Waveform-Domain Adaptive Matched Filtering for Suppressing Interrupted-Sampling Repeater Jamming

The ISRJ signal can be expressed as the product of the transmitted waveform $s(t)$ and the interrupted-sampling function $g(t)$, denoted as $\jmath(t)=g(t)s(t)$. By utilizing the ambiguity function, the output of the matched filter for the ISRJ signal can be represented by the following equation [2]:

where $\operatorname{Sa}(x)=\frac{\sin(x)}{x}$, $T_{\jmath}$ represents the duration of the jamming slice. The interrupted-sampling frequency is denoted by $f_{J}=\frac{1}{T_{J}}$, and $T_{J}$ corresponds to the interrupted-sampling repeater period. The function $\mathcal{X}(t,f_{d})=s(t+\tau)s^{\ast}(\tau)e^{j2\pi f_{d}\tau}\mathrm{d}\tau$ represents the ambiguity function of the transmitted waveform. It is worth noting that for self-defensive repeater interference, due to the utilization of a time-sharing transmit-receive antenna by the jammer, the condition $f_{J}T_{\jmath}\leqslant 0.5$ holds.

Let’s assume that a monostatic pulsed Doppler radar transmits a pulse compression waveform $s(t)$. When the self-defensive jammer transmits a jamming signal, the received signal can be expressed as:

Here, $A_{s}$ represents the amplitude of the target echo signal, $\tau_{s}$ denotes the propagation delay of the target. On the other hand, $A_{\jmath}$ represents the amplitude of the jamming signal, and $\tau_{\jmath}$ denotes the delay of the interference signal.

When $x(t)$ traverses a generalized matched filter $h(t)$, characterized by the system function $\mathrm{H}[\cdot]$, within the framework of ISRJ suppression methods based on matched filtering systems, a predefined mapping is employed for both $x(t)$ and $h(t)$. Consequently, an output is generated, which can be expressed as follows:

Here, $\mathrm{KH}[\cdot]$ symbolizes the mapping resulting from the application of $\mathrm{K}[\cdot]$ to the system function $\mathrm{H}[\cdot]$. Furthermore, $\mathrm{G}[\cdot]$ represents the mapping in relation to $x(t)$. Additionally, $\varrho(t)$ can be defined as the convolution of $\mathrm{G}[\jmath(t-\tau_{\jmath})]$ and $\mathrm{K}[h(t)]$. Hence, an effective strategy to counteract the interference from ISRJ involves maximizing the amplitude of the main lobe in $\mathrm{KH}\{G[s(t-\tau_{s})]\}$, while simultaneously minimizing the amplitude of the side lobes in $\mathrm{KH}\{G[s(t-\tau_{s})]\}$ and inhibiting the peak amplitude of $\varrho(t)$.

In the context of pulsed Doppler radar systems, assuming that the transmitted waveform is a LFM signal. Consequently, the baseband signal can be mathematically expressed as follows:

Here, the function $\operatorname{rect}(\cdot)$ represents the rectangular window function, $T$ corresponds to the pulse width here, and $k$ corresponds to the chirp rate. The impulse response of the matched filter can be expressed as follows:

Combining (6), we can obtain the expression for CWCF of $s(t)$:

Here, $\alpha_{s}^{(t)}=\operatorname{max}\left\{-\frac{T}{2},-\frac{T}{2}+t\right\}$ and $\beta_{s}^{(t)}=\operatorname{min}\left\{\frac{T}{2},\frac{T}{2}+t\right\}$. $c^{(t)}(\rho)$ and $c_{0}^{(t)}$ are defined as follows:

The derivation of (10) and (11c) can be found in (A1) to (A6). By focusing solely on the magnitudes of $y_{s}^{(t)}(\rho)$, it becomes evident that for $t\neq 0$ and $\alpha_{s}^{(t)}\leqslant\rho\leqslant\beta_{s}^{(t)}$,

which represents a half-wave rectified function. The magnitude is $\left|\frac{1}{\pi kt}\right|<\left|\beta_{s}^{(t)}-\alpha_{s}^{(t)}\right|=T$, and it decreases as $\left|t\right|$ increases. Based on the properties of the $\mathrm{Sa}(\cdot)$ function, it can be inferred that when $\rho\rightarrow\infty$, $\left|y_{s}^{(t)}(\rho)\right|\rightarrow 0$.

In particular, when $t=0$, $\left|y_{s}^{(0)}(\rho)\right|$ takes the form of a first-order linear function:

where the minimum value of $\left|y_{s}^{(0)}(\rho)\right|$ is 0, and the maximum value is $T$.

For ISRJ, the interference signal can be regarded as:

Here, $\jmath_{n}(t)$ represents the $n$-th interference slice. Then, CWCF of of $\jmath(t)$ can be expressed as:

where $y_{\jmath_{n}}^{(t)}(\rho)$ represents the CWCF of the $n$-th individual interference slice:

Here, $\alpha_{\jmath_{n}}^{(t)}=\max\left\{-\frac{T}{2},-\frac{T}{2}+t,-\frac{T_{\jmath}}{2}+nT_{J}+t\right\}$ and $\beta_{\jmath_{n}}^{(t)}=\min\left\{\frac{T}{2},\frac{T}{2}+t,\frac{T_{\jmath}}{2}+nT_{J}+t\right\}$. $b_{n}^{(t)}(\rho)$ and $b_{n_{0}}^{(t)}$ are defined as follows:

The derivation of (16) and (17c) can be found in (B1) to (B14). It is straightforward to infer that when $\beta_{\jmath_{m-1}}^{(t)}<\rho\leqslant\beta_{\jmath_{m}}^{(t)}$, the following equation holds true:

Hence, $y_{\jmath}^{(t)}(\rho)$ can be expressed as a piecewise function. Specifically, when $t=0$, we have:

which is a stepped function. The minimum value of $\left|y_{\jmath}^{(0)}(\rho)\right|$ is 0, and the maximum value is $\frac{T_{\jmath}}{T_{J}}\cdot T$.

When $t\neq 0$, the expression for $\left|b_{n}^{(t)}(\rho)\right|$ is given as follows:

which has a period of $T_{b}^{(t)}=\left|\frac{1}{kt}\right|$. In particular, when $t=t_{\jmath_{n}}=-\frac{nf_{J}}{k}$, we have:

and $T_{b}^{(t_{\jmath_{n}})}=\frac{1}{nf_{J}}=\frac{T_{J}}{n}$. If $T_{\jmath}\leqslant\frac{T_{J}}{2n}$, then $\left|b_{n}^{(t_{\jmath_{n}})}(\rho)\right|$ is a monotonically increasing function. Therefore, $|y_{\jmath}^{(t_{\jmath_{n}})}(\rho)|$ can also be thought of as a stepped function similar to $|y_{\jmath}^{(0)}(\rho)|$:

where $Q<1$ arises from the fact that the matched filter is the maximum output SNR filter, and the minimum value of $\left|y_{s}^{(t_{\jmath_{n}})}(\rho)\right|$ is 0, while the maximum value is $f_{J}T_{\jmath}\operatorname{Sa}\left(n\pi f_{J}T_{\jmath}\right)\mathcal{X}\left(t_{\jmath_{n}},-nf_{J}\right)$.

When $t\neq t_{\jmath_{n}}$, the effective accumulation of $|b_{n}^{(t)}|$ is lacking, resulting in $|y_{\jmath}^{(t)}(\rho)|$ being represented as a piecewise envelope with a smaller maximum amplitude. As the difference $\left|t-t_{\jmath}\right|$ increases, $|b_{n_{0}}^{(t)}|$ becomes smaller, and $|y_{\jmath}^{(t)}(\rho)|$ becomes closer to $|y_{s}^{(t)}(\rho)|$, which can be approximated as a stable half-wave rectification function:

Through the above analysis, it can be noted that for $s(t)$, its CWCF is a monotonically increasing linear function with a slope of 1 only when $t=0$, and at all other times, its CWCF is a periodic function. As for $\jmath(t)$, when $t=0$, its CWCF is a piecewise linear function with a linearly increasing segment having a slope of 1. When $t=t_{\jmath_{n}}$, its CWCF is an approximately piecewise linear function with an approximate slope less than 1. At all other times, its CWCF is approximately a periodic function.

Upon careful examination of the aforementioned information, we can proceed to delve into the pertinent characteristics and effectiveness of $y^{(t)}(\rho)$. It is possible to express $y^{(t)}(\rho)$ as follows:

Furthermore, $\left|y^{(t)}\right|$ exhibits the following relation:

Especially, when $t=\tau_{s}$, we have:

Based on (13), (23), (24), and (26), it can be deduced that $\left|y^{(\tau_{s})}\left(\rho\right)\right|$ can be approximated as the combination of an autoterm, $\left|y_{s}^{(0)}\left(\rho\right)\right|$, and a crossterm, $\left|y_{\jmath}^{(\tau_{s}-\tau_{\jmath})}\left(\rho\right)\right|$. In cases where $\tau_{s}-\tau_{\jmath}=t_{\jmath_{n}}$, the crossterm exhibits characteristics resembling those of a step function, thus significantly impeding the similarity between $\left|y^{(\tau_{s})}\right|$ and $\left|y_{s}^{(0)}\right|$. However, when $\tau_{s}-\tau_{\jmath}\neq t_{\jmath_{n}}$, the crossterm can be approximated as a half-wave rectified function, with its magnitude diminishing as $|\tau_{s}-\tau_{\jmath}|$ increases. At this point, due to the periodic characteristics of the half-wave rectification function, we observe that $\left|y^{(\tau_{s})}(\frac{T}{2})\right|\approx\left|y_{s}^{(0)}(\frac{T}{2})\right|$.

Similarly, when $t=t_{\jmath_{n}}+\tau_{\jmath}$, we have:

From (12), (19), (24), and (27), it becomes apparent that the magnitude of $\left|y_{\jmath}^{(t_{\jmath_{n}}+\tau_{\jmath})}\left(\rho\right)\right|$ can be approximated as an auto-term $\left|y_{\jmath}^{(t_{\jmath_{n}})}\left(\rho\right)\right|$, augmented by a cross-term $\left|y_{s}^{(t_{\jmath_{n}}+\tau_{\jmath}-\tau_{s})}\left(\rho\right)\right|$. And there is $\left|y^{(t_{\jmath_{n}}+\tau_{\jmath})}(\frac{T}{2})\right|\approx\left|y_{\jmath}^{(t_{\jmath_{n}})}(\frac{T}{2})\right|$.

For values of $t$ that do not satisfy $t=\tau_{s}$ or $t=t_{\jmath_{n}}+\tau_{\jmath}$, it can be deduced from (12), (23), and (24) that the magnitude $\left|y^{(t)}(\rho)\right|$ corresponds to a complex envelope, devoid of the distinctive amplitude characteristics exhibited by $\left|y^{(\tau_{s})}\left(\rho\right)\right|$ and $\left|y^{(t_{\jmath_{n}}+\tau_{\jmath})}\left(\rho\right)\right|$. Furthermore, its maximum magnitude is significantly inferior to them.

After the above analysis, we describe the variation of the integrated energy of $v^{(t)}(\mu)$ in the waveform domain through $|y^{(t)}(\rho)|$. It is evident that only when $t=\tau_{s}$, $|y^{(\tau_{s})}(\rho)|$ can be approximately viewed as a linear function. In comparison, when $t=\tau_{\jmath_{n}}$, $|y^{(\tau_{\jmath_{n}})}(\rho)|$ can be approximately seen as a stepwise linear function. At all other times, $|y^{(t)}(\rho)|$ can be regarded as a periodic function. Therefore, through statistical methods, we can determine whether each element on $|y^{(t)}(\rho)|$ is equivalent to the corresponding element of a linear function defined in the waveform domain. This helps identify effective and ineffective integration elements in the waveform domain at that moment. Consequently, we can establish the objective function as follows:

where $o^{(t)}=\frac{\left|y^{(t)}\left(\frac{T}{2}\right)\right|}{T}$.

It is evident that when the cross-term becomes zero, and when $t=\tau_{s}$, the objective function $O^{(\tau_{s})}(\rho)$ is equivalent to $\left|y^{(\tau_{s})}(\rho)\right|$ itself. Conversely, when $t=t_{\jmath_{n}}+\tau_{\jmath}$, the approximate linear segments in $\left|y^{(t_{\jmath_{n}}+\tau_{\jmath})}(\rho)\right|$ have a slope at least $\frac{T_{J}}{T_{\jmath}}$ times that of $o^{(t_{\jmath_{n}}+\tau_{\jmath})}$, thereby exhibiting a distinct and discernible characteristic.

In the scenario where the cross-term is non-zero and there is additive Gaussian white noise, the problem of suppressing ISRJ can be formulated as a hypothesis testing problem, aiming to evaluate the equality between $\left|y^{(t)}(\rho)\right|$ and $O^{(t)}(\rho)$.

Assuming the existence of Gaussian white noise, which possesses additive characteristics represented as $\xi(t)\sim(0,\sigma^{2})$ in the time domain, when substituting $\xi(t)$ into (5) and (6), the following results are obtained:

Clearly, the term denoted as $wgn^{(t)}(\mu)$ maintains its characteristic as additive Gaussian white noise, adhering to the properties of $wgn^{(t)}(\mu)\sim(0,\sigma^{2})$. Conversely, $bn^{(t)}(\rho)$ represents a standard Brownian noise, adhering to the properties of $bn^{(t)}(\mu)\sim\left[0,\left(\mu+\frac{T}{2}\right)\sigma^{2}\right]$, which is a Gauss-Markov random process.

Based on the analysis in the preceding sections, the anti-ISRJ problem can be mathematically formulated as a hypothesis testing problem to examine the equality between $\left|y^{(t)}(\rho)\right|$ and $O^{(t)}(\rho)$. However, the complex structure of $|y^{(t)}(\rho)|$ implies that different analytical solutions exist at different times $t$, making it unfeasible to use a single criterion for mathematical modeling.

To overcome this issue, an equivalent mathematical model is proposed as a probabilistic hypothesis testing problem of whether $|v^{(t)}(\rho)|=o^{(t)}$.

If we denote the set of instances without jamming signal on $\mu$ as $\mathrm{U}^{(t)}_{s}$, when only a slice jamming is present on $\mu$ as $\mathrm{U}^{(t)}_{\jmath}$, and when both an echo signal and a slice jamming are present on $\mu$ as $\mathrm{U}^{(t)}_{s+\jmath}$, then the following relationships hold:

where $\mathcal{M}^{(\tau_{s})}(\mu)\in\left[\left|1-\frac{A_{\jmath}}{A_{s}}\right|,1+\frac{A_{\jmath}}{A_{s}}\right]$, and

where $\mathcal{M}^{(t)}(\mu)\in\left[\left|\mathcal{A}^{(t)}-\mathcal{C}^{(t)}\right|,\mathcal{A}^{(t)}+\mathcal{C}^{(t)}\right]$.

(30) and (31) imply that if $|v^{(t)}(\mu)|=o^{(t)}$, then $t=\tau_{s}$ and $\mu\in\mathrm{U}^{(t)}_{s}$. Additionally, for self-defensive forwarding interference, we observe $\mathcal{A}^{(t_{\jmath_{n}}+\tau_{\jmath})}>2$, meaning that when $\mu\in\mathrm{U}^{(t)}_{\jmath}$, we have $|v^{(t_{\jmath_{n}}+\tau_{\jmath})}(\mu)|>2o^{(t_{\jmath_{n}}+\tau_{\jmath})}$. As previously stated, our focus centers around $\left|v^{(t)}(\mu)\right|$ for $\mu\in\mathrm{U}^{(t)}_{\jmath},\mathrm{U}^{(t)}_{s+\jmath}$. Specifically, our objective is to preserve $v^{(\tau_{s})}(\mu)$ to the greatest extent while minimizing $v^{(t_{\jmath_{n}}+\tau_{\jmath})}(\mu)$.

It is evident that when $A_{\jmath}\gg A_{s}$, we have $\left|\mathcal{A}^{(t)}-\mathcal{C}^{(t)}\right|>2$, leading to $\left|v^{(t)}(\mu)\right|>2o^{(t)}$ for $\mu\in\mathrm{U}^{(t)}_{\jmath},\mathrm{U}^{(t)}_{s+\jmath}$. However, in scenarios where $A_{\jmath}$ is relatively small, it is possible to encounter $\left|\mathcal{A}^{(t)}-\mathcal{C}^{(t)}\right|<2<\mathcal{A}^{(t)}+\mathcal{C}^{(t)}$, and thus the condition $\left|v^{(t)}(\mu)\right|>2o^{(t)}$ for $\mu\in\mathrm{U}^{(t)}_{\jmath},\mathrm{U}^{(t)}_{s+\jmath}$ is not universally valid. Considering that $\mathcal{M}^{(t)}(\mu)$ is a continuous function within the range $\left[\left|\mathcal{A}^{(t)}-\mathcal{C}^{(t)}\right|,\mathcal{A}^{(t)}+\mathcal{C}^{(t)}\right]$, and the intervals $\mathrm{U}^{(t)}_{\jmath}$ and $\mathrm{U}^{(t)}_{s+\jmath}$ are relatively short, we can extend $\mu$ to obtain $v^{(t)}(\mu\pm\gamma\cdot d\mu)>2o^{(t)}$, for $\mu\in\mathrm{U}^{(t)}_{\jmath},\mathrm{U}^{(t)}_{s+\jmath}$, where $\gamma$ deones the scaling factor.

Therefore, the anti-ISRJ problem can be reformulated as a probabilistic hypothesis testing problem, determining whether $|v^{(t)}(\mu\pm\gamma\cdot d\mu)|>2o^{(t)}(\mu)$. Consequently, the anti-ISRJ problem can be transformed into an unbiased estimation problem for $v^{(t)}(\mu)$ and $y^{(t)}(\rho)$. We denote their estimates as $\hat{v}^{(t)}(\mu)$ and $\hat{y}^{(t)}(\rho)$, respectively.

In scenarios where $t=\tau_{s}$ and $t=\tau_{\jmath}$, and the impact of crossterm can be deemed insignificant, it is feasible to approximate $y^{(t)}(\mu)$ as locally exhibiting a linear relationship of first order. This characteristic enables us to model $\hat{y}^{(t)}(\mu)$ by employing a linear function model complemented by two impulse function models.

In the subsequent steps, our objective is to establish models for $\hat{y}^{(t)}(\mu)$ and $\hat{v}^{(t)}(\mu)$ using the Interactive Multiple Model Kalman Filter algorithm (IMM-KF)[23]. Within the framework of the IMM-KF algorithm, the interdependent state of $\hat{y}^{(t)}(\mu)$ can be precisely defined as a weighted combination of three distinct model states, given by the expression:

Here, the probabilities $u_{1}$, $u_{2}$, and $u_{3}$ are determined for each model based on the residuals and residual covariance obtained through the utilization of the Kalman filter. $\hat{M}^{(t)}_{1}(\mu|\mu)$, $\hat{M}^{(t)}_{2}(\mu|\mu)$, and $\hat{M}^{(t)}_{3}(\mu|\mu)$ represent the estimated states of their respective models, and their one-step prediction state equations are expressed as follows: