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

308	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

309

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

p2	ðtÞ for 1 ¼6 2 as possible.
	The most often used measure of similarity between two waveforms p1ðtÞ and
p2	ðtÞ is the Euclidian distance given by
	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

310	COMPUTERIZED IMAGING TECHNIQUES I: SYNTHETIC APERTURE RADAR
significant for optimization. It should be minimized for		1 ¼6	2	in order to maximize
D2. The integral in the last term is rewritten as
	Rð 1; 2Þ ¼ ð pðt 1Þpðt 2Þdt			ð17:4-5Þ
which is the same as
	Rð Þ ¼ ð pðtÞpðt þ Þdt			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

p t	Þ ¼	rect	t T0		x t	Þ	ð	17:4-11	Þ
ð			T		ð

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

THE MATCHED FILTER

311

The width of the main lobe of this function is approximately equal to 1=gT. The rectangular windowing function is often replaced by a more smooth function such as a Gaussian function, as discussed in Section 14.2. Because of the properties discussed above, a chirp signal within a finite duration window is often the pulse waveform chosen for good range resolution.

17.5THE MATCHED FILTER

One basic problem addressed by matched filtering is to decide whether a signal of a given form exists in the presence of noise. In the classical case, the filter is also constrained to be a LTI system. Matched filters are used in many other applications as well, such as pulse compression as discussed in the next section, and image reconstruction as discussed in Sections 17.8–17.10.

Suppose the input consists of the deterministic signal xðtÞ plus noise NðtÞ. Let the corresponding outputs from the linear system used be x0ðtÞ and N0ðtÞ, respectively. This is shown in Figure 17.3. The criterion of optimality to be used to determine whether xðtÞ is present is the maximum SNR at the system output. It is shown below that the LTI filter that maximizes the SNR is the matched filter.

If n0ðtÞ is assumed to be a sample function of a wide-sense stationary (WSS) process, the SNR at time T0 can be defined as

	SNR		¼	jx0ðT0Þj2	ð	17:5-1	Þ
				E½N02ðT0Þ&
	The output signal x0ðtÞ is given by
		1
	x0ðTÞ ¼	ð	Xðf ÞHðf Þej2pfT0 df		ð17:5-2Þ
		1

The output average noise power is given by

	1
E½N02ðtÞ& ¼	ð	jHðf Þj2SN ðf Þdf	ð17:5-3Þ
	1

where SN ðf Þ is the spectral density of NðtÞ.

Figure 17.3. The matched filter as a LTI filter to optimally reduce noise.

312

COMPUTERIZED IMAGING TECHNIQUES I: SYNTHETIC APERTURE RADAR

Now, the SNR can be written as

2

1

X

ð

f

Þ

H f ej2pfT0

ð

ð Þ

¼

1

2

ð

Þ

SNR

17:5-4

1

ð

jHðf Þj SN ðf Þdf

1

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

1

Aðf ÞBðf Þdf

2

1 jAðf Þj2df

1 jBðf Þj2df

ð17:5-5Þ

ð

ð

ð

1

1

1

with equality iff

Aðf Þ ¼ CB ðf Þ

ð17:5-6Þ

C being an arbitrary real constant. Let

Aðf Þ ¼ SN ðf ÞHðf Þ

2 fT

17:5-7

pj p

0

ð

Þ

B f

Þ ¼

Xðf Þe

ð

SN ðf Þ

Then, the Schwarz inequality gives

p

1

2

2

1

SN ðf ÞjHðf Þj2df 32

1

X

f

2

3

Xðf ÞHðf Þej2pfT df

ð

ð

j ð

Þj

df

ð17:5-8Þ

S

f

ð

N

ð Þ

1

1

1

54

5

4

or

SNR

1

jXðf Þj2

df

17:5-9

ð

ð

Þ

SN ðf Þ

1

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

H f

Þ ¼

H

f

Þ ¼

C

X ðf Þ

e j2pfT0

ð

17:5-10

Þ

ð

optð

S

f

Þ

N ð

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

313

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

	1
yðTÞ ¼ K	ð	x2ð Þd
	1

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

p

2

ð

2=F2

j

½

g

T2

f

f 2

=F2

1 tan 1

g

T2

&

ð

Þ

X f

4

f

f

17:6-2

FB m=p e

p

0Þ

B e

ð

0

B

2

where

m ¼ ½1 þ g2T4&21

ð17:6-3Þ

FB ¼

m

ð17:6-4Þ

pT

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

	1
yðtÞ ¼	ð	Xðf ÞHðf Þej2pftdf							17:6-7	Þ
	1							ð
¼				t2	ð	Þ	e	j2pf0t
	pme				=	T=m 2

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

315

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	0	¼ 0.	It is	observed from Eq. (6.11.7) that the	matched filter	generates
							2
a spectrum			which	has zero phase, and an amplitude	which is jXðf Þj			when

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		E		ð17:6-8Þ
		A2
		max
Similarly, the spectral width F is defined by
F ¼	E			ð17:6-9Þ
	B2
		max

where Bmax is the maximum amplitude of the spectrum.

The input signal energy is the same as yð0Þ. The output signal energy is

	1
Ey ¼	ð	jXðf Þj4df			ð17:6-10Þ
	1
The compression ratio m is given by
	m ¼		Tx		ð17:6-11Þ
			Ty

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	TxE2	¼ aTxF	ð17:6-12Þ
	Ey

316 COMPUTERIZED IMAGING TECHNIQUES I: SYNTHETIC APERTURE RADAR

where

1

E

ð

jXðf Þj2df

a ¼

2

2

1

E

y

Bmax

¼ Bmax

1

ð17:6-13Þ

ð

jXðf Þj4df

1

a can also be written as

1

ð

D2ðf Þdf

a ¼

11

ð17:6-14Þ

ð

D4ðf Þdf

1

where

D f

Þ ¼

jXðf Þj

ð

17:6-15

Þ

ð

Bmax

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
a	1	1.67	2.22

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

	L=2
Eð Þ ¼ A	ð	ej ðyÞdy	ð17:7-1Þ

L=2

A SIMPLIFIED THEORY OF SAR IMAGING						317

Figure 17.4. Geometry for estimating cross-range resolution.
where A is the incident amplitude assumed constant, and
2p			ð17:7-2Þ
ðyÞ ¼		y sin
	l

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

sin

p

L sin

ð

Þ ¼

l

L

ð

17:7-3

Þ

l L sin

E

p

Normalizing to unity at ¼ 0, the antenna power gain becomes

2

sin2

p

L sin

ð Þ

l

2

G

Eð Þ

17:7-4

ð Þ ¼

E 0

¼

p L sin

ð

Þ

Setting Gð Þ ¼ 21 at half-power points yields, after some simplified computations,

l

3dB ’ :0:44

ð17:7-5Þ

L

For a target at a range R, this

translates

to the following cross-range resolution:

’ 0:88

Rl

ð17:7-6Þ

L

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.