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

Figure 17.2. A SAR image generated in September, 1995 [Courtesy of Center for Remote Imaging, Sensing and Processing, National University of Singapore (CRISP)].

systems carried on the space shuttles are similar SAR imaging systems. Examples of airborne SAR systems carried out with airplanes are the E-3 AWACS (Airborne Warning and Control System), used in the Persian Gulf region to detect and track maritime and airborne targets, and the E-8C Joint STARS (Surveillance Target Attack Radar System), which was used during the Gulf War to detect and locate ground targets.

An example of a SAR image is shown in Figure 17.2. This image of South Greenland was acquired on February 16, 2006 by Envisat’s Medium Resolution Imaging Spectrometer (MERIS) [ESA].

17.3RANGE RESOLUTION

In SAR as well as other types of radar, range resolution is obtained by using a pulse of EM wave. The range resolution has to do with ambiguity of the received signal due to overlap of the received pulse from closely spaced objects.

In addition to nearby objects, there are noise problems, such as random fluctuations due to interfering EM signals, atmospheric effects, and thermal variations in electronic components. Hence, it is necessary to increase signal-to- noise ratio (SNR) as well as to achieve large range resolution.

The distance R to a single object reflecting the pulse is tc=2, where t is the interval of time between sending and receiving the pulse, and c is the speed of light,

3 108 m/sec. Suppose the pulse duration is T seconds. Then, the delay between two objects must be at least T seconds so that there is no overlap between the two pulse echoes. This means the objects must be separated by cT=2 meters (if MKS units are used). Reducing T results in better range resolution. However, high pulse energy is also required for good detection, and short pulses mean lower energy in practice. In order to avoid this problem, matched filtering discussed in Section 17.5 is often used to convert a pulse of long duration to a pulse of short duration at the receiver. In this way, the received echoes are sharpened, and the overall system possesses the range resolution of a short pulse. The peak transmitter power is also greatly reduced for a constant average power. In such systems, matched filtering is used both for pulse compression as well as detection by SNR optimization. This is further discussed in Section 17.5.

17.4CHOICE OF PULSE WAVEFORM

The shape of a pulse is significant in order to differentiate nearby objects. Suppose that pðtÞ is the pulse signal of duration T, which is nonzero for 0tT. The returned pulse from one object can be written as

where 1 is the attenuation constant, and 1 is the time delay. The returned pulse from a second object can be written as

The shape of the pulse should be optimized such that p1ðtÞ is as dissimilar from

D2 can be written as

ð ð ð

D2 ¼ 21 p2ðt 1Þdt þ 22 p2ðt 2Þdt 2 1 2 pðt 1Þpðt 2Þdt

ð17:4-4Þ

The first two terms on the right-hand side above are proportional to the pulse energy, which can be separately controlled by scaling. Hence, only the last term is

where equals 1 2 or 2 1. It is observed that Rð Þ is the autocorrelation of pðtÞ.

A linear frequency modulated (linear fm) signal, also called a chirp signal has the property of very sharp autocorrelation which is close to zero for 6¼ 0. It can be written as

or more generally as

The larger g signifies larger variation of instantaneous frequency fi, which is the derivative of the phase:

It is observed that fi varies linearly with t. The autocorrelation function of xðtÞ can be shown to be

Suppose that a pulse centered at T0 and of duration T is expressed as

The autocorrelation function of pðtÞ is given by

R X ¼ ej2p Ð ½f þ2gðT0þT2Þ&tri ðT j jÞ sin c pg2 ðT j jÞ

To optimize the SNR, the Schwarz inequality can be used. If A(f) and B(f) are two possibly complex functions of f, the Schwarz inequality is given by

The SNR is maximized and becomes equal to the right-hand side of Eq. (17.5-9) when the equality holds according to Eq. (17.5-6), in other words, when

The filter whose transfer function is given by Eq. (17.5-10) is called the matched filter. It is observed that Hoptðf Þ is proportional to the complex conjugate of the FT of