Файл: Ersoy O.K. Diffraction, Fourier optics, and imaging (Wiley, 2006)(ISBN 0471238163)(427s) PEo .pdf

Figure 17.5. SAR imaging geometry [Courtesy of Soumekh, 1999].

SAR imaging is usually discussed in one of two modes. In the squint mode, the center of the target area is at ðxc; ycÞ where yc is nonzero. This is what is shown in Figure 17.5. In the spotlight mode, yc equals zero such that the target region appears perpendicular to the direction of flight.

Assume that the target region consists of stationary point reflectors with reflectivity n at coordinates ðxn; ynÞ, n ¼ 1; 2; 3; . . . . A pulse signal pð Þ is used to illuminate the target area. A radar receiver at ð0; yÞ receives the echo signal sðt; yÞ reflected back from the targets as

n

n

So the problem is how to obtain an approximation of uðx; yÞ from the measured signal sðt; yÞ. The 2-D Fourier transform of uðx; yÞ is given by

n

The 1-D Fourier transform of sðt; yÞ with respect to the time variable t is given by

where k ¼ 2pf =c is the wave number, and Pðf Þ is the Fourier transform of pðtÞ. The 2-D Fourier transform of sðt; yÞ with respect to both t and y can now be

computed as the 1-D Fourier transform of Sðf ; yÞ with respect to y, and is given by

This operation is called source deconvolution where the source is Pðf Þ. However, using Eq. (17.8-9) usually yields erroneous results since it is an ill-conditioned

operation, especially in the presence of noise. Instead matched filtering is used in the form

n

In the space-time domain, it can be shown that this operation corresponds to the

Equation (17.8-10) as compared to Eq. (17.8-4) shows that the spectrum of the reconstructed signal is different from the spectrum of the desired signal by the factor of jPðf Þj2. Since this is typically a slowly varying amplitude function of f with zero phase, reconstruction by matched filtering usually gives much better results than source deconvolution in the presence of noise.

17.9 IMAGE RECONSTRUCTION WITH FRESNEL APPROXIMATION

It is possible to simplify the reconstruction algorithm by using the Fresnel approximation. Suppose that the center of the target area is at ðxc; 0Þ. The Taylor series expansion of rn around ðxc; 0Þ with xn ¼ xc þ xn yields

Fresnel approximation corresponds to keeping the terms shown. Sðf ; yÞ of Eq. (17.8-6) can now be written as

n

In order to recover uðx; yÞ, matched filtering is performed with respect to both Pðf Þ and the chirp signal expð jky2=xcÞ. The reconstruction signal spectrum can be written as

In general, the radar bandwidth is much smaller than its carrier frequency kc so that jk kcj kc is usually true. This is referred to as narrow bandwidth. In addition, the y-values are typically much smaller than the target range. This is

Note that there are two filtering operations above. The first one is with the filter transfer function P ðf Þ, which corresponds to p ð tÞ in the time domain. The second one is with the filter impulse function expðjkcy2=xcÞ.

We recall that the correspondence between the time variable t and the range variable x with the stated approximations is given by

In summary, when narrow bandwidth and narrow beamwidth assumptions are valid, image reconstruction is carried out as follows:

1.Filter with the impulse response p ð tÞ in the time direction.

2.Filter with the impulse response expðjkcy2=xcÞ in the y-direction.

3.Identify x as ct=2 or t as 2x=c.