Analysis

The radar transmits pulses using a LFM chirp signal with chirp rate γ, period T, and center frequency fc. The transmission pulse signal

( ) ( ) cos[2 ( ) ]2

t c

s t = A t π f t +πγ t nT− (3-1)

n is pulse number, and t is fast time

It is convenient to select a rectangular pulse shape of amplitude A₀ for A(t) and to use the complex form to represent this real signal. Defining , the transmitted signal ：

t t nT= −

_x( ) ₀ ( ) exp{ [2 ²]}

T_p is the transmitted pulse length. The rect() function ： 1 | | 1

The expression for transmitted signal in (3-2) does not incorporate the effects of transmits amplifier and waveguide on the signal. It does not include the time delay for the signal to reach the antenna. These effects affect the understanding and image formation techniques.

Assuming an ideal scatterer of radar at (X_t, Y_t, Z_t), its amplitude and phase characteristics do not change with frequency and aspect angle. This assumption is realistic for normal situation. The received signal from a single point at (X_t, Y_t, Z_t) at pulse n is the target and back to the antenna.

The derivation of (3-4) ignores any attenuation due to the two-way propagation of transmitted energy to the scene and back to the radar. In fact, the amplitude of the received signal will be a function of many factors including the amplitude of the transmitted signal, distance to the target, and gain of the antenna, in addition to the range cross-section of the target.

It is common practice to make the simplifying assumption that the sensor is stationary during pulse transmission and reception while moving in discrete

increments between pulses. In this situation, an adequate approximation for t_d is 2 _t

d

t R

≈ c (3-5)

R_t is the distance from the Antenna Phase Center (APC) to the target. Using the approximation, (3-4) becomes

2 / 2 2 2

Figure 3.2 illustrates the geometry.

The last analog signal conditioning step before sampling the received signal is demodulation. A spotlight mode system, occasionally, and stripmap or scan mode systems, often, will not perform this dechirp. We choose to include the dechirp operation in the model development here. The radar demodulates the received signal by mixing it with a replica of the transmitted signal delay by 2R₀/c.

Stripmap and spotlight modes differ in selection of a motion compensation signal. Figure 3.3 shows the difference. Spotlight compensates each pulse exactly for the scatterer at scene center to convert this scatterer’s signal to a constant phase. In Figure 3.3, the spotlight mode compensates for the distance AC and treats the data as if it came from B in subsequent processing.

The radar incorporates a real-time motion compensation operation by mixing the received signal s_r(n,t) of (3-6) with a reference function s_ref(n,t) to produce the intermediate frequency(IF) signal s_if(n,t). The reference signal is

0 2

R₀(n) is reference range from the planned APC location to the scene center. The model does not include the effects of any receiver waveguide dispersion on the signal nor any receiver filter and waveguide time delay. The IF signal that results from mixing (3-6) with (3-8) is

0 It is convenient to write this equation in the form

( , )

− . A cross-correlation of the range-compressed returns from successive pulses determines the pulse-to-pulse changes in peak location. At this point, it is acceptable to either subtract off a range frequency term in the original

signal history or move the compressed pulses in range using an interpolation routine to align successive pulses.The necessary frequency adjustment to (3-10) is to multiply by a complex exponential having a phase Φ_adj given by

0

The next step is to measure the phase of the peak response. The phase is

2

The result of multiplying the frequency adjustment described Φ_adj and a phase that is the negative of Φ_pk is stabilizes a scene center scatter (for which R_t = R₁) gives any signal from this scatterer a constant phase and zero frequency.

The final step is to compress this signal in azimuth and in range dimension via the two-dimensional Fourier transform. The result image is in Figure 3.1(c).

If the target’s relative rotation rate is essentially constant over the coherent data processing interval, this single point compensation alone provides an approximately focused image, especially when the target is small enough that polar formatting is not required. However, when the incremental change in rotation angle is not constant from pulse to pulse, a position-dependent blurring or defocusing of scatterers generally occurs.

The azimuth scale, relating image and target distances in the azimuth direction, depends on this rotation rate； therefore, the azimuth scale accuracy depends on auxiliary data accuracy. However, the range scale, relating image and target distances in the range direction, is independent of the relative motion.

Thus, the range scale accuracy in PPP is equivalent to that in a motion measurement implementation

3.2 Implementation

Figure 3.4 is the flowchart of PPP algorithm. Unlike PGA, PPP algorithm does not need iteration step. Figure 3.5 is a spotlight image with the random phase error that simulates aircraft sway. Next, Figure 3.6 is the corrected image removed the linear phase. It is obviously that the quality of Figure 3.6 is much better than the Figure3.5. Because of PPP algorithm can’t remove linear phase term, the linear phase removal method use the regressive method^[26]. Figure 3.7 is the image without adding random phase error. We can compare Figure 3.6 with Figure 3.7. The brightness of corrected image is smaller than the original.

We see that the point spread in range is more convergent, in additional the most important is the spread in azimuth is also converge. In Figure 3.8, the red dotted line is the random phase error we added, and the blue solid line is the estimated phase. We see the two lines are close, and their trends are similar. From the two viewpoints of the corrected image and the estimated phase, PPP algorithm works well in spotlight image.

Next, we discuss that the larger random phase error influences PPP phase estimation accuracy or not. There, we consider that the added phase error is 10 times larger than above. Figure 3.9 is the information with the larger error. First,

we see the top right image; it is similar with the original image, the top left one.

Following, the bottom block show the added random phase error and the estimated phase where we also see the trends are similar. Although the curve is not so match, the final destination --- image correction is good. The linear phase estimated by regressive method seems not good. But we could say it works successfully with larger phase error.

PPP can also apply to stripmap data. The following implementations are all for stripmap data. First, we see Figure 3.10 that is real image with random error, and the image is distortion. Figure 3.11 is corrected image via PPP algorithm.

Obviously, it looks much better than Figure 3.10. Comparing it with the original image without added error, Figure 3.12, the corrected image can be restored as similar as original one. Figure 3.13 is the plot of the random phase error and the estimated phase. The simulation result is same as spotlight.

Next, we implement the algorithm with the actual data without given phase error. The implementation is what we really care about. Figure 3.14 is the implementation result. We observe that the corrected image is not better than original image. Many reasons may cause the result. We think the most probable reason is that the motion compensation is so good that there is not much aircraft sway information for PPP to correct. And some little error may be caused by the hardware like electric circuits, mechanism vibration, thermal noise, and so on.

Because of the failure of above implementation, we decide to use image without conventional motion compensation (mocomp) to understand what reasons cause the failure of PPP with stripmap data. Figure 3.15 is a no mocomp image, and it is obviously worse than the image using mocomp, Figure 3.7. Here, we see how important mocomp is. Figure 3.16 is the no mocomp image corrected via PPP. We carefully observe the two images, and we don’t get a

much better or worse image. In some regions of the image, the scatters are much convergent, but in other places are not. We can’t judge it is better or worse. And it also has the similar situation with other stripmap images.

There are many simulations, and most of them show satisfying result except the last two simulation. PPP does not design for stripmap SAR but for ISAR.

That’s why PPP doesn’t have such good result for stripmap images.