Ersoy O.K. Diffraction, Fourier optics, and imaging (Wiley, 2006)(ISBN 0471238163)(427s) PEo
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COMPUTERIZED IMAGING TECHNIQUES I: SYNTHETIC APERTURE RADAR |
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,
CHOICE OF PULSE WAVEFORM |
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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
p1ðtÞ ¼ 1pðt 1Þ |
ð17:4-1Þ |
where 1 is the attenuation constant, and 1 is the time delay. The returned pulse from a second object can be written as
p2ðtÞ ¼ 2pðt 2Þ |
ð17:4-2Þ |
The shape of the pulse should be optimized such that p1ðtÞ is as dissimilar from
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ðtÞ for 1 ¼6 2 as possible. |
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The most often used measure of similarity between two waveforms p1ðtÞ and |
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ðtÞ is the Euclidian distance given by |
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D2 ¼ ð ½p1ðtÞ p2ðtÞ&2dt |
ð17:4-3Þ |
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
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COMPUTERIZED IMAGING TECHNIQUES I: SYNTHETIC APERTURE RADAR |
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significant for optimization. It should be minimized for |
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D2. The integral in the last term is rewritten as |
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Rð 1; 2Þ ¼ ð pðt 1Þpðt 2Þdt |
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which is the same as |
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Rð Þ ¼ ð pðtÞpðt þ Þdt |
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ð17:4-6Þ |
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
xðtÞ ¼ A cosð2pðft þ gt2ÞÞ |
ð17:4-7Þ |
or more generally as
xðtÞ ¼ ej2pðftþgt2Þ |
ð17:4-8Þ |
The larger g signifies larger variation of instantaneous frequency fi, which is the derivative of the phase:
fi ¼ f þ 2gt |
ð17:4-9Þ |
It is observed that fi varies linearly with t. The autocorrelation function of xðtÞ can be shown to be
ð |
ð17:4-10Þ |
Rð Þ ¼ ej2p ðf þg Þ ej4pg tdt |
Suppose that a pulse centered at T0 and of duration T is expressed as
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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
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Now, the SNR can be written as |
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ð Þ |
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jHðf Þj SN ðf Þdf |
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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
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Aðf ÞBðf Þdf |
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1 jAðf Þj2df |
1 jBðf Þj2df |
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ð17:5-5Þ |
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with equality iff |
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Aðf Þ ¼ CB ðf Þ |
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ð17:5-6Þ |
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C being an arbitrary real constant. Let |
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Aðf Þ ¼ SN ðf ÞHðf Þ |
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pj p |
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Then, the Schwarz inequality gives |
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SNR |
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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
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e j2pfT0 |
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N ð |
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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
PULSE COMPRESSION BY MATCHED FILTERING |
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the input signal, and inversely proportional to the spectral density of the input noise. The factor
e j2pfT0
serves to adjust the time T0 at which the maximum SNR occurs.
EXAMPLE 17.1 If the input noise is white with spectral density equal to N0, find the matched filter transfer function and impulse response. Also show the convolution operation in the time-domain with the matched filter.
Solution: Substituting N for SN ðf Þ in Eq. (17.5-10) yields
Hoptðf Þ ¼ KX ðf Þe j2pfT0
where K is an arbitrary constant. The impulse response is the inverse FT of Hoptðf Þ, and is given by
hoptðtÞ ¼ KxðT0 tÞ |
ð17:5-11Þ |
The input signal is convolved with Hoptðf Þ to yield the output. Thus,
1ð
yðtÞ ¼ xð Þhoptðt Þd
1 1ð
¼ K xð ÞxðT0 t þ Þd
1
The peak value occurs at t ¼ T0, and is given by
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x2ð Þd |
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which is proportional to the energy of the input signal.
17.6PULSE COMPRESSION BY MATCHED FILTERING
In applications such as pulse radar and sonar, it is important to have pulses of very short duration to obtain good 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 is often used to convert a pulse of long
314 |
COMPUTERIZED IMAGING TECHNIQUES I: SYNTHETIC APERTURE RADAR |
duration to a pulse of short duration at the receiver. In this way, the received echos 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.
The input pulse waveform is chosen such that the output pulse is narrow. For example, the input can be chosen as a chirp pulse in the form
xðtÞ ¼ et2=T2ejð2pf0tþgt2Þ |
ð17:6-1Þ |
where T is called the pulse duration. In practice, jgjT is much less than 2pf0. The spectrum of the pulse is given by
ð Þ ¼ |
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T2 |
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f 2 |
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1 tan 1 |
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FB m=p e |
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where |
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m ¼ ½1 þ g2T4&21 |
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FB ¼ |
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FB is the effective bandwidth of the spectrum since the spectrum is also Gaussian with center at f0.
Disregarding constant terms, the matched filter for the signal xðtÞ (assuming noise is white and T0 ¼ 1) is given by
Hðf Þ ¼ e 4p2ðf f0Þ2=FB2 e j½gT2ðf f0Þ2=FB2 21 tan 1 gT2& |
ð17:6-5Þ |
In practice, the amplitude of Hðf Þ in Eq. (17.6-5) actually reduces the amplitude of the final result. This can be prevented by using the phase-only filter given by
Hðf Þ ¼ e j½gT2ð f f0Þ2=FB2 21 tan 1 gT2& |
ð17:6-6Þ |
Then, the output of the matched filter is
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yðtÞ ¼ |
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Xðf ÞHðf Þej2pftdf |
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pme |
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It is observed that the output signal is again a Gaussian pulse with the frequency
modulation removed, and the pulse duration compressed by the factor m. In addition, p
the pulse amplitude is increased by the factor m so that the energy of the signal is unchanged.
PULSE COMPRESSION BY MATCHED FILTERING |
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If Eq. (17.6-5) is used instead of Eq. (17.6-6), the same results are valid with the p
replacement of m by m= 2.
In pulse radar and sonar, range accuracy, and resolution are a function of the pulse duration. Long duration signals reflected from near targets blend together, lowering the resolution. The maximum range is a function of the SNR, and thereby the energy in the pulse. Thus, with the technique described above, both high accuracy, resolution, and long range are achieved.
Pulse compression discussed above is a general property rather than being dependent on the particular signal. The discussion below is for a general pulse signal
with T |
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It is |
observed from Eq. (6.11.7) that the |
matched filter |
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Eq. (17.11-5) is used. The output signal at t ¼ 0 is given by
1ð
yð0Þ ¼ jXðf Þj2df ¼ E
1
which is large.
In order to define the degree of compression, it is necessary to define practical measures for time and frequency duration. Let the pulse have the energy E and a maximum amplitude Amax. The input pulse duration can be defined as
Tx ¼ |
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ð17:6-8Þ |
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Similarly, the spectral width F is defined by |
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where Bmax is the maximum amplitude of the spectrum.
The input signal energy is the same as yð0Þ. The output signal energy is
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Ey ¼ |
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ð17:6-10Þ |
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The compression ratio m is given by |
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m ¼ |
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where Ty is the pulse width of the output, which is also given by
Cmax2 Ty ¼ Ey
Cmax is the maximum amplitude of the output which is yð0Þ. Since yð0Þ is the same as E, Eq. (17.6-11) can be written as
m ¼ |
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ð17:6-12Þ |
Ey |
316 COMPUTERIZED IMAGING TECHNIQUES I: SYNTHETIC APERTURE RADAR
where
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a can also be written as |
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where |
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D f |
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Since Dðf Þ is less than or equal to 1, a is greater than or equal to 1. For example, three types of spectra and corresponding a are the following:
Spectrum |
Rectangular |
Triangular |
Gaussian |
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1.67 |
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It is seen that the Gaussian spectrum has the best pulse compression property. Equation (17.6-12) shows that the compression ratio is proportional to the product of the signal duration Tx and the spectral width F. This is often called the timebandwidth product.
17.7CROSS-RANGE RESOLUTION
In order to understand the cross-range (azimuth) resolution properties of an antenna of length L, consider the geometry shown in Figure 17.4. It is assumed that the target is so far away that the echo signal impinges on the antenna at an angle at all positions on the antenna.
Assuming that the antenna continuously integrates incident energy, the integrated antenna response can be written as
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Eð Þ ¼ A |
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ej ðyÞdy |
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L=2
A SIMPLIFIED THEORY OF SAR IMAGING |
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Figure 17.4. Geometry for estimating cross-range resolution. |
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where A is the incident amplitude assumed constant, and |
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is the phase shift due to a distance d ¼ y sin . The imaginary part of Eð Þ integrates to zero, and the real part gives
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17:7-4 |
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ð Þ ¼ |
E 0 |
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p L sin |
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Setting Gð Þ ¼ 21 at half-power points yields, after some simplified computations, |
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3dB ’ :0:44 |
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For a target at a range R, this |
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’ 0:88 |
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Thus, improved cross-range resolution occurs for short wavelengths and large antenna lengths.
17.8A SIMPLIFIED THEORY OF SAR IMAGING
The imaging geometry is shown in Figure 17.5. A 2-D geometry is used for the sake
of simplicity. A 3-D real-world geometry would be obtained by replacing x by p
x2 þ z2, z being the height.