Doppler Coefficient Estimation for Synthetic Aperture Radar Using Sub-Aperture Interferogram
Let Ut=y, the magnitude in (4) can be rewritten as followers:
Using equation (1) and (5), the received signals can be written as:
This is a signal of the linear FM form. Where A(y) will be considered a nuisance parameter and will not be estimated.
3. Estimation of the Chirp Signal Coefficient
From section II, we know SAR signal is a typical chirp signal and in general, is blurred by noise. We use a discrete-time polynomial phase signal model for the source signal, the measured data are assumed to consisted of signal and noise,
where
y(y) is the phase of the signal, y0 is the initial point, T denote the total number of samples and r is sample spacing on the synthetic aperture of SAR, and n(y) denote multiply noise, it can be assumed as a zero mean white Gaussian noise[4,5,6]. The phase of the chirp signal is
y(y)= f0 + f1y + f2y2
(8)
where f
0, f
1, f
2 are its initial phase, initial frequency and frequency rate, respectively. The principle value of the phase function of the measured signal can be expressed as
where -
p<=
Ðg(y)<=
p, it can be computed from the data.
With the recorded data set of M elements, we generate a finite number of subsets with the same size N, what we call sub-apertures, by sequentially not overlapping each other.
SA
j denote the j-th sub-aperture; and sub-aperture sizes is express as
DY for convention, it is equal to samples N multiply sample spacing,
DY=N*r . Based on the above-mentioned, the interferometric phase is come from (j+1)-th sub-aperture signal interfere with j-th sub-aperture signal, which is denote as
fj(y) :
Where
f¢jn(y) is the phase noise sequence and 0<y
£N*r for a sub-aperture. Follow equation (10) we can obtain:
We use ERS SAR as the simulation system, to illustration it. We will assume that the electromagnetic behavior of the moving targets is similar to one point scatter response. The magnitude of the slant range is 8.4848e+005km, aircraft altitude is 800000m, azimuth sample spacing is 3.990574m and total samples is 1068, aircraft ground speed is 6.699028e+03m/sec, platform heading is 192.036610832947° and radar wavelength is 0.056666m, target heading is 161°, target speed is 9m/sec.
Some of the interferometric phase
fj are shown in Figure 1. By this procedure the phase history from curve transform to linear and then the phase shift between
fj+1(y)
and
fj(y) are constant. Compare equation (10) and (11), we can obtain the phase shift:
From equation (12) we can estimate a frequency rate f
2i in every subsets of the measured signal:
Using equation (13) and after some manipulations, one can obtain the following estimate for the frequency rate
:
Once
is available, then substitute it to the below equation
Follow the above-procedure of (10), we can obtain
Some of the interferometric phase
are shown in Figure.2. Compare equation (10) with (16), we know the phase difference between
and
fj is
From the equation (17), we can estimate the value of the frequency f
li:
The last result of
is
In the next section we show how to use this method for estimating a moving target speed in SAR.
