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

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

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

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

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

where T is called the pulse duration. In practice, jgjT is much less than 2pf0. The spectrum of the pulse is given by

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

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

Then, the output of the matched filter is

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.

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

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

where Bmax is the maximum amplitude of the spectrum.

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

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

316 COMPUTERIZED IMAGING TECHNIQUES I: SYNTHETIC APERTURE RADAR

where

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:

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

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

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.