Noisy Blind Signal-jamming Separation Algorithm Based on VBICA

Abstract

Aiming at the blind signal-jamming separation (BSJS) in wireless communication environment, we propose a noisy BSJS based on Variational Bayesian Independent Component Analysis algorithm to separate the communication signal from jamming signals and noises. This algorithm takes the Kullback–Leibler divergence between the true post distributions of source signals and the approximate ones as objective function, models sources using mixture of Gaussians, and updates parameters of the model using variational-Bayesian learning method, so as to make the estimated approximate posterior distributions close to the true ones and recover source communication signals finally. The simulation results show that the proposed algorithm is effective for the BSJS in noisy environment.

Appendix A: The Update Equations of the Parameters in VBICA Model

In $p^{\prime }(\mathbf{S}|q)$,

$$\begin{aligned} \hat{{\mu }}_{i,q_i }^{t}&= \frac{1}{\hat{{\beta }}_{i,q_i }^t }\left[ {\left\langle {\beta _{i,q_i } } \right\rangle \left\langle {\mu _{i,q_i } } \right\rangle +\sum _{j=1}^M {\left\langle {\Lambda _j } \right\rangle \left\langle {a_{ji} } \right\rangle \left( {x_{j}{(t)}-\left\langle {\hat{{x}}_{j,k\ne i}{(t)}} \right\rangle } \right) } } \right] \end{aligned}$$

(25)

$$\begin{aligned}&= \hat{{\beta }}_{i,q_i }^t =\left\langle {\beta _{i,q_i } } \right\rangle +\sum _{j=1}^M {\left\langle {\Lambda _j } \right\rangle \left\langle {a^{2}_{ji} } \right\rangle } \end{aligned}$$

(26)

In $p^{\prime }(q)$,

$$\begin{aligned} \gamma _{i,q_i }^{t}&= \tilde{\pi }_{i,q_i } \tilde{p}_{i,q_i } \end{aligned}$$

(27)

$$\begin{aligned} \hat{{\gamma }}_{i,q_i }^{t}&= \frac{\gamma _{i,q_i }^t }{\sum _{q^{{\prime }}_i } {\gamma _{i,q^{{\prime }}_i }^t }} \end{aligned}$$

(28)

$$\begin{aligned} \tilde{\pi }_{i,q_i }&= \exp \left[ {\Psi \left( {\tilde{\lambda }_{i,q_i } } \right) -\Psi \left( {\sum _{q^{{\prime }}_i } {\hat{{\lambda }}_{i,q^{{\prime }}_i } } } \right) } \right] \end{aligned}$$

(29)

$$\begin{aligned} \tilde{p}_{i,q_i }&= \left( {\frac{\tilde{\beta }_{i,q_i } }{\hat{{\beta }}^{t}_{i,q_i } }} \right) ^{\frac{1}{2}}\exp \left[ {\frac{1}{2}\left( {\hat{{\beta }}^{t}_{i,q_i } \hat{{\mu }}_{i,q_i }^{t^{2}} -\left\langle {\beta _{i,q_i } } \right\rangle \left\langle {\mu ^{2}_{i,q_i } } \right\rangle } \right) } \right] \end{aligned}$$

(30)

$$\begin{aligned} \tilde{\beta }_{i,q_i }&= \hat{{b}}_{i,q_i } \exp \left[ {\Psi \left( {\hat{{c}}_{i,q_i } } \right) } \right] \end{aligned}$$

(31)

In (5) and (7),$\Psi ({\bullet })$ is Digamma function.

In $p^{\prime }(\mu )$,

$$\begin{aligned} \hat{{m}}_{i,q_i}&= \frac{1}{\hat{{\tau }}_{i,q_i } }\left( {\tau _{i0} m_{i0} +\left\langle {\beta _{i,q_i } } \right\rangle \sum _{t=1}^L {\hat{{\gamma }}_{i,q_i }^t \left\langle {s_i (t)|q_i^t } \right\rangle } } \right) \end{aligned}$$

(32)

$$\begin{aligned} \hat{{\tau }}_{i,q_i }&= \tau _{i0} +\left\langle {\beta _{i,q_i } } \right\rangle \sum _{t=1}^L {\hat{{\gamma }}_{i,q_i }^t} \end{aligned}$$

(33)

In $p^{\prime }(\beta )$,

$$\begin{aligned} \hat{{b}}_{i,q_i}&= \left( {\frac{1}{b_{i0} }+\frac{1}{2}\tilde{\sigma }_{i,q_i } } \right) ^{-1} \end{aligned}$$

(34)

$$\begin{aligned} \hat{{c}}_{i,q_i}&= c_{i0} +\frac{1}{2}\sum _{t=1}^L {\hat{{\gamma }}_{i,q_i }^t} \end{aligned}$$

(35)

$$\begin{aligned} \tilde{\sigma }_{i,q_i}&= \sum _{t=1}^L {\hat{{\gamma }}_{i,q_i }^t \left( {\left\langle {s_{_i }^2 (t)|q_i^t } \right\rangle -2\left\langle {\mu _{i,q_i } } \right\rangle \left\langle {s_i (t)|q_i^t } \right\rangle +\left\langle {\mu _{_{i,q_i } }^2 } \right\rangle } \right) } \end{aligned}$$

(36)

In $p^{\prime }(\mathbf{A})$,

$$\begin{aligned} \hat{{m}}_{a_{ji}}&= \frac{\left\langle {\Lambda _j} \right\rangle }{\hat{{\alpha }}_{ji} }\sum _{t=1}^L {\left\langle {s_i (t)} \right\rangle \left( {x_j (t)-\left\langle {\hat{{x}}_{j,k\ne i} (t)} \right\rangle } \right) } \end{aligned}$$

(37)

$$\begin{aligned} \hat{{\alpha }}_{ji}&= \alpha _{ji} +\left\langle {\Lambda _j} \right\rangle \sum _{t=1}^L {\left\langle {s_i^2 (t)} \right\rangle } \end{aligned}$$

(38)

$$\begin{aligned} \left\langle {s_i{(t)}} \right\rangle&= \sum _{q_i =1}^{m_{i}} {p^{{\prime }}\left( {q_i^{t} =q_i} \right) } \left\langle {s_{i}{(t)}|q_{i}^{t}} \right\rangle \end{aligned}$$

(39)

$$\begin{aligned} \left\langle {s_i^{2}{(t)}} \right\rangle&= \sum _{q_i =1}^{m_i} {p^{{\prime }}\left( {q_i^{t} =q_i} \right) } \left\langle {s_i^{2} (t)|q_{i}^{t}} \right\rangle \end{aligned}$$

(40)

$$\begin{aligned} p^{{\prime }}\left( {q_{i}^{t} =q_i } \right)&= \hat{{\gamma }}_{i,q_i }^t \end{aligned}$$

(41)

$$\begin{aligned} \left\langle {s_{i}{(t)}|q_{i}^{t}} \right\rangle&= \hat{{\mu }}_{i,q_i }^t \end{aligned}$$

(42)

$$\begin{aligned} \left\langle {s_{i}^{2} (t)|q_{i}^{t}} \right\rangle&= \left( {\hat{{\mu }}_{i,q_i }^t } \right) ^{2}+\frac{1}{\hat{{\beta }}^{t}_{i,q_i }} \end{aligned}$$

(43)

In $p^{\prime }({\varvec{\Lambda }})$,

$$\begin{aligned} {\hat{b}}_{\Lambda j}&= \left[ {\frac{1}{b_{\Lambda j} }+\frac{1}{2}\sum _{t=1}^{L} {\left\langle {\left( {x_{j}(t)-\hat{{x}}_{j} (t)} \right) ^{2}} \right\rangle }} \right] ^{-1} \end{aligned}$$

(44)

$$\begin{aligned} {\hat{c}}_{\Lambda _{j}}&= {c_{\Lambda _{j}}} +\frac{L}{2} \end{aligned}$$

(45)