Clutter positioning for electronically agile multi-beam radars

An electronically agile multi-beam radar including a clutter positioning system for positioning the band of clutter signals in the derived doppler frequency spectrum of each of the individual beams is disclosed. The radar is operative to switchedly transmit a plurality of beams directionally separated by time sharing the illuminating power thereof. Each beam includes at least one transmission of a plurality of R.F. pulses constituting a radar look. The radar is also operative to receive echo R.F. pulses of the look from each transmitted beam dispersed in time with echo R.F. pulses of the looks of the other transmitted beams of the plurality. The clutter positioning system operates to maintain substantially a desired pulse-to-pulse phase relationship for the received plurality of echo pulses of each look of each beam in a time-shared manner in order to derive a substantially representative pulse doppler spectrum of signals associated with each look of the transmitted beams and to adaptively position an identified group of clutter signals about a prespecified dopper frequency in a derived doppler frequency spectrum for each look of each beam. The clutter positioning system may be disposed in either the transmission portion or the reception portion of the radar in which case it effects substantially the desired pulse-to-pulse phase relationship at each transmitted or received R.F. pulsed beam, respectively.

BACKGROUND OF THE INVENTION 
The present invention relates to electronically agile multi-beam radars in 
general, and more particularly to clutter positioning apparatus for 
disposition therein to adaptively position an identified group of clutter 
signals about a prespecified doppler frequency in the doppler frequency 
spectrum for each of the beams of the radar. 
A typical radar of the pulse doppler-coherent type is depicted in the block 
diagram schematic of FIG. 1. In the exemplary radar of FIG. 1, the 
transmission portion includes the conventional units of a mixer/filter 
circuit 10, an amplifier A1, another mixer/filter circuit 12 and a 
transmitter unit 14, all cascadedly coupled together to effect a pulsed 
R.F. signal over signal line 16 coupled to one input of a conventional 
microwave circulator 18. An antenna system shown at 20 may be coupled to 
another port of the circulator 18. Still another port of the circulator 18 
couples the antenna system 20 to the receiving portion of the exemplary 
radar which includes an R.F. amplifier, a mixer/filter circuit 24, a first 
IF amplifier IF1, another mixer/filter circuit 26 and a second IF 
amplifier IF2, all cascadedly coupled together. A stable local oscillator 
(STALO) 28 may provide a fixed IF signal denoted as LO2 to both of the 
mixer/filter circuits 10 and 26. An R.F. signal may be generated by the 
STALO 28 and provided to a frequency synthesizer 30 over signal line 32. 
The synthesizer 30 may alter the generated R.F. signal so as to provide a 
desired sequence of R.F. frequency signals, denoted as LO1, to the 
mixer/filter units 12 and 24. 
Downstream of the IF2 amplifier may be a pair of a conventional 
mixer/filter circuits 36 and 38 for effecting the in phase (I) and 
quadrature (Q) signal components from the receiver signals. The STALO 28 
may generate a fixed frequency signal denoted as LO3 which is supplied in 
phase to the mixer/filter circuit 36 and 90 degrees out-of-phase to the 
mixer/filter circuit 38. The I and Q components of the receiver signals 
may be supplied to a conventional doppler signal processing section 40 
which derives a substantially representative doppler frequency spectrum of 
signals therefrom. 
Generally within the doppler frequency spectrum of signals there may be an 
identified group of clutter signals, commonly referred to as the main beam 
clutter, which may be positioned about an undesirable doppler frequency or 
within an undesirable range of doppler frequencies. Under these 
conditions, any moving targets having a doppler frequency characteristic 
within the clutter range would be masked by the clutter signals. In 
addition, as a result of practical unbalance of hardware components, such 
as an unbalance between the mixer/filter circuits 36 and 38, for example, 
which effect the I and Q signal components of the received signals, an 
image of the main beam clutter may be produced in the doppler spectrum 
mirrored about the baseband doppler frequency which in the present case is 
zero doppler frequency. An illustration of these clutter signal bands both 
main beam and image, is shown in the exemplary graph of FIG. 2A. As has 
been explained above, any moving targets having doppler frequencies within 
the ranges of the clutter signals could be masked and pass undetected by 
the radar. 
Present radars are using clutter positioning techniques to tune the main 
beam clutter to the doppler baseband frequency and remove it from the 
moving target detection ranges in the doppler frequency spectrum. For 
example, in an airborne radar employing downlook air-to-air modes, the 
doppler frequencies of the main beam clutter are a function of primarily 
the velocity of the aircraft and the angle position of the antenna radar 
beam with respect to the earth. Thus, with these two factors being either 
known or measurable, a compensating frequency signal may be injected into 
the radar at a prespecified location to tune the main beam clutter signals 
to a new, more desirable, position in the doppler frequency spectrum, 
normally about the doppler baseband frequency. 
One way of accomplishing this is using a conventional clutter position 
computer 42, a digital-to-analog (D/A) converter 44 and a voltage 
controlled crystal oscillator (VCXO) 46 as depicted in the schematic of 
FIG. 1. For example, a signal 48 representative of the measured aircraft 
velocity and a signal 50 representative of the angle of the radar beam may 
be provided to the clutter position computer 42 which may then compute 
therefrom a control word 52 which may be converted to an analog signal 54 
via the D/A converter 44 to govern the output frequency of the VCXO 46. 
The output frequency signal 56 in the present example is provided to the 
mixer/filter circuit 10. 
For an operational example, let us assume that the VCXO 46 generates a 
frequency signal 56 with a nominal frequency of 40 megahertz. If the fixed 
frequency signal LO2 is on the order of 1460 megahertz, then the mixer 10 
may effect an IF frequency of on the order of 1500 megahertz which is, in 
turn, amplified by A1 and provided to mixer 12. Should the frequency LO1 
be generated at 7500 megahertz, then the mixer 12 provides an R.F. carrier 
on the order of 9 gegahertz which is pulsed by the transmitter unit 14 and 
transmitted to the antenna system 20 via line 16 and circulator 18. 
Accordingly, an echo signal may be received by the antenna system 20, 
passed through circulator 18 and provided to the R.F. amplifier. The 
received signal may be beat down in mixer 24 by the signal LO1, which is 
set at 7500 megahertz. The first IF signal may be conditioned by the IF1 
amplifier and further beat down in mixer 26 to the second IF level by the 
fixed frequency signal LO2. The second IF signal which is now on the order 
of 40 megahertz may be conditioned by the amplifier IF2 and conducted to 
the I-Q mixers 36 and 38. If the STALO 28 generated frequency signal LO3 
is set at 40 megahertz also, then any doppler frequency spectrum derived 
from the received echo signals or components thereof will include a main 
beam and image groupings of clutter signals as illustratively shown in 
FIG. 2A. 
However, the present system includes the clutter position computer 42 which 
is provided with the signals 48 and 50 to derive a compensating frequency 
to reposition the main beam clutter to the baseband level within the 
doppler frequency spectrum. It does this by deviating the frequency signal 
56 of the VCXO 46 from its nominally chosen value, in this case 40 
megahertz, as governed by the control word 52 via D/A 44. Of course, as 
the main beam clutter grouping of signals is positioned to baseband, its 
mirror image in the doppler frequency spectrum is likewise tuned to the 
same new position. The graph of FIG. 2B illustratively shows the main beam 
and image clutter groupings of doppler signals positioned about the 
doppler baseband frequency (i.e. zero doppler frequency). 
More sophistication in clutter positioning may be provided in some radars 
as the application demands. Techniques such as clutter tracking, for 
example, are presently employed in some radar systems to stabilize the 
main beam clutter about baseband. In these more sophisticated radars, a 
"servoing" or additional governing signal 60 may be supplied to the 
clutter position computer 42 to alter the control word 52 and eventual 
frequency signal 56 in a timely manner to stabilize the clutter frequency 
signal grouping about the doppler baseband frequency in the doppler 
frequency spectrum. 
These types of clutter positioning techniques have been found adequate for 
conventional radars using slow moving mechanically scanned antennas 
wherein the signal beam is scanned through a spatial region very slowly. 
However, the more contemporary radars are designed for electronically 
agile multibeam operation where the antenna beam is electronically and 
rapidly switched between a multiplicity of targets, that is the antenna 
power is time shared through a variety of antenna beam directions. Thus, 
the main beam clutter doppler frequency groupings will be at various 
doppler frequencies dependent primarily on the direction of the antenna 
beam and aircraft velocity. 
In the clutter positioning operation, each main beam clutter grouping in 
the doppler spectrums of the plurality of antenna beams will have to be 
tuned to the baseband doppler independently. If the clutter positioning is 
mechanized using the VCXO embodiment for clutter positioning as described 
in connection with the embodiment of FIG. 1, a clutter positioning circuit 
comprising the elements 42, 44 and 46 would be needed for each 
directionally effected antenna beam of the radar. Rapidly tuning a VCXO is 
not suitable since phase memory must be maintained in a coherent radar. 
To better understand the problem of phase memory, let us assume that each 
beam of the antenna includes at least one transmission of a plurality of 
R.F. pulses constituting a radar look and that the radar is operative to 
receive echo R.F. pulses of the look for each transmitted beam 
interspersed in time with echo R.F. pulses of the looks of the other 
transmitted beams of the plurality of beams. The interpulse period of the 
pulse doppler coherent radar may be divided into transmissionable time 
slots with each time slot corresponding to a potential transmission time 
for an R.F. pulse of a different beam look. Under these conditions, the 
radar must keep a fixed phase relationship from pulse-to-pulse in order to 
derive a substantially representative doppler frequency spectrum of 
signals associated with each look of the radar beams. Thus, with the 
requirement of a phase memory another problem is introduced further 
complicating the clutter positioning operation of a electronically agile 
multibeam radar. 
SUMMARY OF THE INVENTION 
An electronically agile multi-beam radar is operative to switchedly 
transmit a plurality of beams directionally separated by time sharing the 
illuminating power thereof. Each beam includes at least one transmission 
of a plurality of R.F. pulses constituting a radar look. The radar is also 
operative to receive echo R.F. pulses of the look from each transmitted 
beam interspersed in time with echo R.F. pulses of the look of the other 
transmitted beams of the plurality. 
In accordance with the present invention, the radar comprises a means for 
maintaining substantially a desired pulse-to-pulse phase relationship for 
the received plurality of echo pulses of each look of each beam in a 
time-shared manner in order to derive a substantially representative 
doppler frequency spectrum of signals associated with each look of the 
transmitted beams and to adaptively position an identified group of 
clutter signals about a prespecified doppler frequency in the desired 
doppler frequency spectrum for each look of each beam. More specifically, 
the radar may be of a pulse-doppler coherent type having an interpulse 
period divided into transmissionable time slots wherein each time slot 
corresponds to a potential transmission time for an R.F. pulse of a 
different beam look. In this embodiment, the maintenance means includes a 
means for deriving in a time-shared manner a relative phase characteristic 
for each of the plurality of R.F. pulse beams in accordance with the 
direction of transmission thereof, means for storing the derived relative 
phase characteristics corresponding to the time slot for which they are 
derived, and means for accessing each stored relative phase characteristic 
once each interpulse period to effect substantially the desired 
pulse-to-pulse relationship corresponding to the R.F. pulse beam 
associated therewith. The deriving means is operative to perform the 
derivations for all the aforementioned beams, once each transmissional 
time slot of the interpulse period. 
In one embodiment, the maintenance beam includes a phase shifter disposed 
in the transmission portion of the radar and updated with the accessed 
relative phase characteristic corresponding to the pulse beam being 
transmitted once each interpulse period to effect the desired 
pulse-to-pulse phase relationships thereof. In another embodiment, the 
maintenance means includes a phase shifter disposed in the reception 
portion of the radar and updated with the accessed relative phase 
characteristic corresponding to the R.F. pulse beam being received once 
each interpulse period to effect the desired pulse-to-pulse phase 
relationship thereof. 
Still further, the radar may be disposed on-board an aircraft which 
includes a means for measuring the velocity of the aircraft in flight. In 
this case, the deriving means may include means for deriving in a 
time-shared manner a relative phase characteristic for each of the 
plurality of R.F. pulse beams in accordance with both the direction of the 
corresponding beam transmission and the measured aircraft velocity 
associated therewith.

DESCRIPTION OF THE PREFERRED EMBODIMENT 
An embodiment for clutter positioning suitable for use in an electronically 
agile multi-beam radar is depicted in the block diagram schematic shown in 
FIG. 3. The embodiment may be disposed at an appropriate place in the 
radar for maintaining substantially a desired pulse-to-pulse phase 
relationship for the received plurality of echo pulses of each look of 
each beam in a time shared manner in order to derive a substantially 
representative doppler frequency spectrum of signals associated with each 
look of the transmitted beams and to adaptively position an identified 
group of clutter signals about a prespecified doppler frequency in the 
derived doppler frequency spectrum for each look of each beam. The 
disposition of the embodiment in the radar system will be described in 
greater detail herebelow. 
Included in the embodiment of FIG. 3 may be a clutter position computer 42 
for deriving in a time shared manner a relative phase characteristic for 
each of the plurality of R.F. pulse beams in accordance with the direction 
of transmission thereof. The computing device 42 may be operative to 
perform the derivations for all the aforementioned beams once each 
transmissionable time slot of the interpulse period of the radar. The 
computing device 42 may be of a conventional variety similar to the type 
described in connection with the embodiment of FIG. 1 but in the present 
example used in a time shared manner to compute the control words for each 
of the beams being switchedly transmitted by the radar. Likewise, the 
information which may be needed by the computing device 42 in its 
derivations of the control words may be provided thereto over signal lines 
48 and 50 much the same as that described in connection with the 
embodiment of FIG. 1. 
The derived relative phase characteristics may be accumulated and stored in 
a storage device corresponding to the time slot for which they are 
derived. In the present embodiment, a plurality of accumulators 70.sub.1 
through 70.sub.N are provided having their inputs coupled to the computing 
device 42, each corresponding to a transmitted radar beam of the radar. 
The output words of the accumulators 70.sub.1 through 70.sub.N may be 
coupled to a digital word multiplexer 72 which may be governed by a beam 
selection code signal 74. The selected digital word output of the 
multiplexer 72 may be coupled to a device which may be used, when disposed 
in the radar at an appropriate position, to effect substantially the 
desired pulse-to-pulse relationship corresponding to the R.F. pulse beam 
associated therewith. 
In the present embodiment, the effecting device 76 may comprise a phase 
shifter of the digital variety governed by a digital word. The phase 
shifter may be disposed in a transmission line of the radar to shift the 
phase of the frequency signal associated therewith. The phase shifter 
element 76 may be of the type described in the U.S. Pat. No. 4,160,958 
issued to James H. Mims et al. on July 10, 1979, and may comprise phase 
shifting circuit elements similar to the type described in U.S. Pat. No. 
4,205,282 issued to John W. Gipprich on May 27, 1980. Both of the 
aforementioned patents are incorporated by reference herein for providing 
a more detailed description of the phase shifter element 76 of the 
preferred embodiment. 
In an exemplary operation, as the antenna of the radar is rapidly switched 
between beam spatial positions, the appropriate digital accumulator 
70.sub.1 through 70.sub.N may be connected to the phase shifting element 
by the multiplexer 72 as governed by the code signal 74. The clutter 
position computing element 42 may be operative continuously to derive the 
control words associated with the plurality of beams in accordance with 
the supplied navigational information and beam position information over 
signal lines 48 and 50, respectively. Because the beam transmitted from 
the radar has a finite beam width, a clutter spread in the derived doppler 
frequency spectrum for each look of the beam is expected. More 
specifically, the echo returns for an angle at one portion of the beam 
will be at a different doppler frequency than that of the echo returns for 
the angle at a different portion of the beam. As a result of this 
phenomena, the main beam clutter has a doppler frequency bandwidth as 
illustrated in the graphs of FIGS. 2A and 2B. Consequently, the computer 
42 cannot derive a specific frequency for the main beam clutter associated 
with the beam. Conventionally, these type derivations which are performed 
in the computer 42 use the average phase progression of the main beam 
clutter, which may be considered as the center frequency f.sub.c thereof. 
It is understood that the intention of the clutter positioning embodiment 
as described in connection with FIG. 3 when disposed in an appropriate 
transmission line of the radar is to impose a phase on each R.F. pulse of 
a radar beam such that when it is finally received, the main beam clutter 
association with that beam appears at essentially the baseband or zero 
doppler frequency in the doppler frequency spectrum derived for the beam, 
thus causing the main beam clutter and its image to fall into a signal 
processor filter notch for post-processing operations. It can be shown 
that only one phase setting per transmission pulse of the radar is needed 
to effect the pulse-to-pulse phase relationship desired, thus the phase 
characteristic derivations and phase settings of the phase shifter 
elements 76 need be done only at clock rates substantially equal to the 
transmitter pulse width. This will be more fully appreciated by the 
description provided in subsequent portions of the instant specification. 
A typical radar which may utilize the clutter positioning embodiment 
described in connection with FIG. 3 may have parameters as follows: 
Tp=Transmit pulse width (1 range cell, i.e., transmissionable slot). 
Tpp=Interpulse period (assumed constant for the present example). 
Nr=Number of range cells (Tpp/Tp) 
Np=Number of pulses coherently integrated (i.e. a look). 
Bf=Doppler filter 
bandwidth=1/(Np.multidot.Tpp)=1/(Nr.multidot.Np.multidot.Tp). 
Fres=1/(10.multidot.Nr.multidot.Np.multidot.Tp) The clutter position 
frequency resolution (Fres) is conventionally computed on the order of 
about 1/10 the doppler filter bandwidth Bf. 
Each storage accumulator 70.sub.1 through 70.sub.N of the preferred 
embodiment may utilize B bits. In this case, the corresponding least 
significant control word phase bit is defined as 2.pi./2.sup.B. The 
control word input to an accumulator may be advanced by W least 
significant bits each time the accumulator is updated or clocked. The 
phase associated with the accumulator, thus advances by 
2.pi..multidot.W/2.sup.B with each update or clock pulse. 
The frequency generated by the phase shifter 76 may be represented by the 
following mathematical relationships: 
##EQU1## 
The resolution frequency Fres may be represented mathematically by the 
following equation: 
##EQU2## 
Thus, the required number of bits B for the accumulator input words may be 
derived from the preceding equations and represented mathematically as: 
EQU 2.sup.B =10Nr.multidot.Np. (7) 
The maximum input control word may correspond to the maximum doppler 
frequency of interest F.sub.DMAX as follows: 
EQU W.sub.MAX =F.sub.DMAX /Fres. (8) 
The above equation (8) represents the maximum number of bits per update 
W.sub.MAX in an input control word of an accumulator 70.sub.i derived by 
the computing element 42. 
A typical radar of the pulse doppler coherent variety may have the 
following parameters: 
Tp=1 microsecond pulse width. 
Tpp=100 microsecond (10 KHz PRF). 
Nr=Tpp/Tp=100 range cells or transmissionable slots. 
Np=64 integrated pulses per look. 
Thus, 2.sup.B =10.multidot.64=64,000, and B=16 bits (i.e. 2.sup.16 =65536). 
EQU Fres=1/(Tp.multidot.2.sup.B)=1/(10.sup.-6 .multidot.2.sup.16)=15.258 Hz. 
For 62.5 KHz maximum doppler frequency, 
EQU W.sub.MAX =62,500/15.258=2.sup.12 (i.e. 12 bits). 
Thus, each of the accumulators 70.sub.1 through 70.sub.N may utilize 12 
input binary bits and 16 output binary bits. On the other hand, if the 
output word of the accumulations is truncated to 12 binary bits, it 
results in spurious of about -6 dB/bit.times.12 bits or -72 dB, in which 
case the system provides on the order of 70 dB dynamic range. In other 
words, the accumulators 70.sub.1 through 70.sub.N should accommodate 
relative phase change updates at 16 bits for the desired frequency 
resolution. Of the 16 bits, the 12 most significant bits may be utilized 
to drive a 12 bit binary phase shifter, like that shown at 76, to provide 
the desired dynamic range. Typically, an agile beam radar might utilize 10 
to 20 beams, each tracking a target. In this case, a like number of 
accumulators would be included with each being switched to govern the 
phase shifter 76 via the multiplexer 72 when the appropriate beam is 
selected as represented by the beam selection code signal 74. 
One radar embodiment suitable for utilizing the clutter positioning system 
as described in connection with FIG. 3 is depicted in the block diagram 
schematic of FIG. 4. In this embodiment, the phase shifter 76 may be 
disposed in the transmission portion of the radar and updated with the 
accessed relative phase characteristic control word 79 corresponding to 
the R.F. pulse beam being transmitted once each interpulse period to 
effect the desired pulse-to-pulse phase relationship thereof. The relative 
phase characteristic control word 79 is derived by the unit 80 which 
comprises the computing element 42, accumulators 70.sub.1 through 70.sub.N 
and multiplexer 72 and operates in a similar manner as that described for 
this combination of elements hereabove. More specifically, the phase 
shifter 76 is disposed in the transmission line between the first 
mixer/filter unit 10 and amplifier A1 with the input thereof coupled to 
the mixer 10 and output coupled to the amplifier A1. In this disposition, 
the phase shifter 76 may effect substantially the desired pulse-to-pulse 
phase relationship in each transmitted R.F. pulse beam using the accessed 
relative phase characteristic control words 79 corresponding associated 
with the beams once each interpulse period. The nominal frequency signal 
supplied to the mixer 10 over the signal line 56, for this example, may be 
generated by the STALO 28 and may be of the same frequency as that used 
for the I and Q mixers 36 and 38, denoted as LO3. 
An alternate radar embodiment utilizing the clutter positioning system 
described in connection with FIG. 3 is depicted in a block diagram 
schematic in FIG. 5. In this embodiment, the phase shifter 76 may be 
disposed in the transmission line carrying the frequency signal LO2 
coupled between the STALO 28 and mixer 10. The nominal frequency supplied 
to the mixer 10. The frequency signal over line 56 may be generated by the 
STALO 28 and be the same frequency utilized by the mixers 36 and 38, 
denoted as LO3. The operation may be similar to that described in 
connection with the embodiment of FIG. 4. 
Still another alternate radar embodiment which includes the phase shifter 
76 disposed in the reception portion of the radar is shown in a block 
diagram schematic in FIG. 6. More specifically, the phase shifter is 
disposed in the transmission line which carries the frequency signal LO2 
coupled between the STALO 28 and mixer 26. In operation, the clutter 
positioning unit comprising elements 76 and 80 effects substantially the 
desired pulse-to-pulse phase relationship in each received R.F. pulse beam 
using the accessed relative phase characteristic control words 79 
corresponding thereto once each interpulse period. 
Another embodiment of a radar may include the aforementioned phase shifter 
76 in one of the transmission lines coupling the frequency synthesizer 30 
to either mixer 12 or mixer 24. However, because of the frequency 
variation nature of the frequency signal LO1 conducted through the 
transmission line of the phase shifter 76 this embodiment is considered to 
be somewhat more sophisticated than the others described in connection 
with FIGS. 4 through 6 hereabove. Nonetheless, it is considered a workable 
embodiment taking into consideration the varying frequencies involved from 
beam to beam or even in some cases from look to look. It is understood 
that some further additions and/or modifications may have to be made to 
the embodiment described in connection with FIG. 3, however, it is felt 
that these additions and/or modifications will not deviate from the broad 
principles of the present invention and may be carried out by someone 
skilled in the pertinent art. 
With regard to updating the phase shifter 76 only once each transmitted 
pulse of a beam independent of the particular embodiment used, the 
following analysis is provided and supplemented with the circuit schematic 
of FIG. 7. FIG. 7 represents a simplified schematic of a doppler signal 
processing unit 40 suitable for operation with a radar embodying the 
clutter positioning system of FIG. 3. Referring to FIG. 7, conventional 
dumped integrators 90 and 92, sample and hold circuits 94 and 96, and A/D 
converters 98 and 100 may be respectively disposed in cascade in their 
corresponding I and Q transmission paths which are coupled to a fast 
fourier transform (FET) signal processor 102. The dumped integrators 90 
and 92 may be operated conventionally by gating signals 104 and 106, 
respectively. Ideally, the inputs to the dumped integrators 90 and 92 may 
be represented mathematically by the following equations for the J'th 
pulse: 
EQU I(J)=cos [.omega..sub.D t+.phi..sub.cp (J)] (9) 
EQU Q(J)=sin [.omega..sub.D t+.phi..sub.cp (J)] (10) 
Where: 
.omega..sub.D is the doppler frequency (i.e. center frequency) of the main 
beam clutter, and 
.phi..sub.CP (J) is the phase shifter setting for the J'th pulse. 
It is understood that in a practical sense, the mixers and amplifiers of 
the radar such as depicted in FIGS. 4 through 6 introduce amplitude 
uncertainties which shall be donated as .epsilon., and phase shift errors 
which shall be denoted as .phi..sub.E, both of which will be assumed here 
to be independent of .omega..sub.D. Thus, the dumped integrator inputs may 
then be represented mathematically including the amplitude uncertainty and 
phase shift error as follows: 
EQU I(J)=cos [.omega..sub.D t+.phi..sub.cp (J)] (11) 
EQU Q(J)=(1+.epsilon.) sin (.omega..sub.D t+.phi..sub.cp (J)+.phi..sub.E) (12) 
The gating times for the dumped integrators as controlled by the gating 
signals 104 and 106 may be the same as the interpulse period Tpp and the 
pulse width Tp as described supra. Accordingly, the I dumped integrator 90 
may have an output (DI.sub.I) at the end of the J'th pulse which may be 
represented mathematically as follows: 
##EQU3## 
Likewise, the Q dumped integrator 92 may have an output at the end of the 
J'th pulse which may be represented mathematically as follows: 
##EQU4## 
The terms of the equations (14) and (16) contain the main beam clutter and 
its image, all of which must be positioned in the doppler frequency 
spectrum within the DC filter or "notch" filter of the FET 102. This may 
happen when the terms of the equations (14) and (16) are independent of J, 
i.e., they are of the same value for each pulse. 
This occurs when: 
EQU .phi..sub.cp (J)=-.omega..sub.D JTpp+.phi..sub.cpo (17) 
where .phi..sub.cpo is some constant phase. .omega..sub.D JTpp is simply 
the phase of the main beam clutter doppler bandwidth at the start of each 
pulse. 
By substituting equation (17) into equations (14) and (16) the outputs of 
the dumped integrators 90 and 92 may be mathematically represented by the 
following equations: 
##EQU5## 
Since the above equations (18) and (19) are now independent of the pulse 
number J, they are therefore within the FFT notch filter or positioned 
substantially about the zero doppler frequency and may be removed by the 
notch filter of the FFT 102, as desired.