A Note on the Mean - Square Quantization Error

This report summarizes the understanding I have gained in my studies for the independent studies course ENEE 699 up to 15th April, 2000. It includes the derivation of an extension of a result of Wong and Brockett on the behaviour of scalar quantizers. 1 Control under limited communication Let, ẋ(t) = f(x(t), u(t)) + n(t) (1) be a controlled dynamical system where the state x(·) ∈ R, the control u(·) ∈ R, and the noise process n(·) ∈ R are all defined on a suitable probability space (Ω,F , P ) and are assumed to be F -measurable. We assume further that f(·, ·) : Rn×Rm → R satisfies regularity properties that ensure that (1) has smooth solutions on at least a finite time interval [T0, T1] for a class U , of admissible controls that includes the set of piecewise continuous (or more specifically, piecewise-constant) controls. We are interested in generating feedback control laws in U based on state observations y(t) = x(t) + n′(t), where the observation noise process n′(·) ∈ Rl is also defined on (Ω,F , P ) and is F -measurable. The observer is physically removed from the plant (1) and so the observation signals are sent to it over a digital communication link of finite capacity (because of finite bandwidth as well as noise). Hence, the observation signals necessarily have to be sampled, quantized and coded for transmission over the digital communication link. Considerations of the complexity of implementation lead us to settle for a uniform sampling rate and allocation of fixed word lengths for the transmitted code-words. 1 The optimal coding and decoding scheme would be in general time-varying and could need to use all of the past observations (y(t) for the encoder and the received binary code-words for the decoder). Again, we opt for a sub-optimal but simpler scheme of coding and decoding that uses not the entire past history of observations but a finitedimensional statistic of it. This is the notion of ‘Finitely recursive state estimation’ proposed in [Wong-Brockett,1] .The simplified system then takes the form: X(i+ 1) = X(i) + F (X(i), U(i)) +N(i), X̂(i) = G (X(i), N ′(i), A(i)) , A(i+ 1) = H (A(i), X(i), N ′(i)) , U(i) = J(i, X̂(i)), where X(i) and U(i) are time-discretized versions of x(·) and u(·). N(i) and N ′(i) are discrete-time noise processes derived from n(·) and n′(·). X̂(i) ∈ R is the sequence of state estimates made by the observer and A(i) ∈ R represents the finite memory of the encoder-decoder scheme. All of the discrete-time variables declared above are defined on a modified probability space (Ω∆,F∆, P∆). The functions G : R × Rl × R → R and H : R × R × Rl → R are maps representing the state estimation process. Hidden in G and J , are the time-delays due to the finite bandwidth of the link. An object that we need to keep track of is {E[|X̂(i)−X(i)|2]}, the sequence of stateestimation error variances. We would desire the convergence to zero or at least the boundedness of this sequence. The next section deals with the study of when this is possible. 2 The quality of state estimation and quantization In the setup of [Mitter-Borkar], the system dynamics is linear i.e. F (x(t), u(t)) = Ax(t) +Bu(t). The sequence of error covariance matrices Rk is kept bounded by assuming that the singular values of the matrix A are less than 1 in magnitude(a condition stronger than the schur-stability of A). [Wong-Brockett,1] study the convergence behaviour of {E[|X̂(i)−X(i)|2]} for some special cases. The main component of their analysis is the derivation of some explicit inequalities governing the error-variance of a single step of encoder-decoder operation. In the remainder of this section, we will study a key equation that leads to the proof of these inequalities. For simplicity, we treat(as in [Wong-Brockett,1]) the case of scalar quantization. Let x be a real valued random variable with a probability density function p(x). Assume 2 that E[x] = μ < ∞ and E[(x − μ)] = σ < ∞. A finite code-book quantizer is one that partitions the real line into a finite collection of sets S = {Si}, members of which are mutually disjoint and which together cover the entire real line (or at least the support of p(x)). To each Si, the quantizer assigns a codeword that represents the quantized value q(x) of x when x ∈ Si. The distortion measure of the quantizer is the expectation value of a given distance function d(q(x), x). We seek a quantizer that minimises the squared error distortion function : E[(q(x)− x)]. It is proved in [Gersho-Gray] that: (i) If the set of values of the quantized levels is prescribed, the optimal partition sets Si are intervals each containing one quantization level, and (ii) If the partition sets are pre-specified, the optimal quantization levels are the conditional means : E[x|x ∈ Si]. Given partition sets, we can find an expression for the minimum variance of the quantization error. Such a result is presented in [Wong-Brockett,1] with a small mistake in notation. We reproduce it in the form of a lemma . Let pi = prob{x ∈ Si}, μi = E[x|x ∈ Si], and σ i = E[(x− μi)|x ∈ Si]. Lemma 1 The minimum variance of the quantization error is