Fourier Transforms in Radar And Signal Processing_ part 8

Biến đổi Fourier được sử dụng hàng ngày để giải quyết các chức năng duy nhất và sự kết hợp của các chức năng được tìm thấy trong các máy radar và xử lý tín hiệu. Tuy nhiên, nhiều vấn đề có thể được giải quyết bằng cách sử dụng biến đổi Fourier đã đi chưa được giải quyết bởi vì họ yêu cầu hội nhập đó là quá tính toán khó khăn. Hướng dẫn sử dụng này thể hiện như thế nào bạn có thể giải quyết những vấn đề hội nhập nhiều với một cách tiếp cận để...
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Fourier Transforms in Radar And Signal Processing_ part 8 Array Beamforming 185separation d wavelengths, by a pseudorandom step chosen within an intervalof width d − 0.5, which ensures that the elements are at least half a wavelengthapart. Figure 7.10(a) shows the response in u space for an array of 21 elementsat an average spacing of 2/3. A sector beam of width 40 degrees centered atbroadside was specified. A regular array would have a pattern repetitive atan interval of 1.5 in u , and this is shown by the dotted response. Theirregular array ‘‘repetitions’’ are seen to degrade rapidly, but the pattern thatmatters is that lying in the interval [−1, 1] in u . This part of the responseleads to the actual pattern in real space, shown in Figure 7.10(b). We notethat the side lobes are up to about −13 dB, rather poorer than for thepatterns from regular arrays shown in Figures 7.6, 7.8, and 7.9, though thislevel varies considerably with the actual set of element positions chosen. Theintegration interval I was chosen to be [−1, 1], to give the least squarederror solution over the full angle range (from −90 degrees to +90 degrees,and its reflection about the line of the array). A second example is given in Figure 7.11 for an array of 51 elements,but illustrating the effect of steering. In Figure 7.11(a, b) the 40-degreebeam is steered to 10 degrees, and again we see the rapid deterioration ofthe approximate repetitions in u space of the beam, and a nonsymmetricside-lobe pattern, though the levels are roughly comparable with those ofthe first array. The average separation is 0.625 wavelengths, giving a repetitioninterval of 1.6 in u . If we steer the beam to 30 degrees [Figure 7.10(c, d)],there is a marked deterioration in the beam quality. This is because one ofthe repetitions falls within the interval I over which the pattern error isminimized, so the part of this beam (near u = −1) that should be zero isreduced. At the same time the corresponding part of the wanted beam (nearu = 1⁄ 2 ) should be unity, so the solution tries to hold this level up. We notethat the levels end up close to −6 dB, which corresponds to an amplitudeof 0.5, showing that the error has been equalized between these two require-ments. We note from the dotted responses that the result would be muchthe same using a regular array. In fact, this problem would be avoided bychoosing I to be of width 1.6 (the repetition interval) instead of 2, preservingthe quality of the sector beam, but in this case the large lobe around −90degrees would be the full height, near 0 dB. Even if this solution (with alarge grating lobe) were acceptable for the regular array, it is not so satisfactoryfor the irregular array as the distorted repetitions start to spread into thebasic least squares estimation interval, as the array becomes more irregular,creating more large side lobes. Thus, although a solution can be found for the irregular array, itsusefulness is limited for two reasons; the set of nonorthogonal exponential 186 Fourier Transforms in Radar and Signal ProcessingFigure 7.11 Sector patterns from a steered irregular linear array: (a) response in u -space, beam at 10 degrees; (b) beam pattern, beam at 10°; (c) response in u -space, beam at 30°; (d) beam pattern, beam at 30°. Array Beamforming 187functions (from the irregular array positions) used to form the requiredpattern is not as good as the set used in the regular case, and if the elementseparation is to be 0.5 wavelength as a minimum, an irregular array musthave a mean separation of more than 0.5 wavelength, leading to grating (orapproximate grating) effects.7.5 SummaryAs there is a Fourier transform relationship between the current excitationacross a linear aperture and the resultant beam pattern (in terms of u , adirection cosine coordinate), there is the opportunity to apply the rules-and-pairs methods for suitable problems in beam pattern design. This has thenow familiar advantage of providing clarity in the relationship betweenaperture distribution and beam patterns, where both are expressed in termsof combinations of relatively simple functions. However, there is the complication to be taken into account that the‘‘angle’’ coordinate in this case is not the physical angle but the directioncosine along the line of the aperture. In the text we have taken the angleto be measured from broadside to the aperture, and defined the correspondingFourier transform variable u as sin , s ...