# Nonlinearity Correction Algorithm Based on Homomorphic Deconvolution

Bogert et al. [16] proposed the concept of homomorphic deconvolution. Its basic idea is: as to signals with echo interruption, the logarithm of its power spectrum contains an additive periodic component, which appears a peak at the time delay of echo signal after IFFT. The original signal can be reverted by eliminating this peak. The time domain after processing IFFT on the logarithm is de**fi**ned as Cepstrum domain. Since time domain and Cepstrum domain are homomorphic about convolution and addition, the algorithm to revert a time domain signal by **fi**ltering in Cepstrum domain is de**fi**ned as homomorphic deconvolution. The concept of homomorphism is: set A and set B are de**fi**ned as homomorphic about transforms **A **and O, if and only if there exists a surjection from A to B, that for any element *a*_{1}, *a*_{2}** 2 ***A* and *b*_{1}, *b*_{2} 2 *B,* if *a*_{1} ! *b*_{1}** **and *a*_{2} ! *b*_{2}*,* then *a*_{1}*Aa*_{2} ! *b*_{1}*Ob*_{2}** **[17].

Similar as multipath interference in communication signal processing, homomorphic deconvolution can be used to revert the transmitted nonlinearity in FMCW SAR. Consider *e**(**t _{r}*

*)*as the original signal, and

**АФ**(

*t*

_{r}*,*

*sf*as the interferential signal, then

**АФ**(

*t*

_{r}*,*

*sf*can be regarded as a convolution of

*s*

*(*

*t*

_{r}*)*and a S function series represented by

*p(t*

_{r}*,*

*s*

_{re}f)

where *p(t _{r}*

*,*

*sf = 8(t*

_{r}**) —**

*8(t*

_{r}**—**s

_{re}/). Perform FFT on Eq. 5.3.11, the spectrum of

**АФ**(

*t*

_{r}*,*

*sf*can be obtained as

Perform logarithm transform on Eq. 5.3.12, then Eq. 5.3.12 can be rewritten as

Expand term ln[1 — exp( *—jmf * in Eq. 5.3.13 into a Taylor series

Substitute Eq. 5.3.14 into Eq. 5.3.13, and perform IFFT on Eq. 5.3.14, it yields

Equation 5.3.15 is the expression of nonlinear error phase in beat signal in Cepstrum domain. 8 series should be eliminated to get the transmitted nonlinearity e(t_{r}). Since time delay *s _{re}f* is known, one direct method is to construct a 8 series according to the second term on the right of (5.3.15). Thus, 8 series can be eliminated by subtracting the constructed 8 series function. This method is a direct and ideal method that has an optimal performance in estimating the transmitted nonlinear error because the energy of

*e(t*

_{r}) is not changed after the subtraction.

However, the 8 series subtraction method can only be used in simulations. 8 function is a pulse signal with infinite amplitude at one frequency spot in analog system. But in digital system and computer simulation, it is represented by a signal with amplitude of 1 at one frequency spot. Therefore, there may be differences between the simulated 8 series and 8 pulses in real data, and it will induce estimation error when using 8 series subtraction method in practical application.

In order to accomplish the same goal of 8 series subtraction method, a comb notch filter is generated in Cepstrum domain to remove 8 series. Comb notch filter is a filter with amplitude of zero at the points of t = *ks _{re}f* and one at other points. The width of the filter is the same as the signal. Given that the time-width of the signal is

*T*and the length of the signal is

_{r}*N*

_{r}, the width corresponding to time delay

*s*can be calculated as

_{re}f

Hence, the number of notches is N_{r}/N_{s} — 1. The waveform of a comb notch filter is shown in Fig. 5.3. After Cepstrum filtering, *e(t _{r})* can be reverted from the interferential signal.

In the case that the energy of the phase error concentrates at t = *ks _{re}f*, energy of the phase error will be filtered out by the comb notch filter as well as 8 series at t =

*ks*. However, since the width of the notch is only 1-2 spots, the majority of phase error energy can still be reserved after the filtering. Therefore, comb notch filter method is also effective in the case that energy of the original phase error

_{re}f**Fig. 5.3 ****Waveform of the comb notch filter**

concentrates at *t = ks _{re}f*. Moreover, comb notch filter is easy to be built comparing with S series subtraction method, and it is reliable in practical use since filtering technique has been widely used in signal processing field. Therefore, comb notch filter is used in our algorithm to eliminate S series in Cepstrum domain.

Throughout the whole algorithm, no approximations are used, and no specific model of *e(t*_{r}) is required. Therefore, this algorithm can give a theoretically unbiased estimation of *e(t*_{r}). Moreover, compared with derivative algorithm, the proposed algorithm has no limitation of time delay *s _{re}f*, thus the effectiveness of the proposed algorithm in practical application is further strengthened. The flowchart of the proposed algorithm is illustrated in Fig. 5.4.

The nonlinearities in the beat signal vary among each range cell. This will make the correction process complicated and not applicable to real-time tasks. To address this problem, a strategy is proposed to eliminate the dependence of nonlinearity on s using RVP removal. After RVP removal, the beat signal is independent of s, and the received nonlinearity can be corrected by a unique correction function

where F{-} and F^{-1}{ } denote FFT and IFFT, respectively.

**Fig. 5.4 ****Flowchart of the proposed estimation algorithm**

Figure 5.5 shows the signal of each steps of the strategy. Figure 5.5a shows the beat signal with nonlinear error in time-frequency domain. The larger the time delay is, the larger the nonlinearity will be. Figure 5.5b shows that after transmitted nonlinearity correction, there remains received nonlinearity that dependent on time delay. Figure 5.5c shows that using RVP removal, the delay-dependence of beat signal is eliminated. After received nonlinearity correction, the ideal time-frequency characters of beat signal are shown in Fig. 5.5d. Beat signal is reverted to a monochromatic signal after the strategy.

Figure 5.6 illustrates the flowchart of FMCW SAR nonlinearity correction strategy combining with homomorphic deconvolution.

To give a full-scale simulation of the proposed algorithm, quadratic, cubic, and sinusoid nonlinearity errors are simulated in this section. In addition, algorithm proposed in [15] is also operated as a comparison.

Figure 5.7a, c, e show the performances of the proposed and derivative algorithms with quadratic, cubic, and sinusoid nonlinear errors, respectively. For each nonlinearity model, the proposed algorithm provides more accurate nonlinearity estimations than the derivative algorithm. Moreover, derivative algorithm introduces erroneous estimations at the beginning of the signal, which should be

**Fig. 5.5 **FMCW SAR nonlinearity correction strategy. **a **Beat signal with nonlinear error. **b **After transmitted nonlinearity correction. **c **After RVP removal. **d **After received nonlinearity correction

**Fig. 5.6 ****Flowchart of FMCW SAR nonlinearity correction with proposed algorithm**

**Fig. 5.7 **Comparison of the estimation accuracy of both algorithms. **a **Quadratic nonlinearity simulation. **b **Estimation error of quadratic errors. **c **Cubic nonlinearity simulation. **d **Estimation error of cubic errors. **e **Sinusoid nonlinearity simulation. **f **Estimation error of sinusoid errors

**Fig. 5.8 **Comparison of the imaging qualities of both algorithms. **a **Quadratic nonlinearity simulation. **b **Cubic nonlinearity simulation. **c **Sinusoid nonlinearity simulation

abandoned. To make a more clear comparison of both algorithms, the estimation errors of each model are shown in Fig. 5.7b, d, f. In addition, in Fig. 5.7e, f, a sinusoidal phase error with oscillation frequency *s _{re}f* is simulated. The energy of the phase error is concentrated at frequency spots

*t = ks*in this simulation. It can be concluded from the results that the proposed algorithm still has a convincing performance in this case.

_{re}fFigure 5.8a shows the result of range compression of FMCW SAR with quadratic nonlinearity. Quadratic nonlinearity causes main lobe broadening, and the imaging quality of derivative algorithm is worse than that of the proposed algorithm. Figure 5.8b shows the imaging result comparison with cubic nonlinearity. Since nonlinearity is not completely removed by derivative algorithm, its side-lobes are raised and appear to be unsymmetrical. Figure 5.8c illustrates the comparison results with sinusoid nonlinearity, in which the proposed algorithm also provides a better resolution. Since azimuth resolution is not affected by nonlinearity problem, simulation in Fig. 5.8 is adequate to prove the imaging resolution improvement of the proposed algorithm.

Another simulation is used to verify the robustness of the proposed algorithm. Quadratic nonlinearity model is used to represent all models since the generality

**Fig. 5.9 **Comparison of estimation accuracies with different delay. **a **Comparison results with delay of 0.2 ps. **b **Comparison results with delay of 0.5 ps. **c **Comparison results with delay of 2 ps

character has been demonstrated. Set PRI as 10 ps, and *s _{re}f* gets the values of 0.2, 0.5 and 2 ps. Figure 5.9a shows that when s

_{re}f

*=*0.2 ps, the proposed and derivative algorithm both provide accurate estimations of the nonlinearity error. When s

*0.5 ps, the derivative algorithm induces an considerable estimation error in Fig. 5.9b. In Fig. 5.9c, since*

_{re}f =*s*2 ps, the estimation of derivative algorithm obviously departs from the actual value. However, the proposed algorithm performs a robust estimation with high accuracy in all three simulations. Therefore, the robustness of the proposed algorithm is proved.

_{re}f =