FIR system identification based on a nonparametric Bayesian model using the Indian buffet process

Abstract

In this paper, we propose a nonparametric Bayesian model combined with the Indian buffet process (IBP) for a finite impulse response (FIR) system. We develop an FIR system identification method that can simultaneously estimate the number of FIR taps and coefficients. In the proposed model, each FIR tap consists of a coefficient and a gain, and the gain is a binary value. An infinite-dimensional binary vector is composed of binary values, and we assume that this binary vector is generated by the IBP. To identify the FIR system, we specify the likelihood function and prior distributions of the parameters and derive their posterior distributions. We can simultaneously estimate the number of FIR taps and coefficients by sampling from posterior distributions using the Gibbs sampler. Our simulations demonstrate that although the number of FIR taps is unknown, the identification performance of the proposed method in a high signal-to-noise ratio environment is similar to or better than that of the conventional least square solution.

Appendices

Appendix 1: Probability distributions used in this paper

We describe specification of the probability distributions used in this paper. For a random variable z that can be zero or one, the Bernoulli distribution is defined for $q\in [0,1]$ as

$$\begin{aligned} \mathrm {Bernoulli}(z|q)=q^z(1-q)^{1-z}. \end{aligned}$$

(30)

The beta and gamma distribution of scalar random variable x are

$$\begin{aligned}&\mathrm {Beta}(x|\,\beta ,\gamma )=\frac{{\varGamma }(\beta +\gamma )}{{\varGamma }(\beta ){\varGamma }(\gamma )}x^{\,\beta -1}(1-x)^{\gamma -1}\;(0\le x\le 1), \end{aligned}$$

(31)

$$\begin{aligned}&{\mathcal {G}}(x|\eta ,b)=\frac{b^\eta }{{\varGamma }(\eta )}x^{\eta -1}e^{-bx}\;(0\le x), \end{aligned}$$

(32)

where $\eta ,b,\beta ,\gamma >0$. The Gaussian distribution of a d-dimensional random variable $\mathbf {x}\in {\mathbb {R}}^d$ is

$$\begin{aligned} {\mathcal {N}}(\mathbf {x}|\mathbf {\mu },\mathbf {\Sigma })= (2\pi )^{-\frac{d}{2}}\Vert \mathbf {\Sigma }\Vert ^{-\frac{1}{2}} \exp \left( -\frac{1}{2}(\mathbf {x}-\mathbf {\mu })^\top \mathbf {\Sigma }^{-1}(\mathbf {x}-\mathbf {\mu })\right) .\nonumber \\ \end{aligned}$$

(33)

Appendix 2: Derivation of the prior distribution of the infinite-dimensional binary vector

The prior distribution $P_{\mathbf {z}}^\infty (\mathbf {z})$ is derived by integrating $\alpha$ out from $P_{\mathbf {z}|\alpha }^\infty (\mathbf {z}|\alpha )$ with $p_\alpha (\alpha )$, that is,

$$\begin{aligned} P_{\mathbf {z}}^\infty (\mathbf {z})=\int _{(0,\infty )} P_{\mathbf {z}|\alpha }^\infty (\mathbf {z}|\alpha )p_\alpha (\alpha )d\alpha . \end{aligned}$$

(34)

Since $p_\alpha (\alpha )$ is conjugate to $P_{\mathbf {z}}^\infty (\mathbf {z})$, this integral is calculated using the constant term of the gamma distribution as (11).

Appendix 3: Derivation of the evidence

To obtain the evidence $p_{\mathbf {d}|\mathbf {X},\mathbf {z}}(\mathbf {d}|\mathbf {X},\mathbf {z})$, we integrate $\mathbf {w}_+$ and s out from (8) as

$$\begin{aligned}&p_{\mathbf {d}|\mathbf {X},\mathbf {z}}(\mathbf {d}|\mathbf {X},\mathbf {z})\nonumber \\&=\int _{[0,\infty )}\int _{{\mathbb {R}}^{K_1}} f(\mathbf {d}|\mathbf {X},\mathbf {w},s,\mathbf {z})p_{\mathbf {w}|s}(\mathbf {w}_+|s)p_s(s)d\mathbf {w}_+ds. \end{aligned}$$

(35)

Equation (20) is obtained by calculating this integral since $p_{\mathbf {w},s}(\mathbf {w}_+,s)=p_{\mathbf {w}|s}(\mathbf {w}_+|s)p_s(s)$ is conjugate to $f(\mathbf {d}|\mathbf {X},\mathbf {w},s,\mathbf {z})$.