3.1. Description of the Proposed RCFT
In the CPF approach proposed in [
17,
18], instantaneous autocorrelation function of (4) is defined as
where
is the lag time variable. The
and
are the cross-terms and the noise terms, respectively, and detailed expressions can be found in [
17].
Taking the Fourier transform (FT) to (5) along the lag time variable
, one can obtain the CPF
where
is the frequency variable corresponding to the lag variable
and
is the Dirac delta function. The
and
are the cross term and the noise term after the FT.
Note also that from (6), the CPF performs the discrete FT (DFT) in terms of the lag time variable
whose sampling grid is non-uniform. As a result, the traditional uniform-sampling based efficient fast FT (FFT) cannot be directly applied. To calculate the CPF, the non-uniform discrete Fourier transform (NUDFT) is usually utilized, and the cost of the direct usage of NUDFT is as high as
with the signal length
. To reduce the computational cost of the NUDFT, the non-uniform FFT (NUFFT) [
19] is preferred, and its computational cost of performing the FT along the
axis is
without the performance loss. For more information, the detailed implementation procedures on NUFFT can be found in [
19].
Applying the NUFFT in (5) produces the CPF that is the same as in (6) as
where
is NUFFT operator along the lag time variable
.
In (6), because of the nonlinear coupling between azimuth slow-time variable
and the lag time variable
, the energy of the auto-terms concentrates along the line of
in the slow time-Doppler frequency
plane. Utilizing this line, the third- and second-order coefficients are extracted by the slope and the y-intercept [
17]. However, in the case of multi-component QFM signal, the identifiability of the CPF is of problem because of cross-terms and spurious peaks [
17,
18]. To visually demonstrate this issue, the CPF of a two-component QFM signal with length
with parameters
,
,
and
,
, and
is shown in
Figure 2. From
Figure 2a,b, a sharp spurious peak is of presence, and the energy of auto-terms is focused along the slope lines (i.e.,
), whereas the energy of cross-terms is diffused in the
domain since their positions vary with time
. Generally speaking, for a
K-component QFM signal, there are
cross-terms and
spurious peaks [
17], which pose a serious interference to the estimation and detection of the auto-terms, unless they are properly reduced.
To overcome the identifiability problem of the CPF, the cross-terms and spurious peaks must be properly reduced. To this goal, two approaches of the Radon-CPF transform (RCT) in [
18] and Hough generalized high-order ambiguous function (Hough-GHAF) method in [
20] have been developed to suppress the cross-terms, spurious peaks for multicomponent QFM signal. To utilize the energy of auto-terms, for the RCT, the integration was performed along the slope line defined by the polar distance
(radius) from the origin and polar angle
formed by the perpendicular to the line in the radon domain. The RCT method
is given by [
16]
where
represents the RCT operator on
. From (8), the integral accumulates all the energy of auto-terms and suppresses the cross-terms and spurious peaks.
The similar idea based on multilinear function of fourth-order GHAF is also adopted by Hough-GHAF method in [
20]. The Hough-GHAF is defined by
where
represents the Hough-GHAF transform operator. The detailed definition of the
can be found in [
18]. It is worth mentioning that, the Hough-GHAF is based on a multilinear function of fourth-order, and thus its SNR threshold is higher than RCT. From both (8) and (9), although the energy of auto-terms is explored, the operations in RCT and Hough-GHAF are not coherent, and therefore, the suppression ability of the cross-terms and noise is still not adequate.
To perform coherent integration, the RCFT is developed that fully exploits the energy of the auto-terms. However, the difficulty of performing coherent integration comes from the fact that the cubic and quadratic power terms of are present in auto-terms of (6). They are must be eliminated first because the peaks of auto-terms would be unfocused if the direct integration along were utilized. In what follows, a novel RCFT algorithm is proposed to coherently integrate the energy of auto-terms along slope lines.
First, to eliminate the effect of the quadratic power terms of
in (6), here, we intelligently exploit the idea of the sampling property of the Dirac delta function, which is
where
is a general function of the variable
, and
denotes a fixed time index.
According to (10), in order to utilize sampling property of the Dirac delta function, an appropriate phase term function should be designed to construct a same expression with the delta function. With this thinking, a modified CPF (MCPF) by utilizing the sampling property of the delta function is designed as
where
. It is now clear from (11) that the negative effects of the quadratic power of
in auto-terms are removed by means of the sampling property of the Dirac delta function. This is the first important step in the proposed RCFT algorithm.
It is also found from (11) that the cubic power term of
just corresponds to the slope of the auto-terms energy distribution in the time-frequency plane. Therefore, to eliminate the effect of the cubic power terms of
in (5), inspired by the Radon-Fourier-transform (RFT) in [
21,
22], a novel RCFT algorithm is defined by
where
denotes the RCFT operator. The
in (12) is a novel transform kernel function that is given by
where
is the frequency variable with respect to
. Note that in the case of zero cubic term, i.e.,
, the proposed RCFT reduces to the coherent integrated CPF (CICPF) approach proposed in our previous work [
9]. Moreover, when both the second- and three-order terms are zeros, i.e.,
, the proposed RCFT becomes the FT, which indicates that FT is a special case of the RCFT.
Substituting (11) into (12) and after reassigning yields
where
. In (14), when the slope of the searching slope line
matches the slope of the auto-terms energy distribution
, namely
, the cubic power of
in auto-terms is eliminated, also demonstrated by RFT method [
21]. Therefore, the proposed RCFT in (14) is capable of realizing the coherent integration for auto-terms while suppressing the cross-terms and spurious peaks. This is the second important step in the proposed RCFT algorithm. Moreover, when the searching slope line fully overlaps with the energy distribution slope line of the auto-terms, namely
and
, the proposed RCFT maximizes the output energy of auto-terms and produces a distinct peak in which the maximal output energy
is calculated by
where
is the FT coherent integration gain. Meanwhile, when
or
, the energy of the auto-terms integration
due to incoherent integration or the fact that only part of the auto-terms energy is accumulated.
Figure 2c,d depicts the coherent integration results of the
Figure 2a obtained by the proposed RCFT method in the Radon domain, where only the auto-terms are accumulated into peaks, while the cross-terms and spurious peaks are almost completely suppressed. Although
Figure 2c shows that there also exist the cross-terms on the RCFT plane, they are much smaller compared to the auto-terms. Unlike the traditional quadratic time-frequency distributions that usually exploit the smoothing, optimal kernel design or nonlinear filtering techniques to moderately tradeoff between the cross-terms and resolution, the proposed RCFT, on its own, can greatly “suppress” the cross-terms without any resolution loss. By “suppress”, we mean a relative suppression is achieved since the RCFT greatly strengthens the energy of the auto-terms instead of suppressing the cross-terms directly.
From (12), interestingly, the proposed RCFT has similar operations as RCT and Hough-GHAF, and they all use the energy of the auto-terms along time-frequency trajectory in
domain. The main difference is the coherent accumulation developed in the proposed RCFT. Therefore, the RCFT will definitely outperform the existing methods in complex environments via coherent integration operation. The detailed procedure of the proposed RCFT is shown in
Figure 3.
The differences and advantages of RCFT compared with others approaches are briefly summarized as follows.
- Remark 1:
The RCFT employs the merits of both RCT and FT, and it not only has the same integration time as RCT but also works well as a useful tool for nonstationary signals.
- Remark 2:
The bilinear cubic phase function in (5) utilizes only one time correlation, which is viewed as a signal energy preservation because each additional one time correlation loses about 4 –5 dB in the SNR threshold [
17]. In addition to that, the 2-D coherent integration realized in the proposed RCFT will further enhance the SNR. Therefore, the proposed RCFT algorithm provides a good performance, especially when the SNR is low, see simulation section.
- Remark 3:
the NUFFT speeds up the Fourier transform along the non-uniformly spaced lag-time axis, which is helpful for algorithm real-time realization.
3.4. Components Computational Complexity Analysis
In this section, we analyze quantitatively the computational complexity of our proposed algorithm. For comparison purposes, the RCD method in [
15] where the high-resolution ISAR image can be obtained based a new quasi-time-frequency transform named Lv’s distribution using the second-order phase model under a low SNR environment. Moreover, the cross-terms suppression in the RCD method is better achieved with no time-frequency resolution loss. On the other hand, to compare with the parameter estimation-based ISAR imaging method, the CIGCPF-CICPF algorithm [
9] where it is recently proposed for maneuvering target ISAR imaging and parameter estimation with third-order motion model in low SNR condition.
In this comparison, for the illustration conveniences, assume that the range compression and translational motion compensation have been completed. The computational complexities of above-mentioned two methods and our proposed algorithm are quantitatively provided. In general, an N-point FFT or inverse FFT (IFFT) needs floating-point operations (FLOPs) and one-time complex multiplication needs 6N FLOPs. In what follows, and are respectively used to denote the number of range cells and the number of azimuth pulses, is the target scatterer number in the range cell, is the signal length , and is used to represent the length of the lag variable.
For the RCD method, its implementation steps mainly include performing the quasi-time-frequency distribution (Lv’s distribution) to each range cell. Take a range cell processing procedure for example, the complex multiplication in constructing the symmetric instantaneous autocorrelation function matrix with computational complexity of
, the FFT operation along the lag-time variable axis with computational cost of
, a keystone transform adopted to remove the coupling terms with complexity
, where
is the length of the interpolation operation kernel, the FFT operation along the scaling slow time variable with computational cost of
, and neglecting other relatively small computational complexity operation steps. Therefore, the total computational complexity of the RCD algorithm [
15] is
Compared with the parameter estimation-based method and our proposed method, which will be discussed later, the time-frequency analysis-based RCD method has a great advantage in terms of computational complexity. However, this method suffers from imaging performance degradation without considering the third-order phase effects.
For the parameter estimation-based CIGCPF-CICPF method where it needs to estimation each scatterer parameter, the computational load consists of the following steps. Take one scatterer estimation for example, to estimate the first- and third-order coefficient using the CIGCPF, the complex multiplication in constructing fourth-order multilinear GCPF function matrix with computational complexity of
, the NUFFT operation along the lag-time variable axis with computational cost of
, one time compensation function multiplication with complexity
, the FFT operation along the slow time variable with computational cost of
. Then one Dechirping operation is required, which needs one
-dimensional complex multiplication. Second, to obtain second-order coefficient using the CICPF, the computational complexity requirement is similar to the CIGCPF operation. Finally, one
-dimensional FFT is needed to estimate amplitude. Therefore, the total computational complexity of the CIGCPF-CICPF method [
9] for maneuvering target imaging with third-order phase model is
Similar to the RCD method, the proposed ISAR imaging algorithm is also based the quasi-time-frequency analysis named RCFT. According to the imaging steps and the flowchart of the proposed algorithm in
Figure 6, the proposed algorithm implementation procedures mainly include applying the proposed RCFT to each range cell. Take a range cell processing procedure for example, in constructing bilinear CPF function matrix with computational complexity of
the NUFFT operation along the lag-time variable axis with computational cost of
, one time compensation function multiplication with complexity
and the auto-terms trajectory extraction in 2-D time-frequency
domain and performing a FFT operation to the extracted data along slow-time variable with computational cost of
with searching point number M. Therefore, the total computational cost of the proposed ISAR imaging method is about
According to the above analysis, the computational complexity of the proposed ISAR imaging is higher than that of the RCD method, but still much lower that of parameter estimation-based CIGCPF-CICPF approach. In conclusion, the proposed method may well achieve a trade-off between the computational complexity and the imaging performance, see performance analysis section.