United States Patent (19)
`Apostolos et al.
`
`4,654,667
`11) Patent Number:
`45
`Date of Patent: Mar. 31, 1987
`
`54
`
`(75)
`
`(73)
`
`(21)
`22)
`51
`52
`58)
`
`56)
`
`SIGNAL-ACQUISITION SYSTEM FOR A
`CIRCULAR ARRAY
`Inventors: John T. Apostolos, Merrimack;
`Robert H. Carrier, Durham, both of
`N.H.
`Assignee: Sanders Associates, Inc., Nashua,
`N.H.
`Appl. No.: 685,175
`Filed:
`Dec. 21, 1984
`Int. Cl* .......................... G01S 5/04; G01S 3/16;
`G06G 7/19
`U.S. C. .................................... 342/445; 342/378;
`342/22; 364/827; 364/516; 364/726; 324/77 B
`Field of Search ..................... 343/5 FT, 378, 417,
`343/443, 445; 324/77 B;364/516, 725,726,827
`References Cited
`U.S. PATENT DOCUMENTS
`4,035,804 7/1977. Overbury .
`4,084,148 4/1978 Koshikawa .
`4,166,980 9/1979 Apostolos et al. .
`4,180,814 12/1979 Barton .
`4,244,037 1/1981 Jelks.
`4,254,417 3/1981 Speiser .
`
`4,305,159 12/1981 Stromswold et al. .
`4,332,016 5/1982 Berntsen.
`Primary Examiner-Theodore M. Blum
`Assistant Examiner-Gregory C. Issing
`Attorney, Agent, or Firm-Louis Etlinger; Stanton D.
`Weinstein; William F. Porter, Jr.
`57)
`ABSTRACT
`A signal-acquisition system (10) for a circular antenna
`array (12) includes a two-dimensional compressive re
`ceiver (18) that performs a two-dimensional Fourier
`transformation in time and position on the outputs of the
`array. Each of the outputs of the compressive receiver
`(18) is fed to input ports of several processing units (24),
`which multiply them by an appropriate time-dependent
`function. The resultant modified signals are then pro
`cessed by Butler matrices (30) that together have a
`matrix of output ports (32). Each output port is associ
`ated with a different combination of azimuth and eleva
`tion angles. A signal source at given azimuth and eleva
`tion angles with respect to the array (12) causes its
`greatest response in the output port (32) associated with
`those angles.
`
`7 Claims, 3 Drawing Figures
`
`
`
`Y4 (a)
`
`16 in
`
`20(n)
`
`2
`COMPRESSW
`RECEWER
`IstN)
`2CNY
`
`
`
`
`
`Page 1 of 8
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`SONOS EXHIBIT 1008
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`

`

`U.S. Patent Mar. 31, 1987
`
`Sheet 1 of 2
`
`4,654,667
`
`
`
`
`
`-003 (N-2) 91
`
`(N-I) 91
`
`CJ-2
`BAISSE (Jd WOO
`
`
`
`Page 2 of 8
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`SONOS EXHIBIT 1008
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`

`

`U.S. Patent Mar. 31, 1987
`
`Sheet 2 of 2
`
`4,654,667
`
`
`
`
`
`-N
`
`(-i)"Wi-NJ-Ned cos(at/2a)
`
`FUNCTION GENERATOR
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`36 (q, 1-N)
`
`:
`
`34(q, n)
`
`(-j)" WnJned.Cos(at/20)
`
`FUNCTION GENERATOR
`
`36 (a,n)
`
`O
`
`34(q,N)
`
`(-i)NWN ded cos(ar/2a)
`
`FUNCON GENERATOR
`
`
`
`
`
`
`
`CIRCULAR
`ARRAY
`ON X-Y
`PLANE 42
`
`X
`
`DIRECTION OF ARRIVAL (DOA)
`
`"R-1
`
`Y
`
`/
`
`FG 3
`
`Page 3 of 8
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`SONOS EXHIBIT 1008
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`

`

`1.
`
`SIGNAL-ACQUESTION SYSTEM FOR A
`CIRCULAR ARRAY
`
`5
`
`O
`
`BACKGROUND OF THE INVENTION
`The present invention is directed to signal-acquisition
`systems. It is concerned specifically with a system for
`processing the output of a circular array of antenna
`elements so as to determine both the azimuth and the
`elevation angles of the source of signals that the antenna
`array receives.
`U.S. patent application Ser. No. 551,664, filed on
`Nov. 14, 1983, by Apostolos, Boland, and Stromswold
`for an ACQUISITION SYSTEM EMPLOYING CIR
`15
`CULAR ARRAY, discloses a powerful system for
`determining the directions of arrival and frequencies of
`many signals simultaneously. An improvement in that
`system is disclosed in U.S. patent application Ser. No.
`536,477, filed on Sept. 28, 1983, by John T. Apostolos
`20
`for a TWO-DIMENSIONAL ACQUISITION SYS
`TEM USING CIRCULAR ARRAY. In both of these
`systems, a spatial Fourier transformation is performed
`on the outputs of a circular antenna array. The resultant
`transform is processed with certain correction factors
`related to the antenna pattern of the array and then
`25
`subjected again to a spatial Fourier transformation. A
`temporal Fourier transformation is also performed. The
`result of each system is an ensemble of signals at a group
`of output ports in which each output port represents a
`different azimuthal direction. Signals from a source in a
`30
`given azimuthal direction result in a maximum output at
`the port associated with that azimuthal direction. Thus,
`the azimuthal direction of each source is readily identi
`fied in real time. The descriptions included in these
`patent applications are helpful in understanding the
`35
`present invention, and they are accordingly incorpo
`rated by reference.
`The assumption on which the design of the systems of
`those two applications is based is that the source has a
`negligible elevation angle. That is, there is only a very
`40
`small angle between the direction of arrival of the signal
`and the plane of the circular antenna array. For a wide
`range of applications, this is an accurate assumption.
`For sources whose angle of elevation is significant,
`however, the direction indications produced by the
`systems of those two applications are inaccurate.
`An object of the present invention is to eliminate the
`inaccuracies that can be caused in such systems by sig
`nificant elevation angles.
`It is a further object of the present invention to deter
`mine the values of the elevation angles of signal sources.
`SUMMARY OF THE INVENTION
`The foregoing and related objects are achieved in the
`method described below and in apparatus for carrying
`55
`out that method. The method includes performing a
`spatial Fourier transformation on an ensemble of input
`signals from a circular antenna array to generate an
`ensemble of input-transform signals, each of which is
`associated with a separate integer index n. A signal
`associated with an index n represents the spatial-fre
`quency component of n electrical degrees per spatial
`degree around the circular antenna array and consists of
`components representing all of the antenna-signal tem
`poral-frequency components that give rise to that spa
`65
`tial frequency. According to the invention, an ensemble
`of modified-transform signals is generated from these
`input signals for each of a plurality of elevation angles.
`
`4,654,667
`2
`Each ensemble includes a modified-transform signal
`associated with each input-transform signal. Each modi
`fied-transform signal consists of modified-transform
`components, each of which represents a value that is
`substantially proportional to an associated component
`in the input-transform signal multiplied by (-1)n times
`the azimuth-independent factor in the antenna pattern
`that would be generated by the antenna array at the
`associated elevation angle if the antenna array were
`driven by signals whose temporal frequency is the fre
`quency with which that input-transform component is
`associated and whose phases vary with element position
`at the spatial frequency represented by that input-trans
`form signal.
`A spatial Fourier transformation is performed on
`each ensemble of modified-transform signals to generate
`an ensemble of output-transform signals for each of the
`plurality of elevation angles. Within a given output
`transform ensemble, each output-transform signal is
`associated with a different azimuth angle. The result of
`this process is that radiation emitted by a source and
`received by the antenna array causes a maximum re
`sponse in the output-transform signal associated with
`the azimuth and elevation angles of that source.
`BRIEF DESCRIPTION OF THE DRAWINGS
`These and further features and advantages of the
`present invention are described in connection with the
`accompanying drawings, in which:
`FIG. 1 is a block diagram of the system of the present
`invention for determining the elevation and azimuthal
`position of the source of signals detected by a circular
`array of antenna elements;
`FIG. 2 is a more-detailed block diagram of a portion
`of the system of FIG. 1; and
`FIG. 3 is a diagram used to define variables employed
`in the mathematical treatment of the invention.
`DETALED DESCRIPTION OF THE
`PREFERRED EMBODIMENTS
`The invention will be described initially by simulta
`neous reference to FIGS. 1, and 2. The system 10 of the
`present invention is a device for determining the angle
`of elevation, the azimuth angles, and the temporal fre
`quencies of radiation received from a plurality of
`sources by a circular antenna array 12 of 2N elements
`14(1-N) through 14(N). The outputs of the antenna
`elements 14 are fed to corresponding input ports 16 of a
`two-dimensional compressive receiver 18. In essence,
`the two-dimensional compressive receiver performs a
`two-dimensional Fourier transformation on the signal
`ensemble that it receives at its input ports. The transfor
`mation is from time to temporal frequency and from
`position to spatial frequency. The compressive receiver
`18 has 2N output ports, each of which is associated with
`a spatial-frequency component, and a spatial-frequency
`component in the input ensemble causes its greatest
`response at the output port 200n) associated with that
`spatial-frequency component.
`Spatial frequency in this context refers to the instanta
`neous phase advance around the elements of the circu
`lar array. For instance, suppose that the signals on all of
`the elements 14 of the circular array 12 are sinusoidal
`signals of the same temporal frequency but having dif
`ferent phases. Suppose further that these phases ad
`vance with element position by n electrical degrees per
`spatial degree, where n is an integer. In such a situation,
`
`45
`
`50
`
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`SONOS EXHIBIT 1008
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`

`

`O
`
`4,654,667
`4.
`3
`the array output has a single temporal-frequency com
`input port. In the modified Butler matrix 30(a), the
`ponent and a single spatial-frequency component. For
`spatial-frequency difference between any two adjacent
`such an ensemble of signals, the output of the compres
`output ports is the same as that for a conventional
`Butler matrix. However, the spatial frequencies repre
`sive receiver 18 is a burst of oscillatory signal whose
`frequency is the center frequency of the compressive
`sented by the output ports 32(q, n) of the modified
`receiver. This output is greatest on the output port 200n)
`Butler matrix 30(a) differ from those of a conventional
`associated with the spatial frequency of n electrical
`Butler maxtrix by one-half of that spatial-frequency
`degrees per spatial degree. The compressive receiver is
`difference. For N=4, therefore, the output ports 32(q)
`correspond to spatial frequencies of 0,
`45, -90,
`repeatedly swept in frequency, and the burst occurs at a
`135, and 180 electrical degrees per input port.
`time within the sweep that is determined by the tem
`poral frequency of the radiation that causes the signal
`The modified Butler matrix can be constructed in a
`ensemble. For the ensemble just described, the response
`number of ways. The most straightforward conceptu
`at any of the other output ports 20 is negligi
`ally is to provide phase shifters (not shown in the draw
`ble-because there are no other spatial-frequency com
`ings) at the input ports of a conventional Butler matrix.
`ponents-and a significant output on output port 200n)
`Each of the phase shifters provides a different phase
`15
`occurs only at the time within the sweep associated
`shift, the phase shifts increasing with input-port position
`with the temporal frequency of the radiation.
`in such a manner that the phase shifts of adjacent phase
`Of course, this signal ensemble, which has only one
`shifters differ by one-half the spatial-frequency spacing
`spatial-frequency component, is extremely artificial;
`of the output ports 32.
`even a single plane-wave signal at a single temporal
`The ultimate result of the system is that each output
`20
`frequency gives rise to many spatial-frequency compo
`port 32(q,n) is associated with an elevation angle of
`nents in a circular array. In ordinary operation, many
`90°Xq/Q and an azimuth angle of 180Xn/N. A plane
`spatial-frequency components, and usually many ten
`wave that arrives at the antenna array 12 at a given
`poral-frequency components, are present in the ensem
`combination of azimuth angle and elevation angle
`ble of signals processed by the two-dimensional com
`causes the greatest response on the output port associ
`25
`pressive receiver 18, which processes all of these com
`ated with that combination of angles, and the time
`ponents simultaneously.
`within a compressive-receiver sweep at which the re
`Those skilled in the art will recognize that the two-di
`sponse occurs is an indication of the temporal frequency
`mensional compressive receiver includes a two-dimen
`of the plane wave.
`sional dispersive delay line and that the position of an
`FIG. 2 shows one of the processing units 24(g) of
`30
`output port on the output edge of the delay line deter
`FIG. 1 in more detail. Associated with each input port
`mines the spatial-frequency component with which that
`22(q, n) and output port 26(q,n) is an analog multiplier
`output port 20 is associated. Therefore, the output ports
`34(a,n) which multiplies the signal from the input port
`of compressive recievers can in general be positioned so
`22(q, n) by a processing factor represented by a signal
`as to be associated with other than integral numbers of
`that a function generator 36(g, n) produces. The value of
`35
`electrical degrees per spatial degree. However, com
`the processing factor is shown in FIG. 2, where Jn is the
`pressive receiver 18 is arranged so that the spatial fre
`nth-order Bessel function of the first kind. The W.'s are
`quencies associated with the output ports 20 are inte
`weighting factors that would be used in most practical
`gral; as was stated before, each output port 200n) is
`applications to improve the dynamic range of the sys
`associated with a spatial frequency of n electrical de
`tem output, as will be described in more detail below.
`grees per spatial degree.
`The weighting factors are constants that differ for dif
`The signal from each compressive-receiver output
`ferent function generators within a processor unit 24 but
`port 200n) is fed to a corresponding input port on each
`are the same for corresponding function generators in
`of Q-1 different processing units 24(2)-24(Q). These
`different processor units.
`processing units 24 multiply the input signals by pro
`The processing factors depend on q, n, and the wave
`45
`cessing factors that are functions of time within the
`number B. The wave number, in turn, is proportional to
`compressive-receiver sweep and depend on the particu
`the antenna-signal temporal-frequency component to
`lar input port 22(qn) to which the signal is applied. The
`which the compressive receiver is responding at the
`signal resulting from multiplication of the signal on each
`current point in the compressive-receiver sweep; that is,
`input port 22(qn) is presented on a corresponding out
`the processing factors are functions of time within a
`50
`put port 26(qn) to a corresponding input port 28(qn) of
`sweep. The processing factor produced by function
`one of Q-1 modified Butler matrices 30(5)-30(Q).
`generator 36(qn), if a factor of 27r is ignored, is the
`Each modified Butler matrix 30(q) performs a spatial
`weighting factor multiplied by (-1) times a quantity
`Fourier transformation but no temporal Fourier trans
`that, as will shortly be explained, can be described as the
`azimuth-independent factor in a particular antenna pat
`formation.
`55
`The Butler matrix 30(a) is a modified version of a
`tern. This antenna pattern is one that is generated by an
`appropriate phasing of the circular array 12. Specifi
`conventional Butler matrix of the type described in U.S.
`Pat. No. 3,255,450, which issued on June 7, 1966, to
`cally, if the antenna elements were used for transmission
`Jesse L. Butler for a Multiple Beam Antenna System
`and driven at the temporai frequency corresponding to
`Employing Multiple Directional Couplers in the Lea
`B but at different phases so that the phases advance
`60
`din. In the conventional Butler matrix, the two adjacent
`around the array at a spatial frequency of n electrical
`central output ports represent opposite phase gradients,
`degrees per spatial degree, then the far-field antenna
`or spatial frequencies, of the same magnitude, and the
`pattern associated with processing unit 24(g) is:
`other output ports represent spatial frequencies that are
`2njhu nerdyad cos(q7/29))
`odd harmonics of these spatial frequencies. For exam
`65
`ple, with N=4, the outputs of a conventional Butler
`matrix would correspond to spatial frequencies of t22
`, -67 , -- 112 , and 157 electrical degrees per
`
`This is the antenna pattern mentioned above. The only
`azimuth-dependent factor in this pattern is is end. The
`
`Page 5 of 8
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`SONOS EXHIBIT 1008
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`

`

`4,654,667
`5
`remaining factors of this pattern-i.e., the azimuth
`independent factor-is included in the processing factor
`in the manner just described.
`The function generator typically includes a read-only
`digital memory containing values of the processing
`factor, which, for any individual function generator
`36(q,n), is a function of radiation temporal frequency
`only. Since the frequency to which a compressive
`receiver output is responding at any given time is a
`function of the time during the compressive-receiver
`10
`sweep, the function generator is synchronized with the
`sweep to achieve the correct timing.
`The function generator typically also includes a digi
`tal-to-analog converter to which the outputs of the
`read-only memory are fed. The outputs of the digital-to
`analog converter are applied to the analog multiplier
`34(g, nThe functions depicted in FIG. 2 are all either
`purely real or purely imaginary. Therefore, although
`the multipliers 34(q,n) perform complex multiplications,
`they can be provided as simple doubly balanced modu
`lators with or without 90 phase shifts; there is no need
`to include a device for adding phase and quadrature
`components.
`The broader teachings of the present invention can be
`practiced without the temporal Fourier transformation
`that the compressive receiver 18 performs. That is, the
`compressive receiver 18 could, in principle, be replaced
`with a modified Butler matrix or similar device. In such
`a system, the processing unit 24(g) would receive the
`results of antenna signals of all received temporal fre
`30
`quencies simultaneously. Accordingly, the multiplier
`and function generator of FIG. 2 would be replaced
`with a filter network for performing the functions de
`picted in FIG. 2. Although synthesis of a network in
`35
`plementing one of the functions of FIG. 2 does not
`appear to be straightforward, a close approximation can
`readily be achieved in a filter having a reasonable num
`ber of poles by employing available network-synthesis
`routines.
`We now turn to a mathematical treatment of the
`operation of the invention. This treatment will proceed
`with the aid of FIG. 3, which includes a circle 38 that
`represents a continuous linear array that the discrete
`array 12 approximates. A plane wave propagates along
`45
`a direction of arrival 40 at an angle of elevation 6 and an
`azimuth angle d with respect to an arbitrary azimuth
`reference 42. To determine the phase, relative to the
`center 44 of the array, of the signal at a given element
`position 46 whose azimuth angle is d', one determines
`50
`the perpendicular distance between the element 46 and
`a plane normal to the direction of arrival 40 through the
`center 44. If the radius of the circular array is d, then the
`perpendicular distance is given by d cos(p'-d)cos6.
`Therefore, the phasor representation of the signal at a
`position d' is given by:
`55
`E(q)-sidcos('-101 )cos 69
`
`-
`
`Substitution of equation (1) into equation (2) yields
`the following expression for the nth Fourier coefficient
`Cn:
`
`T
`c pe f ejnd'gigacos(d'-d)cosedd
`
`-
`
`(3)
`
`5
`
`Evaluation of the integral in equation (3) can be per
`formed with the aid of Hansen's integral formula for an
`nth-order Bessel function of the first kind:
`
`25
`
`1
`Jn(z) = -- S
`
`t
`
`--
`
`sizcostsjn(-7/2)d
`
`(4)
`
`This results in the following expression for the signal
`at output port 200n):
`(5)
`C=2arjeh.J.(3d cos 8)
`Inspection of equation (5) reveals that this output signal
`is a function of the azimuth and elevation angles of the
`source and is also a function of the wave number, which
`is proportional to temporal frequency.
`In general, the radiation may come from more than
`one direction, and the output cn is the sum of the signals
`given by equation (5) for sources at different elevation
`and azimuth angles. At a given time within the com
`pressive-receiver sweep, however, the output at port
`200m) responds to signals of only a single frequency.
`Therefore, the outputs in response to antenna signals
`that are sufficiently separated in temporal frequency are
`not added, because they occur at different times.
`As was stated above, each processing unit 24(g) mul
`tiplies each of the cr's by a different factor and provides
`the output to the modified Butler matrix 30(a), which
`generates a spatial Fourier transform of the signal en
`semble that it receives. This output is given by the foll
`lowing phase shifting and summation of discrete inputs:
`
`N
`
`E -
`
`(6)
`
`Crdane"
`Fan =
`where Fn is the signal at output port 32(a,n), dan is the
`value of the factor applied by multiplier 34(a,n), and
`6 searn/N. For present purposes, it will be assumed that
`the value of the weighting factor Win is unity. In most
`practical embodiments, Wn will differ from unity, but
`the assumption of a unity value will simplify the discus
`sion, and the effect of the weighting factors will be
`considered at the conclusion of the mathematical devel
`opment.
`If Wn=1, the value of dn is given by the following
`expression
`
`dan=(-j)"J(Rd cose)
`where 6=g ar/2O.
`Substitution of equation (5) and equation (7) into
`equation (6) yields the following expression for the
`signal on Butler-matrix output port 32(q,n):
`
`(7)
`
`(1)
`
`It will be recalled that the compressive receiver 18
`generates an output ensemble at any given time that
`60
`represents a spatial Fourier transformation of the input
`signal components in its input ensemble having the
`temporal frequency associated with that particular time
`in the compressive-receiver sweep. Accordingly, at a
`time associated with the frequency for which the wave
`65
`number is equal to B, the signal at a compressive
`receiver output port 200n) is represented by the spatial
`Fourier coefficient with which it is associated:
`
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`SONOS EXHIBIT 1008
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`

`

`7
`
`Fan F
`
`2nd,(8dcos0).J.,(Bdcos0)e(den)
`N
`
`S
`Evaluation of this expression can be made by applica
`tion of Graf's addition theorem for Bessel functions:
`
`(8)
`
`J(p) = S J (6dcos0).J.(6dcose)gn(dn)
`
`c
`
`- do
`
`w
`
`where
`
`9
`
`(9)
`
`(10)
`
`15
`
`4,654,667
`8
`fied Butler matrix 30 could be replaced with a beam
`forming surface-acoustic-wave delay line, which, like
`the modified Butler matrix, performs a spatial Fourier
`transformation. Furthermore, equivalent devices for
`processing the signals digitally could be substituted: A
`two-dimensional fast-Fourier-transform device could
`be substituted for the compressive receiver 18, for ex
`ample, and a plurality of one-dimensional fast-Fourier
`transform devices could replace the Butler matrices.
`The processing units would then use digital multipliers.
`Other variations of the device illustrated in the pres
`ent invention will also suggest themselves to those
`skilled in the art in light of the foregoing disclosure.
`We claim:
`1. A method of determining the azimuth and eleva
`tion angles of a source of electromagnetic radiation to
`which a circular array of antenna elements responds by
`generating an ensemble of input signals, the method
`comprising the steps of:
`A. performing a spatial Fourier transformation on the
`ensemble of input signals to generate an ensemble
`of input-transform signals, each input-transform
`signal being associated with a different integer
`index n, representing the spatial-frequency compo
`nent of n electrical degrees per spatial degree, and
`consisting of input-transform components, each of
`which is generated in response to a different input
`signal temporal-frequency component;
`B. for each of a plurality of elevation angles, generat
`ing an ensemble of modified-transform signals asso
`ciated therewith, each modified-transform signal
`being associated with an input-transform signal and
`consisting of modified-transform components, each
`of the modified-transform components being asso
`ciated with an input-transform component and
`representing substantially the value of its associ
`ated input-transform signal multiplied by a process
`ing factor equal to (-1) times the azimuth
`independent factor of the far-field antenna pattern
`that would be generated by the antenna array at
`that elevation angle if the antenna array were
`driven by signals whose temporal frequency is the
`frequency with which that input-transform compo
`nent is associated and whose phases advance with
`element position at the spatial frequency repre
`sented by that input-transform signal;
`C. for each of the plurality of elevation angles, per
`forming a spatial Fourier transformation on the
`ensemble of modified-transform signals associated
`therewith to generate an output-transform ensem
`ble of output-transform signals associated there
`with, each output-transform signal in an output
`transform ensemble also being associated with a
`different azimuth angle so that radiation detected
`by the antenna array results in a maximum in the
`output-transform signal associated with the azi
`muth and elevation angles of the radiation source.
`2. A method as defined in claim 1 wherein the step of
`performing a spatial Fourier transformation further
`includes the step of performing a temporal Fourier
`transformation so that the input-transform signals are
`responses to only one temporal-frequency component
`at a
`3. An apparatus for determining the azimuth and
`elevation angles of a source of electromagnetic radia
`tion to which a circular array of antenna elements re
`
`p = {3d \ (sinöcosd - sinöcosd) -- (sinesinds - sin6.sind.)
`Comparison of equations (8) and (9) reveals that, with
`the exception of a factor 27, the only difference be
`tween the expressions to the right of the equals signs is
`20
`that equation (8) includes only a finite number of ad
`dends. With the exception of the 2nt factor, therefore,
`the expression in equation (8) is a good approximation
`for the expression of equation (9) whenever Bd is less
`than N, i.e., whenever the inter-element spacing is less
`than a wavelength. If this requirement is met, the signal
`on Butler-matrix output port 32(q,n) is given by the
`following expression:
`
`25
`
`Fan as
`
`(11)
`
`30
`
`2.J(gd N (sin6cosdb - sin9cosd) -- (sin6sind - sin0.sindb.)
`
`)
`
`The value of an output of the form set forth in equa
`tion (11) can be understood if it is recognized, first, that
`35
`the zero-order Bessel function of the first kind has its
`overall maximum when its argument is zero and, sec
`ond, that the Bessel-function argument in equation (11)
`goes to zero for a given output Fan only when the ele
`vation angle of the source is equal to 6 and, at the same
`time, the azimuth angle of the source is equal to dhere
`fore, each output port 32(qn) is associated with a direc
`tion (0-dn), i.e., the direction that provides the maxi
`mum response at the output port 32.
`Although the overall maximum of the zero-order
`45
`Bessel function occurs for an argument of zero, it has
`local maxima for other arguments, and it may prove
`desirable to "smooth' the output to reduce these local
`maxima. It is for this purpose that the weighting factors
`W are shown in the function generators 36 of FIG. 2.
`The weighting factors that are best for a particular
`situation can be determined empirically. The resultant
`smoothing can increase the dynamic range of the sys
`tem, although this increase in dynamic range is usually
`accompanied by some loss in resolution.
`It is interesting to note that equation (11) is the output
`that one would obtain from a phased circular array
`aimed at (6 bn) for a wave number of B. Thus, the
`system of the present invention is equivalent to a large
`number of phased circular arrays, each one phased for a
`60
`different combination of direction and temporal fre
`quency.
`It will be apparent to those skilled in the art that the
`teachings of the present invention can be employed in
`systems that differ somewhat from that illustrated in the
`65
`foregoing description. As was stated above, the two-di
`mensional compressive receiver 18 could be replaced
`with a modified Butler matrix. Additionally, the modi
`
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`SONOS EXHIBIT 1008
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`

`

`4,654,667
`10
`unit, for performing a spatial Fourier transforma
`sponds by generating an ensemble of input signals, the
`apparatus comprising:
`tion on that modified-transform ensemble to gener
`ate an ensemble of output-transform signals associ
`A, a first Fourier-transform device, adapted to re
`ated with that elevation angle, each modified-trans
`ceive the ensemble of input signals, for performing
`form signal in a modified-transform ensemble being
`a spatial Fourier transformation on the ensemble of 5
`input signals to generate an ensemble of input
`associated with a different azimuth angle so that
`transform signals, each input-transform signal
`radiation received by the antenna array from a
`radiation source results in a maximum in the out
`being associated with a different integer index n,
`representing the spatial-frequency component of n
`put-transform signal associated with the azimuth
`electrical degrees per spatial degree, and consisting 10
`and elevation angles of the radiation source.
`of input-transform components, each of which is
`4. An apparatus as defined in claim 3 wherein the first
`generated in response to a different input-signal
`Fourier-transform device is a two-dimensional com
`temporal-frequency component;
`pressive receiver whose output ports are positioned to
`correspond to integer spatial frequencies, the two-di
`B. a plurality of processing units, each processing unit
`mensional compressive receiver simultaneously per
`being associated with a different one of a plurality 15
`forming a temporal Fourier transformation so that the
`of elevation angles and connected to receive the
`input-transform ensemble and generate a modified
`input-transform signals are responses to only one tem
`poral-frequency component at a time.
`transform signal associated with each input-trans
`form signal to produce a modified-transform en
`5. An apparatus as defined in claim 4 wherein each
`semble associated with that elevation angle, each 20
`processing unit includes means for performing a com
`plex multiplication of the incoming signal by a time
`modified-transform signal consisting of modified
`transform components, each of the modified-trans
`dependent factor equal to the processing factor evalu
`form components being associated with an input
`ated at the temporal frequency to which the input-trans
`transform component and representing substan
`form output is a response at that time.
`tially the value of its associated input-transform 25
`6. An apparatus as defined in claim 5 wherein each
`signal multiplied by a processing factor equal to
`processing factor is proportional to
`(-1) times the azimuth-independent factor of the
`far-field antenna pattern that would be generated
`(-j)"WJOBd cos6),
`by the antenna array at that elevation angle if the
`antenna array were driven by signals whose tem- 30
`poral frequency is the frequency with which that
`input-transform component is associated and
`whose phases advance with element position at the
`spatial frequency represented by that input-trans
`form signal;
`C. a plurality of second Fourier-transform devices,
`each of which is associated with a different pro
`cessing unit and thus with a different elevation
`angle and is connected to receive the modified
`transform ensemble from its associated processing 40
`
`k
`
`k
`
`k
`
`sk
`
`where J is the nth-order Bessel function of the first
`kind, B is the wave number at the temporal frequency to
`which the input-transform output is a response at that
`time, d is the radius of the circular array, 6 is the eleva
`tion angle with which the procesing unit is associated,
`and W is a weighting factor.
`7. An apparatus as defined in claim 3 wherein the
`second Fourier-transform devices include Butler matri
`ces modified to produce integer-spatial-frequency com
`ponents.
`
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`
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`SONOS EXHIBIT 1008
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