 Z< (A8

and Z(z) is positive, Q. E. D.

111-43

APPENDIX B

Our proof of theorem I made use of the following interesting result

long known to network theorists.

The Renormalization Theorem. Let C be any constant rnxm matrix0

such that 1jColl < 1. Let R and Rb denote, respectively, any pair of

hermitean mxrn matrix solutions of the two equations,

Rz =I -C C* (B1)a m 0 B

and

Rb 2 -C 'C (BZ)b 0 0 o

Then, if W(z) is an mxrn bounded matrix, so is

r(z) = Ra(1 - WC)(W-C )Rb (B3)

a m 0 0 b (

Proof. The obvious identity

C( I-C C*) 1 (1 - C C (B40 m 0 Ml 0 0 0 (4

yields,

0 R C- 2 (B 5)

Hence, upon multiplying both sides of (BS) on the left by C we obtain0

R za = I + C oR Co (B6)

Similarly,

R - = I + C*R a ZC (B7)b m o a a

To establish the boundedness of r(z) it is necessary to prove that

1nm- (z) F"(z) >0 , Iz I < (B8)

IIX-44

-T 44

From (B3), (BI), and (BZ) we obtain,

Im - w c *)R - I (l m r r , )R - I ( i m c o W*)=(1 m Wc *)R2 (1-Co w , ) . (w -c o )R ' (w , . c o )o n a m a 0 o a o o'b 0

= 2 W * - c o CC*+wC*Rzc W* +=Ra Wpt2*Cobb+C

+CARZW*-R-C W*+WRQ'-C*-WC*R ° 2OD a o b 0 o a

1 -WW* (B9)

Thus,

I- rr*=Ra(1m-WCo)- I (I mWW*)(1m-CoW*)-IR > 0 (BIO)

since Im -WW* > 0 (by hypothesis), Q. E. D.

The proof of the following impedance version is considerably simpler

and is therefore omitted [8 ].

Corollary. Let C be a constant mxm matrix such that

C +C*R 0 0 > 0 Ii.1)

O 2

Let R1/2 equal the (unique) hermitean positive-definite square-:eot of Ro.0

Then, if W(z) is an mxm positive matrix,

r(z) = Ro'/-(W-Co)(W + C* ) - I R / (B 12)

is an mxm bounded matrix.

NOTE: By inverting (B12) we obtainC -C*

W(W) = Ro/0(Im + r(. - r) + (B13)

In particular, if CO is also hermitean, Ro= C and

0/ 0

W(z) = C0 /2(im + )(m- r)-I C 1/2 (B 14)0 m 0

By starring both sides of (BS) we get Ra C Co Rb2

a-45 b

' __

PART IV

Some observations andSuggestions for New Work

by

Haywood E. Webb Jr.

IV. SOME OBSERVATIONS AND SUGGESTIONS FOR NEW WORK

The discussion on The Interpolatory Cascade, in Part I indicatesthat every admissible spectral estimator can be interpreted as the realpart of the driving point impedance of a cascade of uniform transmissionlines of characteristic impedances given by (63) terminated in an arbitraryload impedance W(Z) as shown in Fig 1. For given sample covariance data, itis a different load impedance that corresponds to a different spectralestimator. The FEE is a particular class of estimators which provides aset of "tuning" parameters. The setting of all the work here is in adiscrete setting. It is the digital computer implementation that drivesthe discrete setting. We recall that a process with a continuous covariancefunction, in order to be non determinstic, must satisfy the continuousanalogue of (7) Part 1.

fILog A(w) < (i)l+w

2

Where A(w) is here the spectrum obtained from the Fourier transform of the

covariance. But note also, that any function which satisfies (I) is not

bandlimited, so that a discrete sampling representation introduces aliasing

errors at the outset, and the problem, from a continuous viewpoint, isimproperly formulated. The concept of entropy does not make sense.

In the network formulation of the spectral estimation problem, thediscrete cascade, can be generalized to a distributed continuous trans-mission line. Again, we conjecture, that every admissible spectrumobtainable by a "continuation" of a segment of the covariance function, canbe obtained by a non-uniform transmission line with a shunt capacitor,terminated in an arbitrary positive real load impedance. It would beinteresting to maximize the continuous analogue (1) in this section to(7) in Part 1.

Note that under these circumstances that (1) of Part IV cannotbecome infinite under the assumptions. After all, (1) of Part IV is anecessary and sufficient condition that the inverse Fourier transform ofA(w) not be quasi-analytic i.e., not be uniquely deterministic over itsdomain from its derivatives at a point.

It is proposed that further exploration of the continuous problem bemade, and that after understanding it, that digital computer implementationsbe considered in the light of the results.

IV-1

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