The bilinear transformation

The bilinear transformation (BLT) is useful to instrumentation and design engineers in many settings. Two examples are: in the modeling of continuous-time (CT) systems via digital computer and in the design of digital infinite-impulse-response filters that are to have response characteristics similar to those of a selected CT filter. An introduction to the BLT is presented along with a discussion of how to obtain the BLT numerically     

Prewarping techniques in filter simulation by bilinear transformation

The prewarping technique in the bilinear transformation is widely described in the literature concerned with the design of digital filters, but it is not always applicable to simulation problems. This paper deals with the application of the prewarping technique in the simulation of Laplace-defined filters. The proposed methods are suitable to simulation, because they are automatic (made by the program without any user action), involve a very low software complexity and practically do not increase the CPU time of the whole simulation process.     

Properties of the Qn-matrix in bilinear transformation

The usefulness of a matrix transformation has been noted in the design of digital filters from analog prototypes. In this letter, a new general property, that relates two such matrices of successive orders, (n - 1) and n, is proved.     

A note on the bilinear transformation matrices for multivariable polynomials

Special symmetry of bilinear transformation matrices for multivariable polynomials is stated. It is shown that reduction in computation in existing algorithm can be achieved by making use of this symmetry. It has been brought out that the number of nonzero terms in the given polynomial of specified order plays a role in the computational efficiency of the ‘Rao-Aatre algorithm’.     

Simple state-space formulations of 2-D frequency transformation and double bilinear transformation

In this paper, an explicit relationship between the two-dimensional (2-D) frequency transformation and the theory of linear fractional transformation (LFT) representation is shown. Based on this relationship, a simple alternative state-space formulation of 2-D frequency transformation for 2-D digital filters is derived by utilizing the well-known Redheffer star product of LFT representations. The proposed formulation is then utilized to establish a simple relationship between the state-space representations of a 2-D continuous system and a 2-D discrete system which are related by the double bilinear transformation. Moreover, the inherent relations among the proposed formulations and the existing results are discussed. It turns out that all the existing results given in the literature can be unified as special or equivalent cases by the new state-space formulation of 2-D frequency transformation in a very concise and elegant form. Numerical examples are given to illustrate the effectiveness of the proposed formulations.     

Alignment of the digital filter frequency axis with the bilinear transformation

The problem of aligning the frequency axes of a digital filter with the reference analog prototype is considered. It is shown that the use of an arbitrary constant in the bilinear transformation results in a particularly simple means for matching the responses of the analog and the digital filters at any reference frequency.     

A general state-space representation of n-variable bilinear transformation

This paper proposes a general formulation of the relationship between the state-space representations of a multidimensional (n-D) continuous system and an n-D discrete system which are related by the n-variable bilinear transformation such that the state-space representations of these two related systems can be directly calculated from each other. The new formulation is derived based on the theory of linear fractional transformation (LFT) and the resultant form is simple and concise. Moreover, the relations among the proposed formulation and the existing 1-D and 2-D results are investigated and it turns out that the new formulation includes these existing results as special cases. A numerical example is presented to illustrate the effectiveness of the proposed formulation.     

Some difficulties with the double bilinear transformation in 2-D recursive filter design

It is shown that the double bilinear transformation approach for designing a 2-D IIR digital transfer function from a predetermined 2-D analog transfer function, may in certain cases lead to unstable solutions.     

A design of a simple structure switched-capacitor filter based on the concept of the bilinear transformation

A design procedure of the SCF based on the concept of the bilinear transformation is proposed. The final SCF structure is the same as that obtained by the LDI transformation, which means a structure simpler than the usual SCF obtained by the bilinear transformation and which requires no sample-and-hold circuits at the front and back of it when it is employed with the prefilter and the postfilter as the analog filter. The filter response does not suffer from the effect of the inexact simulation of the terminated resistance of the prototype LC filter which appears in the SCF realization based on the LDI transformation and the aperture effect is compensated. The filter performance realized by the proposed design procedure is tested by an experimental circuit with discrete elements.     

On the stability preserving property of the double bilinear transformation on a class of 2-D transfer functions

This paper addresses the BIBO (bounded-input bounded-output) stability of a class of discrete 2-D quarter-plane filters in the presence of nonessential singularities of the second kind (NSSK's) on the unit bidisk. Conditions under which the double bilinear transformation (DBT) preserves stability are derived. The results presented here also extend the class of systems whose stability can be predicted. Use of the inverse DBT to produce a continuous equivalent of the discrete 2-D transfer function allows easy application of a continuous-domain equivalent of a criterion developed by Dautov. The necessary and sufficient condition for stability derived in this work provides a simple check for the class of systems under consideration. From this class of systems, it is also possible to construct stable pairs of mutually inverse transfer functions.     

基于小波的双线性插值误差补偿算法的图像放大

为了尽可能地保持低分辨率图像的基本信息,提高图像的视觉效果和空间分辨率,本文提出一种基于小波的双线性插值误差补偿算法。该算法增加双线性插值的误差补偿项,利用Sobel算子设定插值点的边缘方向,得到初始放大图像。利用小波提取高频成分,原始图像幅值增强充当低频部分,再经过小波逆变换得到高分辨率图像。实验结果表明,相对于传统的图像放大算法,该算法考虑到全局相关性,得到更加清晰的边缘信息。     

Multidimensional bilinear transformations

The bilinear transformation is extended to transform multi-variable polynomials, using discrete convolution and the Kronecker product. This approach is very simple, easy for computer implementation, and useful in complex curve fitting and stability studies of discrete systems and the design of digital filters etc.     

Photorefractive solitons

We present a review of recent works on spatial optical solitons in photorefractive media. We discuss fundamental properties of screening solitons and their interaction.     

On the stability of bilinear filters

We consider stability properties of discrete-time bilinear filters. Simple sufficient conditions are given for bounded-input bounded-output stability (with not necessarily zero initial conditions),l _p stability, and three other important types of stability. In particular, conditions are given under which asymptotically periodic inputs produce asymptotically periodic outputs with the same period.     

Current–voltage curve translation by bilinear interpolation

By means of bilinear interpolation and four reference current–voltage (I–V) curves, an I–V curve of a photovoltaic (PV) module is translated to desired conditions of irradiance and PV module temperature. The four reference I–V curves are measured at two irradiance and two PV module temperature levels and contain all the essential PV module characteristic information for performing the bilinear interpolation. The interpolation is performed first with respect to open‐circuit voltage to account for PV module temperature, and second with respect to short‐circuit current to account for irradiance. The translation results over a wide range of irradiances and PV module temperatures agree closely with measured values for a group of PV modules representing seven different technologies. Root‐mean‐square errors were 1·5% or less for the I–V curve parameters of maximum power, voltage at maximum power, current at maximum power, short‐circuit current, and open‐circuit voltage. The translation is applicable for determining the performance of a PV module for a specified test condition, or for PV system performance modeling. Copyright © 2004 John Wiley & Sons, Ltd.     

Signal transmission by optical solitons in monomode fiber

A transmission rate of ≃1 Tbits/sec (≃0.1 Tbits/s) per 30 km can be achieved using envelope solitons with peak power of ≃10 W (≃10 mW) in a monomode optical fiber, respectively. Unlike the linear pulse in which the bit rate is limited by the group dispersion, the bit rate of soliton transmission is limited by the fiber loss and the input power. Conditions for achieving optimum transmission rate using solitons are theoretically obtained including the effects of fiber loss and second order group dispersion.     

Adaptive lattice bilinear filters

Two fast least-squares lattice algorithms for adaptive nonlinear filters equipped with bilinear system models are presented. The lattice filter formulation transforms the nonlinear filtering problem into an equivalent multichannel linear filtering problem, thus using multichannel lattice filtering algorithms to solve the nonlinear filtering problem. The computational complexity of the algorithms is an order of magnitude smaller than that of previously available methods. The first of the two approaches is an equation error algorithm that uses the measured desired response signal directly to compute the adaptive filter outputs. This method is conceptually very simple, but results in biased system models in the presence of measurement noise. The second is an approximate least-squares output error solution; the past samples of the output of the adaptive system itself are used to produce the filter output at the current time. Results indicate that the output error algorithm is less sensitive to output measurement noise than the equation error method     

SAW convolvers using the transverse-horizontal bilinear field

Conventional SAW convolvers rely on the transverse-vertical polarisation field induced by crystal nonlinearity in the presence of two contraflow input waves. The letter presents results for single-track SAW convolvers fabricated on LiNbO3 using the transverse-horizontal (in-plane) electric polarisation. The new electrode arrangement involved reduces the possibility of unwanted 'fold-over convolution from fabrication defects and increases design flexibility.     

Covariant and contravariant bilinear transformation of quadripole scattering matrix and matching strategy for a quadripole type domestic microwave oven

The author obtains two formulations for calculating the image scattering matrix of a quadripole through a linear octopole circuit, which become equivalent when the octopole circuit is reciprocal. Thus bilinear transformation can be used for quantifying that a magic tee can match perfectly any quadripole type applicator or any domestic microwave oven to two generators.