We study reconstruction of time-varying sparse signals in a wireless sensor network, where the bandwidth and energy constraints are considered severely. A novel particle filter algorithm is proposed to deal with the coarsely quantized innovation. To recover the sparse pattern of estimate by particle filter, we impose the sparsity constraint on the filter estimate by means of two methods. Simulation results demonstrate that the proposed algorithms provide performance which is comparable to that of the full information (i.e., unquantized) filtering schemes even in the case where only 1 bit is transmitted to the fusion center.

In recent years, wireless sensor networks (WSNs) have been widely applied in many areas. A WSN system is composed of a large number of battery-powered sensors via wireless communication. Reconstruction of time-varying signals is a key technology for WSNs and plays an important role in many applications of WSNs (see, e.g., [

On the other hand, due to sparseness of signals exhibited in many applications, recently developed compressed sensing techniques have been extensively applied in WSNs [

In this paper, we study reconstruction of time-varying sparse signals in WSNs by using quantized measurements. Our contributions are as follows:

We propose an improved particle filter algorithm which extends the fundamental results [

We propose a new method to impose sparsity constraint on estimator by the particle filter algorithm. Compared to the iterative method in [

This paper is organized as follows. Section

Compressive sensing is a framework for signal sensing and compression that enables representation of a sensed signal with fewer samples than those ones required by classical sampling theory. Consider a sparse random discrete-time process

However, [

Unfortunately, the above optimization problem is NP-hard and cannot be solved effectively. Fortunately, as shown in [

This is a fundamental result in compressed sensing (CS). Moreover, for reconstructing a

For the system given in (

As shown in [

Consider a WSN configured in the star topology (see Figure

Network topology.

The goal of the WSN is to form an estimate of sparse signal

System model.

Most of the earlier works for estimation using quantized measurements concentrated upon using numerical integration methods to approximate the optimal state estimate and make an assumption that the conditional density is approximately Gaussian. However, this assumption does not hold for coarsely quantized measurements, as demonstrated in the following.

Firstly, suppose

The state

See Appendix.

It should be noted that the difference between (

Under an environment of high rate quantization, it is apparent that

At time

Measurement update is given by

where

Resample

For

Consider

Assume that the channel between the sensor and fusion center is rate-limited severely, and the sign of innovation scheme is employed (i.e.,

In addition, note that the quantizer output,

In order to develop a particle filter to propagate

The likelihood ratio between the conditional laws of

The random variable

This result should be rather obvious. Here, one can observe that

Hence, we substitute the information form for (

Together with (

From the above, the particle filter using coarsely quantized innovation (QPF) has been derived for individual sensor. The extension of multisensor scenario will be described in Section

To ensure that the proposed quantized particle filtering scheme recovers sparsity pattern of signals, the sparsity constraints should be imposed on the fused estimate, that is, (

As stated in Section

Suppose a novel refinement method based on cubature Kalman filter (CKF). Compared to the iterative method described above, the resulting method is noniterative and easy to implement.

It is well known that unscented Kalman filter (UKF) is broadly used to handle generalized nonlinear process and measurement models. It relies on the so-called unscented transformation to compute posterior statistics of

By the iterative PM method, it should be noted that (

We now summarize the intact algorithm as follows (illustrated in Figure

Initialization: at

The fusion center transmits

The

On receipt of quantized innovation (see (

Run measurement updates in the information form (see (

Resample the particles by using the normalized weights.

Compute the fused filtered estimate

where

Impose the sparsity constraint on fused estimate

iterative PM update method;

sparse cubature point filter method.

Determine time updates

Illustration of the proposed reconstruction algorithm.

The complexity of sampling Step (4) in the general algorithm is

In this section, the performance of the proposed algorithms is demonstrated by using numerical experiment, in which sparse signals are reconstructed from a series of coarsely quantized observations. Without loss of generality, we attempt to reconstruct a 10-sparse signal sequence

Figure

Nonzero component tracking performance.

Figure

Instantaneous values at

Finally, the error performance of the algorithms is shown in Figure

Normalized RMSE.

In the second scenario, we verify the effectiveness of the proposed algorithm for sparse signal with slow change support set. The simulation parameters are set as

Support change tracking.

In addition, we study the relationship between quantization bits and accuracy of reconstruction. Figure

Normalized RMSE versus number of quantization bits.

Moreover, it is noted that the information filter is employed to propagate the particles in our algorithms. Compared with KF, apart from the ability to deal with multisensor fusion, the IF also has an advantage over numerical stability. In Figure

Support change tracking.

The algorithms for reconstructing time-varying sparse signals under communication constraints have been proposed in this paper. For severely bandwidth constrained (1-bit) scenarios, a particle filter algorithm, based on coarsely quantized innovation, is proposed. To recover the sparsity pattern, the algorithm enforces the sparsity constraint on fused estimate by either iterative PM update method or sparse cubature point filter method. Compared with iterative PM update method, the sparse cubature point filter method is preferable due to the comparable performance and lower complexity. A numerical example demonstrated that the proposed algorithm is effective with a far smaller number of measurements than the size of the state vector. This is very promising in the WSNs with energy constraint, and the lifetime of WSNs can be prolonged. Nevertheless, the algorithm presented in this paper is only suitable for the time-varying sparse signal with an invariant or slowly changing support set, and the more general methods should be combined with a support set estimator which will be discussed in our future work further.

In order to prove Lemma

Consider the pseudo-measurement equation

The authors declare that they have no competing interests.

This work was supported by the fund for the National Natural Science Foundation of China (Grants nos. 60872123 and 61101014) and Higher-Level Talent in Guangdong Province, China (Grant no. N9101070).