Principles of Modern Radar - Volume 2 1891121537

442 CHAPTER 9 Adaptive Digital BeamformingFIGURE 9-43Wideband antennapattern response.Normalized Frequency1.11.081.061.041.0210.980.960.940.920.9–1 –0.8–0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1Angle (sines)throughout this section. Over wider bandwidths, the difference in the propagation delay betweenthe digital channels leads to jammer dispersion and frequency response mismatcheslimit the depth of nulls on the jammers. The frequency response mismatches arise fromboth the frequency response variations of the hardware components in the receive chainand antenna pattern variations over frequency.The steering vector is a function of frequency as well as angle:[d sin θv (θ,f ) =j2π f1 e ( c ) ...ej2π f (K −1)( d sin θc ) ] (9.52)and therefore the antenna pattern response will vary as a function of frequency.F (θ,f ) = w H v (θ,f ) (9.53)A contour plot of a wideband antenna pattern response as a function of angle and normalizedfrequency is shown in Figure 9-43. As the electrical size of the antenna changeswith frequency, the locations of the sidelobe peaks and nulls move in angle. As a result,to cancel jammers over a wide bandwidth, it is necessary to jointly optimize the patternresponse in both angle and frequency.Several possible approaches to achieving cancellation over a wide bandwidth areillustrated in Figure 9-44. The figure represents a region in the sidelobes of the antennapattern with angle varying along the horizontal axis and frequency along the vertical axis.The shaded areas indicate the null width for the various wideband cancellation options.The dark diagonal line shows the null that would be produced if a narrowband beamformerwere applied with weights computed from data only at the center frequency. In this case,the null will be in the correct angular location only at the center frequency and will sweepaway from the jammer location for other frequencies across the band. This would allownearly all the jammer power to leak through the adaptive filter except for a very narrowband around the center frequency. The broad gray region shows the null that would beproduced by a narrowband (spatial only) beamformer with weights computed using thefull wideband data. In this case, the beamformer uses multiple spatial degrees of freedomto form a broad null that covers the jammer position over the full band.The light gray regions show the nulls produced if the data are filtered into subbandswith separate adaptive beamformers in each subband to cancel the jammer. This effectively