When a radar images a scene and produces a one-dimensional (range com- pressed) range profile, the voltage in each range bin is the sum of all of the voltages due to all of the scatterers in that range bin for the whole beam. This means that the voltage in each range cell is actually the beam-weighted in- tegral over the range of beam angles. This is illustrated in Figure 6.2, from which it can be seen that the contents of the beam in each range cell are ‘added’ together to give an overall single value of the reflected power. The range profile gives no indication of the cross-range distribution of the reflec- tors. This is often the case for high range resolution compact mm-wave radars operating at long range, where the cross-range resolution is coarse compared with the range resolution[13].
range bins
range amplitude
scatterers
this response is the coherent sum of the responses to each of the red scatterers
Figure 6.2: The range profile resulting from a radar illuminating a patch of scene con- taining multiple scatterers. The range profile does not describe the cross-range distribu- tion of the scatterers.
There are several very different approaches to SAR processing, but they all have the same goal, which is to translate a coherent set of one-dimensional
radar measurements into a two-dimensional map of the RCS of the scene, i.e. the SAR image. N.B. here the word ‘image’ does refer to a two-dimensional map of reflected power.
Various algorithms exist to perform the inversion from radar data to image. The various algorithms prioritize different aspects of the problem. Mostly, though, the balance is among precision, generality, and computational load.
In the following discussion, it will be useful to refer to fast and slow time. This was discussed in Section 5.3 in the context of MTI. In this context, the fast time still refers to the time during the chirp (or the IF signal), and slow time still refers to the time between successive chirps. In fact, it is thedistance between the IF signal collections that is important, not the time. However, it is usual to assume a constant along-track velocity for the radar, so slow time translates directly into along-track position.
There are essentially two types of SAR processor: those that operate in the time domain and those that operate in the wavenumber domain[104].
A popular time-domain method is called the Back Projection Algorithm (BPA). This approach operates by reversing the projection of received power along constant range curves onto the range profiles. The complex amplitude in each cell from each range profile collected is smeared out in the corresponding arc and added to an image that is built up by repeating this process for every range cell from every range profile. This can also be understood from the reverse point of view: for every image pixel, the value in the corresponding range bin of every range profile is added; effectively a voting system (i.e. a spatially variant matched filter) for the presence of a reflector in each image cell. The returns are coherent, so if there is a scatterer in a resolution cell, the corresponding contributions from all of the range profiles will add up. If there is no scatterer in that scene location, the contributions will tend to cancel. A filtering step finishes the process. The location of every image cell and the location of the radar at every range profile can be arbitrary. Consequently, this method has the potential to be very versatile, being able to cope with non-planar images (e.g. a hillside) and nonlinear trajectories. However, this is also a computationally very intensive method, having to sum over the range profiles for every resolution cell. Methods have been developed (e.g. the Fast Factorized BPA,[68, 105]) that attempt to accelerate the BPA. However, these are difficult to implement and require careful factorization of the problem if the resulting image quality is not to degrade unacceptably.
phase corrections and distortions to the collected radar signals to obtain the Fourier conjugate (i.e. the spatial frequency response) of the scene. A compu- tationally efficient two-dimensional FFT can then be used to form the image. This is one way of understanding how an FMCW radar forms a normal range profile.
Generally, matched filtering is a correlation search of the reflected signal (i.e. the echoes) for delayed and attenuated copies of the transmitted signal. This could be done directly as a time-domain correlation but would be very inefficient. However, it can also be done computationally efficiently in the frequency domain using FFTs and the following relationship:
f ∗g ≡FT−1
FT(f)FT(g)
, (6.1)
where FT signifies Fourier transformation.
This allows the convolution of the received signal and the complex-con- jugate of the time-reversed transmitted signal (i.e. the matched filter) to be done in the frequency domain, where the computationally efficient FFT can be used.
It is useful to think of the signal in this domain. When variously delayed signals (from scatterers at different ranges) are Fourier transformed, their Fourier transforms are all aligned. The only difference between them is a linear phase ramp with gradient proportional to their delay. This means that the same (multiplicative) matched filter can be applied simultaneously to can- cel out the phase variation due to the transmitted (chirped signal). After the filter has been applied, each echo is just a pure signal tone, with frequency (i.e. phase ramp gradient) proportional to the range of the scatterer that re- flected it. The signal is now the range-wavenumber-domain radar range pro- file. Fourier transforming it translates it back into the range domain, i.e. giving the range profile. This also applies for two-dimensional SAR signals.
The above description of matched filtering is true for signals that are not linearly chirped, for instance, the phase modulation could be a piece of mu- sic, or noise. However, in the case of linear FM, the radar illuminates the scene with a range of frequencies in continuous succession. Effectively, the first Fourier transform to convert the received echoes into a frequency repre- sentation has been done. It is then possible to remove the phase variation due to the chirp by down converting against the current transmitting signal, effec- tively applying the matched filter. After this, the signal only has linear phase variations, with the phase gradient being proportional to the delay, i.e. it is
the range-wavenumber-domain radar range profile. A single Fourier trans- form translates the signal back into the image domain, giving the required radar range profile.
So, in the case of the linear chirp, much of the signal processing can be done in analogue devices, principally the mixer, reducing the computational load dramatically compared to a general matched filter pulse compression sys- tem. As discussed in the introduction, this is a very important reason that (linear) FMCW is a powerful technique.
A particular wavenumber-domain processor, the Polar Format Algorithm (PFA), forms an image by taking the IF signals collected along the synthetic aperture and subtracting the phase of the scene centre from the phase histories of all the scatterers. A mapping is calculated to transform the signal from the f-t domain (the data collection coordinates) to the kx-ky domain, which
is the Fourier conjugate of the image domain. The data is projected (and interpolated) onto the ground plane (doing this removes the change in phase due to the elevation of the radar) and can then be efficiently two-dimensional FFTd into the image. A final ‘keystone’ correction removes a range-dependent cross-range scaling in the image caused by the mapping of the data for all of the scene being done with a mapping calculated for the scene centre. This is essentially a range dependent stretch on the width of the image.
The PFA can produce excellent results from suitable data, is computation- ally efficient, and does not require excessive along-track sampling rates. How- ever, owing to the approximate cross-range dechirping, which is only true for the scene centre, phase errors begin to degrade the resolution as the scene size is increased relative to the stand off range. The limit on the scene size forπ/2 quadratic phase errors has been shown to be given by[67, 106]
L=ρ
r
2R
λ , (6.2)
where ρ is the cross-range resolution, R is the stand off range, and λ is the carrier wavelength.
This method of processing the data is highly suitable for far-off, low-resolu- tion, or small scene cases. However, for the experimental test bench, in which relatively large, short standoff range, fine resolution images will be processed, it is not suitable.
In an extreme experiment, the SAR test bench might be used with the fol- lowing parameters: ρ = 0.05 m, R = 3 m, and λ = 0.003 m. This gives a
maximum scene size of 2.2 m, which is not sufficient. Doren [106]describes a technique for correcting the phase errors that limit the scene size. However, for the test bench, a transparent and exact approach with reasonable compu- tational load is required.