GB2182177A - Simplified pre-processor for a constrained adaptive array - Google Patents
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Abstract
An adaptive combiner (1) is preceded by a data pre-processor (2) which acts to impose a linear weight constraint on the adaptive process. The pre-processor subjects the input data to a transformation described by a matrix A which satisfies the condition A S = C,where W<T>S = ??? defines the overall constraint and @<T>C = ??? defines the constraint on the adaptive combiner. The one or more look direction constraints can be applied by means of a simple form of array feed network. S is the "look direction" vector, C being a vector having only one non-zero component, of valve unity, e.g. the last. ??? is the gain constant, the components of vector W being weighting factors. <IMAGE>
Description
SPECIFICATION
Simplified pre-processorfor a constrained adaptive array
This invention relates to a simplified pre-processor for a constrained adaptive array.
The fundamental operation of a constrained adaptive array system is to minimise the total output power from the array subject to a linearweight constraint which can be arranged, for instance, to fixthecomplex response of the antenna in some pre-defined direction. The use of constrained adaption offers considerable advantages for those systems, for example NAVSTAR, where the angle of arrival of the wanted signal is known and where it is desirable to maintain the antenna response in the look direction to prevent phase and amplitude modulation ofthe signal.
It has previously been shown (0. L. Frost "An algorithm for linearly constrained adaptive array processings", Proc. IEEE, 1971, Vol.60, pp661-675) that the steepest descent algorithm, conventionally used to control array weighting coefficients, can be modified to include a projection matrix which ensures weight updates only occur over a surface defined by the constraint equation.
More recently, we have shown (GB PatentApplication No. 8136510 (Serial No. 21 11311at. Ward 1) how constraints can be implemented by transforming (pre-processing)the received data prior to adaptive combination in an adaptive processor.
According to one aspect of the present invention there is provided a constrained multi-element adaptive array system including an adaptive combiner preceded by a data preprocessorwhich acts to impose a linear weight constraint on the adaptive process, which data pre-processor subjects the input data to a transformation described by a matrixA which satisfies the condition A S = C, where WTS = E definesthe overall constraint and WT C = E defines the constraint on the adaptive combiner.
According to another aspect of the present invention there is provided a data preprocessorfor use with a subsequent multi-element adaptive array combiner, which pre-processor acts to impose weight constraints on the adaptive process, by subjecting the in put data to transformation by a matrix A which satisfies the conditionA S = C, where WTS = edefinesthe overall constrain and \AiTC= E definesthe constraint onthe adaptive combiner.
Embodiments of the invention will now be described with reference to the accompanying drawings, in which:
Figure 1 shows schematically a conventional adaptive combiner,
Figure2 and 2a shows schematically a basic form of the modified adaptive combiner to be employed with a pre-processor of the present invention and an alternative structure therefor, respectively,
Figure 3 shows schematically a pre-processor of the present invention, together with an adaptive combiner ofthe Figure 2a form, for one constraint, Figure4shows schematically a beamspace pre-processor, comprising a Discrete FourierTransform (DFT), together with an adaptive combiner of the Figure 2 form,
Figure 5shows schematically one specific embodiment of pre-processortransformation networktogether with an adaptive combiner ofthe Figure 2 form, Figure 6shows schematically another specific embodiment of pre-processortransformation network together with an adaptive combinerofthe Figure 2form, Figure 7shows schematically a further specific embodiment of pre-processor transformation network together with an adaptive combiner ofthe Figure 2 form,
Figure 8shows schematically a pre-processor of the present invention, together with an adaptive combiner, fortwo constraints,
Figure 9 shows schematically a pre-processor transformation network for two constraints,
Figure 10 shows schematically another pre-processortransformation networkfortwo constraints, and
Figure 11 shows schematicallyfurther pre-processortransformation networkfortwo constraints.
The complex weighting vector, W,that minimises the output power from an N element array subject to the constraint WTS = 1 is given by W = R-1 S*
ST R-1 S* where S is a vector of complex exponentialswith phase factors characterising a wavefront arriving atthe elements of the array from the direction of the desired signal (space vector), and R is the covariance matrix.
One method of achieving the effective weighting described by Wis to subjectthe element outputs to a matrix transformation A, and then to combine the new outputs with a weighting vector Wselected to minimise the resulting power subjectto the constraint WTC= C = 1,with C = (00 1 XT. The required value for W is then given by RC
CTR 1 C where R = A* RA is the covariance matrix characterising the transformed outputs.
Figure 1 shows a conventional adaptive combiner where each input data X1 to X4 is subjected to respective weighting coefficients W1 toW4 before combination in summing means 1. Figure 2 shows the basic form of the multi-element adaptive array combiner required to combine the new outputs, auxiliary input data X, with the weighting factor W. Since there is one constraint there is one reference channel for a reference signal Y.
Figure 2a shows such an adaptive combiner using the same notation of Figure 1 ,X, representing the auxiliary data and E4the reference channel. Figure 3 shows a preprocessor 2 for performing the matrix transformation
A in combination with the adaptive combiner of Figure 2a and using the notation of Figures 1 and 2a.
The overall effect of the transformation, A, and the weighting vector, W, is equivalentto a weighting vector
W applied directly to the element outputs which is given by
W = ATW = AT R1 C
CT R-1 C AT(AT)- R-1 (A*)-1 C = CT(AT)-1R-1 (A*)-1 C
= R-1 [A-1 C+]
[A-1 C]T R-1 [A-1 C*]
If we selectA- C = Swhich is equivalentto C=A Swewill thus obtain the desired weighting. Wethen require
where S = (S1S2... SN)T
Figures 4,5,6 and 7 show examples oftransformation networks which satisfy this relationship and therefore allow implementation of the constrained adaptive weight solution represented mathematically by WTS = 1.
Many different forms of pre-processortransformation network can, in fact, be used to implement a given linearweightconstraint.
Figure 4 shows a common form of constraint pre-processor using a Discrete Fourier Transform (DFT) 3. The effective transformation applied by the DFT 3 creates an orthogonal set of beam responses with one ofthese constituting the required look direction beam. The outputs from the DFT3 are then applied to the adaptive combiner 4 with the look direction beam connected to the reference channel.
The DFT matrix transform can be expressed as
where5iH5j=i fori=j
=0 fori=j and where S* is the weighting appropriate to the reference channel beam response. The DFT matrix also ensures that SiHS* foralli This transformation obviously satisfies the relationship
thereby confirming that the DFT preprocessor implements the constraint represented mathematically by
WTS= 1.
By inspection it can be shown that a particularlysimpleformforA is given by
Such a transformation can be realised by the feed network illustrated in Figure 5. In this network the end element signal is weighted successively by the factors - (Si/SN) where i = 1 to N - I,and com bined with each of the otherelement signals. The resultant signals arethen applied to the adaptive combiner Swhich hasthe format of Figure 2.
A different form of constraint pre-processor is shown in Figure 6. Signals from all elements are combined with a weighting vector 5N* to produce a mainbeam response pointed in the required look direction. This response is applied to the reference channel input of the adaptive combiner 6 which has the format of Figure 2. Neighbouring pairs of element signals are combined so that the effective directional response of each pair of elements is zero in the look direction. The resultant signals are applied to the auxiliary inputs ofthe adaptive combiner.The form of A forthis system is given by
where
W12 = -S1
S2
W23 = -S2
S3 S3
W(N1)N = 1
SN For the case of a linear array having identical, equally spaced elements, W12 = W23 = ... W(N~1)N = ei+, that is, equivalent to a common phase shifting element.
Figure 7 shows a simplified version of the Figure 6 arrangement. The auxiliary inputs to the adaptive combiner7 are again derived by combining signals from neighbouring pairs of elements. Assuming the use of a linear array the two element combiners would again imply a common weighting coefficient. The reference channel inputtothe adaptive combiner7 is derived by scaling the end element signal by 1/SN.
The form of matrix A is now
where, by assumption, W = S1/S2 = S2/S3 = ...SN-1/SN = e1. In the scheme of Figure 6 the auxiliary outputs to the adaptive processor are obtained by combining neighbouring pairs of element signals and steering control of the look direction beam is given by varying a ganged set of phase shifters.
It has so far been shown how an adaptive combinerwith a simple constraint, such as one clamped element, can be preceded by a matrix of weighting constants chosen such that an overall (linear weight) constraint is applied to provide a single "lookdirection" beam. This matrixobeystheformulaA S = Cwheretheconstraint is, in general terms, WT S = e, where W, Sand E are the overall equivalent weights, the "look direction" space vector and the gain constant, respectively. Whereas onlythe constraintWTS = 1 was considered above, the principle is also applicable to multiple constraints.If independent control of multiple "look direction" beams is available, then it is possible to steer broad beams in the direction of the wanted signal thereby avoiding cancellation when jammers are present closeby.
The full analysis for a single constraint is as follows.
The constraint is defined:- WTS= g (1)
The costfunction to minimisethe output residue is: # = WH M W + (WT S - #)# + (WT S - #)#* where M is the covariance matrix and A is an undetermined complex Lagrange multiplier.
The complex gradient is zero ata minimum of:- ## = ## = 2[MW + S*]#* =0 (2)
#W The optimal weights are then found from:
W = -M-1 S* #* (3)
ST W = -ST M-1 S*#* = # from (1) #* = -[ST M-1 S*]-1# W= M-1 S* [ sTM-l S* ] - e from (3)
This is the general expression for a constrained least squares solution. If the adaptive combiner is used then Sabove becomes C,where C = [ 00...1 ] , say, and WC= E implying thatthe spacevectoris omnidirectional (i.e. one clamped element).The covariance matrixforthe adaptive combiner is: # = E(#*#T) = A*E(X*XT)AT = A* M AT where # = A X and M = E(X*XT) have been used to express the combiner input vector and the overall covariance matrix.
The inverse of M is: #-1 = (AT)-1 M-1 (A*)-1 (5) using the general formula (4) the optimum weights for the combiner are:
W= M-1 C* [ CTM-1 C* ] -1 (6) where the constraintwas:
WTC=e
Nowisincay = WTX=WTX=TWAX
W = A'W (7) then the overall weights are, using (6) (5) and (7):
W=M-1 (A-1C)* [(A-1C)TM-1 (A-1C)*]-1 (8)
This must equate with (4), so comparing (4) with (8) we have
S = A-1C or
AS=C (9) as already found above. There are many ways of selecting the constants oftheA matrix to satisfy (9).
Thefull analysisformultipleconstraintsisasfollows:
If the space vectors of the desired look directions are written as column vectors of a matrix Sand the constraint vectors as column vectors of a matrix Cthen: forexample, WTS = j (10) [W1 W2W3W4]
= [e1 E2 represents m constraints with N elements. In the example N = 4, m = 3. This equation is the matrix generalisation of (1). The cost function to minimise the output residue is:
# = WHMW + (WTS=ET) # + (WTS-ET)#* where # is now a vector of Lagrange multipliers. The gradientis :
V # = 2 [MW + S*#*] and the optimum weights are
W + -M-1 S* #* STW + -STM-1 S* #* = E from (10) # * = -[STM-1S*]-1 E noting thatA and e have the same dimensions
W = M-1 S* [STM-1 S*]-1 E (11)
The covariance matrix for the adaptive combiner is:
M + E(#*#T) = A* E(X*XT)AT = A*MAT where X = AX and M = E(X* XT) as above.
M-1 - (AT)-1 M-1 (A*)-1 (12) weightsforthecombinerare: W= M-' C* [CTM-' C*]-1 e (13) where the constraintwas:
CTW=E
Now since y = WT X = WT X = WT A X W=ATW (14) then the overall weights are, using (12, (13) and (14) W= M-' (A-1C)* [ A-iC)TM-l (A-1C)* ] -1 e (15)
Note that as before the C matrix of the adaptive combiner has replaced the S matrix in (4) i.e.WC = e Now comparing (11)with (15) we have that: S =A-1 C or
AS=C (16) so this formula is quite general and it is again found thatthere are many ways of selecting the constraints of theA matrix to satisfy (16). In the following three examples of preprocessor networks illustrating the practical implementation ofthe above are given.
An explicit formulation of (16) with theA matrix on the left hand side can be made as follows. Suppose N = 4 and m = 2 (N elements, m constraints) then:
where S and C have been augmented: Then A = C S-1. To illustrate this, suppose N = 2 and m = 1:
which gives All in terms of A12, and A21 in terms of A22. Generally values of theA matrix should be chosen to simplify the network orto make adjustments easy when it is implemented.
The general resultA S= Ccan befound bya more direct route. WTS= E defines the overallconstraint. WTC = s defines the constraint on the adaptive combiner only.
Thus WT C = WT S (17)
The output is given by:
y= WTk= WTX= WT A X
Thus WT= WTA
Hence from (17)
WT C = WT S = WT A S
i.e.AS=C
The adaptive combiner and preprocessorfortwo constraints is illustrated schematically in Figure 8 using similar notation to that employed in Figure 3.
Afirst example of a preprocessor network for the constraints will now be described.
Consider the constraint equation
WT[S1 S2] = [#1,#2] or
WTS = #T or [W1, W2, W3, W4]
= [#1, #2] TheA matrix is determined by:
X=AX
AS=C
Consider the network shown in Figure 9. The A matrix will satisfy:
where 81 = e2 = 1
The calculation for row 1 of A are
a1S11 + 12S13 + S14 =0 a1 S21 + a2 S23 + S24 = 0 which can be rewritten as
Hence a1 and a2 can be solved, similarly rows 2,3 and 4.
The constraints are then a1 = (-S23 S14 -S13 524)/(S23 S11 -S13 S21) a2 = ( S21 S14 +S11 S24)/(S23 S11 -S13 S21 ) b1 = (-S23S14 - S13S14)/(S23S12 - S13S22) b2 = ( S22 S14 +S12 S24)/(S23 S12 -S13 S22) c1 = S24/(S24S13 - S14S23) C2 = - S23/(S24S13 - S14 S23)
d1 = -S14/(S24S13 - S14S23)
d2 = S13/(S24S13 - S14S23) A second example will now be considered with the constraint equation WT [ S1 S2 ] = [ 81,82 ]
TheA matrix where X = A Xis determined by the equation (choosing e1 = #2 = 1):-
Consider the network shown in Figure 10.This is chosen to be like a Davies nulling tree where f6r a linear array a null is placed in the beam pattern by each row of constants, the beam output appearing at the apex of the inverted triangle. The constraint equation above fixes the two (overall) weights in the adaptive combiner to the left and right i.e. W1 = 1/s and W3 = 1/S23. The centre (overall) weight is the only degree offreedom available in this example to adapt against jamming sources.
TheA matrix can be computed directly but to show howthe Davies principle works it is done in stages as follows:
The equation
Corresponds to the matrixA1,where X = A1Xand shows that S1 has been nulled by the second and third row ofA
at =
a2 = -S11/S13
a3 = -S12/S13
The equation :
corresponding to the matrixA where k=A2k=A2A1X=AX is the full expression for theA matrix.
The bconstantsare nowcalculated as: b1 = -ar S21/(S21 -a2 S22)
b2 = (-S21 -a2S22)/(S22 +a3S23) b3 = a/(S23 -a3 S23) by row products with 52. Row products with S2 are confirmed equal to 81.
The method extends to N elements and m # N constraints.
Afinal example will now be considered with the constraint equations
WT[AB] = [ 1, 2] [ W1 W2....WR....WN-1 WN ]
= [ 1, 2] which corresponds to two constraints with N elements. Now consider the network of Figure 11. The M matrix
X= MXis related bythe equation (N = 5):
where an extra row has been augmented to make M square. This allows XN.
From the equation :
M [ A B ] = [ C1 C2 ] we calculate the processor constants as follows. Taking a typical row Mand multiplying with A then B:
kAR + P1( )AN-1 + P2(R)AN = 0
kBR + P1(R)BN-1 + R2(R)BN = 0
Let the denominator = k
P1(R) = ANBR - BNAR
P2(R) = BN-1AR - AN-1BR
Now using row N-1 and N AN=ILi
AN-1 + AN = 1
k k
f1RN-1 + f2RN = 2
T so by a similar procedure
f1 = BNu1 -AN 112
f2 = BN-1 1 - AN-1 2 again with k = An 1 BN -1
It has thus been shown that there is a class of data pre-processor, with a simple form of array feed network, which allows the application of one or more look direction constraints to an adaptive sensor array such as in spread spectrum communications and navigation, particularly NAVSTAR, and sonor applications. The preprocessor subjects the element outputs to a matrix transformation A to provide a linearweight constraint on the adaptive process, the matrix obeys the relationshipA S = Cwhere WTS = 8 defines the overall constraint and WTC = 8 defines the constraint on the adaptive combiner. There are many ways of selecting the contents of theA matrix to satisfy the condition AS=C.
Claims (10)
1. A constrained multi-element adaptive array system including an adaptive combiner preceded by a data preprocessor which acts to impose a linear weight constraint on the adaptive process, which data preprocessor subjects the input data to a transformation described by a matrixA which satisfies the conditionA
S = C, where WTS = # defines the overall constraint and WTC = # defines the constraint on the adaptive combiner.
2. A constrained multi-element adaptive array system as claimed in claim 1 wherein the overall constraint provides a single look direction beam, the adaptive combiner having a reference channel, the overall con straintbeing WTS = 1.
3. A constrained multi-element adaptive array system as claimed in claim 2 wherein input data on an end element is weighted successively by a plurality of factors and applied to the reference channel of the adaptive combiner, each other channel being weighted by a respective one of said plurality offactors.
4. A constrained multi-element adaptive array system as claimed in claim 2 wherein the array of elements is linear and input data on all elements is combined with a common weighting vector to produce a main beam response pointed in the required look direction, which response is applied to the reference channel ofthe adaptive combiner and wherein neighbouring pairs of elements are combined to produce auxiliary inputs for the adaptive combiner.
5. A constrained multi-element adaptive array system as claimed in claim 4wherein steering control of the look direction beam is given by varying a ganged set of phase shifters.
6. A constrained multi-element adaptive array system as claimed in claim 2 wherein the array of elements is linear and wherein neighbouring pairs of elements are combined to produce auxiliary inputsforthe adaptive combiner, a common weighting coefficient being applied to the pairs, the reference channel input to the adaptive combiner being derived by scaling an end element signal.
7. A constrained multi-element adaptive array system as claimed in claim 1 wherein multiple constraints are applied whereby to be able to control multiple look direction beams independently.
8. A constrained multi-element adaptive array system as claimed in claim 1 and substantially as herein described with reference to one or more of Figures 2 to 11 of the accompanying drawings.
9. A data preprocessorfor use with a subsequent multi-element adaptive array combiner, which preprocessor acts to impose weight constraints on the adaptive process, by subjecting the input data totransformation by a matrixA which satisfies the condition A S = C, where WTS = e defines overall constraint and
WTC = e defines the constraint on the adaptive combiner.
10. A data pre-processor, for use with a subsequent multi-element adaptive array combiner, as claimed in claim 9 and substantially as herein described with reference to any one of Figures 3 to 11 of the accompanying drawings.
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| GB8526406A GB2182177B (en) | 1985-10-25 | 1985-10-25 | A simplified pre-processor for a constrained adaptive array |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| GB8526406A GB2182177B (en) | 1985-10-25 | 1985-10-25 | A simplified pre-processor for a constrained adaptive array |
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| GB2182177A true GB2182177A (en) | 1987-05-07 |
| GB2182177B GB2182177B (en) | 1989-10-11 |
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| Application Number | Title | Priority Date | Filing Date |
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| GB8526406A Expired GB2182177B (en) | 1985-10-25 | 1985-10-25 | A simplified pre-processor for a constrained adaptive array |
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Cited By (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| GB2410872A (en) * | 2004-02-06 | 2005-08-10 | Nortel Networks Ltd | Adaptive and constrained weighting in SDMA receivers |
| GB2410873A (en) * | 2004-02-06 | 2005-08-10 | Nortel Networks Ltd | Adaptive and constrained weighting for multiple transmitter and receiver antennas |
| DE4039153B4 (en) * | 1989-12-08 | 2006-09-07 | Thomson - Csf | Method and device for generating a radiation pattern at rest in a group antenna |
Citations (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| GB2111311A (en) * | 1981-12-03 | 1983-06-29 | Standard Telephones Cables Ltd | Adaptive antenna arrays |
| GB2143378A (en) * | 1983-07-06 | 1985-02-06 | Secr Defence | Constraint application processor for signals from antenna arrays |
-
1985
- 1985-10-25 GB GB8526406A patent/GB2182177B/en not_active Expired
Patent Citations (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| GB2111311A (en) * | 1981-12-03 | 1983-06-29 | Standard Telephones Cables Ltd | Adaptive antenna arrays |
| GB2143378A (en) * | 1983-07-06 | 1985-02-06 | Secr Defence | Constraint application processor for signals from antenna arrays |
Non-Patent Citations (1)
| Title |
|---|
| J.E. HUDSON, }ADAPTIVE ARRAY PRINCIPLES}, PETER PEREGRINUS 1981, PP77-81; * |
Cited By (4)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| DE4039153B4 (en) * | 1989-12-08 | 2006-09-07 | Thomson - Csf | Method and device for generating a radiation pattern at rest in a group antenna |
| GB2410872A (en) * | 2004-02-06 | 2005-08-10 | Nortel Networks Ltd | Adaptive and constrained weighting in SDMA receivers |
| GB2410873A (en) * | 2004-02-06 | 2005-08-10 | Nortel Networks Ltd | Adaptive and constrained weighting for multiple transmitter and receiver antennas |
| GB2410872B (en) * | 2004-02-06 | 2006-10-18 | Nortel Networks Ltd | Signal processing method |
Also Published As
| Publication number | Publication date |
|---|---|
| GB2182177B (en) | 1989-10-11 |
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Effective date: 20031025 |