On the identification of FIR systems with binary input and output observations

Abstract

This paper considers the identification of FIR systems, where information about the inputs and outputs of the system undergoes quantization into binary values before transmission to the estimator. In the case where the thresholds of the input and output quantizers can be adapted, but the quantizers have no computation and storage capabilities, we propose identification schemes which are strongly consistent for Gaussian distributed inputs and noises. This is based on exploiting the correlations between the quantized input and output observations to derive nonlinear equations that the true system parameters must satisfy, and then estimating the parameters by solving these equations using stochastic approximation techniques. If, in addition, the input and output quantizers have computational and storage capabilities, strongly consistent identification schemes are proposed which can handle arbitrary input and noise distributions. In this case, some conditional expectation terms are computed at the quantizers, which can then be estimated based on binary data transmitted by the quantizers, subsequently allowing the parameters to be identified by solving a set of linear equations. The algorithms and their properties are illustrated in simulation examples.

Recursive system identification method from binary input/output measurements

“…Note also that these algorithms are developed in the noise-free case. The algorithms proposed in [18] and [19] are dedicated to real-time identification in a framework like ours. These algorithms are based on an analysis of the distribution function of signals and require the real-time design of thresholds on sensors.…”

Learning Dynamical Systems From Quantized Observations: A Bayesian Perspective

“…In this section, we consider the problem of iteratively reconstructing the non-quantized output x (k) at time k given: (i) past estimate x (k − 1) at time k − 1 inferred from new sequences (not used for training) of quantized outputs V k−1 and inputs; (ii) new input u (k) and quantized output v (k); (iii) uncertain parameters θ and τ described by the posterior distribution p(θ, τ |V, U), obtained by marginalizing either the Laplace in (16) or the MCMC approximation in (17) w.r.t. x 0 .…”

“…System identification from binary sensors is addressed in [15] for Finite Impulse Response (FIR) models under the (strong) assumption that that noise distribution is exactly known. Algorithms for identification of FIR models from binary input and output measurements have been recently presented in [16], [22]. Both of these contributions are based on the assumption that the (hidden) input is normally distributed.…”

Impulse response identification from input/output binary measurements

“…Considering the identification with only binary measurements on the input and output signals, [15] and [9] proposed algorithms for the identification of a gain system. They are extended in [7] and [8] where it is assumed that the thresholds of the one-bit quantizers can be adapted.…”

“…2-Compute ρ (y+v y )(u+v u ) (i) and ρ (u+v u )(u+v u ) (i) from ( 2) and ( 4) and then compute ρ yu (i) and ρ uu (i) from ( 3) and ( 5). 3-Compute θ N from (7). Remark 2 For C = 0, from a result presented in [12], it can be shown that ρ…”