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Fourier Transforms in Radar And Signal Processing_7

Tham khảo tài liệu 'fourier transforms in radar and signal processing_7', công nghệ thông tin, kỹ thuật lập trình phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Interpolation for Delayed Waveform Time Series 119 The FIR filter coefficients from the sampled impulse response are given by hr h rT exp -4pSr2 T2 5.49 where T 1 F is the sampling interval. If we take coefficients to the 40-dB level then we have 8p2s2r m T2 4 ln 10 or rm ln _10 2 F 0.342 F 5.50 m p s s where rm are the indexes of the first and last coefficients. We can now estimate the amount of computation required to produce the simulated clutter directly. With F 104 Hz and s 10 Hz we see that rm 342 so there are 685 taps and this is the number of complex multiplications needed for each output sample in addition to generating the inputs from a normal distribution . 5.4.2 Efficient Clutter Waveform Generation Using Interpolation In this case we generate Gaussian clutter with the required bandwidth but at a much lower sampling rate fs and then interpolate to obtain the samples at the required rate F Figure 5.20 . Thus we will need F fs times as many interpolations as samples. From Section 5.2 above we know that with moderate oversampling rates we can achieve good interpolation with very few taps. Figure5.20 Gaussian waveform generation with interpolation. 120 Fourier Transforms in Radar and Signal Processing Let the number of taps in the interpolation filter be m and the number in the Gaussian FIR filter is from 5.50 0.684 la 1 which we neglect so that the average number of complex multiplications per output sample is n m 0.684fs la l Flfs m 0.684f lap 5.51 In Figure 5.12 we see that with an oversampling factor of 3 we need only four taps weighted above the 40-dB level to interpolate up to the maximum time shift of half the sampling interval. Using these figures we have m 4 and fs 24a as the effective bandwidth of the waveform is taken to be 8ain Section 5.3.1 above and from 5.51 we obtain n 4.4 a factor of over 150 lower than in the direct sampling case. There will have to be F 2fs sets of four weights or 21 sets in this example to interpolate from 1l2f to 1l2f.