Principles of Modern Radar - Volume 2 1891121537

410 CHAPTER 9 Adaptive Digital BeamformingTABLE 9-1 Number of Complex Multipliesand Adds for Adaptive ProcessingEstimate covariance matrix O(N 2 K)Invert covariance matrix O(N 3 )Formulate adaptive weights O(MN 2 )Apply adaptive weightsO(MS)In the beamforming processor, the digital data can be combined into beams, subarrays,auxiliary channels, or combinations of these as dictated by the choice of adaptivebeamforming architecture (e.g., beamspace, subarray space, sidelobe canceller). Any nonadaptivecombining that precedes the adaptive beamforming involves relatively simplevector–matrix multiplies that are highly parallelizable and can be efficiently implementedon FPGA or GPU hardware.Adaptive beamforming is discussed in more detail in Section 9.3, but the computationalload can generally be broken down into four basic steps:1. Estimate a covariance matrix.2. Invert the covariance matrix.3. Formulate the weights.4. Apply the weights to the data.The covariance matrix is estimated from the spatial channels being used for adaption and isnumerically inverted. Next, some matrix–vector mutliplies formulate the adaptive weights,and finally the weights are applied to the data to form beams. The number of complexoperations (1 complex multiply/add) for each of these steps are shown in Table 9-1,whereN = number of spatial channelsS = number of samples in waveformK = number of samples for covariance matrix (K ≤ S, K ≥ 10N)M = number of simultaneous beams (three for monopulse)For shorter pulses and larger numbers of DBF channels the computational load will bedominated by the covariance matrix inverse, and for longer pulses with fewer DBF channelsthe computational load will be dominated by the application of the weights to the data.All of these implementation challenges are made easier if the receiver channel count isreduced. The trend for implementing DBF into radar systems has been to start at the lowerfrequency bands and progress into the higher bands over time. At the lower frequencies,the larger cross sectional areas and lower bandwidths make it easier to build low-costreceivers, and the lower data rates reduce the I/O throughput required from the receiversto the beamforming computer. The other trend has been to start with one-dimensionaldigital beamforming, usually in elevation, and progress toward two-dimensional digitalbeamforming, usually at a subarray level. Restricting the DBF to a single dimension andemploying conventional RF analog beamforming in the other dimension keeps the receivercount to a manageable level while still providing the advantages of DBF, albeit in thatone dimension only. This approach has been adopted for surveillance radars that rotate inazimuth and do digital beamforming in elevation. The SMART-L surveillance radar builtby THALES (shown in Figure 9-7) is an example of this type of system. The MESAR 2