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Signal Model for Synthetic Aperture Radar Images

Abstract This chapter introduces the SAR fundamental ideas and analysis methods employed in the subsequent chapters of the book. First, the geometrical configuration of the sensor relative to the imaged scene is described, followed by a brief overview of the time–frequency interpretation of the acquired signals. Afterwards, the principle of SAR image formation and the SAR tomography framework are introduced as the main classical processing tools.

2.1 SAR Acquisition Geometry For the developments from this chapter, the considered raw signal model for synthetic aperture processing is described in the following. The geometry under consideration is presented in Fig. 2.1. The unit vector u describes the azimuth direction of the sensor. The position of the sensor’s antenna phase center (APC) at a certain azimuth (slow) time t will be written as ra (t) = ra,0 + v0 tu,

(2.1)

where ra,0 is the APC position vector at t = 0 and v0 the sensor’s speed in zeroDoppler geometry. For a given target i having the position vector ri , the closest approach distance to the synthetic aperture is given by   r0,i = (ra,0 − ri ) − (ra,0 − ri ) · u u

(2.2)

and the azimuth time at which this distance is attained can be expressed as   ri − ra,0 · u . tr i = v0

(2.3)

The distance APC-target i as a function of azimuth time can be written in terms of the previously defined variables as follows:

© The Author(s) 2017 A. Anghel et al., Infrastructure Monitoring with Spaceborne SAR Sensors, SpringerBriefs in Signal Processing, DOI 10.1007/978-981-10-3217-2_2

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2 Signal Model for Synthetic Aperture Radar Images

Fig. 2.1 SAR acquisition geometry for a point having the position vector ri

©EADS Astrium

u

r0,i

ra(t)

ra,0

ri O

Δri (t) = ra (t) − ri   2 = r0,i + [v0 (t − ti )]2 .

(2.4)

After demodulation, the response from a set of N point scatterers located at ri is a function of two variables (slow time and fast time): s(t, τ ) =

N  i=1





 2Δri (t) 4π fc t − ti , Δri (t) rect exp −j Ai p0 τ − c c Tap

(2.5)

where Ai is the complex amplitude of scatterer i, fc is the central frequency, c is the speed of light, Tap is the synthetic aperture duration, and p0 (τ ) is the complex envelope of the transmitted signal as a function of the fast time τ . In (2.5), the function rect(t/Tap ) is a gate with length Tap centered in the origin.

2.2 Azimuth Time–Frequency Representation The azimuth phase history of an imaged target is actually a nonstationary signal whose location in time domain is given by the interval when the antenna beam illuminates the target. For this type of signals, the Fourier transform cannot provide information about the frequency content at certain time instants, but only an overview of the spectral composition on the whole analyzed period. This happens because the base on which a signal is decomposed in a Fourier transform has a theoretically infinite support which is not compatible with time-localized signals. A straightforward

2.2 Azimuth Time–Freq