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## SAR Processing (One-Dimensional Fourier Transform, Multiple Range Bins)

The extension of Fourier Transform processing from a single range bin (see SAR Processing (One-Dimensional Fourier Transform, Single Range Bin)) to multiple range bins can be pictorially visualized as follows:

The locus of the points in Range-Doppler space is similar to that of the points in x-y space, that is, there is a migration of the frequency points in range corresponding to the motion of the aircraft. The correction process, by which points are shifted from curved range lines to straight ones, results in the discarding of points in the near range (by having those ranges include points which are not of interest in the swath being mapped) and the adding of points in the far range (which would not normally be considered as being in the swath of interest, but which in fact
are).

In addition to correcting the Range-Doppler map, an appropriate correlation function must also be constructed. Because of the geometry, the Doppler also changes as a function of range, meaning that a different correlation function is required for each range. By constructing the Doppler correlation function in x-y space (before being transformed), it can be constrained to be in the correct range bin simply by the manner in which it is generated, i.e., no Range-Doppler correction is necessary. A typical set of correlation functions might look as follows in the x-y domain:

Assuming a map with 1024 points in range and 512 points in azimuth, a block diagram for the overall process would look as follows:

Note that, for successive passes, the correlation function does not change, only the raw data points. Therefore, the construction and forward transforming of the correlation is a one-time only task, providing that the flight geometry (or the swath being looked at) does not change. Note also, that, due to the circular correlation properties of the Fourier transform, the data must be padded in azimuth with 256 points in both the left and right azimuth positions (plus and minus Doppler). Zero padding can also be used, but will result in approximately a 3-dB loss in SNR (see Coherent vs. Non-Coherent Integration).

For information on using interpolation to select points intermediate to those obtained from an evenly sampled data set, see Synthetic Aperture Radar (SAR) Processing Overview.

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