Signal estimation and filtering from quantized observations via adaptive stochastic resonance

Using a gradient-based algorithm, we investigate signal estimation and filtering in a large-scale summing network of single-bit quantizers. Besides adjusting weights, the proposed learning algorithm also adaptively updates the level of added noise components that are intentionally injected into quantizers. Experimental results show that minimization of the mean-squared error requires a nonzero optimal level of the added noise. The process adaptively achieves in this way a form of stochastic resonance or noise-aided signal processing. This adaptive optimization method of the level of added noise extends the application of adaptive stochastic resonance to some complex nonlinear signal processing tasks.

Figure 1

Block diagram representations of (a) the summing threshold network model $G$ for the noise-enhanced estimator $\widehat{\theta }$ and (b) the adaptive noise-enhanced transversal filter ${\widehat{d}}_n$.

Figure 2

MSE ${\mathcal{R}}_{\infty }$ in Eq. (7) as a function of the weight $w$ and the noise level ${\sigma }_{\eta }$ for a given quantizer threshold $\gamma =0$. The minimum ${\mathcal{R}}_{\infty }=0.2498$ ($▴$) at the optimal weight $w^{\mathrm{opt}}=3.1$ and the optimal level ${\sigma }_{\eta }^{\mathrm{opt}}=4.7$ of the added Gaussian noise is marked.

Figure 3

Learning curves of (a) MSE ${\mathcal{R}}_{\infty }$ in Eq. (7), (b) the level ${\sigma }_{\eta }$ of the added noise, (c) the weight $w$, and (d) the quantizer threshold $\gamma$ for estimating an uniformly distributed parameter $\theta$ buried in the Gaussian white noise $\xi$. The learning rate $\mu$ in Eq. (14) is chosen as 1 for the weight $w$, 0.1 for the noise level ${\sigma }_{\eta }$, and 0.01 for the threshold $\gamma$.

Figure 4

Learning curves of (a) MSE $\mathcal{J}$ of the 5-order filter of Eq. (17) and (b) the level ${\sigma }_{\eta }$ of the added noise. The learning rate $\mu =5$ for updating the weight $w$, 0.01 for the noise level ${\sigma }_{\eta }$, and 0.001 for the threshold $\gamma$. Here, the initial noise level ${\sigma }_{\eta }\left(0\right)=2$, the initial weight vector $\mathit{w}\left(0\right)={[1,-1,-1,-1,-1]}^{\top }$, and the initial threshold $\gamma \left(0\right)=0$.

Figure 5

A realization of the output of the designed filter ${\widehat{d}}_n$ (circles plus solid line) and the desired signal $d_n=2cos(2\pi n/N)$ (dashed line). At each iteration number $k$ (i.e., the time $n$), the output of the designed filter ${\widehat{d}}_n$ is calculated by Eq. (17) with the iterated weight vector $\mathit{w}\left(k\right)$, the level ${\sigma }_{\eta }\left(k\right)$ of the added noise and the threshold $\gamma \left(k\right)$, and $M={10}^5$ quantizers in each network model $G_{\ell }$. The other parameter are the same as in Fig. 4.

Figure 6

Learning curves of (a) MSE $\mathcal{J}$ of the 5-order filter of Eq. (17) and (b) the level ${\sigma }_{\eta }$ of the added noise. Here, the initial level ${\sigma }_{\eta }\left(0\right)=1$ of the added noise and the other parameters are the same as in Fig. 4.