Processing of SAR-data

Phase history of a point target

Figure 3.1: Imaging geometry of a SAR-system

Fig. 3.1 pictures the illumination of a point target by a SAR sensor during data acquisition. The sensor is moved along the x-axis (azimuth) and emmits, perpendicular to flight direction, the radar pulses to the ground. The distance between the sensor at position $x$ and the target can be expressed as

$$r=\sqrt{x^2+r_0^2}$$ (8)

where $r_0$ denotes the minimum distance between both at $x=0$. As the extension of the radar footprint on the ground is (usually) much smaller than the target distance ($x\ll r_0$), the following approximation can be made:

$$r=r_0\sqrt{1+\frac{x^2}{r_0^2}} \approx r_0 + \frac{x^2}{2r_0}$$ (9)

The phases of the received echos, resulting from the two-way distance $r$, are:

$$\varphi(x)=2\frac{2\pi}{\lambda}\left( r_0+\frac{x^2}{2r_0} \right) = \frac{2\pi x^2}{\lambda r_0} + \mbox{const.}$$ (10)

Assuming a constant sensor velocity $v$ and the abbreviation $k= 2\pi v^2 / \lambda r_0$, a quadratic phase behaviour in time is resulting, neglecting the constant phase term, which has no time dependency.

$$\varphi(t) = kt^2$$ (11)

The quadratic phase behaviour corresponds to a linear change in the received azimuth frequency $f(t)$, the so-called DOPPLER-effect.

$$f(t)=\frac{1}{2\pi} \frac{\partial\varphi(t)}{\partial t} = \frac{k}{\pi}t$$ (12)

This linear DOPPLER-effect is only present as long as $x$ is really small in comparison to $r_0$. Otherwise high order components are occuring and the correct hyperbolic phase history has to be taken into account. Particularly, this is the case for SAR sensors with very long apertures and for those operating not exactly perpendicular to the flight direction but under a so-called squint-angle.

The maximal illumination time of a point target is defined by the extension of the antenna footprint in azimuth. This length, equal to the length of the synthetic aperture, is determined by:

$$t_{max} = \frac{l_{sa}}{v} = \frac{\theta_{ra} r_0}{v}$$ (13)

The bandwidth of the signal in azimuth $B_a$ is, therefore,

$$B_a = f(-t_{max}/2) - f(+t_{max}/2) = \frac{2v \theta_{ra}}{\lambda}$$ (14)

This bandwidth in azimuth sets also the lower limit of the pulse repetion frequency (PRF) of the radar, with which the radar pulses are emmitted to the ground. After eliminating the carrier frequency (demodulation in the receiver hardware), frequencies between $-B_a/2$ and $+B_a/2$ are present in the complex signal. According to the NYQUIST-criterion, a sampling frequency of two times the maximum frequency is necessary for an unambigous recording of the data. Here, the sampling frequency is given by the PRF.

Processing in azimuth

The echo of a single point target is contained in may received radar pulses and appears therefore defocused. The aim of SAR processing, also called compression, is to focus all the received energy of a target, distributed over the illumination time, on one point at $t=0$. To achieve this, the typical phase history, coming from the data acquisition process, is used. Assuming the backscattering of a point target to be time- and angular-independent, and also dominant to other signals like noise and background reflections, the received signal in azimuth direction can be written as

$$S_a(t)=A_0 \exp(i\varphi(t)) = A_0 \exp(ikt^2)$$ (15)

with $A_0$ denoting the backscattering amplitude of a point target (a complex value). The idea of azimuth compression is now to adjust all these phase value to the same value, followed by a coherent summation. To achieve this, a correlation of $S_a(t)$ with a reference function $R(t)=\exp(-ikt^2)$ is performed. This reference function is constructed in a way that it has in every point exactly the opposite phase of the ideal impulse response in Eq.3.8.

As the length of the synthetic aperture and with that also the length of the signal is limited, it makes sense to limit also the length of the reference function by a box-like wheighting function $W(t)$:

$$W(t) = \left\{ \begin{array}{ll} 1 & \mbox{for $-t_{max}/2 < t < +t_{max}/2$} \\ 0 & \mbox{otherwise} \end{array} \right.$$ (16)

$$R(t) = W(t)\exp(-ikt^2)$$ (17)

The result of the correlation is then

$$V(t) = \int_{-\infty}^{\infty} S_a(\xi)R(t+\xi)d\xi$$ (18)

$$= \int_{-\infty}^{\infty} A_0\exp(ik\xi^2)\exp(-ik(t+\xi)^2)W(t+\xi)d\xi$$ (19)

$$= A_0\exp(-ikt^2)\int_{-\infty}^{\infty} W(t+\xi)\exp(-2ik\xi t)d\xi$$ (20)

Using that only small values of $t$ are important, the approximation $W(t+\xi)\approx W(\xi)$ can be made. In the following $FT[\dots]$ should denote a FOURIER-transform. With this, the correlation result can be written as

$$V(t) = A_0\exp(-ikt^2)\sqrt{2\pi}\cdot FT_{2kt}\Big[W(\xi)\Big]$$ (21)

$$FT_{2kt}\Big[W(\xi)\Big] = \int\limits_{-t_{max}/2}^{t_{max}/2} \exp(-2ik\xi t)d\xi$$ (22)

$$= \frac{\exp(-ikt_{max}t) - \exp(ikt_{max}t)}{-2ikt}$$ (23)

$$= \frac{-2i\sin(kt_{max}t)}{-2ikt} = t_{max}\left[\frac{\sin(kt_{max}t}{kt_{max}t}\right]$$ (24)

$$\Longrightarrow\qquad V(t) = A_0t_{max}\sqrt{2\pi}\exp(-ikt^2) \left[\frac{\sin(kt_{max}t)}{kt_{max}t}\right]$$ (25)

The result of this correlation is the image. The principal shape of the resulting impulse response corresponds thereby to the FOURIER-transform of the weigthing function. Is the weigthing function box-like, as above, the impulse response is a sinus cardinalis function (sinc or $\sin(x)/x$). In Fig. 3.2 this process is illustrated. The received signal, also called 'chirp', has a constant amplitude and a parabolic phase behaviour (shown is only the real part of the complex signal). The reference function has an amplitude of one and exactly the opposite phase than the signal itself. After the correlation with $R(t)$ the signal appears well located at $t=0$. Its maximum amplitude increased from $\vert A_0\vert$ to $\sqrt{2\pi}t_{max}\vert A_0\vert$ and the peak phase is zero. In reality, the neglected phase term proportional to the two-way sensor object distance as well as the object phase appear here.

Figure 3.2: Signal compression. Real part of the complex signal of an ideal point target response (left) and amplitude of the compressed signal (right).

It can be recognized, that the bigger $t_{max}$ gets, i.e. as longer the syntetic aperture gets, the more $V(t)$ appears as a DIRAC'S delta function. Defining the resolution as the half distance between the first minima of the main peak at $t=\pm \pi / kt_{max}$, a synthetic aperture consequently has an azimuthal resolution of:

$$\delta_{sa} = \frac{\pi v}{kt_{max}} = \frac{v}{B_a} = \frac{l_{ra}}{2}$$ (26)

Using the more correct definition of the resolution as the half width at half maximum, a 14% bigger value is resulting. The first sidelobes are -13dB lower than the main peak. This can cause problems, if a strong target is near to some weaker targets. Therefore, instead of using a box-like weigthing function, often instead other shapes are used, whose FOURIER-transform shows a better Peak-Sidelobe Ratio (PSLR). A very common function for this is the so-called HAMMING-weighting:

$$W(t) = \left\{ \begin{array}{ll} \alpha +(1-\alpha)\cos\lef... ...t_{max}/2$} \\ 0 & \qquad\mbox{otherwise} \end{array} \right.$$ (27)

Figure 3.3: Signal compression using a HAMMING weighting function.

Choosing $\alpha=0.54$, the first sidelobes of the FOURIER-transform are completely suppressed (Fig 3.3). The PSLR is now much better and has a value of only -43dB. Indeed, the height of the maximum is lowered and also the resolution is main peak is decreased by almost 30%. Despite of these disadvantages, images processed using a HAMMING-weighting often appears to be better focused.

The process of azimuth focussing, as presented here, is comutationally very intensiv, as for every single pixel a correlation has to be calculated, consisting out of a great number of additions and multiplications. It can be significantly accelerated by utilizing the convolution theorem [58]:

$$\int f(\xi)g(t-\xi)d\xi=f(t)\otimes g(t)=FT^{-1}(FT(f(\omega))FT(g(\omega)))$$ (28)

According to this theorem, the convolution of two functions is equal to the multipliction of its FOURIER-transforms in the spectral domain. A correlation represent a convolution with a time-inverted function [58]:

$$f(t)\otimes g(-t) = \int f(\xi)g(t+\xi)d\xi$$ (29)

Therefore, the desired compressed signal can also be obtained as in the following:

$$V(t) = FT^{-1}\bigg[ FT\Big[S_A(t)\Big] FT\Big[R(-t)\Big]\bigg]$$ (30)

In practice, the occuring DOPPLER-rates of the signal are dependent on $r_0$ and are variing with the target distance. It is therfore necessary to adapt the reference functions to the respective data line under investigation. Is the correct refernce function calculated, it can be used to focus a whole azimuth line in one step, using the convolution theorem.

Another problem of the here described, conventional way of processing, are the signal contributions with higher DOPPLER-rates. They occur under larger angles and consequently have also a larger time delay. It might happen, that these signal parts are recorded in later range cells ('Range Cell Migration'). The echo energy is then distributed over several range line, and the SAR azimuth focussing process becomes a two-dimensional operation. A conventional processing whould therefore not be able anymore to focus the whole energy. In this case more advanced processing methods are necessary, which are able to take into account this effect [59]-[63].

Processing in range

In range direction a SAR can work just like a conventional radar. To achieve a high resolution in the direction perpendicular to the flight direction, only a short pulse duration $\tau$ is necessary. In practice, it can be problematic to generate a very short and high power pulse, as the resulting energy densities are hard to handle. In the spectral domain, with short pulse duration a higher signal bandwidth can be observed. A high resolution is therefore tantamount with a high signal bandwidth. A second possibilty to generate a high signal bandwidth is to use a long, but frequency modulated pulse. It is common to use for this a linear frequency modulation (called 'chirp'):

$$f(t) = \frac{B_r}{\tau} t \qquad\mbox{for}\quad -\tau/2 < t < \tau/2$$ (31)

with $B_r$ denoting the bandwidth of the emmited pulse. Like in azimuth this introduces a 'typical' phase history in the signal, which can later be used to compress the signal. The cirprate $k$ is now

$$k = \frac{B_r \pi}{\tau}$$ (32)

In order to compress the extended signal, a new reference function has to be constructed, which takes into account the typically much faster frequency variation compared to the azimuth case. The signal compression itself takes part like part exactly in the same way, i.e. a correllation of the signal with the new reference functiom has to be calculated. The result is, similar to Eq. 3.18:

$$V_r(t) = A_0\tau\sqrt{2\pi}\exp(i k t^2) \left[\frac{\sin(k\tau t)}{k\tau t}\right]$$ (33)

The resulting resolution in range direction is:

$$\delta_{sr} = \frac{c}{2 B_r}$$ (34)

If SAR raw data is processed in azimuth and range, a two-dimensional impulse response is resulting, which represents the product of the two individual one-dimensional impulse responses (see Fig.3.4). This function represents the intensity distribution of a point-like target in the final SAR image.

Figure 3.4: Two-dimensional point target response (without weighting)

Structure of a SAR-processor

A SAR processor is the technical realization of the signal compression in range and azimuth. Its purpose is to derive from the SAR raw data, as recorded by the sensor, the high resolution image result. Starting from optical techniques, over analog electronics up to modern digital SAR processors, several possibilities are existing to realize the necessary computational steps. Nowadays, in the time of very powerful digital hardware, mostly digital methods are used, either realized in software or by using hardware signal-processing.

The principal sequence of processing SAR raw data is shown in Fig. 3.5. The input is the complex signal, as recorded by the SAR sensor. After an one-dimensional FOURIER-transform in range direction, each range line is multiplied with the FOURIER-transform of the reference function in range. After the inverse FFT back to time domain, the data are compressed in range, but are still defocused in azimuth. At this point a correction of the range-cell-migration can happen. Then a FOURIER-transform in azimuth is performed, followed by a multiplication of the FOURIER-transform of the reference function in azimuth. This fuction has to be adapted to the current range distance under investigation. After the back-transformation, the complex image result is derived.

In Fig. 3.6 a simple SAR processing scheme is shown, on the basis of an ideal point target response. I can be observed very good how the initially defocused signal first is compessed in range and after that in azimuth direction.

Figure 3.5: Block-diagram of a simple SAR processor

Figure 3.6: Processing of an ideal point target response (no range cell migration)