3.20.2.2.1 Flexible transmit beampattern synthesis

The probing waveforms transmitted by a MIMO radar system can be designed to approximate a desired transmit beampattern and also to minimize the cross-correlation of the signals reflected from various targets of interest—an operation that would hardly be possible for a phased-array radar.

The recently proposed techniques for transmit narrowband beampattern design have focused on the optimization of the covariance matrix of the waveforms. Instead of designing , we might think of directly designing the probing signals by optimizing a given performance measure with respect to the matrix of the signal waveforms. However, compared with optimizing the same performance measure with respect to the covariance matrix of the transmitted waveforms, optimizing directly with respect to is a more complicated problem. This is so because has more unknowns than and the dependence of various performance measures on is more intricate than the dependence on .

There are several recent methods that can be used to efficiently compute an optimal covariance matrix , with respect to several performance metrics. One of the metrics consists of choosing , under a uniform elemental power constraint (i.e., under the constraint that the diagonal elements of are equal), to achieve the following goals:

Another beampattern design problem is to choose , under the uniform elemental power constraint, to achieve the following goals:

It can be shown that both design problems can be efficiently solved in polynomial time as a semi-definite quadratic program (SQP).

We comment in passing on the conventional phased-array beampattern design problem in which only the array weight vector can be adjusted and therefore all antennas transmit the same differently-scaled waveform. We can readily modify the MIMO beampattern designs for the case of phased-arrays by adding the constraint that the rank of is one. However, due to the rank-one constraint, both of these originally convex optimization problems become non-convex. The lack of convexity makes the rank-one constrained problems much harder to solve than the original convex optimization problems. Semi-definite relaxation (SDR) is often used to obtain approximate solutions to such rank-constrained optimization problems. The SDR is obtained by omitting the rank constraint. Hence, interestingly, the MIMO beampattern design problems are the SDRs of the corresponding phased-array beampattern design problems.

We now provide a numerical example below, where we have used a Newton-like algorithm to solve the rank-one constrained design problems for phased-arrays. This algorithm uses SDR to obtain an initial solution, which is the exact solution to the corresponding MIMO beampattern design problem. Although the convergence of the said Newton-like algorithm is not guaranteed, we did not encounter any apparent problem in our numerical simulations.

Consider the beampattern design problem with transmit antennas. The main-beam is centered at , with a 3 dB width equal to . The sidelobe region is . The minimum-sidelobe beampattern design is shown in Figure 20.11a. Note that the peak sidelobe level achieved by the MIMO design is approximately 18 dB below the mainlobe peak level. Figure 20.11b shows the corresponding phased-array beampattern obtained by using the additional constraint . The phased-array design fails to provide a proper mainlobe (it suffers from peak splitting) and its peak sidelobe level is much higher than that of its MIMO counterpart. We note that, under the uniform elemental power constraint, the number of degrees of freedom (DOF) of the phased-array that can be used for beampattern design is equal to only ; consequently, it is difficult for the phased-array to synthesize a proper beampattern. The MIMO design, on the other hand, can be used to achieve a much better beampattern due to its much larger number of DOF, viz. .

The radar waveforms are generally desired to possess constant modulus and excellent auto- and cross-correlation properties. Consequently, the probing waveforms can be synthesized in two stages: at the first stage, the covariance matrix of the transmitted waveforms is optimized, and at the second stage, a signal waveform matrix is determined whose covariance matrix is equal or close to the optimal , and which also satisfies some practically motivated constraints (such as constant modulus or low peak-to-average-power ratio (PAR) constraints). A cyclic algorithm for example, can be used for the synthesis of such an , where the synthesized waveforms are required to have good auto- and cross-correlation properties in time.

3.20.2.2.2 Array design

For a phased-array radar system, the transmission of coherent waveforms allows for a narrow mainbeam and, thus, a high signal-to-noise ratio (SNR) upon reception. When the locations of targets in a scene are unknown, phase shifts can be applied to the transmitting antennas to steer the focal beam across an angular region of interest. In contrast, MIMO radar systems, by transmitting different, possibly orthogonal waveforms, can be used to illuminate an extended angular region over a single processing interval, as we have demonstrated above.

Waveform diversity permits higher degrees of freedom, which enables the MIMO radar system to achieve increased flexibility for transmit beampattern design. The assumptions used in the discussions above are that the positions of the transmitting antennas, which also affect the shape of the beampattern, are fixed prior to the construction of followed by the synthesis of . At the receiver, sparse, or thinned, array design has been the subject of an abundance of literature during the last 50 years. The purpose of sparse array design has been to reduce the number of antennas (and thus reduce the cost) needed to produce desirable spatial receiving beampatterns. The ideas behind sparse receive array methodologies can be extended to that of sparse, MIMO array design. For example, cyclic algorithms can be used to approximate desired transmit and receive beampatterns via the design of sparse antenna arrays. These algorithms can be seen as extensions to iterative receive beampattern designs.

3.20.2.2.3 Adaptive array processing at radar receivers

Adaptive array processing plays a vital role at radar receivers, including those of MIMO radar. Conventional data-independent algorithms, such as the delay-and-sum approach for array processing, suffer from poor resolution and high sidelobe level problems. Data-adaptive algorithms, such as MVDR (Capon) receivers, have been widely used in radar receivers. These adaptive signal processing algorithms offer much higher resolution and lower sidelobe levels than the data-independent approaches. However, these algorithms can be sensitive to steering vector errors and also require a substantial number of snapshots to determine the second-order statistics (covariance matrices). To mitigate these problems, diagonal loading has been used extensively in practical applications to make adaptive algorithms feasible. However, too much diagonal loading makes the adaptive algorithm degenerate into data-independent methods, and the diagonal loading level may be hard to determine in practice. Parametric methods tend to be sensitive to data model errors and are not as widely used as the aforementioned data-adaptive algorithms.

In MIMO radar, adaptive array processing is essential, especially because many of the simple tricks used to achieve the longer virtual arrays, such as randomized antenna switching (also called randomized time-division multiple access (R-TDMA)) and slow-time code-division multiple access (ST-CDMA), provide sparse random sampling. Because of such sampling, the high sidelobe level problem suffered by data-independent approaches are exacerbated. Moreover, most of the radar signal processing problems encountered in practice do not have multiple snapshots. In fact, in most practical applications, only a single data measurement snapshot is available for adaptive signal processing. For example, in synthetic aperture radar (SAR) imaging, just a single phase history matrix is available for SAR image formation. Moreover the phase history matrix may not be uniformly sampled. In MIMO radar applications, including MIMO-radar-based space-time adaptive processing (STAP), synergistic MIMO SAR imaging and ground moving target indication (GMTI), and untangling multiple paths for diverse radar operations such as those encountered by MIMO over-the-horizon radar (OTHR), we essentially have just a single snapshot available at the radar receiver, especially in a heterogeneous clutter environment.

Fortunately, the recent advent of iterative adaptive algorithms, such as the iterative adaptive approach (IAA) and sparse learning via iterative minimization (SLIM), obviate the need of multiple snapshots and the uniform sampling requirements but retain desirable properties, including high resolution, low sidelobe level, and robustness against data model errors, of the conventional adaptive array processing methods. Moreover, for uniformly sampled data, various fast implementation strategies of these algorithms have been devised to exploit the Toeplitz matrix structures. These iterative adaptive algorithms are particularly suited for signal processing at radar receivers. They can also be used in diverse other applications, such as in sonar, radio astronomy, and channel estimation for underwater acoustic communications.

3.20.3.1.1 The imaging equation

While the signals of interest are broadband, processing typically takes place in frequency subchannels so that narrowband models can typically be used. Further, since deep space sources are typically seen through line-of-sight propagation, multipath scattering is limited and occurs only locally as reflections off antenna support structures. Thus the propagation channel can be considered to be memoryless (zero delay spread). The synthesis imaging equations relate the observed cross correlation between pairs of array elements to the expected electromagnetic source intensity spatial distribution over a patch of the celestial sphere. Figure 20.14 illustrates the geometry, signal definitions, and coordinate systems for one of the baseline pairs of antennas used to develop the imaging equations.

Consider the electric field observed by the array at frequency due to a narrowband plane wave signal arriving from the direction pointed to by the unit length 3-space vector . We consider only the quasi monochromatic case where a single radiation frequency is observed by subband processing. To simplify discussion, polarization effects are not considered so is treated as a scalar rather than vector quantity, though working synthesis arrays typically have dual polarized antennas and receiver systems to permit studying source polarization. Since distance is indeterminate to the array, in our model the observed and its corresponding intensity distribution are projected without time or phase shifting onto a hypothetical far-field celestial sphere that is interior to the nearest observed object. The goal of synthesis imaging is to estimate from observations of sensor array .

Define the image coordinate axes to be fixed on the celestial sphere and centered in the imaging field of view patch. Since is unit length, we may use these coordinates to express it as . Let point to the origin, thus . For small values of p and q, such as being contained within a field-of-view limited by the narrow beamwidth of array antennas, . Time delays are inserted in the signal paths for receiver outputs to compensate for the differential propagation times of a plane wave originating from the origin. The most distant antenna is arbitrarily designated as the element, and . Thus the array is co-phased for a signal propagating along .

Receiver output voltage signal , is given by the superposition of scaled electric field contributions from across the full celestial sphere surface S, plus local sensor noise:

(20.23)

where represents the known antenna element directivity pattern and downstream receiver gain terms, is the phase shift due to differential geometric propagation distances for a source from relative to a co-phased source from as shown in Figure 20.14, and is the noise seen in the mth array element. For simple imaging algorithms, it is assumed that all elements (e.g., dish antennas) have identical spatial response patterns and that each is steered mechanically or electronically to align its beam mainlobe with , so does not depend on m and sources outside the elemental beams are strongly attenuated. The beamwidth defined by determines the maximum imaging field of view, or patch size. Considering the full array, (20.23) can be expressed in vector form as:

(20.24)

where .

Consider the vector distance between two array elements, , where is the location of the mth antenna. This is known as an interferometric “baseline,” and it plays a critical role in synthesis imaging. Longer baselines yield higher resolution images by increasing the synthetic array aperture diameter, and using more antennas provides more distinct baseline vectors which will be shown to more fully sample the image in the angular spectrum domain. In the following all functions of element position depend only on such vector differences, so it is convenient to define a relative coordinate system in the vicinity of the array to express the difference as . Align with , and w with . Scale these axes so distance is measured in wavelengths, i.e., so that a unit distance corresponds to one wavelength , where c is the speed of light. In this coordinate system we have by simple geometry

(20.25)

At array outputs , after the inserted delays , the effective phase difference between two array elements is then

(20.26)

Using the signal models of (20.23) and (20.26), the cross correlation of two antenna signals as a function of their positions is given by:

(20.27)

(20.28)

(20.29)

(20.30)

(20.31)

where . We have assumed zero mean spatially independent radiators for and , a narrow field of view so , and that . The quantity is known by radio astronomers as a “visibility function” where arguments and are replaced by u and v since the final expression depends only on these terms. A cursory inspection of (20.31) reveals that it is precisely a 2-D Fourier transform relationship, so the inversion method to obtain from visibilities suggests itself:

(20.32)

(20.33)

where is the inverse 2-D Fourier transform. This is the well known synthesis imaging equation. Since only cross correlations between distinct antennas are measured by this imaging interferometer, the self power terms are not computed or used in the Fourier inverse. The d.c. level in the image which normally depends on these terms must rather be adjusted to provide a black, zero valued background.

3.20.3.1.2 Algorithms for solving the imaging equation

The geometry of the imaging problem described in (20.32) and illustrated in Figure 20.14 is continually changing due to Earth rotation. The fixed ground antenna positions rotate relative to the axis, which remains aligned to the axis fixed on the celestial sphere. On one hand, this is a negative effect because it limits the integration time that can be used to estimate under a stationarity assumption. On the other hand, rotation produces new baseline vectors with distinct orientations, filling in the Fourier space coverage for and improving image quality. To exploit rotation, imaging observations are made over long time periods, up to 12 h, to form a single image.

Receiver outputs are sampled as at frequency , and sample covariance estimates of the visibility function (assuming zero mean signals) are obtained as

(20.34)

where N is the number of samples in the long term integration (LTI) window over which the imaging geometry and thus cross correlations may be assumed to be approximately stationary, and k is the LTI index. (We will later introduce a short term integration window length over which moving interference sources appear statistically stationary.)

Since covariance estimates are only available at discrete time intervals (one per LTI index k), and the antennas have fixed Earth positions, only samples of are available with irregular spacing in the plane, so (20.32) must be solved with discreet approximations. However, noting that due to Earth rotation, the corresponding antenna position vector orientations depend on time through k, a new set of samples with different locations is available at each LTI. Index k is thus added to the notation to distinguish distinct baseline vectors for the same antenna pairs during different LTIs. So the (l, m)th element of relates to the sampled visibility function as

(20.35)

where and where as in (20.26) and (20.31), due to inserted time delays we may take to be zero. For simplicity we will use a single index to represent unique LTI-antenna index triples to specify vector samples in the plane, so and . Thus elements of the sequence of matrices provide a non-uniformly sampled representation of the visibility function, or frequency domain image. Consistent with the treatment of in (20.32), diagonal elements in are set to zero.

Figure 20.15a presents an example of a certain VLA geometry, and Figure 20.15b shows where the samples would lie, with each point representing a unique sample . This plot includes 61 LTIs (i.e., ) over a 12 h VLA observation for the Cygnus A radio galaxy of Figure 20.12a. This sample pattern would change for sources with different positions on the celestial sphere (expressed by astronomers in right ascension and declination).

With this frequency domain sampling and including noise effects (20.32) becomes

(20.36)

(20.37)

where is known as the “dirty image,” the sampling function , and represents sample estimation error in the covariance/visibility. Since the plane is sparsely sampled, introduces a bias in the inverse which must be removed by deconvolution as described below. This also means that (20.37) is not a true inverse Fourier transform due to the limited set of basis functions used. It is referred to as the “direct Fourier inverse” solution.

There are two common approaches to solving (20.36) or (20.37) for given a set of LTI covariances . The most straightforward though computationally intensive method is a brute force evaluation of (20.37) given knowledge of the sample locations (e.g., as in Figure 20.15). Alternately, the efficiencies of a 2-D inverse FFT can be exploited if these samples and corresponding visibilities are re-sampled on a uniform rectilinear grid in the plane. “Cell averaging” assigns the average of visibility samples contained in a local cell region to the new rectilinear grid point in the middle of the cell. Other re-gridding methods based on higher order 2-D interpolation have also been used successfully. When large fields of view are required, or array elements are not coplanar, then any of these approaches based on (20.31) will not work and a solution to the more complete expression of (20.30) must be found. Cornwell has developed the W-Projection method to address these conditions [27].

An alternate “parametric matrix” representation of (20.31) and (20.37) has been developed. This is particularly convenient because it models the imaging system in a familiar array signal processing form that lends itself readily to analysis, adaptive array processing and interference canceling, and opens up additional options for solving the synthesis imaging and image restoration problems. Returning to the indexing notation of (20.34), note that since one may express as . Let be the desired image as scaled (i.e., vignetted) by the antenna beam pattern, and sample it on a regular 2-D grid of pixels . The conventional visibility Eq. (20.31) then becomes

(20.38)

(20.39)

which in matrix form is

(20.40)

(20.41)

(20.42)

and where is the diagonal image matrix representation of sampled , M is the total number of array elements, and though noise is independent across antennas, the self noise terms have been included to allow for the case that contributes to the diagonal of full matrix . The matrix discrete “direct Fourier inverse” relationship corresponding to (20.37) is

(20.43)

where K is the number of available LTIs. Equations (20.40) and (20.43) are well suited to address synthesis imaging as an estimation problem, facilitating use of Maximum Likelihood, maximum a posteriori, constrained minimum variance, or robust beamforming techniques. Note that (20.43) is not a complete discrete inverse Fourier transform, indeed, often so a one-to-one inverse relationship between and does not exist and is significantly blurred.

By the Fourier convolution theorem, the effect of frequency sampling by in (20.36) is to convolve the desired image with the “dirty beam response” . Neglecting the effect of individual antenna directivity pattern can be interpreted as the point spread function, or synthetic beam pattern of the imaging array for the given observation scenario. Significant reduction of this blurring effect can be achieved by an image restoration/deconvolution step. The dirty image of (20.36) may be expressed as

(20.44)

where is due to sample estimation error in the visibilities. Since antenna locations in the rotating plane are known precisely over the full observation, is known to high accuracy, and with well calibrated dish antennas so is . Thus (20.44) may be solved as

(20.45)

where “” denotes deconvolution with respect to the right argument. Due to the spatial lowpass nature of dirty beam this problem is ill conditioned and must be regularized by imposing some assumptions about the image. The most popular reconstruction methods impose a sparse source distribution model and use an iterative source subtraction approach related to the original CLEAN algorithm [32]. The sparse model is justifiable for point-source images of star fields, and works well even with more complex distributions of gas and nebular structures given that much of the field of view is expected to be dark. Several variants and extensions to CLEAN have been proposed, some applying source subtraction in the spatial domain, and some in the frequency domain. Typically these have performance tuning parameters which astronomers adjust for most pleasing results. Thus the effective regularization term or mathematical optimization expression is often not known precisely and the process is a bit ad hoc, but solutions with higher contrast and resolution, and with reduced noise and reconstruction artifacts are preferred. Maximum entropy reconstruction has also been used effectively.

3.20.3.2.1 Signal model

After analog frequency down conversion, sampling, and complex baseband bandshifting, the array signal time sample vector of Figure 20.16 is modeled as

(20.46)

where is the array response vector for signal of interest (SOI) is the time varying array response for the dth independent interfering source , and is the noise vector. Source response is assumed to be constant, even for observation times on the order of an hour because the dish mechanically tracks a point in the sky. Even fixed ground interference sources must be modeled as moving (thus depends on i) due to this tracking motion of the dish. Approaches to address man–made interference are discussed in Section 3.20.3.3. This model for can also include natural deep space sources which are bright enough to overwhelm the SOI even when seen in the beam sidelobe pattern. Their apparent rotational motion about the SOI is due to Earth rotation. When the corresponding is known accurately, these can be removed through a successive subtraction algorithm known as peeling. As with synthesis imaging, broadband processing for PAFs is accomplished by FFT based subband decomposition, often with thousands of frequency bins. So in the following we consider only a single frequency channel and adopt the standard narrowband array processing model.

Any array signal processing, including beamforming, must take into account the fact that, unlike synthesis imaging, the PAF noise vector is correlated across the array. Even with cryogenic cooling, first stage amplifier LNA noise is correlated due to electromagnetic mutual coupling at the elements. Another major component, spillover noise from warm ground black body radiation as seen by the feed array, is spatially correlated because it is not isotropic since it stops above the horizon and is blocked over a large solid angle by the dish.

In a practical PAF scenario the beams are steered in a rectangular or hexagonal grid pattern with crossover points at the −1 to −3 dB levels. The total number of beams, J, is limited by the maximum steering angle which is determined by the diameter of the array feed and the focal properties of the dish, by the acceptable limit for beamshape distortion, and by the available processing capacity for real-time simultaneous computation of multiple beams. As illustrated in Figure 20.17, the time series output for a beam steered in the jth direction is given by

(20.47)

where is the vector of complex weights for beamformer . Weights are designed based on array calibration data and the desired response pattern constraints and optimization as described in the following two sections. Separate beamformers with their own sets of J distinct weight vectors are computed for each frequency channel, though we consider only a single channel in this discussion.

3.20.3.2.2 Calibration

Since multiple simultaneous beams are formed with a PAF as shown in Figure 20.16, a calibration for the signal array response vector must be performed for each direction, , corresponding to each formed beam’s boresight direction, and any additional directions where point constraints in the beam pattern response will be placed. Periodic re-calibration is necessary due to strict beam pattern stability requirements, to correct for differential electronic phase and gain drift, and to characterize changes in receiver noise temperatures. Calibration is based on sample array covariance estimates as described in (20.34) while observing a dominant bright calibration point source in the sky. For example, in the northern hemisphere, Cassiopeia A and Cygnus A shown in Figure 20.12 are the brightest continuum (broadband) sources, and with a typical single dish telescope aperture they are unresolved and appear as point sources. Both have been used as calibrators.

3.20.3.2.3 Beamformer calculation

Since discovery of the weakest, most distant sources is a primary aim of radio astronomers, it is paramount to design a dish and feed combination to achieve high sensitivity, which has been derived for a phased array feed to be

(20.48)

where (in ) represents directivity in terms of the effective antenna aperture collecting area, is system noise power at the beamformer output expressed as a black body temperature, is Boltzmann’s constant, B is system bandwidth, , (in Watts/m²) is the signal flux density in one polarization, and and are the signal and noise components of respectively. Here we have assumed , i.e., that there are no interferers. For a reflector antenna with a traditional horn feed, maximizing sensitivity involves a hardware-only tradeoff between aperture efficiency, which determines the received signal power, and spillover efficiency, which determines the spillover noise contribution. With a PAF, sensitivity is determined by the beamforming weights as well as the array and receivers. Adjusting controls both the PAF illumination pattern on the dish which affects , and the response to the noise field, which affects . Noting that all other right hand side terms in (20.48) are constant, sensitivity can be maximized with the well known maximum signal to noise ratio (SNR) beamformer

(20.49)

To date all hardware demonstrated PAF telescopes have used this maximum sensitivity beamformer. However, a hybrid beamformer design method for PAFs that parametrically trades off sensitivity maximization with constraining mainlobe shape and sidelobe levels has been proposed.

3.20.3.2.4 Radio camera results

In 2008, ASTRON and BYU/NRAO independently demonstrated the first radio camera images with a PAF fed dish. Figure 20.18 presents an example of the BYU work as a mosaic image of a complex source distribution in the Cygnus X region. As a comparison, the right image is from the Canadian Galactic Plane Survey image, but blurred by convolution with the equivalent beam pattern of the 20-Meter Telescope to match resolution scales. We expect that the image artifacts caused by discontinuities at mosaic tile boundaries could be eliminated with more sophisticated processing. The Cygnus X radio camera image contains approximately 3000 pixels. A more practical coarse grid spacing of about half the HPBW would require about 600 pixels. A single horn feed would require 600 pointings (one for each pixel) to form such an image, compared to 25 (one for each mosaic tile) for the radio camera. Thus for equal integration times per pixel, this radio camera provides an imaging speed up of 24 times.

3.20.3.3.1 Challenges and solutions to radio astronomical spatial filtering

Many of the well-known adaptive beamforming algorithms appear at first glance to be promising candidates for interference mitigation in astronomical array processing, including maximum SNR, minimum variance distortionless response (MVDR or Capon), linearly constrained minimum variance (LCMV), generalized sidelobe canceler (GSC), Wiener filtering, and other algorithms. Robust canceling beamformers which are less sensitive to calibration error have also been considered for aperture arrays used as stations in large low frequency imaging arrays like LOFAR. However, due to several challenging characteristics of the radio astronomical RFI problem, most of these approaches are less successful here than they would be in typical radar, sonar, wireless communications, or signal intercept applications. These problems have made many astronomers reluctant to adopt the use of adaptive array processing methods for regular scientific observations. We note though that the intrinsic motivations to observe in RFI corrupted bands are becoming strong enough that rapid progress toward adoption is necessary and is anticipated by most practitioners. New algorithm adaptations are being introduced which are better suited for radio astronomical spatial filtering. We consider below some of the significant aspects of radio astronomy that complicate spatial filtering.

The typical astronomical SOI power level is 30 dB or more below the system noise over comparable bandwidth, even when cryogenically cooled LNAs are used with instruments located in radio quiet zones. Canceling nulls must therefore be deep enough to drive interference below the SOI level, i.e., below the on-source minus off-source detection limit, not just down to the system noise level. Most algorithms require a dominant interferer to form deep nulls because minimum variance methods (MVDR, LCMV, max SNR, Wiener Filtering, etc.) which balance noise variance with residual interference power cannot drive a weaker interferer far below the noise floor. The residual will remain well above the SOI level.

Another promising solution to limited null depth is a zero forcing beamformer like subspace projection (SP) where the null in the estimated vector subspace for interference is theoretically infinitely deep. A number of proposed radio astronomical RFI cancelers have adopted the SP approach and some experimental demonstration results have appeared. Figure 20.20 illustrates the first use of subspace projection RFI mitigation with a PAF as reported in [53]. Data were collected from a 19 element PAF mounted on the 20-Meter Telescope at the NRAO Green Bank, West Virginia observatory while observing the deep space Hydroxl Ion (OH) maser radiation source designated in star catalog as “W3OH.” An FM-modulated RFI source overlapping the W3OH spectral line at 1665 MHz was created artificially using a signal generator. The RFI was removed using the subspace projection algorithm. Snapshot radio camera images (see Section 3.20.3.2) of the source with and without RFI mitigation are shown in Figure 20.20. The source which was completely obscured by interference is now clearly visible.

Typically interference subspace estimation is poor in SP and all other cancelers without a dominant RFI signal so null depth suffers at lower INR levels. Short integration times, needed to avoid subspace smearing with moving interference, increase covariance sample estimation error which also limits null depth. To address these issues, an SP canceler using auxiliary antennas as in Figure 20.19 and a new parametric model-based SP approach for tracking low INR moving interferers have been proposed which significantly improves null depth [50].

Adaptive beamformers must distort the desired quiescent (interference free) beam pattern in order to place deep nulls on interferers. For astronomy, even modest beamshape distortions can be unacceptable. A small pointing shift in mainlobe peak response, or coma in the beam mainlobe can corrupt sensitive calibrated measurements of object brightness spatial distribution. Due to strict gain stability requirements it has been preferable to lose some observation time and frequency bands to interference rather than draw false scientific conclusions from corrupted on-sky beam patterns.

For PAF beamforming a potential solution is to use one of several classical constrained adaptive beamformers. Due to the inherent tendency for off-axis steered beams with a parabolic dish reflector to develop a mainlobe coma distortion, it would be necessary to employ several mainlobe point constraints to maintain a consistent symmetric beampattern. It has also been demonstrated that without multiple mainlobe constraints, RFI canceling nulls in the beampattern sidelobes can cause significant distortion in the mainlobe.

A more subtle undesirable effect for both PAF and synthesis imaging arrays is that variations in the effective sidelobe patterns due to moving RFI nulling can translate directly to an increase in the minimum detectable signal level for the radiometer. Weak astronomical sources can only be observed by integrating the received power for a long period to obtain separate low variance estimates of signal plus noise power (on source), and noise only (off source). Both signal and noise (including leakage from other deep space source through beam sidelobe patterns) must be stable to an extreme tolerance requirement over the full integration time. Even small variations in the sidelobe structure can significantly perturb background source and noise signal levels, causing intolerable time variation. This sidelobe pattern rumble due to adaptive cancelation increases the “confusion limit” to detection since unstable noise and background are not fully canceled in the on-source minus off-source subtraction. This occurs even if the beam pattern mainlobe is held stable using constrained or robust beamformer techniques.

1. an unstructured spatial signature (i.e., each is an arbitrary complex vector). In this case, there is an inherent ambiguity between the definition of and , which can be simply avoided by defining as the overall spatial signature. One element of the spatial signature is identified as , and hence the carrier phase of the LOS signal is given by the argument of that element of the spatial signature.

2. a steering vector (or also referred to as structured spatial signature), which is a function of the DOA.

3. a weighted sum of steering vectors: , where each term corresponds to the amplitude and the steering vector of one of the reflections of the cluster. In this case, the ambiguity between and can be handled in the same way as in the first model.

3.20.4.3.1 Adaptive beamforming

As outlined below, several different types of adaptive beamforming algorithms have been proposed for GNSS. Some of these are adaptations of standard algorithms, others have been designed specifically for conditions specific to positioning applications.

Algorithms employing a spatial reference: These approaches are based on knowledge of the steering vector of the LOS signal, . Assuming this a priori information is reasonable in some GNSS applications since the satellite position can be known thanks to the navigation message (transmitted by the satellite itself) or to assistance from ground stations, and a rough estimate of the receiver position may be available from previous position fixes or from the application of a basic positioning algorithm (e.g., using only one antenna and not exploiting the antenna array). Moreover, the accuracy of the satellite and receiver positions is not important in determining the DOA of the signal; errors of several hundreds of meters can be tolerated without affecting the satellite DOA estimate, given that the satellite-receiver distance is more than 20,000 km. However, the assumption of a known relies on the availability of array calibration and especially on the knowledge of the receiver orientation (also known as attitude in the GNSS literature). Errors or uncertainty in the array response correspond to the standard calibration problem found in many applications of antenna arrays, and robust methods developed for generic applications are also applicable here. On the other hand, the need to know the receiver orientation is a feature more specific to GNSS receivers. Assuming that is known, the use of the MVDR beamformer (and variants) is possible, and it is most appropriate to apply them in a pre-despreading scheme. If these beamformers are computed with the post-despreading correlation matrix and multipath components are present, they will suffer from the cancellation of the desired signal.

Algorithms employing a temporal reference: These methods are based on knowledge of the GNSS signal waveform. Knowledge of the waveform can be exploited to design a beamformer that minimizes the difference between its output and the reference signal (e.g., as in a Wiener filter). In practice, the situation is not that straightforward because even though the shape of the signal is known, the delay and frequency shift are not, so the beamformer weights and the signal parameters have to be computed jointly or iteratively. The expression for the beamformer is

(20.51)

where is the cross-correlation between the array output and a local replica of the LOS signal generated using estimates of its delay and frequency shift, and , respectively. In this case, it only makes sense to work with correlations computed after the despreading, otherwise the contribution of the GNSS signals is hardly present in the correlations. This beamformer is able to cancel interference, but its performance in the presence of multipath is not satisfactory, although not as bad as with spatial-reference beamformers. The temporal reference beamformer combines the multipath and the LOS signal in a constructive manner, so as to increase the SNR. This is useful behavior in communications but not in navigation systems, since the increase in SNR comes at the price of a bias in the estimation of the delay and phase due to the presence of strong multipath at the beamformer output.

Hybrid beamformers: The opposite behavior of the spatial-reference and temporal-reference beamformers suggests that their combination may have good properties. Both of them provide the LOS signal at the output, but the former changes the phase of the multipath so that it is roughly in counter-phase with the LOS signal, whereas the later modifies the multipath phase to align it with that of the LOS signal. Therefore, if the output of both beamformers is added together, the multipath will tend to cancel. This observation has led to the proposal of a hybrid beamformer that can be expressed as

(20.52)

where is a spatial-reference beamformer, and and are two scalars weighting the contribution of each beamformer. When is chosen as the MVDR beamformer, it can be shown that the optimal weights are

(20.53)

Since the optimal weights depend on the unknown parameters to be estimated, a practical way to proceed is to use an iterative algorithm where the calculation of the beamformer according to (20.52) is done using the previous estimates of , and next these estimates are updated using the output of the just computed beamformer.

Blind algorithms: This class of methods refers to techniques that do not exploit a priori knowledge of the exact signal or the steering vector, and hence are more robust to errors in these assumptions. Examples of such methods include those based on the constant modulus (CM) assumption, cyclostationarity and the power inversion approach. The civil GPS signal in current use, referred to as the C/A signal, has constant modulus because it is formed by almost rectangular chips. Most other GNSS signals also satisfy the CM property. However, this property cannot be exploited before despreading since the array cannot provide enough SNR gain to bring the signal above the noise. Therefore, the CM beamformer has to be applied after despreading, but in order to do so the despread samples corresponding to the LOS signal have to be CM. This happens when only one sample per integration period is used. However, the presence of multipath does not alter the constant-modulus property of the signal, so the CM beamformer is not useful in combating multipath.

GNSS signals are obviously cyclostationary since the repeated use of the PN spreading sequence introduces periodicity into the statistics of the signal. The fact that several repetitions of the PN code are present during a bit time (a property sometimes referred to as self-coherence) can also be exploited, as depicted in Figure 20.26. Interference will not have in general this structure, so it is possible to design the beamformer by imposing that its output should be as similar as possible to a version of itself delayed by a time equal to the PN code duration. Because of the same reasons as in the case of the CM beamformer, this technique should be applied to the despread signals and it will only be effective against interference and not multipath.

A very simple but rather effective approach is the power inversion beamformer. The weights are obtained as the beamformer vector that minimizes the total output power subject to a simple constraint to avoid the null solution. The constraint is chosen without using any information about the signal, typically forcing a given beamformer coefficient to be equal to one. This method has to be applied to the signals before despreading and, since the response is independent of the GNSS signals, it may happen that some nulls of the reception pattern are near to the DOAs of some of the GNSS signals. However, this situation can often be accepted since it is assumed that many GNSS satellites will be visible (around 10 satellites), and if a few of them are lost due to the coincidence of the pattern nulls with their DOAs, there will still be a sufficient number of satellite signals available (i.e., four or more) to compute the position. In this method, all satellites are received through the same beamformer, so it offers the possibility of being deployed as an add-onto existing single-antenna receivers. This is an important advantage of this method; more sophisticated beamformers that require information provided the receiver (e.g., the DOA or the delay of the LOS signal) or that generate one beam per satellite cannot be coupled with existing single-antenna receivers and require the development of a completely new receiver. The number of antennas used with power inversion should be large enough to cancel the existing interference sources, but not much larger in order not to increase the number of nulls in the pattern and thus the probability that a GNSS signal is canceled. The power inversion approach is popular in military systems where jamming from highly maneuverable sources (fighter jets) is a pivotal concern. The fast maneuvers of these vehicles makes the use of spatial-reference beamformers virtually impossible, and therefore a simple and robust method like power inversion, that does not need any reference or calibration procedure, is an excellent option.

3.20.4.3.2 Deterministic beamforming

Although it is recognized that data-dependent beamformers are more powerful in general than deterministic versions, there are some situations where the latter may be advantageous. Deterministic beamformers are clearly more robust against calibration errors and other uncertainties in the signal parameters. Moreover, if the desired and non-desired signals are known to be confined to distinct spatial regions, the deterministic design may offer an adequate solution since the problem reduces to designing a spatial filter with given pass and stop bands. This a priori spatial separability occurs in several circumstances in GNSS, particularly in GNSS ground stations. In this case, the interference is normally ground-based, and the multipath normally arises from ground-based scatterers, so both interference and multipath impinge on the receiver from relatively low elevation angles. This is contrasted with the satellite signals, which originate from the entire upper hemisphere. Thus, as illustrated in Figure 20.27, a fixed beamformer can be designed to minimize reception of signals from these low elevations. The complicating factor here is that an upwards-facing array typically cannot provide a sharp stop-band to pass-band transition for directions near end-fire. Another advantage of deterministic beamformers is that they allow an easier control of the trade-off between array gain (understood here as the increase of the ratio between the desired signal power and the white noise power) and interference cancellation. In adaptive beamformers, these two characteristics are tightly coupled. For instance, with MVDR, the presence of a strong interference gives rise to a deep null in the pattern, and this null necessarily increases the beampattern in other directions, thus degrading the array gain.