Principles of Modern Radar - Volume 2 1891121537

6.5 Image Reconstruction 239FIGURE 6-15 The truncated sinc interpolator operates by centering the sinc kernel (blackcurve) on the desired output sample location (diamonds) and then summing all of the inputsamples f (x n ) that fall under the kernel after first weighting them by the corresponding valueof the kernel. It is common practice to apply a weighting function to the sinc kernel (dashedcurve).and is shown in Figure 6-15. Each interpolated output f (x out ) is computed by (1) centeringa truncated sinc function at that location, (2) weighting the input samples f (x n ) by thevalues of the overlapping sinc function, and (3) summing the result. Often, a weighting, orwindowing, function w(x) is applied to the sinc kernel to control the level of aliasing in theinterpolated output. The quality of the interpolation improves as the length of the truncatedsinc grows. It is general practice to specify the interpolation by indicating the numberof zero crossings on one side of the truncated sinc. Figure 6-15 shows a 10-point sincinterpolator kernel, and in the diagram the interpolation kernel covers six input samples.It cannot always be assumed that the input sample locations will be uniformly spaced. Forexample, the angle between pulses (the second interpolation stage in PFA) is unlikely tobe uniform because radars typically operate at a fixed PRF during the collection, whilethe aircraft may change speed over this time due to wind conditions.Figure 6-16 shows the performance of several common types of interpolators. Thehorizontal axis shows the normalized frequency of the sinusoid being interpolated, where0.5 corresponds to one-half of the sampling rate. Thus, at ω = 0.5 the sinusoid passesthrough π radians between samples, exactly satisfying the Nyquist criterion. The verticalaxis represents the summed magnitude-squared interpolation error divided by the totalinput signal energy. The plot compares linear and piecewise cubic spline interpolationto unweighted and weighted truncated sinc interpolators, both with eight single-sidedzero crossings. Recall that the function being interpolated is a superposition of complexsinusoids of various frequencies and amplitudes, where the frequency is proportional to thedistance from scene center. A few examples were shown in Figure 6-2. For a fixed samplingrate, higher frequency sinusoids are more difficult to interpolate accurately. Thus, one resultof using a better interpolator is to improve the image quality toward the scene edges.6.5.3 Other Reconstruction AlgorithmsWhile we have emphasized PFA as being the primary reconstruction technique for SAR,its continued dominance is far from ensured. The algorithm was born in an era whencomputing resources were scarce, and the limitations of PFA were seen as a fair trade forits efficiency. Remember that the FFT was hailed as being the most influential algorithm ofthe 20 th Century. Moving forward, more demanding SAR applications will likely outstrip