-
`TRANSACTIONS ON
`A TENNASAND
`-....
`ROPAGATION
`
`JANUARY 1982
`
`VOLUME AP-30
`
`NUMBER 1
`
`(ISSN 0018-926X)
`
`A PUBLICATION OF THE IEEE ANTENNAS AND PROPAGATION SOCIETY
`
`UNIVERSITY OF CALIFORNIA
`LOS ANGELES
`
`JAN 2 0 1982
`
`EM,,;c.c.nlm.> &
`MATHEMATICAL
`SCIENCES LIBRARY '{IS
`
`PAPERS
`Space-Time Integral Equation Approach to Dielectric Targets ... . .. . . . .. . .... . ..... . .... . . . H. Mieras and c. L. Bennett
`A Low Elevation Angle Propagation Measurement of 1.5-GHz Satellite Signals in the Gulf of Mexico . .... . .. . ......... . .. .
`· · · · · · · · · · · · · · · · · · · · · · : · · · · · · · · · · · · · · : · · · · · · · · · : · · · · · .. . . .. ... . . ... . . D. J. Fang; F. T. Tseng, and T. 0 . Ca/vit
`Diakoptic Theory for Multielement Antennas . : . .. .. ...... . . . . ... . . .. ... . .. . G. Goubau, N. N. Puri, and F. K. Schwering
`. . ...... . .... . . .. . .. L. J. Griffiths and C. W. Jim
`An Alternative Approach to Linearly Constrained Adaptive Beamforming
`Transmission into Staggered Parallel-Plate Waveguides . .. .. . . . . .. . .. .. . . . .... . .... . .. . ...... ... L. Grun and S. W. Lee
`Radiation from an Open-Ended Waveguide with Beam Equalizer-A Spectral Domain Analysis ... . ...... . .. . . . ... . .. . . . .
`. . . . . . . . . . . . . · · · · · · · · · · · · · · · · · · · · · · · . · . .... .. . .. . . ... .... . . .. . W. L. Ko, V. Jamnejad, R. Mittra, and S. W. Lee
`Dielectric Tapered Rod Antennas for Millimeter-Wave Applications .. . . . .. . .. .. . .. . S. Kobayashi, R. Miitra, and R. Lampe
`Optimization Techniques and Inverse Problems: Reconstruction of Conductivity Profiles in the Time Domain .. ... . D. Lesselier
`Feed Region Modes in .Dipole Phased Arrays . . . . . . .... . . .. .. ... . . . .. . ..... .. . . ... . . ... . .. . E. D. Mayer and A. Hessel
`Scattering of a Dipole Fi'i:ld by a Moving Plasma Column ... ... .. . . . .... .. . .. ....... . ... . . ... .. . . ... . . . K. Nakagawa
`Surface-Curvature-Induced Microwave Shadows .. . .. . . .... ..... . . . . . . . . . . ... .. . . M. H. Rahnavard and W. V. T. Rusch
`A Discussion of Various Approaches to the Identification/ Approximation Problem ...... . . . . . . . ... . .. .. . .. . . . . . . .. . ... .
`. . .. ... . ... . ......... . . .... .. . .. . ... .. . .. .... . . . ... . . . . . ... . . T. K. Sarkar, D. D. Weiner, J. Nebat, and V. K. Jain
`Unified Theory of Near-Field Analysis and Measurement: Nonmathematical Discussion . . . ... . . ... . ... . ... .. . P. F. Wacker
`Precision Experimental Characterization of the Scattering and Radiation Properties of Antennas . . . . . . .. .... . . .. . . . ... . .. .
`........ . . . . ........ . . . .. . . . . .. .. .. . ... . J. J. H. Wang, C. W. Choi, and R. L. Moore
`Effi~i·e~~ ·C~~;p~.t~~i~~-~f· A~~~~~~ Coupling and Fields Within the Near-Field Region .......... . . . . .. . . . . . . A. D. Yaghjian
`. .. . ...... . .. . · · · · · · · .. · · · · · · . .... · · · . . . . ... S. Zohar
`Adaptive Arrays: A New Approach to the Steady-State Analysis
`
`COMMUNICATIONS
`
`I
`
`. .. . . . . ... . .. . . D. A. Hill and J. R. Wait
`'th n Elevat'lon Change
`M' d p th
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`· · · · · · · · · ·
`1xe
`a WI
`Ground Wave Propagation over a
`a
`Howard and M. Gero iokas
`s. w. Lee and L~ Grun
`A Statistical Raindrop Canting Angle Model · · · · · · · · · ·: · · · · · · · · · · · · · · · · ·: · · · · · · · · · · · · · · J.
`Radiation from Flanged Waveguide: Comparison of Solutions · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
`A A Mohsen
`On the Integral Representation of Electromagnetic Field Vectors · · · · · · · · · · · · · · · · · · · · · · ~ ·~ · ~: ~r;s~h· ~~d ~- ~- ~anselow
`Boresight-Gain Loss and Gore-Relateq Sidelobes of an Umbrella Re~ectord: ·S· · · · · · · · · · · ·
`· ·
`T Satoh and A . Ogawa
`. · · · · · · · · · · · · · · ·
`U · g Celestial Ra 10 ources
`A
`Exact Gain Measurement of Large Aperture ntennas sm
`.. . . R. C. Hansen
`· 'i · ·M·.j1: · t -Wave Applications" .. . .. . · · · · · · · · · · · · · · · · · · · · · · · · · ·
`formation Ratio
`· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
`h D. 1 T
`d T M
`F ld d
`o e an
`-
`ate
`1po e
`rans
`Correction to "Dielectric Rod Leaky-Wave Antennas or 1 lme er
`.S. Kobayashi, R. Lampe, R. Mittra, and S. Ray
`... . ........... . ....... · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · : .. . .. . .. . .. . . .. .. .. . . . . ..... W.-M. Boerner
`Comments on the Bojarski Identity · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
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`- i -
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`Sony v. Jawbone
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`U.S. Patent No. 11,122,357
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`Sony Ex. 1005
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`Sony v. Jawbone
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`

`

`-- ..._ _ _.....
`
`N':CJM B E.R 1
`
`(ISS N· 0 0 1 8- 9 2 6 X)
`
`lN SOCIETY
`
`UNIVERSITY OF CALIFORNIA
`LOS ANGELES
`
`I/\ 1\f 2 0 '"1°82 .
`
`.__, i-\ I ~ .
`
`..,;
`
`Ef~ \.=111--4~i.:.KII'i\:a &
`MATHEMATICAL
`SCIENCES LIBRARY "'(I$/
`
`2
`
`.
`. . ..... ~ ... · .. . · ... . ......... H. Mieras and C. L. Bennett
`ite Signals in the Gulf of Mexico ................. . . .. .
`. . . . . . . . . . . . . . . . D. J. Fang: F. T. Tseng, 'and T. 0 . Calvit _ ---10
`. ............ G. Goubau, N. N. Purl, and F. K. SchM1e.ri.~ig.
`15
`L J G ; rr,· h
`. d-. ~ .... J ..
`. ning
`. . . . . . . . . . . . . . . . . .. . . . . .
`rz11 z t- s ana ~-: Vt: qn
`_ 27
`.................. L. Grun. and :.S. -riV ... Lee
`35
`~ .. . . . . . . . . . . .
`. ,
`, Spectral Domain Analysis · ·: · · · · · · ·: · · · · :\_:_;~; : . - _;: : ; ·
`IU 1 Kn. V. Jamnejad, R. Mzttra, ana 0. V(. Lee . 44
`
`~
`
`.~
`
`- ii -
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`

`

`•
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`IEEE ANTENNAS AND PROPAGATION SOCIETY
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`- iii -
`
`

`

`IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP·30, NO. I, JANUARY 1982
`An Alternative Approach to Linearly Constrained Adaptive Beamforming
`LLOYD J. GRIFFITHS, SENIOR MEMBER, IEEE, AND CHARLES w. JIM
`
`27
`
`A/uiTGCI-A beamformJaa structure Is presented wblcll can be used
`to Implement a wide variety or linearly constrained adaptive array
`proceaon. Tbe stradare Is deslped for use wltb arrays wblcb bave
`been time-delay steered sacb that the desired slpal or Interest
`appean approximately Ia phase at the steered outputs. One m~or
`adnntap or tbe new stracture Is the constraints can be Implemented
`uslnaslmple hardware dllferendna amplmers. Tbe strac:ture Is shown
`to lllcorporate alaorltbms wbkh ban been suapsted previously ror
`use Ia adaptive beamformJaa as we~ as to Include new approaches. It
`Is also parttcalarly useful for studybiatbe effects or steerlna errors 00
`array performance. Numerical elWIIples lllustratlnatbe performance
`or tbe stracture are presented.
`
`INTRODUCTION
`
`THIS PAPER describes a simple time-varying beamfonner
`
`which can be used to combine the outputs of an array of
`sensors. The beam former is constrained to filter the "desired"
`sign~.wi~ a :~t~r ha~g. a pr~scribed gain and phase response.
`The demed Signal 1S Identified by time-delay steering the
`senso~ outputs s~ that any signal incident on the array from
`the direction of mterest appears as an identical replica at the
`outputs of the steering delays. All other signals received by the
`array which do not have this property are considered to be
`noise and/or interference. The purpose of the beamfonner is
`to minimize the effects of noise and interference at the array
`output while simultaneously maintaining the prescribed fre-
`quency response in the direction of the desired signal.
`Beamfonners of this type are termed linearly constrained
`array processors and have been studied by several authors
`including Levin [II. Lacoss [2). Kobayashi [ 3). Booker and
`Ong [4), Frost [5], and Applebaum and Chapman [6f. The
`last five of these authors describe iterative or continuously
`adaptive beamfonners in which the beamforming coefficients
`adjust to new values as each new set of samples of array sensor
`outputs are received. Adaptive methods are of particular
`interest in those problems in which the interference properties
`are either spatially or temporarily time varying.
`The purpose of this paper is to present the linearly con-
`strained adaptive algorithm, due to Frost [5) , using an alter-
`native beamfonning model. This presentation illustrates the
`fundamental properties of the algorithm in an exceedingly
`simple fashion. It also allows for generalizations not available
`with Frost's method. The basic structure of the beamfonning
`model has been suggested by Applebaum and Chapman [6).
`In this paper we describe the structure in detail and give exact
`algorithm comparisons for a variety of linearly constrained
`
`Manuscript received May 19, 1980; revised March S, 1981. This
`work was supported in part by the Office of Naval Research, Washing-
`ton, DC, under Contract N00014-17.C-OS92 and by the Electronics
`System Division (AFSC), Hanscom AFB, MA under Subcontract
`14029 with SRI International, Menio Park, CA.
`L. J. Griffiths and C. W. Jim are with the Department of Electrical
`Engineering, University of Colorado, Boulder, CO 80309.
`
`beamfonners. The structure is shown to be a direct conse-
`quence of Frost's method. One major advantage of our ap-
`proach is an assessment of the performance degradation caused
`by the steering and/or gain errors in the array sensors. In most
`practical situations the theoretically ideal requirement of an
`"identical replica" of the desired signal, at the output of each
`steering delay, is seldom met. The effects of these errors on
`overall beamfonner performance is easily modeled using our
`approach. For example, it is shown that these effects are
`particularly detrimental under conditions of high signal-to-
`noise ratio (SNR).
`A second reason for this presentation is to enumerate cer-
`tain difficulties which may arise with the use of constrained
`adaptive array processors which do not incorporate Frost's
`error-correction feature. Of the papers referenced above,
`four (see [2)-[4) and [7)) use an algorithm based on the
`gradient projection approach [8) . (Levin's approach was
`nonadaptive and utilized matrix inversion techniques.)
`In this paper we first review Frost's algorithm which is not
`susceptible to roundoff error and requires relatively few addi-
`tional computations per adaptive cycle. A simple geometric
`interpretation illustrating the effects of roundoff errors on his
`algorithm and on gradient projection is presented. The error-
`correcting properties of the approach are identified using this
`illustration.
`We then show that the algorithm can be interpreted using a
`new beamfonning model, termed the adaptive sidelobe cancel-
`ing beamfonner. This structure illustrates the constraint fea-
`tures of the algorithm and shows how additional constraints
`can be added. The error-correcting features are also elucidated.
`Sidelobe canceling is shown to be closely related to the method
`of adaptive noise canceling described by Widrow et al. (9] .
`As a consequence results derived in adaptive noise canceling
`can be applied directly to the linearly constrained adaptive
`beam former.
`
`LINEARLY CONSTRAINED ADAPTIVE BEAM FORMING
`
`We denote the sampled output of the mth time-delayed
`sensor by Xm(k). A total of M sensors are assumed to be
`present in the assumptions of ideal steering:
`Xm(k) = s(k) + nm(k).
`
`(I)
`
`In this express~on s(k) i~ the desired signal and nm(k) repre-
`sents the totality of noiSe and interference observed at the
`output of the mth steered sensor. A beamformed output
`signal Y(k) is formed as the sum of delayed and weighted
`Xm(k). Specifically, if am,l is used to represent the weight used
`for the mth channel at delay l, then
`
`M
`Y(k) = 1; ~ am,txm(k -1).
`
`/C
`
`mcJ 1=-/C
`
`(2)
`
`0018-926X/82/0100-0027$00.75 © 1981 IEEE
`
`- 27 -
`
`

`

`IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-30, NO. I, JANUARY 1982
`
`28
`Note that a total of 2K + 1 samples are used from each chan-
`nel and that the zero time reference is at the filter midpoint.
`Matrix notation can be used to simplify this notation. We
`I) represent the filter coefficient and signal
`let A1 and X(k -
`vectors at the lth delay point, i.e.,
`
`A{= [a 1,,, a2,1 ··· aM, II
`XT(k -1) = [x1 (k -1), .x2(k -I),···, XM(k -1))
`(4)
`where superscript T denotes transpose. The output signal of
`(2) then becomes
`
`(3)
`
`this paper we are concerned with Frost's procedure [ S ], in
`which
`
`/1,(k) = gy(k)[qx(k -1)1- X(k -I)]
`
`1
`- Qa,l(k)1 + M/(1)1
`
`and
`
`1
`q x(k - 1) = M xT (k - 1) 1
`
`01)
`
`02)
`
`1
`q0 ,(k) =- A?(k)l.
`'
`M
`The adaptive step size JJ. is a scalar which controls both the
`convergence rate and steady-state noise behavior of the algo-
`rithm (9] and is normalized .bY the total power contained in
`the beamformer. Thus
`
`(13)
`
`(14)
`
`(15)
`
`M
`
`K
`
`P(k) = L L Xm
`2(k -1).
`m=l 1=-K
`Convergence of either algorithm is assured if 0 < a < 1.
`Other power estimates involving time averaging may be em-
`ployed without significantly affecting performance.
`Frost's procedure differs from that used in gradient projec-
`tion [7] by the addition of the last two terms in (11). These
`terms involve a total number of additional (2K + 1)M adds
`and 2K + 1 multiples. They are necessary, however, in that
`they prevent the accumulation . of computational errors which
`may occur on any iteration of the algorithm.
`E"or Effects in Linearly Constrained Beamforming
`The effects of errors may be illustrated by examining the
`constraints (7) for the adaptive algorithm in (1 0) and (11).
`We assume that in the algorithm implementation, the com-
`putation of the signal sum qx(k - 1) and the weight sum
`aa,l(k) in (13) introduced the following errors:
`
`1
`Q x(k -1) = - XT (k -I) 1 + Ex(k)
`M
`
`(16a)
`
`(16b)
`
`or equivalently, the current weight vector A 1(k) is presumed
`to be slightly off the constraint, i.e.,
`
`(16c)
`
`The degree to which the next weight vector fails to meet
`the constraint can then be computed by solving for A{(k +
`1 )1 in (1 O) and (11 ). Thus, using (16),
`A{ (k + 1 )l = f(l) + E~ (k) + p.My(k)Ex(k)
`+ { -f(l) - EA (k) + f(l)}.
`
`(17)
`
`](
`
`y(k)= L AT(l)X(k-1).
`
`(S)
`
`(6)
`
`1=-/C
`Under the ideal steering assumption in ( 1 ), the signal vector
`x(k -I) becomes
`X(k -I) = s(k -1) 1 + N(k -1)
`where 1 is a column vector of M ones and N(k - 1) is a vector
`of noise and interference defmed in a manner analogous to
`(4).
`Prescribed gain and phase response for the desired signal is
`ensured by constraining the sums of channel weights at each
`delay point to be specific values. Thus if f(l) is used to denote
`the sum for the set of weights at delay 1 then
`
`Under this constraint the portion of the output due to desired
`signal reduces to
`
`(7)
`
`K
`
`Ys(k) = L f(/)s(k -1).
`
`1=-K
`
`(8)
`
`\ Thus the f(l) represent the impulse response of a finite-dura-
`p on impulse-response (FIR) filter having length 2K + 1. One
`~mmonly used constraint is that of zero distortion in which
`f(/) = 6(1), where 6(1) is the discrete impulse function. The
`FIR filter constraint function is normalized such that
`
`FT1 = l,
`
`FT = [f(-k), ···,f(k)).
`
`(9a)
`
`(9b)
`
`The objective of linearly constrained adaptive beamforming
`is then to fmd filter coefficients A(l) which satisfy (7) and
`simultaneously reduce the average value of the square of the
`output noise component. This is equivalent to finding those
`coefficients which result in minimum output noise power
`subject to the constraint of the prescribed desired signal.
`filtering.
`In adaptive beamforming the filter coefficients are time
`varying and change as each new set of samples of sensor out-
`puts is received. Thus if A1(k) is used to denote the values at
`time k the values at the next sampling instant k + 1 are com-
`puted as
`A,(k + 1) = A1(k) + .11(k)
`(10)
`where a,(k) is determined by the specific algorithm in use. In
`
`- 28 -
`
`

`

`OIUf'FITHS AND JIM: LINEARLY CONSTRAINED ADAPTIVE BEAM FORMING
`
`29
`
`The terms enclosed in { ·} are produced by error correction
`position of Frost's algorithm while the first three are due to
`· the gra~ient projection operator. Thus if a gradient projection
`adaptahon algorithm is employed-as was the case in (2]-(4]
`and [7]-the constraint error at step k + 1 is
`eA (k + I)= EJt {k) + J,J.My(k)ex(k)
`and with Frost's procedure
`
`(18)
`
`(19)
`
`The cumulative error effects of gradient projection ob-
`served by Shen (7] are due to the first~rder difference rela-
`tionship in (18). If we assume that the driving term iJ.My(k)
`Ex(k) can be modeled as a zero-mean white random process
`'th
`.
`2
`WJ • v:mance ae , and that fJt (0) = 0, then the gradient
`projechon constraint error (18) is a Brownian motion ( 10] or
`random walk process. Although the mean of the error remains
`zero, its variance a A 2(k) grows linearly with the number of
`steps, i.e.,
`
`Fig. 1. Geometrical interpretation for gradient projection adaptive
`algorithm.
`
`Fig. 2. Geometrical interpretation for linearly constrained error-
`correcting adaptive algorithm.
`
`sensor
`number
`
`H
`
`I
`I
`I
`I
`J~N~~Y:E_ON~~N~
`AOAPT I VE AlGOR I THH
`Fig. 3. Direct form implementation of linearly constrained adaptive
`array processing algorithm.
`
`the beamformer is updated by the adaptive processor, which
`computes new values using the algorithm. An alternative
`implementation which achieves precisely the same overall
`processor can be derived in a simple manner directly from this
`algorithm. The resulting structure is termed the generalized
`sidelobe canceling form and is depicted in Fig. 4.
`This processor consists of two distinct substructures which
`are shown as the upper and lower processing paths. The upper
`or conventional beamformer path consists of a set· of flxed
`amplitude weights Wet, Wez, ···, WeM which produce non-
`adaptive-beamformed signal y e(k),
`
`(22)
`
`where
`
`(20a)
`for gradient projection. With the correction terms howe\'er
`th
`,
`,
`e error at each step has constant variance at each iteration ,
`
`(20b)
`
`A simple geometric interpretation [5] can also be given for
`these effects. Consider the geometry associated with the
`gradient p~ojection algorithm shown in Fig. 1. Coefficient
`vectors meeting the desired constraint must lie on the planar
`subspace C defined by the vector F(9b ). It is assumed that the
`coefficient vector ~(k) at time k is too long and that the
`gradient vector produced by the data is g,(k) given by
`
`g,(k) = iJY(k)X(k -I).
`
`(21)
`
`In the gradient projection method the new coefficient vector
`A,(k) is obtained by finding the projection of g1(k) in the
`direction of the plane C, and then by adding this projection
`to the previous vector. As shown by Fig. 1 the resulting new
`coefficient vector will not lie on the constraint plane, even
`with an error-free projection operation.
`Fig. 2 illustrates the geometry for Frost's approach. In
`this case the new coefficient vector is found by projecting the
`sum of the former vector and the gradient in the direction of
`the constraint plane C. The new coefficient vector A1(k) is
`then the sum of this projected vector and the vector F, which
`defines C. As shown in the diagram the new coefficients will
`lie on the constraint plane regardless of the previous error
`provided that the projection operation is error free. The net
`error induced by this method is then restricted to the machine
`quantization error of a single projection operation and accu-
`mulation does not occur.
`
`GENERALIZED SIDELOBE CANCELING MODEL
`The linearly constrained adaptive algorithm defined by
`(10)-(13) may be implemented using the structure shown in
`Fig. 3. Time-delay steering elements T1 , T2, ···, TM are used to
`point the array in the direction of interest. We will refer to
`this implementation as the direct form. Each coefficient in
`
`WeT = (w 1 w 2
`c '
`c '
`
`... w Ml
`,
`c
`
`.
`
`(23)
`This conventional array beamforming system is identical
`to that traditionally used to process sensor array outputs with
`
`- 29 -
`
`

`

`30
`
`1cnsor
`"~.,.
`
`IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-30, NO.1, JANUARY 19&2
`
`where X' and A' are the M - 1 dimensional signal and coeffi-
`cient vectors.
`The overall output of the generalized sidelobe canceling
`structure y (k) is
`
`(29)
`y(k) = Y e'(k)- Y A (k).
`Because y A (k) contains no desired signal terms, the response
`of the processor to the desired signal s(k)l is that produced
`only by y /(k). Thus from (22)-{25) the output due to the
`presence of only the desired signal satisfies the constraint
`defmed by (9), regardless of We. In addition, since Y A (k)
`contains only noise and interference terms, finding the set of
`filter coefficients A/(k) which minimize the power contained
`in y(k) is equivalent to finding the minimum variance, lin-
`early constrained beamformer. The unconstrained lc:ast-mean-
`square (LMS) algorithm [ 12) can be employed to adapt the
`filter coefficients to the desired solution,
`A,'(k) = Az'(k) + 1-!Y(k)X'(k -1).
`(30)
`The step size J.l. is normalized by the total power contained in
`I) using methods analogous to those described
`the X'(k -
`above.
`The algorithm in (30), together with conditions (24) and
`(27), completely defines the operation of the generalized side-
`lobe canceling structure. Although it is not obvious, this
`structure can provide exactly the same filtering operation as
`the constrained beamformer in Fig. 3, which uses Frost's
`algorithm. In addition, it can also provide filtering operations
`which are not the same as Frost's procedure. The key lies
`with the structure of the blocking rna~ W 1 and the conven-
`tional beamformer We. If the rows of Ws are orthogonal (in
`addition to satisfying (27)) and if all conventional beanlformer
`weights equal 1/M, then Frost's method is obtained. Non-
`orthogonal rows and/or other conventional beamformers
`produce a processor having the same steady-state performance
`in a stationary environment, but one which uses a different
`adaptive trajectory.
`The generalized sidelobe canceler separates out the con-
`straint as element "iii, and an FIR filter. In addition, it provides
`a conventional beamformer as an integral portion of its struc-
`ture. Coefficient adaptation is reduced to its simplest possible
`form: the unconstrained LMS algorithm.
`
`Relationship with Linearly Constrained Beam[orming
`The structure of the generalized sidelobe canceler can
`readily be related to the adaptive linearly constrained ~am­
`former. We begin by defining an invertible M X M matrix T as
`
`(31)
`
`Fig. 4. Generalized sidelobe canceling fonn of linearly constrained
`adaptive array processing algorithm.
`
`fiXed nonadaptive coefficients. In typical applications the
`weights We are chosen so as to trade off the relationship be-
`tween array beam width and average sidelobe level [ 11).
`(One widely used method employs Chebyshev polynomials
`to find the We.) For the purpose of this paper, however, any
`method can be used to choose the weights as the performance
`of the overall beamformer will be characterized in terms
`of the specific values chosen. (All Wet are assumed nonzero.)
`In order to simplify notation the coefficients in We are nor-
`malized to have a sum of unity. That is
`
`(24)
`
`~)
`'r
`i
`
`(25)
`
`The signal y/(k) is obtained by filtering Ye(k) and the FIR
`operator containing the constraint values f(l),
`K
`Y /(k) = ~ f(l)y e(k -1).
`1=-K
`The lower path in Fig. 4 is the s!_delobe canceling path.
`It consists of a matrix preprocessor W, followed by a set of
`tapped-delay lines, each containing 2K + 1 weights. The pur-
`pose of "iii, is to block the desired signal s(k) from the lower
`path. Since s(k) is common to each of the ~ered sensor
`outputs (1) blocking is ensured if the rows of W1 sum up to
`zero. SpecificallLif X'(k) is used to denote the set of signals
`at the output of W 1 , then
`X'(k) = w ,X(k).
`(26)
`In addition, if bm T is used to represent the mth row of W,,
`we require that the bm T satisfy
`
`for all m,
`
`(27)
`
`and that the bm are linearly independent. As a result X'(k)
`can have at most M - 1 linearly independent components.
`Equivalently, the row dimension of W, must beM- 1 orless.
`The lower path of the generalized sidelobe canceler gen-
`erates a scalar output y A (k) as tli.e sum of delayed and weighted
`elements of X'(k). Following the notation used to describe
`the linearly constrained beamformer,
`K
`Y...t(k)= ~ (A,'(k)]TX'(k-1),
`P-IC
`
`(28)
`
`The inverse of Tis guaranteed for W c and W3 satisfying (24)
`and (27). In addition, the product Tt is a simple unit vector,
`(32)
`Tl=[l,O,O, ···, O]T.
`Multiplying Frost's algorithm by this invertible transformation
`yields
`Bz(k + 1) = Bz{k) + 1-!Y(k)[qx(k -l)Tl- TX(k - 1)]
`1
`-
`- qa z(k)Tl +- f(l)Tl.
`'
`M
`
`(33)
`
`- 30 -
`
`

`

`GRIFFITHS AND JIM: LINEARLY CONSTRAINED ADAPTIVE BEAMFORMING
`
`31
`
`The transformed weight vector B1(k) can be partitioned in a
`manner analogous to (31) as follows
`
`B,(k) = [~[~~)]·
`B,'(k)
`
`(34)
`
`With this partitioning, and (32), the transformed algorithm
`(33) is recognized as two algorithms: one in the scalar b,'(k)
`and one in theM- 1 dimensional vector B,'(k),
`b,'(k + 1) = b,'(k) + W(k)(qx(k -1)-Yc(k -1)]
`B/(k + Ij = B,'(k) + J.IY(k)X'(k -I).
`
`(35b)
`
`(35a)
`
`These equations may be viewed as an alternative imple-
`mentation of Frost's procedure. Since Tis invertible, the out-
`puty(k) may be expressed as
`Y(k)= k [T 1B,(k)]TX(k-1).
`
`/(
`
`1=-/C
`
`(36)
`
`/(
`
`1=-/C
`
`b,'(k)Yc(k -I)
`
`Tiius if· (35) is used to update the B,(k) and the output is
`computed using (36), this procedure is indistinguishable from
`the original. Many more computations would be required,
`however; and the transformed system offers no advantages.
`We now consider the simplification which arises when T is
`an orthogonal transformation, i.e., when r- 1 = r. The out-
`put equation (36) simplifies to
`y(k)= k
`/( -k [B'(k)) Tx'(k -I).
`
`1=-/C
`
`(:f7)
`
`Inspection of (35)-{37) shows that the transformed linearly
`constrained beamformer in this case is identical to the adap-
`tive-sidelobe canceling beamformer, provided that the b,'(k)
`satisf