IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-30, NO. 1, JANUARY 1982
`An Alternative Approach to Linearly Constrained Adaptive Beamforming
`
`27
`
`Abstrocr-A beamforming structure is presented which can be used
`to implement a wide variety of linearly constrained adaptive array
`processors. The structure is designed for use with arrays which have
`been time-delay steered such that the desired signal of interest
`appears approximately in phase at the steered outputs. One major
`advantage of the new structure is the constraints can be implemented
`using simple hardware differencing amplifiers. The structure is shown
`to incorporate algorithms which have been suggested previously for
`use in adaptive heamforming as well as to include new approaches. It
`is also particularly useful for studying the effects of steering errors on
`array performance. Numerical examples illustrating the performance
`of the structure are presented.
`
`1
`
`T
`
`INTRODUCTION
`HIS PAPER describes a simple
`time-varying beamformer
`which can be used to combine the outputs of an array of
`sensors. The beamformer is constrained to filter the “desired”
`signal with a fiiter having a prescribed gain and phase response.
`The “desired” signal is identified by time-delay steering
`the
`sensor outputs so that any signal incident on the array from
`the direction of interest 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 beamformer 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.
`Beamformers of this type are termed linearly constrained
`array processors and have been studied
`by several authors
`including Levin [ 1 1, Lacoss [ 21, Kobayashi [ 31, Booker and
`Ong [ 41, Frost [5 1, and Applebaum and Chapman
`[ 61. The
`last five of these authors describe iterative or continuously
`adaptive beamformers 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-
`due to Frost [ 5 ] , using an alter-
`strained adaptive algorithm,
`native beamforming 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 beamforming
`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 5, 1981. This
`work was supported in part by the Office of Naval Research, Washing-
`ton, DC, under Contract N00014-77-C-0592 and by
`the Electronics
`AFB, MA under Subcontract
`System Division (AFSC), Hanscom
`14029 with SRI International, Menlo Park, CA.
`L. J. Griffiths and C. W. Jim are with the Department of Electrical
`Engineering, University of Colorado, Boulder, CO 80309.
`
`is shown to be
`beamformers. The structure
`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 beamformer performance is easily modeled using our
`approach. For
`example, it is shown that
`these effects are
`of high signal-to-
`particularly detrimental under conditions
`noise ratio (SNR).
`for this presentation is to enumerate cer-
`A second reason
`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 [ 21 -[41 and [ 71 ) use an algorithm based on the
`[8]. (Levin’s approach was
`gradient projection approach
`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 beamforming model, termed the adaptive sidelobe cancel-
`ing beamformer. 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. [91.
`As a consequence results
`derived in adaptive noise canceling
`can be applied directly
`to the linearly constrained adaptive
`beamformer.
`
`LINEARLY CONSTRAINED ADAPTIVE BEAMFORMING
`
`time-delayed
`sampled output of the mth
`We denote the
`sensor by xm(k). A total of M sensors are assumed to be
`present in the assumptions of ideal steering:
`Xm(k) = s(k) + n,(k).
`
`In this expression s(k) is 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 2, then
`
`0018-926X/82/0100-0027$00.75 0 1981 IEEE
`
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`Petitioner Waves Audio Ltd. 607 - Ex. 1011
`
`

`
`
`
`IEEE TRANSACTIONS ON ANTENNAS
`
`
`
`
`
`28
`Note that a total of 2K + 1 samples are used from each chan-
`ne1 and that the zero time reference is at the filter midpoint.
`Matrix notation can be used to simplify this notation. We
`let Az and X(k - I ) represent the filter coefficient and signal
`vectors at the Zth delay point, i.e.,
`
`AND PROPAGATION, VOL. AP-30, NO.'l, JANUAqY 1982
`
`this paper we are concerned with Frost's
`which
`Az(k) = py(k)[ q,(k -[)I - X(k - I ) ]
`
`procedure [ 5 ] , in
`
`The adaptive step size I-( is a scalar
`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
`
`I
`
`fY
`
`XT(k - I ) = [x1 (k - Z), x2(k - I), --, x&& - I)]
`where superscript T denotes transpose. The output signal of
`( 2 ) then becomes
`
`(4)
`
`and
`
`1
`q,(k - I) = - XT(k - I ) 1
`M
`
`K
`
`AT(I)X(k - I).
`
`y ( k ) =
`I=-K
`ideal steering assumption
`Under the
`x(k - Z) becomes
`
`in (l), the signal vector
`
`X@--)= s(k-Z)l + N ( k - I )
`( 6 )
`where 1 is a column vector of M ones and N(k - Z) is a vector
`of noise and interference defined 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(I) is used to denote
`the sum for the set of weights at delay I then
`
`AT(I) 1 = f(Z).
`
`( 7 )
`
`Under this constraint the portion of the output due to desired
`signal reduces to
`
`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 (1 1). These
`terms involve a total number of additional (2K + 1)nl 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.
`tion impulse-response (FIR) filter having length 2K + 1. One
`Thus the f(l) represent the impulse response of a finitedura- Error Effects in Linearly Constrained Beamforming
`The effects of errors may be illustrated by examining the
`commonly used constraint is that of zero distortion in which
`(10) and (1 1).
`constraints (7) for the adaptive algorithm in
`f(Z) = S(I), where S(1) is the discrete impulse function. The
`We assume that in the algorithm implementation, the com-
`FIR filter constraint function is normalized such that
`putation of the signal sum qx(k - I ) and the weight sum
`a,,z(k) in (1 3) introduced the following errors:
`
`
`
`FTl = 1,
`
`The objective of linearly constrained adaptive beamforming
`find filter coefficients A(Z) which satisfy (7) and
`is then to
`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.
`coefficients time are
`
`
`
`the filter
`
`
`
`In adaptive beamforming
`varying and change as each new set of samples of sensor out-
`puts is received. Thus if Al(k) is used to denote the
`values at
`time k the values at the next sampling instant k + 1 are corn-
`puted as
`
`(1 6b)
`
`1
`4a,Z(k) = - [AzT(k)1 +
`M
`or equivalently, the current weight vector A d k ) is presumed
`to be slightly off the constraint,
`i.e.,
`1 = f(l) + EA (k).
`to which the next weight vector fails to meet
`The
`the constraint can then be computed by solving for ~ [ * ( k +
`
`1)l in (1 0) and (1 1). Thus,
`using (1 6),
`
`( 1 6 ~ )
`
`WAVES607_1011-0002
`
`Petitioner Waves Audio Ltd. 607 - Ex. 1011
`
`

`
`GRIFFITHS AND JIM: LINEARLY CONSTRAINED ADAPTIVE BEAMFORMING
`
`i'
`
`29
`
`Fig. 1. Geometrical interpretation for gradient projection adaptive
`algorithm.
`
`Fig. 2. Geometrical interpretation for linearly constrained error-
`correcting adaptive algorithm.
`
`The terms enclosed in {*} are produced by error correction
`position of Frost's algorithm while the first three are due to
`the gradient projection operator. Thus if a gradient projection
`adaptation algorithm is employed-as was the case in [ 21 -[ 41
`and [ 71 -the constraint error at step k -I- 1 is
`
`EA (k -b 1) = EA ( k ) @fy(k)ex(k)
`
`and with Frost's procedure
`
`EA (k -I- 1) = W Y ( k ) E x ( k ) .
`
`(1 8 )
`
`(1 9)
`
`of gradient projection ob-
`The cumulative error effects
`served by Shen [7] are due to the first-order difference rela-
`tionship in (18).
`If we
`assume that the driving term pM,,(k)
`can be modeled as a
`zero-mean white random process
`E.&)
`E A ( O ) = 0, then the gradient
`with variance ue2, and that
`projection constraint error (1 8) is a Brownian motion [ lo] or
`random walk process. Although the mean of the error remains
`zero, its variance O A ' ( ~ ) grows linearly with the number of
`steps, i.e.,
`
`U A ' ( k ) = koc2
`
`( 2 0 4
`
`for gradient projection. With the correction terms, however,
`the error at each step has constant variance at each iteration,
`
`UA 2 ( k ) = 0~'.
`
`(20b)
`
`A simple geometric interpretation [ 51 can also be given for
`these effects.
`Consider the geometry associated with the
`gradient projection 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 Al(k) at time k is too long and that the
`gradient vector produced by the data is gz(k) given by
`
`gdk) = PY(k)X(k - 0.
`
`(21 1
`
`In the gradient projection method the new coefficient vector
`Al(k) is obtained by finding the projection of gl(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 Al(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
`by
`The linearly constrained adaptive algorithm defined
`(lo)-( 13) may be implemented using the structure shown in
`Fig. 3. Timedelay steering elements T ~ ,
` 72, -.e, 7~ are used to
`point the array in the direction of interest. We will refer to
`this implementation as the direct form. Each coefficient in
`
`sensor
`number
`
`1
`
`2
`
`H
`
`TAPPED
`DELAY
`c
`
`TAPPED
`
`-T---
`
`Y,(k)
`
`DELAY *
`g-dtp
`
`TAPPED DELAY
`
`I I
`I
`I
`I
`I _ - - - - - -
`LI:IEARLY-CONSTRAINED1
`ADAPTIVE ALGORITHM
`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 fixed
`amplitude weights wC1, w C 2 , --, W,M which produce non-
`adaptive-beamformed signal y , ( k ) ,
`
`where
`
`is identical
`This conventional array beamforming system
`to that traditionally used to process sensor array outputs with
`
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`Petitioner Waves Audio Ltd. 607 - Ex. 1011
`
`

`
`30
`
`IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-30, NO. 1, JANUARY 1982
`
`form of linearly constrained
`Fig. 4. Generalized sidelobe canceling
`adaptive array processing algorithm.
`
`applications the
`In typical
`fixed nonadaptive coefficients.
`weights W, are chosen so as to trade off the relationship be-
`tween array beamwidth and average sidelobe level [ l l ] .
`(One widely used method employs Chebyshev polynomials
`to find the W,.) 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 wci are assumed nonzero.)
`In order to simplify notation the coefficients in w, are nor-
`miriized to have a sum of unity. That is
`
`W,T1= 1
`
`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
`
`Because y ~ ( k ) contains no desired signal terms, the response
`of the processor to the desired signal s(k)l is that produced
`to the
`only by y,‘(k). Thus from (22)-(25) the output due
`presence of only the desired signal satisfies the constraint
`(9), regardless of W,.
`defined by
`In addition,
`since y A ( k )
`contains only noise and interference terms, finding the set of
`filter coefficients Al‘(k) which minimize the power contained
`in y ( k ) is equivalent to finding the minimum variance, lin-
`early constrained beamformer. The unconstrained least-mean-
`square (LMS) algorithm [ 121 can be employed to adapt the
`filter coefficients to the desired solution,
`
`>
`
`The step size 1-1 is normalized by the total power contained in
`the X‘(k - 2) using methods analogous to those described
`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 fiitering operations
`which are not the
`same as Frost’s procedure. The key lies
`with the structure of the blocking matr& W, and the conven-.
`tional beamformer W,. If the rows of W , are orthogonal (in
`addition to satisfying (27)) and if all conventional beamformer
`weights equal l/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 W, 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 Beamforming
`The structure of the generalized sidelobe canceler can
`readily be related to the adaptive linearly constrained beam-
`former. We begin by defining an invertible M X llrl matrix T as
`
`The inverse of T i s guaranteed for_Wc and z, satisfying (24)
`and (27). In addition, the product T1 is a simple unit vector,
`T1 = [ 1, 0, 0, -., 01 T .
`(32)
`Multiplying Frost’s algorithm by this invertiblg transformation
`yields
`Bl(k + 1) = Bik) + py(k)[q,(k - Z)T1- TX(k - I ) ]
`
`The signal y,’(k) is obtained by filtering y,(k) and the FIR
`operator containing the constraint valuesf(Z),
`K
`
`(25)
`
`r(k - 0.
`
`Y ,‘(IC) =
`I=-K
`in Fig. 4 is the sidelobe canceling path.
`The lower path
`It consists of a matrix preprocessor W, followed by a set
`of
`tapped-delay lines, each containing 2K 4- 1 weights. The pur-
`pose of W, is to block the desired signal s(k) from the lower
`path. Since s(k) is common
`to each of the steered sensor
`outputs (1) blocking is ensured if the rows of W , sum up to
`zero. Specificallyif X‘(k) is used to denote the set of signals
`at the output of W,, then
`
`for all m,
`
`(26)
`X’(k) = FJ(k).
`In addition, if bmT is used to represent the mth row of F,,
`we require that the b,
`satisfy
`bmT1 = 0,
`and that the b,
`are linearly independent. As a result X‘(k)
`can have at most M - 1 linearly&dependent components.
`Equivalently, the row dimension of W, must be M - 1 or less.
`The lower path of the generalized sidelobe canceler gen-
`erates a scalar output y~ (k) as the sum of delayed and weighted
`elements of X’(k). Following the notation used to describe
`the linearly constrained beamformer,
`
`(27)
`
`K
`
`I= -K
`
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`Petitioner Waves Audio Ltd. 607 - Ex. 1011
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`

`
`
`
`
`
`GRIFFITHS AND
`
`The transformed weight vector Bl(k) can be partitioned in a
`manner analogous to (31) as follows
`
`(34)
`
`algorithm
`With this partitioning, and (32), the transformed
`(33) is recognized as two algorithms: one in the scalar bz'(k)
`and one in the M - 1 dimensional vector Bl'(k),
`bi(k + 1) = hI'(k) + w ( k ) [ q x ( k - 2 ) -Y& - 111
`Bl'(k + 1) = Ez'(k) + p ~ ( k ) X ' ( k - I).
`
`(35b)
`
`(35a)
`
`These equations may be viewed a s a n alternative imple-
`I mentation of Frost's procedure. Since T is invertible, the out-
`put y ( k ) may be expressed as
`
`x
`
`K
`
`~ ( k ) =
`I=-K
`
`[ T Bl(k)] T X ( k - I).
`
`c
`
`Thus if (35) is used to update the Bl(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 T-' =TT. The out-
`put equation (36) simplifies to
`
`(37)
`
`linearly
`Inspection of (35)-(37) shows that the transformed
`constrained beamformer in this case is identical to the adap-
`tive-sidelobe canceling beamformer, provided that the bl'(k)
`satisfy
`
`for all values of k . Since the b ~ ( k ) must satisfy (35a), this
`will occur only if they are initialized to the values in (38) and
`if
`
`4x(k - 2) = Y,(k - I).
`
`(39)
`
`This condition is equivalent to the requirement that
`w, = - 1
`1
`IM
`or, equivalently, that all beamformer weights have equal
`values of l/M.
`In summary the above discussion has shown that the adap-
`tive-sidelobe canceler will be identical to Frost's algorithm
`- provided that the conventional weights satisfy (40) and that
`T is an orthogonal transformation. (From (31)
`and (4), this
`latter condition is equivalent to requiring that the rows of W ,
`sum up to zero and be mutually orthogonal.) It is to be noted
`
`31
`
`that this is a sufficient condition only, and necessity has not
`been considered.
`Jim [ 131 has studied
`the comparison in detail and shown
`that steady-state performance of the twoprocessors is identi-
`cal regardless of the structure of W, and W,, provided that the
`system operates at full rank. He has also shown that different
`eigenvalue spectra will be encounteredby the adaptive filters
`systems unless W, and W, meet the sufficient
`in the two
`equality conditions previously described. As a result the coeffi-
`cient trajectories and adaptive learning curves will differ.
`
`PROPERTIES AND EXTENSIONS OF ADAPTIVE
`CONSTRAINED BEAMFORMERS
`The previous section has presented a generalized sidelobe
`canceling structure which can be used to implement the error-
`correcting linearly constrained adaptive algorithm in (10)-(12).
`This structure can also be used to both analyze the perform-
`ance of the algorithm and to suggest generalizations of con-
`strained beamforming. We begin by summarizing the perform-
`ance characteristics of the algorithm which are readily delin-
`eated by the sidelobe canceling model. These properties are
`then used to extend the concept of linearly constrained adap-
`tive beamforming and to develop new methods for use in array
`processing.
`sidelobe canceler is the signal-
`One key elem_ent in the
`blocking matrix W,. As shown by (27), this matrix is required
`to have llf - '1 linearly independent rows which sum up to
`zero. Of the many matrices which can be generated with this
`property, two possibilities which involve only addition opera-
`tions are shown below for the case M = 4:
`
`[I I; -; :]
`[; ; -; ;].
`
`1
`
`-1
`
`-1
`
`-1
`
`0 0
`
`- W,(l)=
`
`- V 2 ) =
`
`In the first matrix the rows are mutually orthogonal and are
`elements of the binary-valued Walsh functions [ 141. The
`second matrix involves fewer operations and consists of taking
`the difference between adjacent sensg outputs.
`of W , as fixed-weight beam-
`One can interpret the rows
`formers which are applied
`to the sensor outputs. The beam-
`of X'(k) and the con-
`formed signals are then the elements
`straints in (27) ensure the presence of a spatial nuJ
`in the
`broadside direction for each beamformer. Note that W,(l) has
`amplitude response for each row while
`- a different spatial
`W,(*) has identical patterns.
`The effects of imperfect sensor steering and/or gain varia-
`tions are easily modeled using the generalized sidelobe cancel-
`ing structure. For
`example, gain differences at the outputs
`of the time-delayed sensors result in a set
`of received signals
`xm(t> given by ,
`~ r n ( t ) = ~ ( f ) ( l + ern) + nm(t)
`
`(43)
`
`where e,
`represents the gain departure from unity at the mth
`sensor output. Because of the nonzero em, the desired signal
`
`WAVES607_1011-0005
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`Petitioner Waves Audio Ltd. 607 - Ex. 1011
`
`

`
`-. . -
`
`32
`
`
`
`
`
`IEEETRANSACTIONS ON ANTENNAS AND PROPAGATION,
`
`-
`VOL. AP-30, NO. 1, JANUARY 1982
`
`sensor
`number
`
`I
`
`2
`
`3
`
`4
`
`5
`
`6
`
`7
`
`8
`
`appears in both the conventional beamformer output y c ( r ) and
`in the sidelobe canceling path. The presence of desired signal
`in the adaptive filters has been termed “signal leak through”
`by Widrow et al. [ 9 ] , and may result in signal distortion and/
`or reduction in output SNR. The distortion is due to the fact
`that the
`scalar y A ( k ) contains a weighted sum of delayed-
`desired signal terms. It can be demonstrated, however, that
`these effects are negligible provided that the power level of
`the signal leak through
`is small compared with
`the power
`contained in the filtered noise vector N’(k). Equivalently, if
`
`us2 tr { w , ~ R , , w , ~ ~ ) Q tr w ~ N N w , ~
`
`(44)
`
`where u:
`is the power level of the desired signal observed at
`a sensor output, tr { - } denotes trace, and R , , and R N N are
`the autocorrelation matrices for the vector of gain errors e
`and the received noise vector N(k), respectively. For the case
`of uncorrelated, equal variance, gain errors, and white receiver
`noise, the I . - d t simplifies to
`
`(45)
`
`where (2,’ and 0,’ are the variance of the gain errors and
`
`white receiver noises. This result demonstrates a well-known
`property of constrained beamformers,
`i.e., that the system is
`much more sensitive to gain errors at high input signal-to-noise
`ratios.
`New methods of adaptive beamforming are suggested by
`the generalized sidelobe canceling structure illustrated in Fig.
`4. These include the following.
`11Additional spatial constraints can be incorporated into
`the MJ, matrix. For example, one can require both a spatial
`null in the desired direction (as in the system discussed above)
`derivative in that direction. The matrix W,(3) for
`and a zero
`M = 4 achieves this result:
`
`Note that the row dimension in this case is M - 2 due to the
`additional spatial constraint. The system sensitivity to point-
`ing errors (time-delay steering errors), however,
`is markedly
`reduced.
`2) Combined spatial/temporal constraint beamfolmers are
`achieved by including delay-storage elements in the W, matrix.
`Equivalently,
`
`N
`_ .
`
`X’(k) =
`n= -N
`
`W,,,X(k - n).
`
`(47)
`
`authors’ knowledge, studies into the advan-
`Thus far, to the
`tages of combined constraints have not been reported.
`3) Power minimization algorithms other than LMS may be
`used to adapt the filter coefficients. Since the constraints have
`been removed from the algorithm, unconstrained accelerated
`convergence techniques such as the conjugate gradient method
`[ 151 may offer significant advantages in tracking time varia-
`tions.
`
`0
`
`200
`
`400
`sample nrmber
`Fig. 5. Synthetic sensor outputs used to demonstrate algorithm per-
`formance.
`
`600
`
`800
`
`I
`
`>
`
`SIMULATION RESULTS
`In order to demonstrate the performance of the generalized
`sidelobe canceling beamformer described above, a synthetic
`set of eight sensor output samples was generated. The results
`are shown in Fig. 5. They consist of two statistically inde-
`pendent narrow-band spatially propagating, random noise
`sources, each assumed to be an incident on the array from a
`different direction. The array has been time-delay steered such
`at about 570
`that the desired signal pulse appears in phase
`samples in all eight traces. At a normalized
`one sample per
`second sampling rate, the narrow-band random noise sources
`had a bandwidth of 0.03 Hz centered at 0.095 Hz. In addition,
`a small amount of independent white noise was added to each
`output to simulate the effects of receiver noise.
`Fig. 6 shows the conventional beamformer output obtained
`by adding the eight outputs and dividing by eight. The narrow-
`band interference completely dominates the output and the
`desired signal is undetectable. Considerably better signal to
`noise ratio can be achieved with a linearly constrained adaptive
`beamformer, as shown by
`the results in Fig. 7. The upper
`waveform is the conventional beamformer
`output depicted
`in Fig. 6 and the lower two were generated with the use of the
`generalized sidelobe canceler and the gradient projection algo-
`rithm without error correction,
`respectively. All three traces
`are plotted using the same amplitude scaling factor and both
`adaptive beamformers
`employed a five-point time operator
`on each channel. The simple differencing technique described
`in (42) was used to generate the sevendifference channels.
`An identical normalized adaptive step size CY = 0.2 was used in
`the two-adaptive beamformers.
`While the two adaptive outputs appear quite similar, small
`differences are readily apparent. As described in the previous
`section, these differences are directly attributable to the fact
`that the generalized sidelobe canceler used a W , matrix in
`which the rows were not mutually orthogond (see (42)).
`Thus, although
`the steady-state performance
`is the same,
`different adaptive paths are employed by the two algorithms.
`In addition, gradient projection incurs accumulated
`roundoff
`
`WAVES607_1011-0006
`
`Petitioner Waves Audio Ltd. 607 - Ex. 1011
`
`

`
`GRIFFITHS AND JIM: LINEARLY CONSTRAINED ADAPTIVE BEAMFORMING
`
`33
`
`I
`
`0
`
`-2.0
`
`1
`
`I
`200
`
`I
`
`I
`
`I
`600
`
`I
`BOO
`
`I
`
`I
`400
`simple nufiler
`Fig. 6. Conventional beamformer output.
`
`I
`
`I
`
`I
`
`I
`
`I
`
`I
`
`I
`
`I
`
`I
`
`CONVENTIOHAL
`
`c
`
`I .o
`0.5 .
`
`0
`
`-0.5
`
`-
`
`0.5
`0
`
`-0.5
`
`0.5
`
`0
`
`- 0 . 5
`
`0
`
`dB
`
`-I
`I
`
`1
`
`I C
`
`5
`
`0
`
`-5
`
`IO
`
`5
`
`0
`
`-5
`
`1
`
`I
`
`I
`
`I
`
`NO1 SE
`
`I MPROVEHENT
`
`- - -
`
`_ _ _ -
`
`- -
`
`SIGNAL REDUCTION
`
`I
`
`0.2
`
`0.1
`
`I
`0.3
`ct
`
`(a)
`
`0.4
`
`0.5
`
`NO1 SE
`
`IMPROVEHENT
`
`REOUCTI ON
`
`/
`
`I
`
`OENERALIZED SIDELOBE
`CANCELLER
`
`dB
`
`GRADIENT PROJECTION -
`
`I
`200
`
`I
`400
`
`I
`600
`
`I
`e00
`
`rawle number
`Fig. 7. Conventional and adaptive beamformer outputs.
`
`I
`0.1
`
`I
`0.2
`
`0.4
`
`0.5
`
`0.3
`a
`(b)
`Fig. 8. Signal and noise power performance. (a) Gradient projection
`algorithm. @) Generalized sidelobe canceling algorithm.
`
`error. Most noticeable of this error is the difference in peak
`signal amplitude. The ideal noise-free signal had an amplitude
`of 0.940. That measured for the two adapters was 0.938 for
`the generalized sidelobe canceler and 0.786 for the gradient
`projection algorithm. The small error in generalized sidelobe
`canceling is presumably due to the presence of the white noise
`component in the output.
`Careful measurements of the average noise power in the
`30-50 s window and of the signal amplitude reduction in
`gradient projection were conducted for values of (Y between
`0.1 and 0.5. Fig. 8 summarizes these findings for the two algo-
`rithms. As described above, the generalized sidelobe canceler
`exhibits negligible signal amplitude degradation over the range
`of studied.
`
`DISCUSSION AND CONCLUSION
`The simulation results presented above illustrate the effects
`of accumulated error which a n be observed with the use of
`the simple gradient projection algorithm. These
`effects are
`readily discernible even though the simulation experiments
`were conducted on a CDC 6400 general purpose computer
`having a 60-bit word length. The purpose of the simulation
`experimentbwas to demonstrate that the generalized sidelobe
`canceler does not incur similar roundoff penalties. No attempt
`has been made to study the well-known noise reduction prop-
`erties of adaptive beamforming. Experiments conducted with
`minicomputers having smaller work
`length-for
`example,
`lead to similar insensitivity to error.
`16 bits-would
`
`The generalized sidelobe canceling structure described in
`this paper can be viewed as an alternative implementation of.
`Frost’s linearly constrained adaptive beamforming algorithm.
`The structure has additional advantages, however, relating to
`both the. development of other related beamformers and to
`the performance analysis of constrained adaptive beamfor-
`mers.
`
`REFERENCES
`[I] M. J. Levin, “Maximum-likelihood array processing,”
`in Seismic
`Discrimination Semi-Annual Technical Summary Report, M.I.T.
`DDC 455