Fourier Transforms in Radar and Signal Processing_5

Tham khảo tài liệu fourier transforms in radar and signal processing_5, công nghệ thông tin, cơ sở dữ liệu phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả
Nội dung trích xuất từ tài liệu:
Fourier Transforms in Radar and Signal Processing_5 Interpolation for Delayed Waveform Time Series 97Figure 5.7 Flat waveform oversampled.flat part of the waveform, with constant value unity, has been sampled atan oversampling rate of q = 3. We see that at the sample points the weightvalue is 5/3, but the contributions from the interpolating sinc functionsfrom nearby sample points are negative, bringing the value down to thecorrect level. The weights given by (5.12) for oversampling factors of 2 and 3 areshown in Figure 5.8 for comparison with the values for the minimumsampling rate ( q = 1) plotted in Figure 5.4. The same set of delays has beentaken. These plots show that the weight for the tap nearest the interpolationpoint (taken to be the center tap here) can be greater than unity, that theweight magnitudes do not necessarily fall monotonically as we move awayfrom this point, and that much the same number of taps is required abovea given weight level, such as −30 dB. At first, this last point might seemunexpected—there is no significant benefit from using the wider spectralgate that is possible with oversampling. However, the relatively slow fallingoff of the tap weight values is a result of the relatively slowly decayinginterpolating sinc function, and this in turn is the result of using the rectangu-lar gate with its sharp, discontinuous edges. This is the case whether wehave oversampling or not. The solution, if fewer taps are to be required, isto use a smoother spectral gating function, and this is the subject of thenext section.5.2.3 Three Spectral GatesTrapezoidalThe first example of a spectral gate without the sharp step discontinuity ofthe rect function is given by a trapezoidal function (Figure 5.9). As illustrated98 Fourier Transforms in Radar and Signal ProcessingFigure 5.8 FIR interpolation weights with oversampling.Figure 5.9 Trapezoidal spectral gate. Interpolation for Delayed Waveform Time Series 99in this figure and also in Section 3.1, this symmetrical trapezoidal shape isgiven by the convolution of two rectangular functions with a suitable scalingfactor. The convolution has a peak (plateau) level of ( q − 1) F (the area ofthe smaller rect function), so we define G by 1 f f G( f ) = ⊗ rect rect (5.13) ( q − 1) F ( q − 1) F qF Thus, on taking the transform, g (t ) = qF sinc qFt sinc ( q − 1) Ft , andthe interpolating function from (5.9) with T ′ = 1/F ′ = 1/qF , is (t ) = sinc qFt sinc ( q − 1) Ft (5.14)From (5.8) we have u (t ) = sinc qFt sinc ( q − 1) Ft ⊗ comb1/F ′ u (t ) (5.15)The interpolating function is now a product of sinc functions, and thishas much lower side lobes than the simple sinc function. To interpolate attime = T ′, where 0 < < 1 (i.e., is a fraction of a tap interval), weconsider the contribution from time sample r , giving wr ( ) = [(r − ) T ′ ] = sinc (r − ) sinc [(r − ) ( q − 1)/ q ] (5.16) Now let x = r − and y = ( q − 1) x /q ; then w r ( ) = sinc x sinc y = sin X sin Y /XY (5.17)where X = x and Y = y . If we take the case of = 1⁄ 2 , the worst case,as in Section 5.2.1, we have sin X = sin (r − 1⁄ 2 ) = (−1)r +1, and if we takeq = 2 (sampling at twice the minimum rate), then sin Y = sin (r − 1⁄ 2 )/2= ± 1/√2 for r integral. So the magnitudes of the tap weights are | || | √2 1 1 = r− = wr (5.18) 2 2 2 1 2 r− 2100 Fourier Transforms in Radar and Signal ProcessingComparing this with (5.5), we see that the weight values now fall very muchfaster, and this is illustrated in Figure 5.10 for comparison with Figures 5.4and 5.8. We see that the number of taps above any given level has been reduceddramatically—above −30 dB, for example, from 20, 15, and 7 at q = 1 forthe three delays chosen, to 4 ...