WO2008115175A1 - Beam design for synthetic aperture position/velocity estimation - Google Patents

Abstract

Phase modulated beam patterns are substituted for the constant-phase versions that have been used in pπor synthetic aperture systems Relative movement between a radar/sonar/ultrasound platform and a point target causes a sequence of echoes from the point target to be phase and amplitude modulated by the beam pattern, as well as by the usual quadratic phase variation caused by range changes Azimuth, range rate, and azimuth rate estimation, as well as detection in clutter, are substantially improved by appropriate beam pattern phase modulation, which is applied to the transmitter and/or receiver beam patterns Phase modulated beam patterns are synthesized with array element weighting functions (1 18) that are designed for high ambiguity function peak-to-sidelobe level, reduction of unwanted ambiguity ridge lines, and adequate spatial sampling Two dimensional beam pattern phase modulation is useful when the relative motion between a transmit-receive array (109) and multiple targets has both azimuth and elevation components

Description

DESCRIPTION

TITLE

Beam design for synthetic aperture position/velocity estimation

TECHNICAL FIELD

Radar, sonar, ultrasound, echo cardiology

BACKGROUND ART

The purpose of the invention is to improve stripmap synthetic aperture radar/sonar (SAR/SAS) and inverse synthetic aperture radar/sonar/ultrasound (ISAR/ISAS) via better clutter rejection, velocity parameter estimation (range rate and azimuth rate), and azimuth position estimation. The invention also is applicable to target movement in three dimensions relative to a radar/sonar/ultrasound array. In this case, the invention improves estimates of range rate, azimuth rate, elevation rate, azimuth position, and elevation position. Applications are to maritime and ground surveillance SAR containing moving objects, missile defense radar (ISAR), sonar mine hunting (SAS)₁ and noninvasive Doppler ultrasound (ISAS) for fluid velocity measurement in two dimensions (parallel and perpendicular to the vessel).

In synthetic aperture processors, echoes are coherently pulse compressed (replica correlated or matched filtered) on reception and used to form in-phase and quadrature components. These components specify the magnitude and phase of a complex-valued echo range sample. A sequence of such components from multiple signal-echo pairs comprise a phase history corresponding to the range variation (measured in wavelengths) associated with relative motion between the radar/sonar platform and each point target.

If point targets at different azimuth locations do not move except for relative platform motion, then their phase histories are azimuth-displaced versions of a predictable phase history, and they can be separated by azimuth compression if the phase history bandwidth B is sufficiently large (CE. Cook and M. Bemfeld, Radar Signals, Academic Press, New York, 1967). If various point targets have different range rates, however, they may not be separable even if B is large. This lack of resolution can occur if range rate and azimuth displacement can compensate for one another. For example, range rate causes a displacement in frequency. If a phase history lies on a long tilted line in the time-frequency plane, a frequency displacement can be compensated by a time shift. A phase history time shift is equivalent to an azimuth shift. A tilted line in the time-frequency plane corresponds to linear frequency modulation (linear FM) and quadratic phase modulation. Quadratically phase modulated (linear FM) phase histories experience degraded range-rate/azimuth resolution with ambiguous receiver outputs, such that objects with nonzero range rate appear at the wrong azimuth. Azimuth compression often utilizes frequency domain matched filtering to correlate the data phase history with reference phase histories. An unknown range rate causes a Doppler shift, which can be hypothesized by a frequency domain shift of a predicted (reference) phase history. Predicted phase histories are correlated with the data phase history for estimation/detection. Range rate estimation accuracy and resolution capability is proportional to the duration T of the phase histories in the absence of ambiguities and error coupling, such as those that occur with linear FM (Cook and Bernfeld, op. cit).

Azimuth rate in the direction of assumed relative platform motion increases the rate at which the beam pattern sweeps across a target, and causes time compression of phase histories; azimuth rate in the opposite direction causes time dilation. Time scaling (compression/dilation) can be included as an additional parameter hypothesis in the azimuth compression process. The effect of nonzero range rate on a wideband radar/sonar waveform also is represented by time scaling. Estimation accuracy of compression/dilation increases with waveform time-bandwidth product (R.A. Altes and E.L. Titlebaum, "Bat signals as optimally Doppler tolerant waveforms," J. Acous. Soc. Am. Vol. 48, 1970, pp. 1014-1020). Azimuth rate estimation accuracy thus is proportional to the time-bandwidth product (TB) of the phase histories, unless ambiguity effects limit estimation and resolution capability.

For the smooth, low time-bandwidth product beam patterns that comprise prior art, the phase history caused by relative target/platform motion in broadside stripmap SAR is closely approximated by a quadratic phase function, corresponding to linear FM. The azimuth compression process is then subject to the well known range-Doppler error coupling phenomenon for linear FM, such that the receiver response to ambiguous pairs of erroneous azimuth displacements and range rates is nearly as large as the receiver response to the correctly hypothesized azimuth and range rate. This error coupling is manifested as a ridge in the phase history azimuth/range-rate ambiguity function, and it severely degrades estimation/detection performance in a cluttered environment relative to a receiver with an ideal (thumbtack) ambiguity function. Linear FM also is relatively insensitive to azimuth rate (compression/dilation), compared to other waveforms or phase histories with the same TB product (R.A. Altes, Optimum waveforms for sonar velocity discrimination," Proc. of the IEEE vol. 39, 1971, pp. 1615- 1617).

In the absence of error coupling, azimuth resolution improves as SAR/SAS beam width is increased, since phase history bandwidth is increased. Increased beam width also increases the phase history time-bandwidth product, resulting in improved azimuth rate resolution. Increased beam width and phase history duration, however, cause extension of the tilted line representation of linear FM in the time- frequency plane, with consequent extension and flattening of the linear FM ambiguity function ridge line. These effects increase the effect of unknown range rate on azimuth estimation error. A tradeoff thus occurs, such that beam widening improves azimuth rate estimation but degrades azimuth and range rate estimation because of linear FM error coupling. Target detection in Doppler-distributed clutter also tends to be degraded when the FM ridge line is extended via beam widening. Synthetic aperture processors that use beams with no phase modulation are geometrically constrained to operate with quadratic phase histories (linear FM) for azimuth compression, despite the drawbacks and tradeoffs associated with such modulation. This constrained operation constitutes the prior art.

DISCLOSURE OF INVENTION

The problem is to mitigate the deleterious effects of the quadratic phase modulation (linear FM) that is imposed by the geometry of synthetic aperture (and inverse synthetic aperture) data acquisition. These effects include high azimuth/range-rate ambiguity, relatively poor azimuth rate estimation, degraded joint estimation of range-rate and azimuth, and relatively poor detection of objects in moving clutter (e.g., a small boat in sea clutter). If the effects of quadratically modulated phase histories can be reduced or eliminated, the perceived location of an object will not be affected by object motion, detection of an object in moving clutter will be improved, and accurate velocity estimation in two directions (range rate and azimuth rate) will be obtained.

The solution is to replace a conventional (prior art) beam pattern, which lacks nonlinear phase modulation, with a beam pattern that has appropriate nonlinear phase modulation or phase coding. An appropriate beam phase modulation function removes the constraints that have been imposed by linear FM phase histories. As the beam is swept past a point target in a SAR/SAS application (or a point target moves through the beam in ISAR/ISAS), beam coding/modulation adds beam-induced phase variation to the quadratic phase that is associated with range variation. The additional beam-induced phase variation is controlled by the system designer rather than by geometry.

A beam pattern with appropriate nonlinear phase modulation is obtained by phase modulating the aperture shading function that is applied to a phased array. Additional array element phase shifts may be applied for beam steering and to compensate for a non-homogeneous propagation medium. These additional phase shifts are considered to be prior art and are outside the scope of the present invention.

FIG. 1 illustrates the geometry for a broadside stripmap SAR. Phase histories are formed from in-phase and quadrature (I,Q) samples at the output of a coherent signal-echo correlator or matched filter that is applied to each echo in a sequence of pulse-echo pairs. Phase histories are functions of range 101 and azimuth 102, and the azimuth variation is of interest here. For a beam pattern 103 transmitted from an array on platform 104, phase variation of an azimuth-dependent target phase history is associated with pulse-to-pulse range variation of the target relative to the platform, which is caused by target motion along the path 105 and is illustrated by crossing of constant-range contours 106. Constant range contours also can be interpreted as constant-phase contours when the range is measured in wavelengths (A.W. Richaczek, Principles of High Resolution Radar, McGraw-Hill, New York, 1969; R.O. Harger, Synthetic Aperture Radar Systems, Academic Press, New York, 1970; W.G. Carrara, R.S. Goodman, and R.M. Majewski, Spoϋight Synthetic Aperture Radar, Artech House, Norwood, MA, 1995). Relative motion of platform and target causes the beam pattern to be swept across the target. The amplitude variation of the resulting phase history function is determined by the beam pattern magnitude. For a broadside beam with no phase modulation (prior art), the phase variation of the phase history function depends strictly on range changes (measured in wavelengths) over multiple pulse-echo pairs. For broadside stripmap SAR (and Doppler ultrasound ISAS with the beam orthogonal to the direction of fluid motion), the phase histories exhibit quadratic phase modulation or linear frequency modulation (linear FM). Linear FM is associated with undesirable range-rate/azimuth ambiguities. Azimuth rate measurements also are relatively inaccurate.

FIG. 2 illustrates the effect of applying phase coding or nonlinear phase modulation to the array shading function and thus to the beam pattern. The phase modulated beam pattern 107 is broadened relative to the unmodulated beam pattern 103. Motion of the beam across a point target as in SAR/SAS (or motion of the target across the beam as in ISAR/ISAS) results in relative movement of a target along path 105. This motion causes the target to cross constant-phase contours 108 of the beam pattern, and such crossings are associated with beam-induced phase modulation of the target phase history. This beam-induced phase variation is added to the range-induced phase variation illustrated in FlG. 1. For suitable beam coding/modulation, the receiver becomes much less ambiguous with respect to joint range rate and azimuth estimates, and much more sensitive to azimuth rate.

The advantageous effects of the invention for broadside stripmap SAR are that appropriate beam phase modulation: (1) dramatically reduces range-rate/azimuth coupling error, (2) greatly improves resolution, (3) improves detection/estimation performance in clutter, (4) significantly reduces estimation errors for joint estimation of azimuth, range rate, and azimuth rate, and (5) eliminates the beam width tradeoffs that occur with unmodulated beam patterns (prior art), since phase histories no longer have quadratic phase variation.

The advantageous effect of the invention for ultrasonic ISAS as in echo cardiology is that appropriate beam phase modulation allows an operator to obtain accurate information about fluid flow as a function of position inside a conduit, including fluid velocity parallel and orthogonal to the conduit wall, with a noninvasive procedure. For three dimensional operation (e.g., blood flow measurement in the heart) a similar advantage applies to estimates of azimuth, elevation, azimuth rate, elevation rate, and range rate, provided that a two dimensional beam pattern is coded/modulated in azimuth and elevation so as to reduce ambiguities.

BRIEF DECRIPTION OF DRAWINGS (drawing sheets should not be numbered in margins)

FIG. 1 is a schematic illustration of the movement of a point target through an unmodulated beam of a synthetic aperture or inverse synthetic aperture system, wherein phase changes are caused by crossing constant-range contours that correspond to. constant-phase contours. FIG. 2 is a schematic illustration of the movement of a point target through a phase modulated beam of a synthetic aperture or inverse synthetic aperture system, wherein phase changes are caused by crossing constant-phase contours within the beam pattern as well as the constant-range contours in FIG. 1.

FIG. 3 is a schematic illustration of the application of a complex-valued array weighting (shading) function to elements of phased array during transmission and reception.

FIG. 4 is schematic diagram of the spherical coordinates that are used to describe a beam pattern.

FIG. 5A is a graph of the magnitude of the azimuth dependent part of a complex array element weighting function for elements with azimuth coordinates between -dM and dM.

FIG. 5B is a graph of a nonlinear phase modulation function that can be used with the magnitude function in FIG. 5A, where γ is the phase modulation factor.

FIG. 5C is a graph of the magnitude of the elevation dependent part of a complex array element weighting function for elements with elevation coordinates between -cW and dN, where N<M.

FIG. 5D is a graph of the zero-valued phase function that is used with the magnitude function in FIG. 5C when nonlinear phase modulation is applied only in azimuth and not in elevation.

FIG. 6A is a graph of the azimuth variation of the combined transmit-receive beam pattern magnitude when the element weighting functions in Figs. 5A-5D are applied to a rectangular phased array and when the phase modulation parameter γ equals zero (no nonlinear phase modulation of the array shading function).

FIG. 6B is a graph of the azimuth variation of the transmit-receive beam pattern magnitude on a decibel scale when no phase modulation of the array shading function is used (γ = 0), as in FIG. 6A.

FIG. 6C is a graph of the azimuth dependent phase variation of the transmit-receive beam pattern (γ = 0) in FIG. 6A.

FIG. 6D is a graph of the elevation variation of the combined transmit-receive beam pattern magnitude when the element weighting functions in FIGS. 5A-5D are applied to a rectangular phased array and when the phase modulation parameter γ equals zero (no nonlinear phase modulation of the array shading function).

FIG. 6E is a graph of the elevation dependent phase variation of the transmit-receive beam pattern (γ = 0) in FIG. 6D.

FIG. 7A is a graph of the azimuth variation of the combined transmit-receive beam pattern magnitude when the element weighting functions in Figs. 5A-5D are applied to a rectangular phased array and when y = 30 (azimuth dependent nonlinear phase modulation of the array shading function). FIG. 7B is a graph of the azimuth variation of the transmit-receive beam pattern magnitude on a decibel scale when azimuth dependent nonlinear phase modulation as in FIG. 5B with γ = 30 is applied to the array shading function.

FIG. 7C is a graph of the azimuth dependent phase variation of the transmit-receive beam pattern (γ = 30) in FIG. 7A.

FIG. 7D is a graph of the real part (amplitude normalized) of the azimuth dependent variation of the transmit-receive beam pattern (γ = 30) in FIG. 7A.

FIG. 7E is a graph of the imaginary part (amplitude normalized) of the azimuth dependent variation of the transmit-receive beam pattern (γ = 30) in FlG. 7A.

FIG. 7F is a graph of the elevation variation of the combined transmit-receive beam pattern magnitude when the element weighting functions in FIGS. 5A-5D are applied to a rectangular phased array and when γ =30 (azimuth dependent nonlinear phase modulation of the array shading function).

FIG. 7G is a graph of the elevation variation of the combined transmit-receive beam pattern phase when the element weighting functions in FIGS. 5A-5O are applied to a rectangular phased array and when γ =30 (azimuth dependent nonlinear phase modulation of the array shading function).

FIG. 8A is a graph of the azimuth/range-rate ambiguity function for a broadside stripmap synthetic aperture radar that uses the transmit-receive beam pattern in FIGS. 6A-6E (no phase modulation).

FIG. 8B is a graph of the azimuth/azimuth-rate generalized ambiguity function for a broadside stripmap SAR with transmit-receive beam pattern as in FIGS. 6A-6E (no phase modulation).

FIG. 9A is a graph of the azimuth/range-rate ambiguity function for a broadside stripmap synthetic aperture radar that uses the transmit-receive beam pattern in FIGS. 7A-7G, which is obtained from Hann function phase modulation of the array shading function as in Fig. 5B with γ =30.

FIG. 9B is a graph of the azimuth/azimuth-rate generalized ambiguity function for a broadside stripmap SAR with transmit-receive beam pattern as in FIGS. 7A-7G (obtained by nonlinear phase modulation of the array shading function with γ =30).

FIG. 10A is a graph of the range rate, azimuth rate distribution of typical sea clutter.

FIG. 10B is a graph of the azimuth, range rate distribution of spatially invariant sea clutter with a typical velocity distribution.

FIG. 10C is a graph of the azimuth, azimuth rate distribution of spatially invariant sea clutter with a typical velocity distribution. FIG. 11 is a graph of predicted broadside stripmap SAR signal-to-clutter ratio as a function of the nonlinear phase modulation parameter γ.

FIG. 12 is a graph of the transmit-receive beam width as a function of the array phase modulation factor γ.

FIG. 13A is a graph of the relative motion in the azimuth direction between the target environment and the azimuth dependent part of the transmit-receive beam pattern.

FIG. 13B is a graph of the relative motion in the elevation direction between the target environment and the elevation dependent part of the transmit-receive beam pattern.

FIG. 14A is a graph of the elevation/range-rate cross ambiguity function between phase histories from FIGS. 13A and 13B, where the azimuth and elevation beam patterns are designed for low cross ambiguity amplitude in order to minimize confusion between the targets in FIGS. 13A and 13B.

FIG. 14B is a graph of the elevation/elevation-rate cross ambiguity function between phase histories from FIGS. 13A and 13B, where the azimuth and elevation beam patterns are designed for low cross ambiguity amplitude in order to minimize confusion between the targets in FIGS. 13A and 13B.

MODES FOR CARRYING OUT THE INVENTION: BEST MODE

A maritime synthetic aperture radar (SAR) embodiment will be used as the best mode. The invention is concerned with accurate joint estimation of range, azimuth, range rate, and azimuth rate, with associated mitigation of ambiguous measurements and improvement of detection in clutter. For nonzero azimuth rate (target motion parallel to the path 105 in FIGS. 1 and 2), the rate at which the point target moves across the beam is changed. This change dilates or compresses the phase histories contributed by both range change (FIG. 1) and beam phase modulation (FIG. 2). For nonzero range rate, the point target has a velocity component that is orthogonal to the path 105 in FIGS. 1 and 2. This component causes a frequency shift of the phase history function that is measured at zero range rate, and is associated with crossing the constant phase contours 106 in FIG. 1 by moving in the range direction 101. The range rate velocity component also may affect the phase history function by crossing constant-phase contours 108 of the beam pattern 106 in FIG. 2, by moving orthogonal to the path 105. At long ranges, however, this effect usually is negligible.

For estimation of azimuth (phase history time shift), lower bounds on resolution bin size and the standard deviation of the time shift estimate are inversely proportional to the bandwidth of the observed phase history. For range rate (phase history frequency shift) estimation, lower bounds on resolution bin size and the standard deviation of the frequency shift estimate are inversely proportional to phase history time width (duration). For cross range rate (compression/dilation) estimation, the lower bounds are inversely proportional to the time-bandwidth product of the observed phase history (Cook and Bernfeld, op. cit.; Altes and Titlebaum, op. cit.; Altes, op. cit.).

Nonlinear phase modulation of an array shading function broadens the beam width, and a point target that moves across the beam along path 105 in FIG. 2 has a phase history with larger time width than in the absence of the modulation (FIG. 1). The range-induced phase modulation (FIG. 1) and the beam pattern phase variation (FIG. 2) increase the bandwidth of the time-extended phase history. The time-bandwidth product is further increased by an appropriate choice of the nonlinear phase modulation or coding that is applied to the array shading function. An appropriate array weighting function has half- wave, cosine-squared amplitude and phase functions, i.e., a Hann array shading amplitude function (H. L. Van Trees, Optimum Array Processing, Wiley, New York, 2002) with nonlinear phase modulation that is also proportional to a Hann function. This array weighting function results in a combined transmit-receive beam pattern with rapid, high-amplitude variation near the beam edges. For a given beam width, this type of variation yields phase histories with relatively large time-bandwidth product.

For a given complex beam pattern, broadside stripmap phase histories can be predicted from hypothesized azimuth, azimuth-rate, range, and range-rate parameters. These predictions can be tested with an estimation/detection process that correlates predicted (reference) phase histories with the data phase history. For multiple point targets at different azimuths, the correlation process is conveniently implemented with a frequency domain matched filter: The Fourier transform of the data phase history is multiplied by the conjugate of the Fourier transform of the hypothesized phase history, and the resulting product is inverse Fourier transformed.

The receiver response to hypothesized azimuth, azimuth-rate, and range-rate parameters can be represented by a generalized ambiguity function that depends on the three parameters. The central peak amplitude of the ambiguity function represents the receiver response corresponding to correct parameter hypotheses. Other ambiguity function samples correspond to various combinations of the hypothesized parameters. Sidelobes are local ambiguity function maxima that are not at the central peak. A ridge sometimes is formed by a set of ambiguous parameter combinations that lie on a line through the peak. For energy normalized phase histories, an ideal ambiguity function has unit amplitude at the central peak and low amplitude elsewhere (low sidelobe and ridge levels).

A complex-valued array shading function with magnitude equal to a Hann function and with phase equal to a Hann function multiplied by a constant (γ) is used in this embodiment of the invention. When elements of a phased array are weighted by this complex shading function, the resulting transrnit- receive beam pattern imparts high time-bandwidth product phase histories to point targets in broadside stripmap SAR. These phase histories result in a sharp central peak and uniformly low sidelobe levels of the azimuth/range-rate/azimuth-rate ambiguity function, relative to no phase modulation and to other phase modulation functions that have been applied to the Hann array shading function magnitude.

Relevant ambiguity functions correspond to combined transmit-receive beam patterns. The transmission beam pattern (radiation pattern) is obtained by applying samples of the complex-valued weighting (shading) function P_trara(x.y) to the transmitter phased array elements, along with phase shifts for beam steering. This complex weighting process is illustrated in FIG. 3. FIG. 3 shows a rectangular phased array 109 consisting of elements 110 that are separated in azimuth and elevation by d meters, where d is usually equal to one-half of the wavelength λ at the center frequency of the transmitted signal 111. During transmission, the signal 111 (which is the same at each element) is weighted by a complex shading function and phase shifted for beam steering. The shading and beam steering operations both depend on the element location x = x_m, y = y_n, which is measured relative to the center of the array; x = y = 0 at the array center.

During transmission, the shading and beam steering operations at each element are implemented by two multiplications, performed respectively by multipliers 112 and 113 in FIG 3. The signal 111 is first multiplied by a complex shading weight Ptrans(Xm.yn) 114. The resulting product is then phase shifted for beam steering via multiplication by the complex factor exp[j(2π/λ)Δr(x_m,y_n)] 115. The beam steering factor 115 depends on Δr(x_m,y_n), which is the distance from the element at x_m, y_π to the focus point minus the distance from the center element of the array to the focus point. For a non- homogeneous propagation medium, the beam steering factor 115 must be corrected for ideal focusing. The beam steering factor 115 is considered to be prior art; the invention pertains to the magnitude and phase of the complex-valued shading functions Ptrans(Jfm.y_n) 114 and PrecfXm.y,,) 116. During reception, the beam steering and shading operations at each element are again represented by two multiplications, performed respectively with two multipliers 117 and 118 in FIG 3. Assuming that the array look direction is the same for transmission and reception, the first multiplication operation corresponds to the beam steering factor exp[j(2π/λ)Δr(x_m,y_n)] 115, and the second multiplication implements complex weighting with Pmc(x_m<yn) 116. The receiver combines all the phase shifted, shaded element outputs with the summing operation 119.

FIG. 4 shows the spherical coordinates that are used to describe the beam pattern generated by the phased array in FIG. 3. The origin 120 of the coordinate system is at the center of the phased array 109 in FIG. 3. The location 121 of a point on a sphere with radius r₀ is specified by: (1) The length r₀ of a vector 122 drawn between the origin 120 and the point location 121 , (2) the angle #123 between the vector 122 and the z axis 124, where the z axis 124 corresponds to the boresight or focus direction for broadside stripmap SAR, and (3) the angle φ 125 between the azimuth (x) axis 126 and the projection 127 of the vector 122 onto the x,y plane. The azimuth dependent behavior of the combined transmit- receive beam pattern can be represented by the beam pattern as a function of θ 122 with φ 125 equal to zero. The elevation dependent behavior can be represented by the beam pattern as a function of θ 122 with (£ 125 equal to π/2, i.e., in the plane defined by the z (boresight) axis 124 and the elevation axis 128.

For a planar array with element m,n located at x_m,y_n. the complex-valued radiation pattern at a point on a sphere of radius r₀ with the center point of the sphere at the array center is

where dand φ are shown in FIG. 4, and

MwJ=K*. -xof +(y_n -y₀)² +4]^m -*. (2)

For the coordinate system in FIG.4,

Xo = f_osin#cosø , Yo - IΌ sin θ sinø, z_o = r_ocos^. (3)

The transmission array weighting function P_trans(Xm.yn) in Equation (1 ) is denoted by 114 in FIG. 3.

If the same array weighting function and steering vector are applied during both transmission and reception, then

Prβc(X.y) = Piracy) (4) where p_rec(x.y) is denoted by 116 in FIG. 3. In this case, the combined complex-valued transmit/receive beam pattern PTR(#,$ is the square of the radiation pattern:

PTR(0,0 = [Ptrans(0.0]². (5) When Equation (4) is true, the transmit/receive beam pattern is obtained from the radiation pattern in Equation (1) by squaring the radiation pattern magnitude and doubling the radiation pattern phase shift at each point where the beam pattern is evaluated.

For the SAR embodiment described here, the two-dimensional transmission weighting function 114 is separable:

PtransOf./) = Ptrans.azOO Ptrans.elfr)- (6)

FIG. 5A shows the magnitude 129 of an azimuth dependent complex weighting function = cos²[πx/(2dM)] exp{j γcos²[πx/(2c/M)]}, -dM≤x≤dM . (7)

The magnitude of Pu_ans.azM is a Hann weighting defined over 2M+1 azimuth (x) locations in the phased array 109 with uniform element spacing d. Since the elements at x = ± dM have zero weight, they are not utilized; the number of functional element locations in the azimuth dimension of the phased array is 2Λf-1. The element spacing d should be less than or equal to one-half wavelength (λ/2) for adequate spatial sampling. For a shading function that corresponds to prior art, the modulation factor γ equals zero, and the shading function in Equation (7) is real-valued. FIG. 5B shows the nonlinear phase modulation function 130 corresponding to Ptranε,az(x) in Equation (7), where the modulation factor γ may be nonzero. The Hann function phase modulation 130 in FIG. 5B and Equation (7) for nonzero γ is used in the embodiment of the invention presented here.

FIG. 5C shows the magnitude 131 of an elevation dependent Hann shading function Ptrans.ei(y) = cos²[πy/(2dΛQ], -dN≤y≤dN , (8) and FIG. 50 shows the phase 132 of the same function, which is defined over 2Λ/+1 elevation (/) locations in a phased array 109 with uniform spacing d ≤ λ/2. The number of functional element locations (with nonzero weight) in the elevation dimension of the phased array is 2Λ/-1 , where N<M for the broadside stripmap SAR embodiment described here. The shading function in Equation (8) has no phase modulation, and phase coding of the resulting beam pattern is expected to occur only in the azimuth (x) direction.

For the SAR embodiment presented here, the same array element weights (and the same array) are used for reception weights 116 as well as for transmission weights 114 in FIG. 3. The array is planar and rectangular with 129 azimuth element locations with nonzero weighting (M=Q5) and 31 elevation element locations with nonzero weighting (Λ/=16). The array 109 contains 129x31 elements in this case. Element spacing is one-half wavelength.

The transmit/receive beam pattern as a function of azimuth for γ = 0 is illustrated in FIGS. 6A- 6C. FIG. 6A shows the magnitude, on a linear scale, of the combined transmit/receive beam pattern P-m{θ,ψ) at φ = 0 and when γ = 0 (133), in the x,z plane in FIG. 4, with the maximum value of |PTR(0, 0)| normalized to unity. P-m(θ,φ) is computed as in Equations (1 )-(8) with γ = 0 in Equation (7), i.e., when nonlinear phase modulation is not used (prior art). FIG. 6B shows |PTR($ 0)| 134 on a decibel scale as a function of θ in the x,z plane. The dotted line 135 in FIG 6B denotes the maximum grating lobe (side lobe) level of the beam pattern when no phase modulation is used. FIG. 6C shows the phase of PTR(#.(*) at φ = 0 and when γ = 0 (136), which is expected to be close to zero.

FIGS. 6D and 6E illustrate the beam pattern as a function of elevation for γ = 0. FIG. 6D shows an amplitude normalized, linear-scale plot 137 of |Pτ_R(0,π/2)| when γ = 0, in the y,z plane in FlG. 4. FlG. 6E shows the phase 138 of P_TR(6>,π/2) when γ = 0, which is expected to be close to zero.

When nonlinear phase modulation is applied to the array shading function in the azimuth direction, an example of the resulting beam pattern as a function of azimuth is illustrated in FIGS. 7A-7C. FIG. 7A shows |PTR(0,0)| 139 when γ = 30 in Equation (7), amplitude normalized and on a linear scale. FIG. 7B shows a decibel scale version 140 of |PTR(# 0)| with γ = 30. The dotted line 141 in FIG 7B denotes the maximum grating lobe (side lobe) level of the beam pattern, which is 50 dB lower than in FIG. 6B. FIG. 7C shows the phase of PTR(0,0) when γ = 30 (142). This function is inverted relative to zero when the sign of γ is reversed. The vertical dotted lines 143 in FIG. 7C indicate beam pattern phase angles at 2π intervals, and correspond to the dotted line phase contours 108 of the beam pattern 107 in FIG. 2.

FIG. 7D shows the normalized real part 144 of the azimuth dependent profile PTR(Ø.O) of the combined transmit-receive beam pattern Pm{θ,φ) forγ = 30. This function is unaffected when the sign of γ is reversed. The time-bandwidth product of corresponding phase histories is increased by rapid, high- amplitude variation near the beam edges, resulting in improved azimuth rate resolution. FIG. 7E shows the corresponding normalized imaginary part 145 of PTR(#0) for γ = 30. This function is inverted with respect to zero when the sign of γ is reversed. When the beam moves along the path 105 in FIG. 2, the phase histories generated by the beam pattern in FIGS. 7D and 7E have relatively high velocity resolution and strong suppression of ambiguity function ridge behavior.

Figs. 7F and 7G describe the beam pattern as a function of elevation for γ = 30. For broadside stripmap SAR, nonlinear phase modulation is not applied to the array shading function in the elevation direction, and the elevation beam pattern for γ = 30 in Equations (7) and (8) is expected to be the same as for γ = 0. In FIG. 7F₁ the magnitude 146 of the combined transmit/receive (TR) beam pattern Pm(θ,Φ) at φ = π/2 radians for γ = 30 is indeed similar to its counterpart 137 in FIG. 6D. In FlG. 7G, the phase function 147 of PTR(0,JT/2) for γ = 30 is similar to its counterpart 138 in FIG. 6E.

Ambiguity functions are obtained from the beam patterns and from the geometry in FIGS. 1 and 2. The computations that are required to construct the ambiguity functions are similar to those required to implement the corresponding receiver. In order to specify these computations, the following definitions are required:

R₀ = the smallest range between the platform (transmit-receive array location) and the target, which occurs at time t = fo and at broadside azimuth (0= 0, φ = 0) Vpaz ≡ platform velocity (relative motion between transmit/receive array and target environment) along the path 105 shown in FIGS. 1 and 2 V₀. ≡ range rate, the target velocity component (measured relative to the platform) that is orthogonal to the path 105 in FIGS. 1 and 2. Vt_BZ ≡ target azimuth rate that is not included in the platform velocity to ≡ time when the target is at broadside azimuth; estimation of target azimuth is equivalent to estimation of <b t ≡ time measured relative to the time when the target is at broadside azimuth;

\t- to\ ≤ W2

Tote ≡ phase history observation time « B_1OciBRolVpaz

Bio_dB ≡ beam width [radians] determined by the interval between the points where the beam pattern drops to one-tenth of its maximum value.

The time-varying range between a point target and the center of the transmit-receive array is R(t I to,V_tr,Vt_az) = [(R₀ + along-range displacement)² + (cross-range displacement)²]¹'²

Assuming that |v_fr(f - t_o)\ ≤ \vώ T₀₆₃ /2 « R₀,

W I^W^J ≡Λo +v^-O + Kv^ +v^)² /2i?J(*-/₀)². (10)

For a signal center frequency with wavelength λ, the range dependent phase shift for two-way propagation is tPrangβC I footer) = (4π/λ) R(t | to,Vtr,V_tαz). (11 )

This phase shift function has a linear component (4πlλ)v_t!(t-to) corresponding to a frequency shift of the corresponding phase history, and a quadratic component (4π/λ)[(Vpa_Z+v_tez)²/(2R_o)](Wo)² that depends on platform velocity and cross-range velocity.

If V_taz is in the same direction as V_p01 (positive v_taz), the phase history is compressed in time, with a corresponding increase in the linear FM chirp rate associated with the quadratic phase variation. Conversely, if v_taz is in the opposite direction from v_mz (negative v_tgz), the phase history is dilated in time, with a corresponding decrease in the linear FM chirp rate associated with the quadratic phase variation. The time-varying azimuth angle between a point target and the center of the transmit-receive array is

W I O.V_/r.O = tan -1 cross-range azimuth displacement

(12)

R₀ + along-range displacement Again assuming that \v_tr{t - t_o)\ ≤ \vu\ T₀*, /2 « R₀, 0(t I to,vtr,v_βzz) = tan-MKVz÷ ^z)IRo] (M_β». (13)

For a combined transmit-receive beam pattern PTR(0,Ø ). the beam-induced time-dependent echo variation is Pjn[θ(t | to,v_tr,Vtaz). Φ\, where θ{t | fo.Vir.vw)]. is given by Equation (13). The beam- induced phase modulation is

<P_bβam(t I to,v_tr,v_taz) = tan-¹{imag{PτR.0 (t | fo.^vw). $}/real{ PTR[0 (t | t_Q,v,r,v_taz), φ ]}} (14) and the beam-induced amplitude modulation is |PTRW I t_o,v_tr,v,_az), φ\\.

The complete phase history function (including amplitude variation) for the point target is h_∞(*|*0,V_fr.V_te2)

(15)

Broadside stripmap SAR ambiguity functions are obtained by correlation of an energy normalized data phase history h_az(f|O,O,O) with a sequence of energy normalized reference (hypothesized) phase histories haz(f|4>. f{r_>V_faz) for different t₀ values (azimuth ≡ v^Jo), range rates v_(rand target azimuth rates v_taz:

I X_az(^o_»v_fr,v_to) |²= | Jh_az(t | O,0,0)h;α U_o,v_<r,v_tø) Λ |². (16)

A frequency domain implementation of the inner product in Equation (16) takes advantage of the fact that v,_r is associated with a frequency shift, and that different t₀ values are easily hypothesized by computing the inverse Fourier transform of the frequency domain product of the Fourier transform of the data phase history and the conjugated Fourier transform of a reference (hypothesized) phase history.

The generalized ambiguity function in Equation (16) represents the response of an estimator/detector receiver that is optimum for additive white, Gaussian noise, when the phase histories for specified f₀. v_tr, v_taz are known except for a constant phase shift, and the additive noise power is zero. The ambiguity function can be constructed from the delay-dependent outputs of a bank of matched filters, where each filter impulse response is a conjugated, time-reversed phase history corresponding to a different v_fr,v_ter pair.

A receiver for SAR data from a phase modulated beam pattern is implemented as in Equation (17) with the noise-free point target phase history replaced by the data phase history:

Estimator/detector receiver response for hypothesized parameters t^.v^v^ dt |² (17)

where haz,ref (f|<b.vv.vw) in Equation (17) equals

Equation (15). The receiver in Equation (17) implements a generalized data-reference cross ambiguity function. When a data phase history is not energy normalized and the reference phase histories are energy normalized, the output of the estimator/ detector is proportional to the squared amplitude of the data phase history and is thus an estimate of relative target strength. In this case, the outputs of the estimator/detector in equation (17) for various hypothesized ranges, azimuths, range rates, and azimuth rates comprise a map of target strength as a function of the four variables range, azimuth, range rate, and azimuth rate. This map is a generalized synthetic aperture image, which conventionally represents target strength as a function of range and azimuth.

FIG. 8A shows the broadside stripmap SAR azimuth/range-rate ambiguity function 148 when the same array weighting functions are used for transmission and reception as in Equation (4), the azimuth and elevation weighting function is separable as in Equation (6), the azimuth dependent part of aperture weighting function is given by Equation (7) with γ = 0, the elevation part is given by Equation (8), and M=65 and Λ/=16 in Equations (7) and (8), respectively. FtG. 8B shows the corresponding azimuth/azimuth-rate ambiguity function 149. The azimuth/range-rate ambiguity function in Fig. 8A is obtained from the azimuth/range-rate/azimuth-rate ambiguity function by assuming zero error in the azimuth rate hypothesis. Similarly, the azimuth/azimuth-rate ambiguity function in Fig. 8B assumes zero range rate error. The corresponding beam patterns are shown in FIGS. 6A-6E.

FIGS. 9A and 9B show ambiguity functions generated under the same conditions as in FIGS. 8A and 8B, except that the azimuth dependent part of the array shading function has nonlinear phase modulation; y = 30 in Equation (7). FIG. 9A shows the azimuth/range-rate ambiguity function 150 for γ = 30, and FIG. 9B shows the azimuth/azimuth-rate ambiguity function 151 for γ = 30. The corresponding beam patterns are shown in FIGS. 7A-7G. Comparison of the ambiguity functions in Figs. 8A and 8B with those in Figs. 9A and 9B indicates that significant improvements are obtained when appropriate phase modulation or coding is applied to the array weighting (shading) function.

Improved detection performance in clutter is associated with increased signal-to-clutter ratio (SCR). The "signal" in the SCR calculation is the expected receiver response to a point target in the absence of noise and clutter, and is represented by the ambiguity function amplitude with perfect parameter hypotheses, at the origin of the ambiguity function coordinates. The "clutter" in the SCR calculation is the expected receiver response to clutter, and is represented by the three dimensional integral of the product of the ambiguity function and the clutter distribution in azimuth/range-rate/azimuth- rate space. For maritime radar, a model for the sea clutter distribution is shown in FIGS. 10A-1 OC. This distribution model was obtained from an average velocity distribution for sea clutter (M.I. Skolnik, "Sea Echo," in Radar Handbook, M.I. Skolnik, ed., McGraw-Hill, New York, 1970), with the additional assumptions that the sea clutter velocity distribution is the same in azimuth-rate and range-rate and is uniformly distributed in azimuth. FIG. 10A shows the range-rate/azimuth-rate sea clutter model distribution 152, FIG. 10B shows the azimuth/range-rate sea clutter model distribution 153, and FIG 10C shows the azimuth/azimuth-rate sea clutter model distribution 154. When the target is moving, the clutter distribution is shifted relative to the origin of the ambiguity plane before computing the three dimensional integral.

SCR has been calculated using the three dimensional ambiguity functions represented by FIGS. 8A, 8B, ΘA, and 9B, and the three dimensional clutter distribution model represented by FIGS. 10A₁ 10B₁ and 10C. In these calculations, a persistent point target is assumed to have a range rate of 2.5 kt and an azimuth rate of 2.5 kt. FIG. 11 shows a function 155 representing the SCR for various modulation factors γ, divided by the SCR for γ = 0, on a decibel scale. FIG. 11 demonstrates the improved detection performance that is expected when the invention is properly applied, i.e., when suitable nonlinear phase modulation as in Equation (7) is applied to the array shading function.

For a broadside stripmap maritime SAR that uses the array weighting function in Equations (7) and (8), SCR becomes larger as the modulation factor |γ| increases, as indicated by FIG. 11. This improvement is limited by a bound on the maximum value of |γ| that is obtained from spatial sampling considerations. The instantaneous frequency corresponding to the phase variation in Equation (7) is finst(x) [rad/m] = (d/dx) γcos²[πx/(2dM)] = -2γ[π/(2c/M)]cos[πx/(2cWW)] sin[πx/(2cHW)]

= -[πγ/(2d/W)] s\n[πxI(dM)}, -dM≤x≤dM. (18)

The phase change between array elements is the frequency in radians per meter multiplied by the element spacing d in meters per element:

Phase change per element = -[πγ/(2M)] sin[πx/(dλf)], -dM≤x≤dM. (19)

The maximum absolute phase change per element in the azimuth (x) direction is max |Phase change per element| = π\y\l(2M) [rad/element]. (20)

-dM≤x≤dM

For adequate spatial sampling, the maximum phase change between elements should be less than or equal to π radians, which implies that

|γ| ≤ 2M = no. of functional array element rows (in the azimuth direction) +1 (21 ) where functional array element rows have nonzero element weighting function magnitude.

For |γ|=30 in Equation (21), the number of functional array element rows in the azimuth direction should be greater than or equal to 59. At X-band (λ « 3 cm), an array that is L meters long in azimuth with element spacing λ/2, contains approximately 67L element rows. If |γ| ≤ 67L + 1 , then |γ|=30 corresponds to an X-band array that is at least 43 cm long in the azimuth direction. A longer array can accommodate a larger value of |γ|.

A second constraint on |γ| pertains to permissible observation time. The phase modulated beam pattern 139 in FIG. 7A is approximately 14 times wider than the unmodulated beam pattern in FIG. 5A. The corresponding observation time is 14 times longer. This increased observation time may violate assumptions concerning constant velocity motion and the persistence/variation of target and clutter echoes. FIG. 12 shows the 10 dB beam width in the azimuth direction 156 as a function of the absolute value of the modulation factor |γ|. The observation time (phase history duration) is approximately equal to the 10 dB beam width in radians multiplied by the minimum range R₀ to the target, divided by the platform velocity V_p92 along the path 105 in FIG. 2.

A third constraint on |γ| is that the phase-modulated beam pattern should not cause under sampling of beam-induced phase histories as a target moves through the beam. The maximum absolute value of the beam-induced phase change between echoes should be less than or equal to π radians. The maximum absolute value of the beam-induced phase change between echoes equals the maximum absolute value of the slope of the phase function 142 in FIG. 7C multiplied by the azimuth change between observations in radians. If R₀ is the target range at broadside, the azimuth change between observations in radians is

tan -I cross-range azimuth displacement between observations

(22) R₀ + along-range displacement between observations

Letting PRI denote the radar pulse repetition interval, the cross-range azimuth displacement between observations equals (the radar platform velocity + azimuth rate) x PRI and the along-range displacement between observations equals (the target range rate) x PRI. If PRI = ζR</c where ζ is a constant (usually greater than ten) and c is the propagation velocity, the argument of the arctangent function is approximately C₁Vp₃Jc where V_p92 is the radar platform velocity. The quantity V_p0Jc is usually very small, and ζV_pgJc « 1. The azimuth change in radians between observations is usually very small, and the third constraint is rarely an important limitation for Jγ|.

Appropriate beam phase modulation improves synthetic aperture resolution in azimuth, range rate and azimuth rate, and suppresses ambiguity function ridges. This improvement implies that a four dimensional output representation is relevant, i.e., a map of target strength as a function of range, azimuth, range rate, and azimuth rate. This dimensionality increase is of no concern for computer analysis of estimator/detector outputs, provided the computer has sufficient memory and processing speed. The increased dimensionality is important, however, for display of the outputs for the benefit of a human observer. A stripmap representation that is suitable for a human observer can be obtained as follows. At a given stripmap (range, azimuth) pixel location, the most likely range rate V₁₁. and azimuth rate v_tta (conditioned on the pixel range and azimuth values) are the values of v,_r and v_tez that maximize I X_az,data-r_ef (O»^v _/r> ^v _tø) f 'ⁿ Equation (17) for the specified pixel azimuth (f₀) and range. The corresponding maximum value | Xaz,_da_ta-r_ef (^o»^_fr»^_Mz) f ^{is tne} target strength estimate at the pixel location. The maximum likelihood range rate V₀. and azimuth rate V₁₁₁₁ at the pixel location can be

combined into a single velocity parameter | v |= y v£ + V^₁ . In the stripmap for human observation, the target strength estimate at the pixel location is represented by pixel intensity (brightness), and the velocity estimate | v | at the pixel location is represented by a pixel color code. Such a generalized display is unnecessary if the observer specifies v_tr and v,_az values that are of interest. In this case, the four dimensional target strength map is evaluated at the specified v^ and v_te2 values, resulting in a conventional stripmap function of range and azimuth.

In practice, an observer often is interested in the difference between the current data image and a reference image. The reference image can represent the average clutter in the surrounding area at the observation time or a registered image of the same area that was viewed at a previous time. The difference image can be obtained by subtracting the reference image from the current image, or by subtracting log (reference image plus a small constant) from log(current image plus a small constant), which is equivalent to creating a normalized image via division of the current image by the reference image. At each range and azimuth, the display shown to the observer represents the maximum over all relevant range rates and azimuth rates of the difference image at the specified range and azimuth. If the range rate and azimuth rate of interest can be specified by the observer, then the range, azimuth map shown to the observer is the difference image evaluated at the given values of vv and v_taz. A nonnegative difference image is obtained by applying a nonlinear operation such as half-wave rectification (the image sample value or zero, whichever is largest), an exponentiation operation, an absolute value operation, or squaring, depending on the application.

MODES FOR CARRYING OUT THE INVENTION: OTHER MODES

The invention can be applied to ultrasonic inverse synthetic aperture sonar (ultrasonic ISAS) for noninvasive measurement of fluid velocity parallel and orthogonal to a conduit or blood vessel, as in echo cardiology. In this case, the relative motion shown in FIG. 2 is interpreted as the expected or average motion of a fluid relative to a beam pattern originating from a stationary phased array. The beam pattern phase modulation function, the array element weighting function, and coherent multi-pulse echo processing are similar to that given for the best mode (the maritime SAR embodiment). The relevant display for a human observer, however, is different. An ultrasound technician specifies the range (distance from the transducer array, orthogonal to the array surface) and azimuth (e.g. the zero-azimuth line orthogonal to the array), and the resulting display represents target strength at the chosen location as a function of fluid velocity components parallel and orthogonal to the conduit. This display can be used to assess turbulence as well as the velocity distribution parallel to the conduit as a function of distance from the vessel wall, which is progressively varied by the operator.

Some other embodiments involve relative motion of the target environment and the transmitter/receiver platform in three dimensions (range, azimuth, and elevation). Examples are ultrasonic ISAS observation of fluid flow in which the orientation of the fluid velocity vector in the azimuth- elevation plane is unknown, or ISAR processing of objects that can cross the beam in more than one direction. Some additional definitions are required for three dimensional analysis:

Vp_r≡ range rate component of platform velocity (relative motion between the transmit/ receive array and the target environment in the direction orthogonal to the azimuth, elevation plane)

Vpei ≡ elevation component of platform velocity (relative motion between the transmit/ receive array and the target environment in the elevation direction)

V_tei s target elevation rate that is not included in the platform velocity.

A combined transmit-receive phase modulated beam pattern P_TR(#.Ø) that is phase coded in both azimuth (φ=0) and elevation (φ=πl2) is used.

For two cross range dimensions with |( v^+v _fc> (t-t_o)\ ≤ I V⁺^_TI T<_HJ2 « R₀, R(t I fc. w_tov«) = -J[R₀ +(v +^vΛt-Of +[(V + O(f -t₀)]² +[(_Vpe; +v_lel)(t-t₀)f

≡ R₀ +OV + V_n-Xf -O + {[0%. +O² +(_Vpe,

, (23)

Θ (t I to.v_tr, v_tβz) == tan^"1{[( V₉₃₁ + v_taz)IR₀] ή ≡ θ (t), (24)

Φ (t I Iowa) s tan \[(_Vpel+ v_teI)!R₀] <} ≡ φ (t), (25) and the range dependent phase shift for two-way propagation is φrangβtf I to,V^V_taz,V,_β,) = (4π/λ) R(t | k,V_tr,Vt_az,Vtβ). (26)

For a combined transmit-receive beam pattern Py_H{θ,φ), the beam-induced time-dependent echo variation is P_TR[6(Γ I t_o,v_lr,v_laz), φ(ϊ | to.Vt_r.Vtei)] where dand φare given by Equations (24) and (25), respectively.

The beam-induced phase modulation is

PτR.0(t).<*(t)J}}. (27) the beam-induced amplitude modulation is |PτR[#(t),ø(t)]|, and the complete phase history function (including amplitude variation) for the point target is

= |PTR[0

(t|*6,Vfr,V_te/)]| x exp{j[φ_rangβ(f|fo.V(_f,v_tez,v_te,) + φb∞_m(Ψo,Vtr,v,_BZ,v_tβι)]}. (28)

The generalized ambiguity function corresponding to an energy normalized data phase history with <b = Vtr = v_taz = Vtβi = 0 and energy normalized reference phase histories with various hypothesized values of to, v_tn v_taz, and v_tβι is a function of four variables:

I

dt I². (29)

The receiver corresponding to Equation (29) is implemented with a generalized data-reference cross ambiguity function as in Equation (17):

Estimator/detector receiver response for hypothesized parameters fo^m^az.^

\ t_Q , V_fr , V_te , V,_e/ ) dt f (30)

where h_az,ei.ref({|<b,v'tr.V_tez,Vte/) in Equation (30) equals

ⁱⁿ Equation (28). If data phase histories are not energy normalized and reference phase histories are energy normalized, estimator/detector outputs are representative of relative target strengths at the hypothesized parameter values. A map of relative target strengths at various hypothesized ranges, azimuths, elevations, range rates, azimuth rates, and elevation rates comprises a further generalization of a conventional synthetic aperture image. The added image dimensions make depictions for a human observer (without loss of information) more difficult than in the lower dimensionality case represented by FIG. 2 and Equation (17). Computer-aided analysis is not adversely affected by the increased dimensionality if memory and processing speed are adequate.

Human observation of the estimator/detector outputs is feasible if difference images are computed as in the lower dimensionality case. At a chosen range, azimuth, and elevation sample, values of V_tr, V_tβ_z, and v_tβ, are automatically selected to correspond with the maximum difference image value; the maximum is computed with respect to range rate, azimuth rate, and elevation rate at the chosen range, azimuth, and elevation. The resulting map of target strength difference as a function of range, azimuth, and elevation can be depicted by a sequence of azimuth, elevation plots corresponding to different range values or by a sequence of range, azimuth plots corresponding to different elevation values, depending on the application. Similar difference representations are used if v_tr, v_taz, and v_te! are specified by an observer.

The many possible combinations of the four variables to, v^ v_toz, v_tel increase the opportunity for high ambiguity sidelobes. One obvious example occurs when the beam pattern is separable and the component azimuth and elevation beam patterns are the same: Pγ_R(0,0) equals Pτ_R(0,π/2). In this case, the phase history of a target that moves along vector v_« 157 relative to the azimuth-coded beam pattern 158 in FIG. 13A is the same as the phase history of a target that moves along vector v_βl 159 relative to the elevation-coded beam pattern 160 in Fig. 13B, and error coupling is likely to occur between azimuth/elevation estimates and between azimuth-rate/elevation-rate estimates. The phase histories in FIGS. 13A and 13B will be different if different nonlinear phase modulation functions are applied to the array weighting function in the x (azimuth) and y (elevation) directions. One way to obtain different phase modulations in azimuth and elevation is to reverse the sign of the modulation factor γ in Equation (7):

Ptran₃.az(x) = Preα∞M = C0S²[πx/(2cHW)] exp{jγcos²[πx/(2c/M)]}, -dM≤x≤dM (31 )

Ptrans.ei(y) = PrcceM = cos²[πy/(2dM)] _e ^χp{-jγcos²[_πy/(2c/M)]}, -dM≤y≤dM . (32)

The γ sign reversal is effective if the maximum value of the cross ambiguity function between the two phase histories from FIG. 13A and FIG. 13B is small. The elevation/range-rate cross ambiguity function 161 is shown in FIG. 14A, and the elevation/elevation-rate cross ambiguity function 162 is shown in FIG. 14B. In these cross ambiguity functions, the data phase history is from FIG. 13A and the reference phase history is from FIG. 13B₁ and the array excitation function is described by Equations (31) and (32) with γ = 30. In the cross ambiguity functions 161 and 162, the maximum cross ambiguity amplitude is approximately one-tenth of the maximum auto ambiguity amplitude. Two phase histories that are generated as in FIGS. 13A and 13B could be confused (in the absence of noise) if their energies differ by more than a factor of ten.

INDUSTRIAL APPLICABILITY

Application of the invention to maritime synthetic aperture radar (SAR) yields improved discrimination of small objects from sea clutter. Application to ground mapping SAR yields improved discrimination of moving vehicles from stationary roadside objects, and reduction of image degradation caused by moving clutter objects. These applications pertain to radar systems for defense and environmental monitoring.

Application of the invention to synthetic aperture sonar (SAS) yields improved discrimination between moving and stationary objects, and between objects that move with different velocities or in different directions. SAS applications include ocean bottom profiling, underwater geophysical prospecting, undersea salvage, mine hunting, and cable/pipeline detection/localization, and fish detection.

An important application of the invention is to improve the use of the environment to provide a position/velocity reference for correction of platform location/motion error. This type of error occurs in applications where sensor position varies and is difficult to track to within a small fraction of a wavelength (e.g., sonar systems on underwater platforms). The invention permits a synthetic aperture processor to resolve reference targets that can be used to measure the average motion (pulse-to-pulse position change) of a radar/sonar/ultrasound platform relative to the environment in which the system operates. Reference targets are resolved by maximizing the receiver response with respect to phase history hypotheses that are conditioned on various uncompensated position and velocity errors. The corresponding phase history corrections are then used for detection, parameter estimation, and imaging of relevant objects.

The invention enables accurate, noninvasive ISAS ultrasound measurement of range, azimuth, range rate, and azimuth rate, thus creating informative representations of fluid flow parallel and orthogonal to a vessel's length, as a function of distance from the vessel wall. Noninvasive ultrasonic ISAS monitoring of velocity in a conduit can be used for monitoring of pipelines, hydraulic systems, water cooling systems, and for manufacturing that involves regulation of fluid flow. ISAS applications also pertain to medical ultrasound equipment for echo cardiology.

The invention includes a generalization that allows synthetic aperture measurement of a six dimensional state vector (range, azimuth, elevation, range rate, azimuth rate, and elevation rate). This generalization can be applied to ISAS flow measurement in the heart and industrial containers, and to ISAR tracking/monitoring of objects that move in different directions through a radar/sonar/ultrasound beam.

Claims

1. Beam pattern phase modulation or coding for improved estimation/detection performance of a synthetic aperture system (SAR, SAS, ISAR, ISAS, monostatic, bistatic, and multistatic) comprising: a transmit, receive, or combined transmit/receive beam pattern that changes phase in a nonlinear manner as an observer moves across the beam in one direction (azimuth), or in two directions (azimuth and elevation); a phased array element weighting (shading) function with nonlinear phase variation, such that the shading function induces nonlinear phase modulation of the beam pattern in one direction (azimuth) or in two directions (azimuth and elevation), independent of element phase shifts that are used for beam steering and independent of phase corrections necessitated by a non-homogeneous propagation environment; a method for constructing generalized target phase history models that incorporate beam-induced phase modulation (in addition to quadratic phase variation caused by range changes) as the beam pattern is moved across the target or the target is moved across the beam pattern, and that incorporate unknown (hypothesized) azimuth, range rate, and azimuth rate parameters when relative motion between an array and the target environment is only in the azimuth direction, and that include hypothesized elevation and elevation rate parameters when relative motion is in both the azimuth and elevation directions; a method for correlating echo data with the generalized target phase history models so as to create a generalized synthetic aperture map that represents target strength as a function of range, azimuth, range rate, and azimuth rate for relative motion in the azimuth direction, and that represents target strength as a function of range, azimuth, elevation, range rate, azimuth rate and elevation rate for relative motion in both the azimuth and elevation directions; a method for depicting the generalized synthetic aperture map for the benefit of a human observer; and a method for depicting a difference image constructed from a generalized data map and a generalized reference map for the benefit of a human observer.

2. Beam pattern phase modulation or coding defined in claim 1, wherein the relative motion between a transmit-receive array and the target environment is in one dimension (azimuth), and the specific phased array element weighting function in the azimuth (x) direction has Hann (half-wave, squared-cosine) amplitude and phase modulation:

PtratK_McM = cos²[πx/(2d/W)] exp{j γcos²[πx/(2α7W)]}, -dM≤x≤dM

where d is the element spacing, the planar, rectangular array contains 2M- 1 rows of elements with nonzero weighting in the x-direction (and two virtual element rows at the ends that have zero weighting), γ is a phase modulation factor with absolute value less than or equal to 2M, and the same array element weighting function is used for transmission and reception. The complete azimuth-elevation element weighting function is Ptrans(x.y) = Ptrans,az(x) Ptrans.ei(y). where ptrarø,ei(y) is a smooth shading function without nonlinear phase modulation, e.g.,

Ptπms.e.M = cos²[πy/(2dΛ/)] , -dN≤y≤dN.

It will be understood that various substitutions with respect to particular complex beam shading functions (e.g., Hamming amplitude functions with phase functions specified by phase-hop codes, frequency-hop codes, or continuous nonlinear phase modulation functions) and non-rectangular, non-planar array configurations may be made by those skilled in the art, without departing from the spirit of the invention.

3. Beam pattern phase modulation or coding defined in claim 1 , wherein the relative motion between the transmit-receive array and the target environment is in two dimensions (azimuth x and elevation y), and the specific two-dimensional phased array element weighting function for transmission can be written as the product

PtransCf.y^Ptrans.azMPtrans.elCK)

for a planar, rectangular array, where

(W_K(X) = cos²[πx/(2cΛW)] exp{jγcos²[πx/(2c«W)]}, -dM≤x≤dM

and

P_tran₃,βi(y) = cos²[πy/(2dM)] exp{-jγcos²[πy/(2d/W)]}, -dM≤y≤dM.

The receiving array has the same element weighting function as the transmitting array. The element spacing is d in both x and y directions, and the array dimensions are (2Λ/M) x (2/W-1) when only elements with nonzero weights are considered. The phase modulation factor γ has absolute value less than or equal to 2M. It will again be understood that various substitutions with respect to particular complex beam shading functions and non-rectangular, non-planar array configurations may be made by those skilled in the art, without departing from the spirit of the invention.