826_SAR_Processing_Algorithms_Overview-F15.pptx

Editor's Notes

• #1: For the few classes, we are going to look at some efficient ways to implement SAR processing. But before we do this, I want to draw your attention to some important details about why we can do this and the motivation for it.
• #2: SAR processing algorithms model the scene as a set of discrete point targets whose scattered EM fields do not interact with each other. This allows us to consider each pixel in the scene independently from the others. Note that this is an assumption since each target in the scene is influenced by all the other targets and to get the exactly correct solution all targets must be considered simultaneously. But it turns out that in many cases, the interaction or scattered fields are much smaller than the incident field so that we can get close to the correct answer with the simplified model. Why this model? Because we can treat the target scattering as linear and we can consider each target individually. In other words, we don’t have to search through all possible target combinations which quickly becomes an intractable problem even for less than ten target pixels and typical SAR images have millions of pixels.
• #3: SAR processing is just the application of a matched filter.
• #4: A matched filter is also known as a correlation filter. In this case: we correlate the raw data with a single point target response. From a linear algebra perspective we call this taking the inner product between the raw data and a single point target response.
• #5: So, SAR processing is a linear filter. It would be convenient if the filter was also space invariant. This would mean every target has the same response only translated based on its location. If that was the case, then we could use an FFT to implement the filter. However, as you know, the point target is not space invariant and depends on the range.
• #6: So why do we care? Recall from your signal processing class that linear time invariant systems can be modelled by convolution and that convolution (which is slow) in one Fourier domain is equivalent to component-wise multiplication (which is fast) in the opposite Fourier domain. Since SAR processing is a two dimensional filter, the speed up from using Fourier methods is very large. Each of the algorithms that we look at will exploit the structure of the data collection geometry to interpolate the data into a domain which is space invariant.
• #7: To understand how to exploit the structure in the SAR signal to do Fourier domain processing, we need mathematical models in each Fourier domain. To avoid complicated derivations that require numerical methods, we use a technique called the principle of stationary phase to evaluate the Fourier transforms to get an approximate solution. The PSOP is used to evaluate integrals of this form. Note that this looks very similar to a 1-D Fourier integral with A and B replaced with –inf and +inf and with the complex Fourier exponential 2 pi f t embedded inside the phase function phi of t. The requirement is that the phase function varies rapidly over the range of integration except at the stationary points and that the envelope is slowly varying relative to the phase function. SAR signals have these properties: The phase function is quadratic in range and is hyperbolic in azimuth: i.e. they vary rapidly everywhere except at the stationary point. Also, the envelope of the SAR signal changes slowly relative to the phase function. For SAR signals, the accuracy of the PSOP scales with the time bandwidth product. The general rule of thumb is that accuracy is not good enough below a time bandwidth product of 100. Since most SAR signals have range and azimuth TBP much larger than this, the PSOP is a good solution for our purposes.
• #8: To illustrate the principle of stationary phase, we show here the Fourier transform of a 20 MHz chirp. The integrand is shown in the bottom panel for each frequency shown in the title. The accumulation of the integrand is shown in the top left panel for each frequency where the red star on the right shows the total integration. The top right panel shows the frequency response at each frequency (this corresponds to the red star on the left). Note how the integral output is dominated by the contribution at the stationary point of the phase in the lower panel. The stationary point is where the phase function is slowly varying. The idea behind the principle of stationary phase is to approximate the phase function at the stationary point with a Taylor expansion and ignore contribution elsewhere.
• #9: With this method, we replace the phase function with a Taylor series approximation expanded around the stationary point as shown in (3). We then note that we can replace the envelope with a constant evaluated at the stationary point because it is effectively constant around the stationary point where effectively all of the integration output comes from. In other words, even though the envelope changes for other values of “x” away from the stationary point x_s, the value of the envelope does not matter because the integration is equal to zero over that range. The Taylor series approximation and constant envelope are inserted into (1) and we end up with an integrand that has a closed form solution which is given in (4). The critical steps are 1) Write out your envelope function and phase function (make sure your phase function includes the 2*pi*f*t term from the Fourier complex exponential). 2) Find the first derivative and solve for the integrand variable at the stationary point. 3) Plug this into (4). Equation is messy at this point, so simplify! Derivation from: my.ece.ucsb.edu/York/Bobsclass/201C/Handouts/StationaryPhase.pdf
• #12: Include animation showing this 1-D contour concept.