Neural Network-Based IIR All-Pass Filter Design

Abstract

This paper presents a neural network-based Lyapunov energy function for the weighted least-squares design of IIR all-pass filters. In the proposed method, the error reflecting the difference between the desired phase response and the phase of the designed IIR all-pass filter is formulated as a Lyapunov error criterion. Based on the neural network architecture and suitable Hopfield parameters, the optimal filter coefficients can be obtained when convergence is achieved. Furthermore, a weight updating function is proposed to achieve accurate approximation of the equiripple response. The simulation results indicate that the proposed technique can achieve high performance in a parallel manner.

Appendix

The tangent expression of (5) is rewritten as

$$ \frac{\sin [ \rho_{d} ( \omega ) ]}{\cos [ \rho_{d} ( \omega ) ]} = \frac{\sum_{n = 1}^{N} a_{n}\sin ( n\omega )}{1 + \sum_{n = 1}^{N} a_{n}\cos ( n\omega )}. $$

(26)

Multiply both sides by $\cos [ \rho_{d} ( \omega ) ] ( 1 + \sum_{n = 1}^{N} a_{n}\cos ( n\omega ) )$; hence,

$$ \sin \bigl[ \rho_{d} ( \omega ) \bigr] \Biggl( 1 + \sum _{n = 1}^{N} a_{n}\cos ( n\omega ) \Biggr) = \cos \bigl[ \rho_{d} ( \omega ) \bigr] \Biggl( \sum _{n = 1}^{N} a_{n}\sin ( n\omega ) \Biggr). $$

(27)

Then, (27) can be formulated as

$$ \sum_{n = 1}^{N} a_{n} \bigl\{ \sin \bigl[ \rho_{d} ( \omega ) \bigr]\cos ( n\omega ) - \cos \bigl[ \rho_{d} ( \omega ) \bigr]\sin ( n\omega ) \bigr\} = - \sin \bigl[ \rho_{d} ( \omega ) \bigr]. $$

(28)

Using the properties of the trigonometric function, (28) can be expressed in a compact form:

$$ \sum_{n = 1}^{N} a_{n}\sin \bigl[ \rho_{d} ( \omega ) - n\omega \bigr] = - \sin \bigl[ \rho_{d} ( \omega ) \bigr]. $$

(29)

This equation is analogous to the general FIR filter design problem, in which the constant amplitude response is replaced by the sinusoidal amplitude. Consequently, there exist several linear algebra-based methods that can be used to solve this optimization problem.