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Signal and noise considerations in multi-channel SAR

One of the most ubiquitous, yet treacherous relations in radar is the equation describing the ratio of signal-to-noise powers, known as the radar equation. It is straight forward to derivation the radar equation for a point scatterer, as commonly done on an undergraduate courses level when radar is introduced. Typically, the point scatterer based approach is extended to derive the radar equation for distributed scatterers as applicable for imaging synthetic aperture radar. There are, however, several hidden pitfalls in the radar equation when applied to imaging radars. The most common one, is, that it appears to include a gain, i.e. an increase in the signal-to-noise ratio, associated to image focusing (also know as range and azimuth compression). A conclusion which is incorrect and misleading. As it turns out, understanding the radar equation requires a profound SAR and signal processing understanding. This paper attempts a rigorous approach interpreting the signal and noise power of a multi-channel digital beam-forming SAR system.