Principles of Modern Radar - Volume 2 1891121537

6.4 Sampling Requirements and Resolution 229mixed down to frequencies that could be displayed by a cathode ray tube to which thefilm was exposed. Phase history data are also sometimes called range history data, sincephase is proportional to range and because plotting this information shows the range to ascatterer as a function of time.One must generally take care with terminology to avoid confusing the domains inwhich the data reside. For example, the raw data coming from a deramp receiver can bethought of as existing over RF frequency (or fast time frequency) and pulse number (slowtime). If the receiver directly samples the incoming signal, then the data exist in range (fasttime) and pulse number (slow time). The two domains are equivalent, being separated onlyby a Fourier transform in the range dimension that can be realized digitally or throughanalog deramp reception.Since deramp reception is so common, spotlight SAR data are often assumed to becollected in the Fourier domain and already matched filtered. We will use this fact later inour discussion of image reconstruction via the polar format algorithm. As a final note wepoint out that it is common parlance to use the phrase pulse compression to refer to the actof inverse Fourier transforming the deramped signal to obtain the range profile. Strictlyspeaking, however, pulse compression is a synonym for matched filtering. A signal maybe pulse compressed regardless of whether it is in the time or frequency domain. The factthat deramp accomplishes both matched filtering and the Fourier transform can make thispoint confusing.6.4.2 Along-Track Sampling RequirementsWe have already seen that a SAR data collection comprises a set of pulses that are transmittedand received as the sensor flies along its trajectory. It stands to reason that thealong-track signal is subject to some sampling rate constraint just as any other signalwould be. This means that the Shannon–Nyquist sampling theorem is lurking nearby. Ineveryday engineering conversation, the sampling theorem states that we need to sample atleast twice the rate of the highest frequency present in the signal. The application to SARspatial sampling becomes clearer if we say this another way: we need to sample at leasttwice per wavelength. The spatial sampling requirement comes from the need to sample ator before the point at which the change in phase (two-way range) to any visible scattererchanges by λ/2. Since λ = c/f , the highest frequency has the shortest wavelength, in turndriving the sampling constraint.Recall the earlier assertion that SAR can be thought of purely in spatial terms. Wenow apply this statement to the problem of determining the along-track spatial samplingrequirement. Consider a simplification of the SAR collection in which the antenna beamfootprint on the ground is idealized as being equal to zero outside of the main lobe.The footprint generally has an elliptical shape, but it is drawn as a circle in Figure 6-8.Either because of the flight path or through motion compensation, a spotlight collection istantamount to flying in a circle about the aim point (i.e, the center of the scene), so we canalso think of the sensor as being stationary while the illuminated scene rotates as if it wereon a turntable. The question we need to answer then becomes, “How much can the scenerotate before we need to record another pulse?” The answer is that the angle δφ betweenpulses must be small enough to ensure that the two-way path from the antenna to anypoint in the scene changes no more than one-half of a wavelength. The sensor effectivelyflies in a circle centered on the aim point, so the distance to scene center never changes(we can think of the reflection from the aim point as a DC signal). The maximum path