20/a high resolution spectrum estimator. (u) 2hhehh~e · 2014. 9. 27. · ad-al14 996 rome air...
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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:
DAVID A. RIMIQ0Y, Lt Co
Coummad and Control Division
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UNCLAS SIFTIEDSECURITY CLASSIFICATION OF THIS PAGE (When Ds,. Entered).RA NSRCIN
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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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.
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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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LU5.
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LL.'4
I- It
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C40. CL4 CS -L
C4 C-18
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L.L
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IT-19
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I~~A I.- I I I
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II I *III
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I act
! * 1 ~C.3I I I I I
II
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----- ---- I ------ ---- r --- 4- ---- cn ~I I ~ II I cc
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--- - - -- -- -- C14I
;1U7 -- -- IA -- - - L -- - - -L - - -- --L -- - -- J C
LAo cm cc co I I-C%-----------4---------- JN C,, C
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4=;;
-- 4----r--- --- , cc-
I I
LLU
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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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