Algorithm design of a combinatorial mathematical model for computer random signals

Definitions and properties of FFs

If an inverse Fourier transform (IFT) is performed on the instantaneous correlation function concerning time t, another two-dimensional function, namely, an FF can be obtained. The definition of the FF is given in Eq. (1). (1)${\displaystyle A\left(\tau ,v\right)={\int }_{-\infty }^{\infty }x\left(t+\frac{\tau }{2}\right)x^{\ast }\left(t-\frac{\tau }{2}\right)e^{jvt}dt}$

Therefore, a non-stationary signal can have two bilinear representations, both of which are the two-dimensional FT of the signal, and Eq. (2) presents two-dimensional interchanges. (2)${\displaystyle A\left(\tau ,v\right)={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }WVD\left(t,f\right)e^{j2\pi \left(tv+\tau f\right)}dtdf}$

For a two-component signal, denoted by x(t) = x₁(t) + x₂(t), the Wigner-Ville distribution of this signal and the positional relationship between the cross term and the self-term of the ambiguity function are shown in Fig. 1. Among them, the label of the ellipse is the cross term (coherent term), and the label of the rectangle is the self-term.

Figure 1 WVD of two-component signal and the positional relationship of self-term and cross-term in FF

The FF of the signal has the following properties:

(1) The time-shift invariance is denoted by Eq. (3). (3)${\displaystyle A_{x\left(t-t_0\right)}\left(\tau ,v\right)=e^{jv{t}_0}{A}_{x\left(t\right)}\left(\tau ,v\right)}$

(2) The frequency shift invariance is represented by Eq. (4). (4)${\displaystyle A_{x\left(t\right)e^{j{\omega }_{0}t}}\left(\tau ,v\right)=e^{j{\omega }_{0}t}{A}_{x\left(t\right)}\left(\tau ,v\right)}$

(3) The maximum value of the FF is always at the origin of the (τ, v) plane, and the maximum value is the energy of the signal characterized by Eq. (5). (5)${\displaystyle max{A}_x\left(\tau ,v\right)=A_x\left(0,0\right)=\frac{1}{2\pi }{E}_x}$

(4) The self-term of the FF is always distributed near the origin of the fuzzy domain (FD), while the coherent term of the FF is always distributed far from the origin. The greater the time or frequency difference between the two components, the further away the coherent term they form is from the origin of the FD.

FD filtering and KF theory

The self-term of the FF of the signal is always concentrated in the center of the origin of the FD, and the coherent term of the FF is always far away from the center of the FD, so it can be filtered out by a two-dimensional low-pass filter.

Two-dimensional filtering is performed on the FF of the signal, and then a two-dimensional FT is performed on the τ and v variables to obtain the bilinear time-frequency distribution of the signal, also known as bilinear or Cohen-like time-frequency analysis. The simulation results show that the better the suppression of the cross-term, the worse the time-frequency resolution of the signal. The two-dimensional filter function in the FD is called a KF, and different KFs determine different time-frequency distributions.

Therefore, a method to realize the bilinear time-frequency distribution is the FD filtering method, which first calculates the instantaneous correlation function r_z(t, τ) of the signal z(t) given in Eq. (6). (6)${\displaystyle r_z\left(t,\tau \right)=z\left(t+\frac{\tau }{2}\right)z^{\ast }\left(t-\frac{\tau }{2}\right)}$

Then, the IFT of the instantaneous correlation function about the variable t is calculated, and the FF is obtained and characterized by Eq. (7). (7)${\displaystyle A_z\left(\tau ,v\right)={\int }_{-\infty }^{\infty }{r}_z\left(t,\tau \right)e^{jvt}dt}$

The characteristic function M(τ, v) isobtained by multiplying the FF and the KF defined in Eq. (8). (8)${\displaystyle M_z\left(\tau ,v\right)=\phi \left(\tau ,v\right)A_z\left(\tau ,v\right)}$

The bilinear time-frequency distribution can be obtained by performing the two-dimensional FT of the product, that is, the characteristic function on the variable τ, v defined by Eq. (9). (9)${\displaystyle C_z\left(t,\omega \right)={\iint }_{-\infty }^{\infty }{A}_z\left(\tau ,v\right)\phi \left(\tau ,v\right)e^{-j\left(vt+\omega \tau \right)}d\tau dv}$

The flow chart of the above FD filtering algorithm is shown in Fig. 2.

If the KF in Eq. (8) selects the exponential kernel denoted by Eq. (10). (10)${\displaystyle \phi \left(\tau ,v\right)=e^{-\left[\alpha {\left(\tau v\right)}^2\right]}.}$

Using the MATLAB 2017 version with the simulation toolbox where α = 0.5, the Choi-Williams distribution (CWD) of the signal will be obtained, as shown in Fig. 3.

In the simulation, the observation duration is assigned to 10 ms, the hopping period is set to 2 ms, the number of hops is assigned to 5, and the hopping frequency is randomly selected ranging between 5 kHz through 50 kHz.

The MATLAB 2017 implementation of the instantaneous correlation function, FF, and Choi-Williams distribution after two-dimensional filtering are shown in Fig. 4.

Through the above calculations and simulations, the FD filtering method can more intuitively reflect the relationship between the signal in the time-frequency domain and the FD.

FF of the conventional FHS

Substituting Eq. (8) into Eq. (1), the FD representation of the conventional FHS is obtained. (11)${\displaystyle \begin{array}{c}{A}_x\left(\tau ,v\right)={\int }_{-\infty }^{\infty }x\left(t+\frac{\tau }{2}\right)x^{\ast }\left(t-\frac{\tau }{2}\right)e^{j2\pi tv}dt\hfill \\ ={\int }_{-\infty }^{\infty }\sum _{n=0}^{N-1}rect\left(t+\frac{\tau }{2}-n{T}_h-\alpha T_h\right)e^{2\pi f_n\left(t+\frac{\tau }{2}-n{T}_h-\alpha T_h\right)}\hfill \\ \sum _{m=0}^{N-1}rect\left(t+\frac{\tau }{2}-n{T}_h-\alpha T_h\right)e^{2\pi f_n\left(t+\frac{\tau }{2}-n{T}_h-\alpha T_h\right)}\hfill \end{array}}$

$t+\frac{\tau }{2}-n{T}_h-\alpha T_h=t\prime$ is set to obtain: (12)${\displaystyle \begin{array}{c}{A}_x\left(\tau ,v\right)={\int }_{-\infty }^{\infty }\sum _{n=0}^{N-1}rect\left(t\prime \right)e^{j2\pi f_n\left(t\right)}\sum _{m=0}^{N-1}rect\left(t\prime -\tau -m{T}_h+n{T}_h\right)\hfill \\ e^{j2\pi f_m\left(t\prime -\tau T_h+n{T}_h\right)}{e}^{j2\pi v\left(t\prime -\frac{v}{2}+n{T}_h+\alpha T_h\right)}dt\hfill \\ =\sum _{n=0}^{N-1}\sum _{m=0}^{N-1}{e}^{j2\pi f_m\left(\tau -n{T}_h+m{T}_h\right)}{e}^{j2\pi v\left(n{T}_h+\alpha T_h-\frac{\tau }{2}\right)}\cdot \hfill \\ {\int }_{-\infty }^{\infty }rect\left(t\right)rect\left(t-\tau -m{T}_h+n{T}_h\right)e^{j2\pi t\left(v-f_m+f_n\right)}dt\hfill \\ =\sum _{n=0}^{N-1}\sum _{m=0}^{N-1}{e}^{j2\pi f_m\left(\tau -n{T}_h+m{T}_h\right)}{e}^{j2\pi v\left(n{T}_h+\alpha T_h-\frac{\tau }{2}\right)}\ast A_{ect}\left(\tau -n{T}_h+m{T}_h,v-f_m+f_n\right)\hfill \end{array}}$

Among them, A_rect(.) denotes the FF of the square wave pulse signal $rect\left(\frac{t}{T_h}\right)=\left\{\begin{array}{c}1,\left|t\right|<\frac{T_h}{2}\hfill \\ 0,others\hfill \end{array}\right.$, and the width is denoted by T_H, namely, (13)${\displaystyle A_{rect}\left(\tau ,v\right)=\left\{\begin{array}{c}\hfill e^{j2\pi v{T}_h}\frac{sin\pi v\left(T_h-\left|\tau \right|\right)}{\pi v},\left|\tau \right|<T_h\hfill \\ \hfill 0,\left|\tau \right|>T_h\hfill \end{array}\right.}$

The self-term of the FF of each component has the same delay direction width. Figure 5 presents a contour plot of the magnitude of the ambiguity function for a conventional FHS. Among them, the darker and thicker color in the center of the FD is the self-term of the FF, which needs to be retained, while the others are coherent terms, corresponding to the cross terms of the time-frequency distribution, which should be eliminated.

When located near the origin of the FD it is the self-term of the FF, and the length of the self-term delay of the FF in the conventional FHS is proportional to the period of the conventional FHS.

KF design criteria and methods

Through the previous analysis, the designed KF should preferably completely suppress the coherent term in the FD, which needs to be appropriately truncated in the frequency-shift direction.

According to the design criteria of the FD-KF, the designed KF is defined in Eq. (14). (14)${\displaystyle \phi \left(\tau ,\mathrm{v}\right)=\frac{sin\left(\alpha \pi \mathrm{v}\left(\beta -\left|\tau \right|\right)\right)}{\alpha \beta \pi v},\alpha \ne 0,\beta \ne 0}$

Figure 6 depicts that the larger the value of β is, the wider the central term of the KF is in the delay direction, and the smaller the value of β is, the narrower the central term of the KF is in the delay direction. Therefore, no matter how β changes, the coherent term at the far end of the delay axis cannot be suppressed. The larger the value of parameter β is, the lower the time resolution of the time-frequency distribution is, and the higher the frequency resolution is, the better the effect of suppressing the cross term is.

The parameter α controls the extension of the signal self-term along the frequency shift direction.

Since the parameter β cannot suppress the cross term at the far end of the delay axis, it is necessary to add a low-pass filter function in the delay direction, so that the KF has a better inhibitory effect on the far end of the delay axis. The improved KF is defined in Eq. (15) by using the rectangular function as the low-pass filter function. (15)${\displaystyle \phi \left(\tau ,v\right)=\frac{sin\left[\alpha \pi v\left(\beta -\left|\tau \right|\right)\right]}{\alpha \beta \pi v}\times rect\left(\frac{\tau }{\beta }\right),\alpha \ne 0,\beta \ne 0}$

The parameter β controls the width of the KF in the delay direction and the width of the rectangular function filter. The width of the low-pass filter function varies with the value of β.

Time-frequency distribution, FF of interference, and noise signals

Noise and interfering signals have a specific time-frequency domain representation. By calculating the FFs of these signals and studying their positional relationship with the FF of the FHS and when KF is given, we can know the degree of inhibition of these signals by the KF itself.

For example, white Gaussian noise refers to noise whose probability distribution obeys the Gaussian distribution and is one of the most commonly employed mathematical models of channel noise. Figure 7 is a blur function of additive white Gaussian noise (AWGN).

Fixed-frequency signal interference means that the frequency of the interference signal is constant during the observation period, and a single fixed-frequency signal can be regarded as a stationary signal. The mathematical forms of the time domain and frequency domain are represented by Eq. (16). (16)${\displaystyle \begin{array}{c}Inte{r_{fixed}}_{-freq}\left(\mathrm{t}\right)={\mathrm{e}}^{\mathrm{j}{\omega }_0\mathrm{t}}\hfill \\ INTE{R}_{fixed-freq}\left(\omega \right)=2\pi \delta \left(\omega -{\omega }_0\right)\hfill \end{array}}$

By calculating its FF, Eq. (17) is attained. (17)${\displaystyle A_I\left(\tau ,v\right)=e^{j{\omega }_0\tau }\delta \left(v\right)}$

Equations (16) and (17) depict the instantaneous frequency of the signal ω₀. The simulation uses a constant-amplitude fixed-frequency interference signal whose observation duration is set to 10 ms, sampling rate is set to 200 kHz, and frequency is a constant amplitude, fixed frequency interference signal 20 kHz. Figure 8 depicts a time-frequency distribution diagram of a fixed-frequency interference signal.

Since the simulated signal is a finite-length signal, the FF of the fixed-frequency signal is an isosceles triangle when viewed from the time-shift direction, and the impulse function is viewed from the frequency-shift direction. The degree of coincidence with the KF is high near the center of the FD. The smaller the KF parameter β, the smaller the overlapping part, and the higher the suppression of fixed-frequency interference.

The frequency sweep signal interference refers to the linear change of the frequency of the interference signal during the observation period, and the mathematical form of the time domain is defined in Eq. (18). (18)${\displaystyle {Inter}_{sweep-freq}\left(t\right)=e^{j\frac{1}{2}m{t}^2}}$

By calculating its FF, Eq. (19) is attained. (19)${\displaystyle A_I\left(\tau ,v\right)=\delta \left(v-m\tau \right)}$

Equation (19) shows that the instantaneous frequency of the signal is mt. The FF is in the form of a constant-amplitude impulse function that changes linearly through the origin. The simulation uses a constant-amplitude swept-frequency interference signal whose observation duration is 10 ms, sampling rate is 200 kHz, and frequency linearly increases from initial 7 kHz to 22 kHz. Figure 9 depicts the time-frequency distribution diagram of the sweep-frequency interference signal.

Since the simulated signal represents a finite-length signal, the FF of the swept-frequency signal is an isosceles triangle when viewed from the time-shift direction, and is also an isosceles triangle when viewed from the frequency-shift direction. The degree of coincidence with the KF is low near the center of the FD, the smaller the KF parameter β, the larger the α, the smaller the overlapping part, and the higher the suppression of frequency sweep interference.

Burst interference means that the frequency of the interference signal suddenly increases in amplitude at a certain moment during the observation period. The mathematical representations of the time domain and frequency domain are defined in Eqs. (20) and (21). (20)${\displaystyle \begin{array}{c}Inte{r}_{burst}\left(t\right)=\delta \left(t-t_0\right)\hfill \\ {INTER}_{burst}\left(\omega \right)=e^{-j\omega t_0}\hfill \end{array}}$

By calculating its FF, Eq. (21) is attained. (21)${\displaystyle A_I\left(\tau ,v\right)=e^{-jv{t}_0}\delta \left(\tau \right).}$

The simulation employs a signal whose observation duration is 10 ms, the sampling rate is 200 kHz, and burst interference occurs randomly at a certain moment.

Since the simulated signal represents a finite-length signal, the FF of the burst signal is an impulse function when viewed from the time-shift direction, and an isosceles triangle when viewed from the frequency-shift direction. The degree of coincidence with the KF near the center of the FD is low, the larger the KF parameter α, the smaller the overlapping part, and the higher the suppression of burst interference.

The non-linear frequency sweep signal interference refers to the non-linear frequency change of the interference signal during the observation period. The mathematical representation of the time domain is defined in Eq. (22). (22)${\displaystyle Inte{r}_{sweep-freq}\left(\mathrm{t}\right)={\mathrm{e}}^{\mathrm{j}\frac{1}{3}mt3}}$

Equation (22) shows that the instantaneous frequency of the signal is mt2, and its FF is in the form of a nonlinear change impulse function through the origin. The simulation employs a constant-amplitude nonlinear swept-frequency interference signal with an observation duration of 10 ms, a sampling rate of 200 kHz, and a non-linear increase in frequency from an initial 1.5 kHz to 25 kHz. Figure 10 depicts a time-frequency diagram of a nonlinear sweep-frequency signal.

Since the simulated signal represents a finite-length signal, the maximum value of the FF of the nonlinear frequency swept signal is located at the origin of the FD, and the FF is symmetrical about the origin. The degree of coincidence with the KF near the center of the FD is low, the smaller the KF parameter β, the larger the α, the smaller the overlapping part, and the higher the suppression of nonlinear frequency sweep interference.