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Instantaneous crosscorrelation function type of WD based LFM signals analysis via output SNR inequality modeling
EURASIP Journal on Advances in Signal Processing volume 2021, Article number: 122 (2021)
Abstract
Linear canonical transform (LCT) is a powerful tool for improving the detection accuracy of the conventional Wigner distribution (WD). However, the LCT free parameters embedded increase computational complexity. Recently, the instantaneous crosscorrelation function type of WD (ICFWD), a specific WD relevant to the LCT, has shown to be an outcome of the tradeoff between detection accuracy and computational complexity. In this paper, the ICFWD is applied to detect noisy single component and bicomponent linear frequencymodulated (LFM) signals through the output signaltonoise ratio (SNR) inequality modeling and solving with respect to the ICFWD and WD. The expectationbased output SNR inequality model between the ICFWD and WD on a pure deterministic signal added with a zeromean random noise is proposed. The solutions of the inequality model in regard to single component and bicomponent LFM signals corrupted with additive zeromean stationary noise are obtained respectively. The detection accuracy of ICFWD with that of the closedform ICFWD (CICFWD), the affine characteristic Wigner distribution (ACWD), the kernel function Wigner distribution (KFWD), the convolution representation Wigner distribution (CRWD) and the classical WD is compared. It also compares the computing speed of ICFWD with that of CICFWD, ACWD, KFWD and CRWD.
Introduction
In recent decades, the linear canonical transform (LCT) has been attracted much attention due to its significance in optics propagation [1], timefrequency analysis [2], and signal processing [3]. Some wellknown integral transformations, including Fourier transform (FT) [4], fractional Fourier transform (FRFT) [5,6,7,8,9], Fresnel transform [10] and Lorentz transform [11], are special cases of the LCT. The generalizability of LCT enables it to be a representative integral transformation. The LCT has three free parameters, which outperforms the FT without any degrees of freedom and the FRFT with only one degree of freedom in nonstationary signal analysis. Indeed, the LCT exhibits more flexibility in signal representation through a socalled linear canonical domain beyond the ordinary time, frequency and fractional domains.
Wigner distribution (WD) began its definition at quantum statistical mechanics [12] and later had found many applications in signal processing [13]. As it is known, the WD is an effective timefrequency analysis tool but subjected to the interference of the crossterm when dealing with multicomponent signals. This kind of issue is the subject which the WD pays attention throughout. As a result, a large number of variations to the WD are currently derived, including pseudo Wigner distribution [14], smoothed pseudo Wigner distribution [15], Smethod [16], general Wigner distribution [17], L Wigner distribution [18], ChoiWilliams distribution [19], and scaled Wigner distribution [20]. These variations are both energy distributions in the timefrequency plane. Take the skywave overthehorizon radar signal detection [21] for example, however, without any degrees of freedom they fail to extract principal features of the target echo signal from the extreme strong noise background. To address the problem of weak signal detection [22], ones need some breakthrough research approaches to extend the conventional WD.
From the perspective of improving signal representation flexibility, it seems a feasible method for the detection problem of weak signals to introduce the parameters of LCT into the conventional WD. There exist various types of parameters embedded technologies, such as linear canonical domain autocorrelation function replacement [2], linear canonical domain kernel replacement [23,24,25], linear canonical domain convolution replacement [26], linear canonical domain instantaneous crosscorrelation function (ICF) replacement [27], and linear canonical domain closedform instantaneous crosscorrelation function (CICF) replacement [28]. The corresponding variations of the WD are referred to as the affine characteristic Wigner distribution (ACWD) [2], the kernel function Wigner distribution (KFWD) [23], the convolution representation Wigner distribution (CRWD) [26], the ICF type of Wigner distribution (ICFWD) [27], and the CICF type of Wigner distribution (CICFWD) [28], respectively.
The numbers of LCT free parameters of the ACWD, KFWD, CRWD, ICFWD and CICFWD are three, three, three, six and nine, respectively. The more the number of parameters the timefrequency distribution has, the more flexible the signal representation is. In this case, the detection accuracy will be better. Therefore, our previous works focused mainly on the CICFWDbased weak signal detection. To be specific, we established an output signaltonoise ratio (SNR) inequality [29] (inequalities system [30]) model or optimization [31] (multiobjective optimization [32]) model of the CICFWD to explain why its detection accuracy improves. Also, we solved the inequality (inequalities system) model or optimization (multiobjective optimization) model for noisy linear frequencymodulated (LFM) signals [33,34,35] to verify the improvement of detection accuracy. However, there exist two problems caused by too many parameters. The parameters selection strategy of CICFWD is not unique so that there is an unstable detection accuracy [36]. The CICFWD’s high complexity and low computation efficiency make it not suitable for realtime applications [37]. For the options with less parameters, there are the ACWD, KFWD, CRWD, and ICFWD. It is therefore wise to choose the ICFWD for maintaining a high level of detection accuracy as possible, because the number of its parameters is the largest among them.
In our latest works, the output SNR optimization [37] and multiobjective optimization [36] models of the ICFWD were formulated respectively. The optimal solutions of the models with respect to noisy single component LFM signal were also derived. It was crucially used there that only one absolute value term can be found in the objective function for the single component LFM signal. Then Lagrangian multiplier method [38] works very well by merely squaring the objective function. This method thereby relies heavily on the fact that the LFM signal is single component which does not seem to work for multicomponent LFM signals, for which there is more than one absolute value term found in the objective function. To overcome this shortcoming, it might be feasible to replace the optimization model with the inequality model as solving the inequality model does not need to use Lagrangian multiplier method.
The main purpose of this paper is to study weak signal detection problem through the output SNR inequality modeling and solving of the ICFWD. We first propose an output SNR inequality model of the ICFWD. We then solve the inequality model in regard to both single component and bicomponent LFM signals under a zeromean stationary noise background. We also compare the detection accuracy of ICFWD, CICFWD, ACWD, KFWD, CRWD and WD, as well as the computing speed of ICFWD, CICFWD, ACWD, KFWD and CRWD. The main notations are summarized in Table 1.
The main contributions of this paper are summarized below:

It formulates the expectationbased output SNR of ICFWD for pure deterministic signal embedded in additive zeromean random noise.

It establishes the expectationbased output SNR inequality model between the ICFWD and WD on the noisy signal.

It deduces the solutions of the inequality model for single component and bicomponent LFM signals added with zeromean stationary noise, respectively.

It demonstrates the advantages of ICFWD in maintaining/improving detection accuracy and saving computing time.
The remainder of this paper is structured as follows. Section 2 reviews the definitions of LCT and ICFWD. Section 3 investigates the expectationbased output SNR inequality modeling and solving of the ICFWD. Section 4 derives the solutions of the inequality model for noisy LFM signals. Section 5 conducts numerical experiments. Section 6 concludes the paper.
Preliminaries
Linear canonical transform (LCT)
From the view point of geometry, the FT and FRFT are rotation transformations with the angles of \(\frac{\pi }{2}\) and \(\alpha\) in the timefrequency plane, respectively. The LCT, a generalization of the FRFT or known as extended FRFT, can be regarded as an affine transformation in the timefrequency plane [39,40,41,42]. The LCT of a signal f(t) relevant to the parameter matrix \({\mathbf {A}}=(a,b;c,d)\) is defined by [43,44,45,46,47,48]
where
denotes the linear canonical domain kernel function. The parameters a, b, c, d are real numbers satisfying the affine condition \(adbc=1\).
Two special cases of LCT are worth emphasizing as follows. Firstly, the LCT with \({\mathbf {A}}=(1,0;0,1)\) reduces to a unit transformation, that is \(F_{(1,0;0,1)}(u)=f(u)\). Secondly, the LCT with \({\mathbf {A}}=(0,1;1,0)\) turns into the conventional FT, regardless of an extra constant factor \(\frac{1}{\sqrt{\mathrm {j}}}\), that is \(F_{(0,1;1,0)}(u)=\frac{1}{\sqrt{\mathrm {j}2\pi }}\int _{\infty }^{+\infty }f(t)\mathrm {e}^{\mathrm {j}ut}\mathrm {d}t\).
The LCT with \(b=0\) is just a combination of a scaling operation \(\sqrt{d}f(du)\) and a chirp multiplication operation \(\text {e}^{\text {j}\frac{cd}{2}u^2}\). The LCT with \(a=0\) is none other than a combination of a scaling FT operation \(\frac{1}{\sqrt{b}}F_{(0,1;1,0)}\left( \frac{u}{b}\right)\) and a chirp multiplication operation \(\text {e}^{\text {j}\frac{d}{2b}u^2}\). Then, the linear canonical domains with \(b=0\) and \(a=0\) become the ordinary time and frequency domains, respectively. Without loss of generality, this paper therefore discusses merely on the LCT with \(b\ne 0\) and \(a\ne 0\). The three free parameters of LCT are a, b, d or a, b, c, because it derives from \(adbc=1\) and \(b\ne 0\) or \(a\ne 0\) that \(c=\frac{ad1}{b}\) or \(d=\frac{bc+1}{a}\).
ICF type of WD (ICFWD)
Let \(F_{{\mathbf {A}}_1}\left( t+\frac{\tau }{2}\right) f^{*}\left( t\frac{\tau }{2}\right)\) denote the linear canonical domain ICF, where \(F_{{\mathbf {A}}_1}\) stands for the LCT of f(t) relevant to the parameter matrix \({\mathbf {A}}_1=(a_1,b_1;c_1,d_1)\), and the superscript \(*\) is complex conjugate. Then, the ICFWD of f(t) is defined by the LCT of \(F_{{\mathbf {A}}_1}\left( t+\frac{\tau }{2}\right) f^{*}\left( t\frac{\tau }{2}\right)\) relevant to the parameter matrix \({\mathbf {A}}\), i.e. [27],
Two parameter matrices \({\mathbf {A}}_1,{\mathbf {A}}\) implies that the ICFWD has six LCT free parameters.
Let \(F_{{\mathbf {A}}_2}\) denote the LCT of f(t) relevant to the parameter matrix \({\mathbf {A}}_2=(a_2,b_2;c_2,d_2)\). By replacing the linear canonical domain ICF \(F_{{\mathbf {A}}_1}\left( t+\frac{\tau }{2}\right) f^{*}\left( t\frac{\tau }{2}\right)\) with the linear canonical domain CICF \(F_{{\mathbf {A}}_1}\left( t+\frac{\tau }{2}\right) F_{{\mathbf {A}}_2}^{*}\left( t\frac{\tau }{2}\right)\), it follows the definition of the CICFWD of f(t) [28,29,30,31,32]
It is none other than the LCT of \(F_{{\mathbf {A}}_1}\left( t+\frac{\tau }{2}\right) F_{{\mathbf {A}}_2}^{*}\left( t\frac{\tau }{2}\right)\) relevant to the parameter matrix \({\mathbf {A}}\). There are three parameter matrices \({\mathbf {A}}_1,{\mathbf {A}}_2,{\mathbf {A}}\) so that the CICFWD has nine LCT free parameters.
The ICFWD exhibits less computational complexity than the CICFWD due to no calculations for \(F_{{\mathbf {A}}_2}\). This is the main reason why the paper use the ICFWD to improve the efficiency of weak signal detection. Moreover, the ICFWD is not a special case of the CICFWD because of an assumption of \(b_2\ne 0\). Then, the ICFWD maybe could share the same level of detection accuracy in comparison with the CICFWD.
The ICFWD with \({\mathbf {A}}_1=(1,0;0,1)\) and \({\mathbf {A}}=(0,1;1,0)\) becomes the conventional WD [13]
regardless of an extra constant factor \(\frac{1}{\sqrt{\mathrm {j}2\pi }}\). Compared with the WD, the ICFWD achieves more degrees of freedom to improve the accuracy of weak signal detection.
Mathematical model
In this section, we first define the expectationbased output SNR of ICFWD for a general noisy signal modeled by a pure deterministic signal f(t) added with a zeromean random noise n(t). We then establish the expectationbased output SNR inequality model between the ICFWD and WD. Finally, we study how to solve the inequality model from the perspective of signal forms including synthetic signals and realworld signals.
Expectationbased output SNR of ICFWD
Let the noisy signal be \(f(t)+n(t)\), where f(t) and n(t) denote a pure deterministic signal and a zeromean random noise, respectively. Then, the expectationbased output SNR of CICFWD is reproduced here as [29], Eq. (32)]
where ‘Mean’ would be the arithmetic mean if \(\mathop {\arg \max }\limits _{(t,u)}\left \text {W}_f^{{\mathbf {A}}_1,{\mathbf {A}}_2,{\mathbf {A}}}(t,u)\right\) were a countable set, while the integral average if it were an uncountable set.
Note that the expectationbased output SNR of CICFWD is welldefined thanks to an important relation \(\text {E}\left[ \text {W}_{f+n}^{{\mathbf {A}}_1,{\mathbf {A}}_2,{\mathbf {A}}}(t,u)\right] =\text {W}_f^{{\mathbf {A}}_1,{\mathbf {A}}_2,{\mathbf {A}}}(t,u)+\text {E}\left[ \text {W}_n^{{\mathbf {A}}_1,{\mathbf {A}}_2,{\mathbf {A}}}(t,u)\right]\) [31, Eq. (5)]. Similarly, this relation can be reduced to \(\text {E}\left[ \text {W}_{f+n}^{{\mathbf {A}}_1,{\mathbf {A}}}(t,u)\right] =\text {W}_f^{{\mathbf {A}}_1,{\mathbf {A}}}(t,u)+\text {E}\left[ \text {W}_n^{{\mathbf {A}}_1,{\mathbf {A}}}(t,u)\right]\) for the ICFWD. Thus, the expectationbased output SNR of ICFWD is welldefined [36, 37]:
Inequality modeling of ICFWD
Since the value of the expectationbased output SNR of ICFWD depends only on the parameter matrices \({\mathbf {A}}_1,{\mathbf {A}}\) for given signals and noises, our latest work formulated an optimization model [37]
and then solved it to obtain the optimal LCT free parameters of ICFWD. However, it seems very complicated to solve the optimization model because there exists an inner optimization problem \(\max \limits _{(t,u)\in {\mathbb {R}}^2}\left \text {W}_f^{{\mathbf {A}}_1,{\mathbf {A}}}(t,u)\right\) embedded in it. Indeed, our latest work demonstrated that the solution of the optimization model for multicomponent LFM signals does not seem to be feasible because there are many absolute value terms found in the objective function need to be taken partial derivatives.
The inequality model is simpler than the optimization model due to no calculations for the inner optimization problem. Then, as an alternative to the expectationbased output SNR optimization model of ICFWD, the expectationbased output SNR inequality model between the ICFWD and WD might be suitable for the case of multicomponent LFM signals.
Let the expectationbased output SNR of WD be [29], Eq. (33)]
It is a constant for given signals and noises. Then, the value of the expectationbased output SNR of ICFWD can be larger than that of the expectationbased output SNR of WD for appropriate parameter matrices \({\mathbf {A}}_1,{\mathbf {A}}\). Thus, the expectationbased output SNR inequality model between the ICFWD and WD is wellestablished:
Inequality solving of ICFWD
The inequality (10) can be rewritten as
There are two optimization problems \(\max \limits _{(t,u)\in {\mathbb {R}}^2}\left \text {W}_f^{{\mathbf {A}}_1,{\mathbf {A}}}(t,u)\right\) and \(\max \limits _{(t,\omega )\in {\mathbb {R}}^2}\left \text {W}_f(t,\omega )\right\) need to be solved firstly. It is clear that the solving methods are different for synthetic signals and realworld signals.
Synthetic signals. The ICFWD \(\text {W}_f^{{\mathbf {A}}_1,{\mathbf {A}}}(t,u)\) can be expressed as a function with the variables t, u and the parameters \(a_1,b_1,d_1,a,b,d\). As for the WD \(\text {W}_f(t,\omega )\), it can be expressed as a function with the variables \(t,\omega\). Thanks to the classical extreme value theory [49], there is an analytic solution to the optimization problem \(\max \limits _{(t,u)\in {\mathbb {R}}^2}\left \text {W}_f^{{\mathbf {A}}_1,{\mathbf {A}}}(t,u)\right\), as well as the optimization problem \(\max \limits _{(t,\omega )\in {\mathbb {R}}^2}\left \text {W}_f(t,\omega )\right\). The former is an algebraic formulation with the parameters \(a_1,b_1,d_1,a,b,d\), while the latter is an algebraic formulation without any parameters. Substituting them into (11) yields an algebraic inequality with the parameters \(a_1,b_1,d_1,a,b,d\). In a word, the solution of the inequality model is an algebraic inequality for the case of synthetic signals.
Realworld signals. Due to peak detection algorithms [50], there is an arithmetic solution to the optimization problem \(\max \limits _{(t,u)\in {\mathbb {R}}^2}\left \text {W}_f^{{\mathbf {A}}_1,{\mathbf {A}}}(t,u)\right\) for given parameters \(a_1,b_1,d_1,a,b,d\). Similarly, it follows an arithmetic solution to the optimization problem \(\max \limits _{(t,\omega )\in {\mathbb {R}}^2}\left \text {W}_f(t,\omega )\right\). Traversing all parameters and checking the inequality (11) gives a point set of parameters \(a_1,b_1,d_1,a,b,d\). In order to narrow down the range of parameters traversal, the technique of uniform design [51] can be used to obtain a set of representative experimental points. In brief, the solution of the inequality model is a point set of parameters for the case of realworld signals.
Methods
This section focuses on solving the wellestablished inequality model for a kind of important synthetic signals, i.e., the LFM signals, including the single component and bicomponent cases.
We first explore the solutions of the optimization problem \(\max \limits _{(t,u)\in {\mathbb {R}}^2}\left \text {W}_f^{{\mathbf {A}}_1,{\mathbf {A}}}(t,u)\right\) for single component and bicomponent LFM signals respectively. We then obtain the solution of the mean problem \(\mathop {\text {Mean}}\limits _{\mathop {\arg \max }\limits _{(t,u)}\left \text {W}_f^{{\mathbf {A}}_1,{\mathbf {A}}}(t,u)\right }\left\{ \left \text {E}\left[ \text {W}_n^{{\mathbf {A}}_1,{\mathbf {A}}}(t,u)\right] \right \right\}\) for zeromean stationary noise. Finally, we formulate the solutions of the inequality model for single component and bicomponent cases respectively.
ICFWD of single component LFM signal
This subsection revisits the optimization problem \(\max \limits _{(t,u)\in {\mathbb {R}}^2}\left \text {W}_f^{{\mathbf {A}}_1,{\mathbf {A}}}(t,u)\right\) for single component LFM signal given by
where the initial frequency \(\alpha\) is arbitrary, and the frequency rate \(\beta \ne 0\).
Let \(\delta\) denote Dirac delta operator. Then, the amplitude of ICFWD of the single component LFM signal f(t) can yield an impulse which is reproduced here as [27]
where \(h_1=\frac{1}{2\beta b_1+a_1}\). Here the LCT free parameters have to satisfy \(2\beta b_1+a_1\ne 0\) and \(\frac{a}{2b}+\frac{d_1h_1}{8b_1}\frac{\beta }{4}=0\).
As it is seen, the solution of the optimization problem reads [37]
ICFWD of bicomponent LFM signal
This subsection studies the optimization problem \(\max \limits _{(t,u)\in {\mathbb {R}}^2}\left \text {W}_f^{{\mathbf {A}}_1,{\mathbf {A}}}(t,u)\right\) for bicomponent LFM signal given by
where \({\widehat{\beta }},{\widetilde{\beta }}\ne 0\) and \({\widehat{\beta }}\ne {\widetilde{\beta }}\).
The bilinearity of the ICFWD implies that the ICFWD of the bicomponent LFM signal f(t) can be expanded as
where \(\text {W}_{{\widehat{g}}}^{{\mathbf {A}}_1,{\mathbf {A}}}(t,u)\) and \(\text {W}_{{\widetilde{g}}}^{{\mathbf {A}}_1,{\mathbf {A}}}(t,u)\) are two auto terms, and
and
are two cross terms, and where \({\widehat{G}}_{{\mathbf {A}}_1}\) and \({\widetilde{G}}_{{\mathbf {A}}_1}\) denote the LCTs of \({\widehat{g}}\) and \({\widetilde{g}}\) relevant to the parameter matrix \({\mathbf {A}}_1\), respectively.
The amplitudes of the auto terms \(\text {W}_{{\widehat{g}}}^{{\mathbf {A}}_1,{\mathbf {A}}}(t,u)\) and \(\text {W}_{{\widetilde{g}}}^{{\mathbf {A}}_1,{\mathbf {A}}}(t,u)\) can generate impulses
and
respectively, where \({\widehat{h}}_1=\frac{1}{2{\widehat{\beta }}b_1+a_1}\) and \({\widetilde{h}}_1=\frac{1}{2{\widetilde{\beta }}b_1+a_1}\), if and only if the LCT free parameters satisfy \(2{\widehat{\beta }}b_1+a_1\ne 0\), \(2{\widetilde{\beta }}b_1+a_1\ne 0\), \(\frac{a}{2b}+\frac{d_1{\widehat{h}}_1}{8b_1}\frac{{\widehat{\beta }}}{4}=0\), and \(\frac{a}{2b}+\frac{d_1{\widetilde{h}}_1}{8b_1}\frac{{\widetilde{\beta }}}{4}=0\).
Due to \({\widehat{\beta }}\ne {\widetilde{\beta }}\), \(\frac{a}{2b}+\frac{d_1{\widehat{h}}_1}{8b_1}\frac{{\widehat{\beta }}}{4}=0\), and \(\frac{a}{2b}+\frac{d_1{\widetilde{h}}_1}{8b_1}\frac{{\widetilde{\beta }}}{4}=0\), it follows that \(\widetilde{{\widehat{l}}}\triangleq \frac{a}{2b}+\frac{d_1{\widehat{h}}_1}{8b_1}\frac{{\widetilde{\beta }}}{4}\ne 0\) and \(\widehat{{\widetilde{l}}}\triangleq \frac{a}{2b}+\frac{d_1{\widetilde{h}}_1}{8b_1}\frac{{\widehat{\beta }}}{4}\ne 0\). Then, the cross terms \(\text {W}_{{\widehat{g}},{\widetilde{g}}}^{{\mathbf {A}}_1,{\mathbf {A}}}(t,u)\) and \(\text {W}_{{\widetilde{g}},{\widehat{g}}}^{{\mathbf {A}}_1,{\mathbf {A}}}(t,u)\) can not generate impulses because the amplitudes of them read
and
respectively. For the proof of the results, ones can refer to “Appendix 1”.
Taking amplitude on each terms in (16), and substituting (19)–(22) gives
Thus, the solution of the optimization problem is
ICFWD of zeromean stationary noise
This subsection discusses the mean problem \(\mathop {\text {Mean}}\limits _{\mathop {\arg \max }\limits _{(t,u)}\left \text {W}_f^{{\mathbf {A}}_1,{\mathbf {A}}}(t,u)\right }\left\{ \left \text {E}\left[ \text {W}_n^{{\mathbf {A}}_1,{\mathbf {A}}}(t,u)\right] \right \right\}\) for zeromean stationary noise.
The stationarity of the noise indicates that \({\text {E}}[n(t_1)n^{*}(t_2)]=D\delta (t_1t_2)\), where D denotes the power spectral density of the noise. By using the sifting property of Delta function, the expectation of the ICFWD of the zeromean stationary noise n(t) can be calculated as
Owing to the wellknown Gaussian integral formula [52]
the amplitude of \(\text {E}\left[ \text {W}_n^{{\mathbf {A}}_1,{\mathbf {A}}}(t,u)\right]\) reads [37]
for \(b(a_1+d_1+2)+4ab_1\ne 0\).
The noise is a uniform distribution in the timefrequency plane as \(\left \text {E}\left[ \text {W}_n^{{\mathbf {A}}_1,{\mathbf {A}}}(t,u)\right] \right\) is independent of the variables t, u. Then, the solution of the mean problem takes
Solutions of the inequality model
This subsection deduces the solution of the inequality model for single component LFM signal added with zeromean stationary noise. And on this basis, it obtains the solution of the inequality model for the bicomponent case.
Single component case
Thanks to the equality \(\frac{a}{2b}+\frac{d_1h_1}{8b_1}\frac{\beta }{4}=0\), there is \(b(a_1+d_1+2)+4ab_1=b\frac{(h_1+1)^{2}}{h_1}\). See “Appendix 2” for the proof of this equation. Then, (28) can be simplified to [37]
for \(h_1+1\ne 0\).
Substituting (14) and (29) into (7) gives the expectationbased output SNR of ICFWD for the single component case
The expectationbased output SNR of WD for the single component case is reproduced here as [29, 30]
By substituting (30) and (31) into (10), it follows the solution of the inequality model for the single component case
Note that this inequality implies \(h_1+1\ne 0\).
Bicomponent case
Due to the continued equality \(\frac{a}{2b}+\frac{d_1{\widehat{h}}_1}{8b_1}\frac{{\widehat{\beta }}}{4}=\frac{a}{2b}+\frac{d_1{\widetilde{h}}_1}{8b_1}\frac{{\widetilde{\beta }}}{4}=0\), there is \(b(a_1+d_1+2)+4ab_1=b\frac{\left( {\widehat{h}}_1+1\right) ^{2}}{{\widehat{h}}_1}=b\frac{\left( {\widetilde{h}}_1+1\right) ^{2}}{{\widetilde{h}}_1}\). See “Appendix 2” for the proof of this continued equation. Then, (28) can be reduced to
for \({\widehat{h}}_1+1\ne 0,{\widetilde{h}}_1+1\ne 0\).
Substituting (24) and (33) into (7) yields the expectationbased output SNR of ICFWD for the bicomponent case
The expectationbased output SNR of WD for the bicomponent case is reviewed as follows [29]:
By substituting (34) and (35) into (10), there is the solution of the inequality model for the bicomponent case
Here this inequality indicates \({\widehat{h}}_1+1\ne 0,{\widetilde{h}}_1+1\ne 0\).
See Table 2 for a summary of the solutions of the inequality model and the associated constraints on LCT free parameters.
Results and discussion
In order to demonstrate the usefulness and effectiveness of ICFWD in noisy LFM signals processing, this section designs some simulations to compare the detection accuracy of ICFWD, CICFWD, ACWD, KFWD, CRWD and WD, as well as the computing speed of ICFWD, CICFWD, ACWD, KFWD and CRWD.
The simulated single component and bicomponent LFM signals \(f_1(t)\) embedded in additive complex white Gaussian noise n(t) are chosen as
and
respectively.
Assume that the observing interval is \([5\text {s},5\text {s}]\), and the sampling frequency is 20 Hz for the single component case and 40 Hz for the bicomponent case. Let the input SNR of the noisy signal f(t) be \(10\text {log}_{10}\frac{\int ^{5}_{5}f_1(t)^2\text {d}t}{\text {Var}[n(t)]}\), where the variance of the noise \(\text {Var}[n(t)]\) equals to the product of the power spectral density of the noise and the bandwidth of the noise. In the simulations, the input SNR is assumed to be \(10\text {dB}\) for the single component case and \(8\text {dB}\) for the bicomponent case.
Figure 1 compares the detection accuracy of ICFWD with that of CICFWD, ACWD, KFWD, CRWD and WD for the single component case. The ICFWD with LCT free parameters satisfying \(\frac{\left b\left( h_1+1\right) \right }{2}=1.9>1\), \(2\beta b_1+a_1=1.1111\ne 0\), \(\frac{a}{2b}+\frac{d_1h_1}{8b_1}\frac{\beta }{4}=0\) and the relevant contour picture are plotted in Fig. 1a, b, respectively. The CICFWD with LCT free parameters selected in [29] and the relevant contour picture are plotted in Fig. 1c, d, respectively. The ACWD with LCT free parameters selected in [2] and the relevant contour picture are plotted in Fig. 1e, f, respectively. The KFWD with LCT free parameters selected in [23] and the relevant contour picture are plotted in Fig. 1g, h, respectively. The CRWD with LCT free parameters selected in [26] and the relevant contour picture are plotted in Fig. 1i, j, respectively. The WD and the relevant contour picture are plotted in Fig. 1k, l, respectively.
Figure 2 compares the detection accuracy of ICFWD with that of CICFWD, ACWD, KFWD and WD for the bicomponent case. The ICFWD with LCT free parameters satisfying \(\frac{\left b\left( {\widehat{h}}_1+1\right) \right +\left b\left( {\widetilde{h}}_1+1\right) \right }{4}=1.3125>1\), \(2{\widehat{\beta }}b_1+a_1=2\ne 0\), \(2{\widetilde{\beta }}b_1+a_1=0.5\ne 0\), \(\frac{a}{2b}+\frac{d_1{\widehat{h}}_1}{8b_1}\frac{{\widehat{\beta }}}{4}=0\), \(\frac{a}{2b}+\frac{d_1{\widetilde{h}}_1}{8b_1}\frac{{\widetilde{\beta }}}{4}=0\) and the relevant contour picture are plotted in Fig. 2a, b, respectively. The CICFWD with LCT free parameters selected in [29] and the relevant contour picture are plotted in Fig. 2c, d, respectively. The ACWD with LCT free parameters selected in [2] and the relevant contour picture are plotted in Fig. 2e, f, respectively. The KFWD with LCT free parameters selected in [23] and the relevant contour picture are plotted in Fig. 2g, h, respectively. The WD and the relevant contour picture are plotted in Fig. 2i, j, respectively. It should be noted here that the CRWD fails to deal with general bicomponent LFM signals unless the two components have opposite frequency rates [26].
It can be observed from the sharpness of energy straight lines found in Figs. 1 and 2 that the ICFWD maintains the same level of detection accuracy as the CICFWD. Moreover, it achieves better detection accuracy than the ACWD, the KFWD, the CRWD, and the conventional WD.
It is wellknown that the Radon transform (RT) [53] can accumulate energy straight lines, giving rise to the output SNR of timefrequency distributions which seems more intuitive than the sharpness of energy straight lines. Figures 3 and 4 compare the output SNR of ICFWD with that of CICFWD, ACWD, KFWD, CRWD and WD for the single component case and the bicomponent case, respectively. For the single component case, Fig. 3a plots the kamplitude distribution of RTbased ICFWD [37] with LCT free parameters satisfying \(\frac{\left b\left( h_1+1\right) \right }{2}=1.9>1\), \(2\beta b_1+a_1=1.1111\ne 0\), \(\frac{a}{2b}+\frac{d_1h_1}{8b_1}\frac{\beta }{4}=0\), Fig. 3b plots the kamplitude distribution of RTbased CICFWD [29, 31] with LCT free parameters selected in [29], Fig. 3c plots the kamplitude distribution of RTbased ACWD with LCT free parameters selected in [2], Fig. 3d plots the kamplitude distribution of RTbased KFWD with LCT free parameters selected in [23], Fig. 3e plots the kamplitude distribution of RTbased CRWD with LCT free parameters selected in [26], and Fig. 3f plots the kamplitude distribution of RTbased WD [29, 31]. As for the bicomponent case, Fig. 4a plots the kamplitude distribution of RTbased ICFWD with LCT free parameters satisfying \(\frac{\left b\left( {\widehat{h}}_1+1\right) \right +\left b\left( {\widetilde{h}}_1+1\right) \right }{4}=1.3125>1\), \(2{\widehat{\beta }}b_1+a_1=2\ne 0\), \(2{\widetilde{\beta }}b_1+a_1=0.5\ne 0\), \(\frac{a}{2b}+\frac{d_1{\widehat{h}}_1}{8b_1}\frac{{\widehat{\beta }}}{4}=0\), \(\frac{a}{2b}+\frac{d_1{\widetilde{h}}_1}{8b_1}\frac{{\widetilde{\beta }}}{4}=0\), Fig. 4b plots the kamplitude distribution of RTbased CICFWD with LCT free parameters selected in [29], Fig. 4c plots the kamplitude distribution of RTbased ACWD with LCT free parameters selected in [2], Fig. 4d plots the kamplitude distribution of RTbased KFWD with LCT free parameters selected in [23], and Fig. 4e plots the kamplitude distribution of RTbased WD.
It is obvious from the amplitude of noises found in Figs. 3 and 4 that the ICFWD maintains the same level of output SNR as the CICFWD. Moreover, it achieves higher output SNR than the ACWD, the KFWD, the CRWD, and the conventional WD.
Tables 3 and 4 record the computing time of ICFWD, CICFWD, ACWD, KFWD and CRWD in four different sampling frequencies 20 Hz, 40 Hz, 80 Hz and 120 Hz by using MATLAB language (version R2021a) and Desktop equipped with Intel(R) Core(TM)i59400F CPU @ 2.90 GHz for the single component case and the bicomponent case, respectively. The computing time is statistically obtained by averaging over 1000 realizations. Figures 5 and 6 plot a comparison of the computing speed of ICFWD, CICFWD, ACWD, KFWD and CRWD for the single component case and the bicomponent case, respectively.
As it is seen, the ICFWD maintains the same level of computation efficiency as the ACWD. Moreover, it exhibits higher computation efficiency than the CICFWD and CRWD while lower computation efficiency than the KFWD.
Conclusion
Since the ICFWD has a significant benefit in the tradeoff between detection accuracy and computational complexity among all of the linear canonical domain WDs, the application of ICFWD in weak multicomponent LFM signals detection problem has been investigated. By modeling and solving the expectationbased output SNR inequality between the ICFWD and WD, the selecting methods of the LCT free parameters of the ICFWD for both the single component and bicomponent cases are derived. A larger number of numerical experiments demonstrate the correctness of theoretical results. It turns out that the detection accuracy of ICFWD is similar to that of CICFWD, and it is better than the detection accuracy of ACWD, KFWD, CRWD and the conventional WD. In addition, the ICFWD has computation efficiency comparable to the ACWD, and it is superior to the CICFWD and CRWD in high computation efficiency while inferior to the KFWD.
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Abbreviations
 LCT:

Linear canonical transform
 FT:

Fourier transform
 FRFT:

Fractional Fourier transform
 WD:

Wigner distribution
 ICF:

Instantaneous crosscorrelation function
 CICF:

Closedform instantaneous crosscorrelation function
 ACWD:

Affine characteristic Wigner distribution
 KFWD:

Kernel function Wigner distribution
 CRWD:

Convolution representation Wigner distribution
 ICFWD:

ICF type of Wigner distribution
 CICFWD:

CICF type of Wigner distribution
 SNR:

Signaltonoise ratio
 LFM:

Linear frequencymodulated
 RT:

Radon transform
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Acknowledgements
The authors are very thankful to all colleagues and reviewers for their help and useful suggestions on this paper.
Funding
This work was supported by the National Natural Science Foundation of China [No. 61901223], the Natural Science Foundation of Jiangsu Province [No. BK20190769], the Jiangsu Planned Projects for Postdoctoral Research Funds [No. 2021K205B], the Natural Science Foundation of the Jiangsu Higher Education Institutions of China [No. 19KJB510041], the Jiangsu Province HighLevel Innovative and Entrepreneurial Talent Introduction Program [No. R2020SCB55], the Macau Young Scholars Program [No. AM2020015], the Startup Foundation for Introducing Talent of NUIST [No. 2019r024], the NUIST Students’ Platform for Innovation and Entrepreneurship Training Program [No. 202110300033Z, No. 202010300235], and the Six Talent Peaks Project in Jiangsu Province [No. SWYY034].
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Contributions
The inequality model proposed in this paper has been conceived by SZQ and ZCZ, SZQ, XJ, XYS, AYW, and YS made the theoretical analysis and numerical experiments. SZQ, XJ, PYH, and ZCZ wrote the initial draft. SZQ and ZCZ edited the revised version. ZCZ, YJC, XJ, and SZQ contributed to the funding support. The authors read and approved the final manuscript.
Authors' informations
ShengZhou Qiang was born in Wuxi City, Jiangsu Province, China, in 2001. He is currently studying as a thirdyear undergraduate in Information and Computing Sciences with the School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, Jiangsu, China.
Xian Jiang was born in Changzhou City, Jiangsu Province, China, in 2000. He is currently studying as a thirdyear undergraduate in Information and Computing Sciences with the School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, Jiangsu, China.
PuYu Han was born in Shenyang City, Liaoning Province, China, in 2001. He is currently studying as a thirdyear undergraduate in Information and Computing Sciences with the School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, Jiangsu, China.
XiYa Shi was born in Pizhou City, Jiangsu Province, China, in 1998. She received a double B.S. degree in Financial Mathematics and Economic Law from Yancheng Normal University, Yancheng, Jiangsu, China, in 2020. She is currently working towards the M.S. degree in Mathematics with the School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, Jiangsu, China.
AnYang Wu was born in Nanjing, Jiangsu Province, China, in 1997. He received the B.S. degree in Applied Statistics from Nanjing University of Information Science and Technology, Nanjing, Jiangsu, China, in 2019. He is currently working towards the M.S. degree in Mathematics with the School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, Jiangsu, China.
Yun Sun was born in Nantong City, Jiangsu Province, China, in 2001. She is currently studying as a thirdyear undergraduate in Information and Computing Sciences with the School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, Jiangsu, China.
YunJie Chen was born in Yancheng city, Jiangsu, China, in 1980. He received the Ph.D. degree in pattern recognition and intelligent system from the Nanjing University of Science and Technology, Nanjing, China, in 2008. He is currently a Full Professor with the School of Mathematics and Statistics, NUIST, Nanjing. His research interests are mainly focused on pattern recognition, image segmentation, and image processing.
ZhiChao Zhang was born in Jingdezhen City, Jiangxi Province, China, in 1991. He received the B.S. degree in Mathematics and Applied Mathematics from Gannan Normal University, Ganzhou, Jiangxi, China, in 2012, and the Ph.D. degree in Mathematics of Uncertainty Processing from Sichuan University, Chengdu, Sichuan, China, in 2018. From September 2017 to September 2018, he was awarded a grant from the China Scholarship Council to study as a visiting student researcher with the Department of Electrical and Computer Engineering, Tandon School of Engineering, New York University, Brooklyn, NY, USA. Since 2019, he has been with the School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, Jiangsu, China, where he is currently a Full Professor and Master’s Supervisor. He is currently working as the Macau Young Scholars Postdoctoral Fellow in Information and Communication Engineering with the Faculty of Information Technology, Macau University of Science and Technology, Macau, China. His research interests cover the mathematical theories, methods and applications in signal and information processing, including fundamental theories such as Fourier analysis, functional analysis and harmonic analysis, applied theories such as signal representation, sampling, reconstruction, filter, separation, detection and estimation, and engineering technologies such as satellite communications, radar detection and electronic countermeasures.
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Appendices
Appendix 1: Derivation of the cross terms \(\text {W}_{{\widehat{g}},{\widetilde{g}}}^{{\mathbf {A}}_1,{\mathbf {A}}}(t,u)\) and \(\text {W}_{{\widetilde{g}},{\widehat{g}}}^{{\mathbf {A}}_1,{\mathbf {A}}}(t,u)\)
The cross term \(\text {W}_{{\widehat{g}},{\widetilde{g}}}^{{\mathbf {A}}_1,{\mathbf {A}}}(t,u)\) is revisited as follows:
where
Thanks to (26), substituting \({\widehat{g}}(t)=\text {e}^{\text {j}\left( {\widehat{\alpha }}t+{\widehat{\beta }}t^{2}\right) }\) and \({\widetilde{g}}(t)=\text {e}^{\text {j}\left( {\widetilde{\alpha }}t+{\widetilde{\beta }}t^2\right) }\) into (39) gives
for \(\frac{1}{{\widehat{h}}_1}=2{\widehat{\beta }}b_1+a_1\ne 0\), and subsequently, there is
for \(\widetilde{{\widehat{l}}}=\frac{a}{2b}+\frac{d_1{\widehat{h}}_1}{8b_1}\frac{{\widetilde{\beta }}}{4}\ne 0\).
Similarly, it follows that
for \(\frac{1}{{\widetilde{h}}_1}=2{\widetilde{\beta }}b_1+a_1\ne 0\) and \(\widehat{{\widetilde{l}}}\triangleq \frac{a}{2b}+\frac{d_1{\widetilde{h}}_1}{8b_1}\frac{{\widehat{\beta }}}{4}\ne 0\).
Appendix 2: Derivation of the equations \(b(a_1+d_1+2)+4ab_1=b\frac{(h_1+1)^{2}}{h_1}\) and \(b(a_1+d_1+2)+4ab_1=b\frac{\left( {\widehat{h}}_1+1\right) ^{2}}{{\widehat{h}}_1}=b\frac{\left( {\widetilde{h}}_1+1\right) ^{2}}{{\widetilde{h}}_1}\)
From the equality
there is
Substituting into \(b(a_1+d_1+2)+4ab_1\) gives
and then it follows that
because of \(2\beta b_1+a_1=\frac{1}{h_1}\).
Similarly, from the continued equality \(\frac{a}{2b}+\frac{d_1{\widehat{h}}_1}{8b_1}\frac{{\widehat{\beta }}}{4}=\frac{a}{2b}+\frac{d_1{\widetilde{h}}_1}{8b_1}\frac{{\widetilde{\beta }}}{4}=0\), and \(2{\widehat{\beta }}b_1+a_1=\frac{1}{{\widehat{h}}_1}\) and \(2{\widetilde{\beta }}b_1+a_1=\frac{1}{{\widetilde{h}}_1}\), there are two equations
and
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Qiang, SZ., Jiang, X., Han, PY. et al. Instantaneous crosscorrelation function type of WD based LFM signals analysis via output SNR inequality modeling. EURASIP J. Adv. Signal Process. 2021, 122 (2021). https://doi.org/10.1186/s13634021008307
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DOI: https://doi.org/10.1186/s13634021008307
Keywords
 Computational complexity
 Detection accuracy
 Instantaneous crosscorrelation function
 Linear canonical Wigner distribution
 Weak signal detection