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Showing papers on "Adaptive beamformer published in 1992"


Journal ArticleDOI
TL;DR: Evaluations of a two-microphone adaptive beamforming system for hearing aids show that in environments with relatively little reverberation modifications of the basic Griffiths-Jim algorithm allow good performance even with misaligned arrays and high input target-to-jammer ratios; and performance is better with a broadside array with 7-cm spacing between microphones than with a 26-cm broadside or a 6-cm endfire configuration.
Abstract: In this paper evaluations of a two-microphone adaptive beamforming system for hearing aids are presented. The system, based on the constrained adaptive beamformer described by Griffiths and Jim [IEEE Trans. Antennas Propag. AP-30, 27-34 (1982)], adapts to preserve target signals from straight ahead and to minimize jammer signals arriving from other directions. Modifications of the basic Griffiths-Jim algorithm are proposed to alleviate problems of target cancellation and misadjustment that arise in the presence of strong target signals. The evaluations employ both computer simulations and a real-time hardware implementation and are restricted to the case of a single jammer. Performance is measured by the spectrally weighted gain in the target-to-jammer ratio in the steady state. Results show that in environments with relatively little reverberation: (1) the modifications allow good performance even with misaligned arrays and high input target-to-jammer ratios; and (2) performance is better with a broadside array with 7-cm spacing between microphones than with a 26-cm broadside or a 7-cm endfire configuration. Performance degrades in reverberant environments; at the critical distance of a room, improvement with a practical system is limited to a few dB.

226 citations


PatentDOI
TL;DR: In this paper, a plurality of linearly arrayed sensors to detect spoken input and to output signals in response thereto, a beamformer connected to the sensors to cancel a preselected noise portion of the signals to thereby produce a processed signal, and a speech recognition system to recognize the processed signal and to respond thereto.
Abstract: Systems and methods for improved speech acquisition are disclosed including a plurality of linearly arrayed sensors to detect spoken input and to output signals in response thereto, a beamformer connected to the sensors to cancel a preselected noise portion of the signals to thereby produce a processed signal, and a speech recognition system to recognize the processed signal and to respond thereto. The beamformer may also include an adaptive filter with enable/disable circuitry for selectively training the adaptive filter a predetermined period of time. A highpass filter may also be used to filter a preselected noise portion of the sensed signals before the signals are forwarded to the beamformer. The speech recognition system may include a speaker independent base which is able to be adapted by a predetermined amount of training by a speaker, and which system includes a voice dialer or a speech coder for telecommunication.

214 citations


Journal ArticleDOI
TL;DR: The effectiveness of the model is demonstrated by designing a beamformer of several hundred weights that duplicates and interpolates the measured external ear response of a cat over broad ranges of frequency and direction.
Abstract: In this article, a beamformer is proposed as a functional model for the spatial and temporal filtering characteristics of the external ear. The output of a beamformer is a weighted combination of the data received at an array of spatially distributed sensors. The beamformer weights and array geometry determine its spatial and temporal filtering characteristics. A procedure is described for choosing the weights to minimize the mean‐squared error between the beamformer response and the measured response of the external ear. The effectiveness of the model is demonstrated by designing a beamformer of several hundred weights that duplicates and interpolates the measured external ear response of a cat over broad ranges of frequency and direction. A limited investigation of modeling performance as a function of array geometry is reported.

53 citations


Journal ArticleDOI
01 Jun 1992
TL;DR: The concept of algorithmic engineering is introduced and discussed in the context of parallel digital signal processing, mainly relating to the use of QR decomposition by square-root-free Givens rotations as applied to adaptive filtering and beamforming.
Abstract: Algorithmic engineering provides a rigorous framework for describing and manipulating the type of building blocks commonly used to define parallel algorithms and architectures for digital signal processing. The concept is first illustrated by means of some fairly simple worked examples. These relate to the use of QR decomposition by Givens rotations for the purposes of adaptive filtering and beamforming. It is then shown how a novel modular architecture for linearly constrained adaptive beamforming has been derived by transforming an established least squares processor design using some simple algorithmic engineering techniques. This novel architecture constitutes a stable and efficient recursive realisation of the modular adaptive beamformer proposed by Liu and Van Veen.

38 citations


Journal ArticleDOI
TL;DR: In this paper, the authors apply the frequency-domain LMS algorithm with the self-orthogonalizing technique, called FLMS, to the Griffiths-Jim adaptive beamformer to accelerate the convergence rate for real-time processing of adaptive array signals.
Abstract: It is well known that the drawback of the time‐domain least‐mean‐square (LMS) algorithm proposed by Widrow [Proc. IEEE 63, 719–720 (1975)] in adaptive filter applications is that its convergence speed decreases as the eigenvalue spread of the input autocorrelation matrix increases. However, this problem can be overcome by employing the transform‐domain LMS method [Narayan et al., IEEE Trans. ASSP 31, 609–615 (1983)] which first transforms input time‐domain signals into another transform‐domain signals through DFT or some orthogonal transforms and then uses the self‐orthogonalizing algorithm [Gitlin and Magee, IEEE Trans. Commun. 25, 666–672 (1977)] to optimize the variable weights of an adaptive filter. This method has been shown to offer great improvement in convergence rate over the time‐domain LMS method. The objective of this paper is to apply the frequency‐domain LMS algorithm with the self‐orthogonalizing technique, called FLMS, to the Griffiths–Jim adaptive beamformer to accelerate the convergence rate for real‐time processing of adaptive array signals. It is shown that the two minimum mean‐square errors of the beamformer implemented in the time and frequency domains are identical. Computer simulations show that the adaptive beamformer using the FLMS exhibits faster convergence behavior and better performance of nulling jammers than that using the LMS, especially for the larger eigenvalue spread.

37 citations


Journal ArticleDOI
TL;DR: The authors treat the problem of finding the Wiener filters for a wideband adaptive beamformer as a challenge and give the possibility of investigating superresolution for wideband signals and of determining sufficient lengths of the finite-impulse-response filters in adaptive arrays.
Abstract: The authors treat the problem of finding the Wiener filters for a wideband adaptive beamformer. Explicit expressions are given for the filters and error power spectrum in the frequency domain. The expressions are simple to program and give the possibility of investigating superresolution for wideband signals and of determining sufficient lengths of the finite-impulse-response filters in adaptive arrays. The major results are an analytical expression for the broadband array Wiener solution given for an arbitrary spectrum for target, interference, and noise and an approximate analytic expression for the angular 3-dB beamwidth for broadband signals. >

32 citations


Book
01 Jun 1992
TL;DR: This paper presents a vector space approach to adaptive filtering adaptive image recovery adaptive time-varying spectral estimation communication channel equalization for digital data adaptive beamforming and the recognition of multicomponent signals.
Abstract: Part 1 The vector space approach: introduction to the vector space approach of signal processing discrete linear systems and signals - a vector space approach. Part 2 Signal recovery: image recovery - a vector space approach image recovery using row action projection methods the restoration of 3-D surfaces from 2-D images. Part 3 Adaptive signal recovery: a vector space approach to adaptive filtering adaptive image recovery adaptive time-varying spectral estimation communication channel equalization for digital data adaptive beamforming. Part 4 Signal recognition: neural networks neural tree networks the recognition of multicomponent signals.

31 citations


Journal ArticleDOI
TL;DR: It is shown that, in general, the smoothing, in addition to decorrelating the sources, can alleviate the effects of finite-data perturbations.
Abstract: The finite-data performance of a minimum-variance distortionless response (MVDR) beamformer is analyzed with and without spatial smoothing, using first-order perturbation theory. In particular, expressions are developed for the mean values of the power gain in any direction of interest, the output power, and the norm of the weight-error vector, as a function of the number of snapshots and the number of smoothing steps. It is shown that, in general, the smoothing, in addition to decorrelating the sources, can alleviate the effects of finite-data perturbations. The above expressions are reduced to the case in which no spatial smoothing is used. These expressions are valid for an arbitrary array and for arbitrarily correlated signals. For this case, an expression for the variance of the power gain is also developed. For a single interference case it is shown explicitly how the SNR, spacing of the interference from the desired signal and the correlation between them influence the beamformer performance. Simulations verify the usefulness of the theoretical expressions. >

29 citations


Book
01 Jun 1992

28 citations


Journal ArticleDOI
TL;DR: An approach to adaptive beamforming (adaptive reconstruction of a desired signal in the presence of interferers and noise) that uses just a single snapshot to calculate the antenna weights is presented andSimulations verify its good performance in comparison with the optimum beamformer.
Abstract: An approach to adaptive beamforming (adaptive reconstruction of a desired signal in the presence of interferers and noise) that uses just a single snapshot to calculate the antenna weights is presented. As in previous studies, a structured and grouped array of sensor elements is assumed. The concept exploits the induced special data structure, which can be described as a generalized rank-one eigenvalue problem and can be solved by means of a linear (overdetermined) system solver. Arbitrary signal statistics are allowed and no difficulties with nonstationary, and coherent interferers arise. Furthermore, the algorithm does not exhibit any transient behavior. Simulations verify its good performance in comparison with the optimum beamformer. >

21 citations


Journal ArticleDOI
TL;DR: In this article, a constrained optimal beamformer based on inversion of the observed (signal plus noise) cross-spectral matrix of the hydrophone outputs is proposed to suppress the received signals.
Abstract: When a thin flexible line array of hydrophones is towed through the sea, the straightness of the array can be affected by transverse motions of the tow vessel, by ocean swells and currents, and by motion‐induced hydrodynamic forces acting on the array. Traditionally, the spatial processing of the data from the hydrophones proceeds on the assumption that the array is always straight. The spatial processor considered here is a constrained optimal (adaptive) beamformer based on inversion of the observed (signal‐plus‐noise) cross‐spectral matrix of the hydrophone outputs. When the actual positions of the hydrophones deviate from their assumed positions, the adaptive beamformer responds by suppressing the received signals, resulting in a decrease in the output signal‐to‐noise ratio of the beamformer. Two methods of overcoming the signal suppression problem are compared and results are presented both for simulated data and for real data collected from an experimental towed array as the tow vessel changed course. Both methods infer the spatial distribution of the hydrophones along the array and require at least one acoustic source to be present in the far field. One method is an optimization technique where a cost function, known as sharpness, is calculated by integrating the product of the beam output power squared and the sine of the beamsteer angle over all beamsteer angles from forward endfire to aft endfire. When the estimated positions of the hydrophones coincide with the actual positions, the sharpness is a maximum. The other method uses the eigenvector corresponding to the largest eigenvalue of the cross‐spectral matrix to extract the phase of the signal at each of the hydrophones and then, after assigning a direction to the source of the signal, uses the relative phase information to estimate the positions of the hydrophones along the array. Both techniques use only data from the hydrophones themselves to estimate the shape of the array and do not require data from additional nonacoustic sensors such as compasses and depth sensors.

Proceedings ArticleDOI
23 Mar 1992
TL;DR: A microphone array for speech recording in car environments, designed for hands-free radiotelephone, and also used as a front-end for an automatic speech recognition system.
Abstract: The authors describe a microphone array for speech recording in car environments. The array is designed for hands-free radiotelephone, and is also used as a front-end for an automatic speech recognition system. The configuration of the array and the adaptive beamforming technique implemented are described and the performance of the array are evaluated. The measure of performance is the score obtained with a speech recognition system. Within the European ESPRIT project ARS (adverse environment recognition of speech), a prototype of this microphone array was built. It has eight microphones and works in real time, using one TMS C30 processor. >

Proceedings ArticleDOI
26 Oct 1992
TL;DR: Algorithms for adaptively steering the null of a cardioid to achieve interference rejection are developed and of particular interest is the nulling of highly directional reverberation when sonobuoys are considered as multistatic active receivers.
Abstract: Algorithms for adaptively steering the null of a cardioid to achieve interference rejection are developed. The algorithms are basically broadband, but by frequency filtering it is easy to steer independently in different frequency bands. Of particular interest is the nulling of highly directional reverberation when sonobuoys are considered as multistatic active receivers. Simple adaptive algorithms are shown to be effective in simulations. >

Patent
11 May 1992
TL;DR: In this paper, a beamformer system is described which includes an array of sensors such asydrophones which produce signals from which information relating to a target is to be determined.
Abstract: A beamformer system is disclosed which includes an array of sensors such asydrophones which produce signals from which information relating to a target is to be determined. The beamformer system includes a channel for each sensor, the channel including a varable length shift register which provides bulk delay to received signals, a FIR interpolation filter which provides vernier delay and shading. The system also includes a summation tree which adds the received signals and a sign extension/bit reversal filter to avoid authmetic overflow.

Patent
06 Oct 1992
TL;DR: In this article, an adaptive beamforming scheme was proposed for processing an echo received by an array of N receiving elements from a target at range R ensonified by an LFM signal of bandwidth B transmitted for a duration T.
Abstract: A device is provided for processing an echo received by an array of N receiving elements from a target at range R ensonified by an LFM signal of bandwidth B transmitted for a duration T. The device receives each elements output fn (t) over the elements total receiving time and selects time window element outputs f'n (t), which are discrete portions of fn (t), each time window extending from an arbitrary initial time T0 to time Tf and is spaced from T0 no more than T and, if there is a target within the window, an echo is received at time Tn. Further the device produces element frequency difference outputs gn (t) from the frequency difference between a replica of at least a portion of the transmitted LFM signal and the time window element outputs f'n (t), the outputs gn (t) being tones with frequencies directly proportional to the time Tn of the target from the time T0 so that the target range R=(c/2)(T0 +Tn)+ΔRn, where c is the speed of sound, and ΔRn is the distance by which R exceeds each element's range to the target. With this arrangement range and bearing of the target from the array can be more easily ascertained by dealing with parameters of cw, rather than broadband signals. This arrangement facilitates the application of adaptive beamforming techniques in a manner similar to their application to narrowband signals.

Journal ArticleDOI
TL;DR: In this article, neural adaptive beamformers (NABFs) utilize neural paradigms to accomplish desired adaptations that are associated with sensory-field-responsive partitioning and selection processes.
Abstract: Neural adaptive beamformers (NABFs) utilize neural paradigms to accomplish desired adaptations that are associated with sensory-field-responsive partitioning and selection processes. Kohonen-type organization and Hopfield-type optimization have been formulated as NABF mechanisms and have been applied to test data. Formulations and results are included. NABFs are also used in conjunction with a learning network for interpretation of weight sets as population codings of direction. An example is included. Desirable qualities of human auditory response are being interpreted in the context of neural adaptive beamforming for the purpose of creating an integrated processing structure that incorporates NABFs, a cochlear model, and an associative memory as part of a total spatiotemporal processing scheme for selective attention. >

Journal ArticleDOI
TL;DR: In this article, an adaptive beamformer whose outputs are processed by using the LMS algorithm to track distinct sources individually is proposed and an algorithm for spatially filtering out, enhancing, and tracking individual directional sources in an adaptive array is investigated.
Abstract: An algorithm for spatially filtering out, enhancing, and tracking individual directional sources in an adaptive array is proposed and investigated. In this algorithm, the sources are separated by using an adaptive beamformer whose outputs are processed by using the LMS algorithm to track distinct sources individually. From the LMS weights used, the source locations can be estimated. Whenever significant changes in these are detected, the beamformer is updated so that its outputs will be due to different sources in the steady state. With this algorithm, the problems of look-direction errors in look-direction constrained arrays and of large signal power in power inversion arrays are eliminated, and the enhancement of multiple moving sources becomes a natural process. Furthermore, because the sources are individually tracked and the beamformer is only updated occasionally, the algorithm possesses fast tracking behavior, and its implementation complexity is comparable to that of beamformer-based adaptive arrays using the LMS algorithm. >

Proceedings ArticleDOI
23 Mar 1992
TL;DR: The authors propose choosing the degrees of freedom to minimize the average mean squared error over a likely set of interference scenarios and an approximate solution to this optimization problem is described.
Abstract: The problem of adaptive beamforming in a coherent interference environment is addressed. A partially adaptive beamformer is proposed to present desired signal cancellation. The mean squared error at the beamformer output is shown to consist of a signal cancellation term associated with the presence of interferers coherent with the desired signal and an interference cancellation term. Both terms are functions of the degrees of freedom employed in the partially adaptive beamformer. Hence, the authors propose choosing the degrees of freedom to minimize the average mean squared error over a likely set of interference scenarios. An approximate solution to this optimization problem is described. Simulations are provided to illustrate the effectiveness of this approach. >

Proceedings ArticleDOI
30 Nov 1992
TL;DR: In this article, the problem of implementing a least square problem on a multiprocessor network comprising of general purpose DSP processors is addressed, and a scheme for an efficient implementation has been described.
Abstract: In this paper we address the problem of implementing a Least Squares problem, like that arising in the adaptive beamforming case, on a multiprocessor network comprising of general purpose DSP processors. Although optimal array processors have been proposed for this problem, an implementation on a generalized network is a flexible and more modular and reconfigurable solution, suited for a highly dynamic environment with changing applications and problem sizes. The parallelization issues have been explored and a scheme for an efficient implementation has been described.

Proceedings ArticleDOI
18 Jun 1992
TL;DR: In this paper, the adaptive array beamforming with steering vector errors is proposed, where a robust algorithm is used in conjunction with the subarray beamforming technique to find the optimal adaptive weight vector.
Abstract: The authors present an efficient method for adaptive array beamforming with steering vector errors. One partitions the adaptive array beamformer into several subarrays with size N for each. Due to the fact that data vectors received by any two subarrays have the same envelope waveform except a difference in phase shift, the corresponding optimal adaptive weight vectors also differ only in a phase shift for these two subarrays. Therefore, one computes the optimal adaptive weight vector for one of these subarrays and copies it to the others. When steering errors exist, a robust algorithm is used in conjunction with the subarray beamforming technique to find the optimal adaptive weight vector. Simulation results show that the proposed method can cure the degradation effect due to steering errors as well as save computing cost. >

Journal ArticleDOI
TL;DR: In this paper, a multiply constrained minimum-variance (MCMV) adaptive beamformer was applied to a large set of Pn arrivals observed on the NORESS array to estimate first and second-order sample statistics of the array gain as a function of frequency and subarray size.
Abstract: Broadband operation of small regional seismic arrays suggests the need for beams whose widths are nearly frequency independent and for an adaptive processor capable of suppressing nonstationary noise and interference. It is shown that multiply constrained minimum-variance (MCMV) adaptive beamforming when trained on signal-and-noise windows can achieve these objectives better than conventional beamforming. This paper presents the theory, implementation, characteristics, and performance of such a beamformer in the context of seismic monitoring for underground nuclear tests. The adaptive beamformer was applied to a large set of Pn arrivals observed on the NORESS array to estimate first- and second-order sample statistics of the array gain as a function of frequency and subarray size. The relative merit of adaptive beamforming was established by comparing the mean gain with that realized by conventional beamforming. These comparative results and other findings showed that the MCMV beamformer can (1) improve average signal-to-noise ratios from 0 to 8.5 dB, (2) better identify and suppress organized components in the noise field, and (3) better suppress interference in comparison to the conventional beamformer. The adaptive beamformer also produced peak narrowband gains on the larger subarrays in the neighborhood of 24 dB for specific evens. When implementational, seismic environmental, and practical aspects are added to the findings, important configurational and operational strategies are identified. The array gain statistics for both the adaptive and conventional beamformers are also useful for modeling the performance of monitoring networks.


Journal ArticleDOI
01 Dec 1992
TL;DR: In this paper, a beamforming algorithm for separating and tracking multiple directional sources in a linear power-inversion array is proposed and investigated, where the sources are separated by using an adaptive beamformer whose responses consist of perfect steerable nulls.
Abstract: A new algorithm for separating and tracking multiple directional sources in a linear power-inversion array is proposed and investigated. In this algorithm, the sources are separated by using an adaptive beamformer whose responses consist of perfect steerable nulls. By using the LMS algorithm for adaptive processing of the beamformer outputs to minimise the array output power and examining the adaptive weights employed, these nulls can be adjusted to track the sources individually so that the beamformer outputs will be due to different sources in the steady state. With this algorithm, the problem of incidental cancellation is eliminated and the enhancement of multiple moving sources becomes a natural process. Also, since the sources are individually tracked and the beamformer is only updated occasionally when significant changes in the environment are detected, the algorithm possesses fast tracking behaviour and its implementation complexity is comparable with that of beamformer-based adaptive arrays using the LMS algorithm.

Proceedings ArticleDOI
07 Oct 1992
TL;DR: Results based on both simulation and real HF field data show that adaptive-beamspace processing can be superior to either element-space or fixed- beamspace processing.
Abstract: Radio direction finding in the high-frequency band is challenging since the complex signal and noise environment is difficult to model, defeating the basis of most high-resolution techniques. Since fixed beams are ineffective with the small apertures common at HF, the authors propose a method for designing beams adaptively (i.e., in a data-dependent manner). Results based on both simulation and real HF field data show that adaptive-beamspace processing can be superior to either element-space or fixed-beamspace processing. >

Proceedings ArticleDOI
11 Aug 1992
TL;DR: In this article, a novel algorithm for digital beamforming in conformal arrays is presented which has specific application to high performance, digital, adaptive beamforming, and the operations are only dealt with real numbers and the resulted weight distributions are real ones.
Abstract: In this paper, a novel algorithm for digital beamforming in conformal arrays is presented which has specific application to high performance, digital, adaptive beamforming. With this algorithm, the operations are only dealt with real numbers and the resulted weight distributions are real ones. The computation results confirm the theoretical analysis.

Journal ArticleDOI
TL;DR: In this paper, a new linear array GSC beamforming structure employing DOA estimation and the null steering technique is presented, which has the advantages that the quiescent response of the beamformer can be closely maintained and that the adaptive dimension of the beacons can be reduced to about half of that of the full processor.
Abstract: In the Letter, a new linear array GSC beamforming structure employing DOA estimation and the null steering technique is presented. It has the advantages that the quiescent response of the beamformer can be closely maintained and that the adaptive dimension of the beamformer can be reduced to about half of that of the full processor. Theoretical justification of the proposed approach is given and the performance achievable is investigated via computer studies. >

Book ChapterDOI
01 Jan 1992
TL;DR: In this paper, the authors investigated a method very similar to the one published by O'Donnel and Flax in 1988, and showed the merits and limits of their method named Adaptive antenna.
Abstract: It is well known from the literature1 that velocity fluctuations across the active aperture of a linear array for B-scan-imaging can give rise to image degradation Various methods have been proposed and tested to compensate for image blurring of this origin We have investigated a method very similar to the one published by O’Donnel and Flax2 in 1988 Results showing up the merits and limits of our method named “adaptive antenna”3 are to be presented in this report

Proceedings ArticleDOI
07 Oct 1992
TL;DR: This paper deals with the optimality of the focused beamformer compared with that of conventional broadband beamformer and an analytical and empirical evaluation of different focusing transformations.
Abstract: The coherent signal-subspace approach to broadband adaptive beamforming involves a focusing preprocessor that aligns signal spaces at different frequencies to a common one by means of transformations and a narrowband beamformer following the preprocessor. The merits of the approach are decorrelation of multipath signals and partial adaptivity due to single-frequency weights. This paper deals with the optimality of the focused beamformer compared with that of conventional broadband beamformer. An analytical and empirical evaluation of different focusing transformations is presented. >

Proceedings ArticleDOI
23 Mar 1992
TL;DR: A fast least squares algorithm is proposed for adaptive filtering with linear constraints that can be used in several applications such as array processing, adaptive beamforming, spectral analysis and telecommunications.
Abstract: A fast least squares algorithm is proposed for adaptive filtering with linear constraints. The algorithm is exact, with a computational complexity proportional to NK+K/sup 2/+N+K where N is the number of the filter coefficients and K is the number of constraints. The approach consists of obtaining the adaptation gain by means of a fast least squares (FLS) algorithm and using this gain in the recursive calculations of the parameters related to the constraints. The recursive procedures are carried out by using the matrix inversion lemma. The good performance of the proposed method is illustrated by some simulation results. The algorithm can be used in several applications such as array processing, adaptive beamforming, spectral analysis and telecommunications. >

01 Dec 1992
TL;DR: In this paper, the effects of various types of imperfections on the radiation pattern of adaptive antenna arrays using the Applebaum and Frost algorithms, are investigated using simulated results and analytical expressions for the expected pattern parameters are given.
Abstract: : The effects of various types of imperfections on the radiation pattern of adaptive antenna arrays. using the Applebaum and Frost algorithms, are investigated. The imperfections considered are nonuniform noise figures in the channels, gain and phase shift error, IQ Imbalance, DC offset, finite number of samples when forming the covariance matrix, quantization of the signals and the weights and saturation of the signals. In particular the effects on the resulting sidelobe level, null depth, and gain are studied, and simulated results and analytical expressions for the expected pattern parameters are given. The effects of some methods to partly correct for the imperfections are also investigated. A method to obtain low sidelobe pattern when using the Frost algorithm is also discussed.... Adaptive arrays, Digital beamforming, Error effects, Sampling effects, Error corrections