Edited by Digital Filters Edited by Fausto Pedro García Márquez Published by InTech Janeza Trdine 9, 51000 Rijeka, Croatia Copyright © 2011 InTech All chapters are Open Access articles distributed under the Creative Commons Non Commercial Share Alike Attribution 3.0 license, which permits to copy, distribute, transmit, and adapt the work in any medium, so long as the original work is properly cited. After this work has been published by InTech, authors have the right to republish it, in whole or part, in any publication of which they are the author, and to make other personal use of the work. Any republication, referencing or personal use of the work must explicitly identify the original source. Statements and opinions expressed in the chapters are these of the individual contributors and not necessarily those of the editors or publisher. No responsibility is accepted for the accuracy of information contained in the published articles. 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ISBN 978-953-307-190-9 free online editions of InTech Books and Journals can be found at www.intechopen.com Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Preface IX Digital Filters for Maintenance Management 1 Fausto Pedro García Márquez and Diego José Pedregal Tercero The application of spectral representations in coordinates of complex frequency for digital filter analysis and synthesis 27 Alexey Mokeev Design of Two-Dimensional Digital Filters Having Variable Monotonic Amplitude-Frequency Responses Using Darlington-type Gyrator Networks 53 Muhammad Tariqus Salam and Venkat Ramachandran Common features of analog sampled-data and digital filters design 65 Pravoslav Martinek, Jiˇr í Hospodka and Daša Tichá New Design Methods for Two-Dimensional Filters Based on 1D Prototypes and Spectral Transformations 91 Radu Matei Integration of digital filters and measurements 123 Jan Peter Hessling Low-sensitivity design of allpass based fractional delay digital filters 155 G. Stoyanov, K. Nikolova and M. Kawamata Integrated Design of IIR Variable Fractional Delay Digital Filters with Variable and Fixed Denominators 179 Hon Keung Kwan and Aimin Jiang Contents Contents VI Complex Coefficient IIR Digital Filters 209 Zlatka Nikolova, Georgi Stoyanov, Georgi Iliev and Vladimir Poulkov Low-Complexity and High-Speed Constant Multiplications for Digital Filters Using Carry-Save Arithmetic 241 Oscar Gustafsson and Lars Wanhammar A Systematic Algorithm for the Synthesis of Multiplierless Lattice Wave Digital Filters 257 Juha Yli-Kaakinen and Tapio Saramäki Chapter 9 Chapter 10 Chapter 11 The new technologies and communications systems are being set up in all areas. It leads to treating data from dierent sources and for several proposes. But it is nec- essary to obtain only the information that is required. Digital lters, together with analogue lters, are used for these objectives. The main advantage of the digital lters is that they can be applied at zero cost and with a great exibility. The mathematical models where they are created have dierent complexity and computational cost. In this book the most relevant lters are described, and with dierent applications. The material covered in this text is crucial for geing a general idea about digital lters. This book also presents some best options for each case study considered. In spite of the mathematical complexity of the digital lters, the text is presented for any reader with a motivation for learning about digital lters. The high level contents are shown with an exhaust introduction, where the most important works in the litera- ture are referenced and it completed with various examples. A discrete lter is presented within a well-known and common framework, namely the State Space with the help of the Kalman Filter (KF) and/or complementary Fixed Interval Smoother (FIS) algorithms. It is presented in several case studies for detecting faults where these models can be adapted to external and internal conditions to the mechanism. All of these models are developed within a well-known common frame- work, namely the State Space (SS). The KF is a powerful algorithm, because it supports estimations of past, present and future states. In this case, it is used for ltering with Integrated Random Walks by seing up a bivariate model composed of two time series, i.e. the reference curve on one hand and each one of the empirical curves obtained on line on the other hand. Other options are to use a model VARMA (Vector autoregres- sive moving-average) class or a local level plus noise but set up in continuous time. Finally, due to the nature of the data, a pertinent class is a Dynamic Harmonic Regres- sion, similar to a Fourier analysis, but with advanced features included to incorporate a time varying period observed in the data. Preface Preface VIII In the case of a linear circuit and frequency lter analysis for sinusoidal and periodical input signals, the spectral representations employing Fourier transform are studied. In that case, Laplace’s transformations are employed in order to consider a complex frequency. The compound nite signal representations are done in the form of the set of damped oscillatory components. It is an ecient method for ltering and it can work with a complex coordinate. In the case of Innite Impulse Response (IIR) lter impulse functions the representation uses this set of damped oscillatory components. Impulse functions of Finite Impulse Response (FIR) lters representation are also based on this set of damped oscillatory components, but with the dierence of a nite duration of the impulse functions. It considers the stationary and non stationary modes, where it can be calculated easily in the spectral representation context. It is possible considering the application of spectral representations in complex frequency coordinates. It leads to consider both spectral approach and the state space method for frequency lter analysis and synthesis. The lter synthesis problem comes to dependence composition for lter transfer function on complex frequencies of input signal components. Complex lters can be namely digital lters with complex coecients. They are em- ployed in complex signal processing compared to the real signal processing (e.g. tele- communications). This can imply real and imaginary inputs and outputs, and these signals need to be separated into real and imaginary parts for being studied as complex signals. The rst- and second-order IIR orthogonal complex sections are synthesized as lters in designing cascade structures or as single lter structures. It leads delay-free loops and has a canonical number of elements. The low-sensitivity 1 and 2 variable complex sections can be used in narrowband band-pass / band-stop structures. The main advantages of these models are the higher freedom of tuning, reduced complex- ity and lower stop-band sensitivity. The response dela in digital circuits should be adjusted to a fraction of the sampling interval and it should be xed or variable in order to control the fractional delay (FD). These circuits are used in telecommunications applications that require speech syn- thesis and processing, image interpolation, sigma-delta modulators, time-delay es- timation, in some biomedical applications and for modeling of musical instruments. Considering the phase-sensitivity minimization of each individual rst- and second- order allpass section in the lter cascade realization, xed and variable allpass-based fractional delay lters are developed and adjusted through sensitivity minimizations. The real and complex-conjugate poles combinations for dierent values of the FD pa- rameter D and of the transfer function (TF) order N are analyzed trying to minimize the overall sensitivity. A two-dimensional (2D) digital lter is employed to aain the desired cut-o fre- quency and the stable monotonic amplitude-frequency responses of this lter. It is developed in accordance with monotonic amplitude-frequency responses employing Darlington-type gyrator networks and doubly-terminated RLC-networks by the ap- plication of Generalized Bilinear Transformation (GBT). The doubly terminated RLC networks are adjusted as second-order Buerworth and Gargour & Ramachandran. It leads low-pass, high-pass, band-pass and band-elimination lters. The transformation between these lters is done by the value and sign of the parameter called g and GBT. It is useful in digital image (video and audio), and for enhancement and restoration in dierent elds, as medical science, geographical science and environment, space and robotic engineering, etc. IX Preface From a 1D lter (low-pass and maximally-at or very selective), a 2D lter can be devel- oped. These are essentially spectral transformations (frequency transformations) via bilinear or Euler transformations followed by mappings. This book analyzes the case of recursive lter approaches in the frequency domain applied in image processing: directional selective lters, oriented wedge lters, fan lters, diamond-shaped lters, etc. The zero-phase case is also considered. All the models are mainly analytical, and in some cases, numerical optimization is employed, in particular - rational approxima- tions. The reason to choose the analytical approach is that the 2D parameters can be controlled by adjusting the prototype. An analytical design method in polar coordi- nates is proposed and dened by a periodic function expressed in polar coordinates in the frequency plane. It can yield selective two or multi-directional lters, and also fan and diamond lters. Finally, two-lobe lters are analysed, selective four-lobe lters with an arbitrary orientation angle, fan lters and diamond lters. Single correction lters or ensembles of correction lters, sensitivity lters, lumbar spine lter, banks of vehicle lters, and road texture lters are presented. They are studied in two examples on safety of trac: road hump analysis and determination of road texture. Digital lters are recommended for low robustness, and this originates from the denition of the feature and/or its incomplete specication instead of a feature which is not robust and questionable. The digital lters employed t into the above mentioned standard linear-in-response nite/innite impulse response (FIR/IIR) form for direct implementation. In this case any lter may be transferred to a state-space form for generalization into a KF. Carry-Save Arithmetic is employed in order to achieve an optimal design of single constant multipliers for coecients with up to 19 bits wordlength. The non-redundant representation is also considered. The proposed techniques are useful when a high- speed realization is required. It is demonstrated in the multiple constant multiplication problems suitable for transposed direct form FIR lters using carry-save representa- tion of intermediate results but non-redundant input. Laice wave digital (LWD) lter (parallel connections of all-pass lters) is a structure implemented in the recursive digital lters. Three cases are considered in this book: primarily the overall lter, constructed as a cascade of low-order LWD lters. Secondly, approximately linear-phase LWD lters are constructed as a single block. The reason for this is the lack of benets for the direct-form LWD lter design in the usage of a cascade of several lter blocks. Finally, it is focused on the design of special recursive single-stage and multistage Nth-band decimators and interpolators. The coecient optimization is performed with following steps: an initial innite-precision lter is designed such that it exceeds the given criteria in order to provide some tolerance for coecient quantization; then, a nonlinear optimization algorithm is employed for de- termining a parameter space of the innite-precision coecients including the feasible space where the lter meets the given criteria; and nally, the lter parameters are found in this space so that the resulting lter meets the given criteria with the simplest coecient representation forms. The realization of these lters does not require the use of a costly general multiplier element. It leads to the fact that the lters are goods in very large-scale integration (VLSI). Preface X The sampled-data and digital lters (i.e. “memory transistor” or “memory transcon- ductor” approaches) are both studied for their eectivity. This case is about biquadratic sections used in cascade design. The switched-current (SI) circuits are also one of the case studies employed, where it can be extended to cases as digital VLSI-CMOS tech- nologies, lower supply voltage and wide dynamic range, considering an SI as “analog counterpart” of the digital lters. The biquadratic realization structures are developed from the rst and second direct forms of the 2nd-order digital lter. The continuous- time biquadratic sections design is also considered. Finally, the optimization of sam- pled-data and digital lters design is solved by using the heuristic algorithm as the dierential evolutionary algorithm. Fausto Pedro García Márquez University of Castilla-La Mancha (UCLM) Spain [...]... R t CT H t Pt |t 1HT t t Kt ˆ P HT Ft 1 t t 1 t |t 1 ˆ ˆ xt 1 t Φt 1 K t HT xt |t 1 K t z t t ˆ ˆ ˆ Pt 1| t Φt 1Pt |t 1 T 1 K t Φt 1Pt |t 1HT t t EQ E T t t T t The backward FIS recursions are: ˆ ˆ ˆ xt N xt |t 1 Pt |t 1st 1 ˆ ˆ ˆ ˆ Pt | N Pt |t 1 Pt |t 1St 1Pt |t 1 ˆ st 1 HT Ft 1 z t H t xt |t 1 ΦtT st t St 1 HT Ft1H t ΦtT St Φt t ˆ... but one fairly general representation is given by equations (1) (see [3] and [ 21] ) In general, much simpler models are sufficient, as later case studies show State Equations : x t 1 t x t Et w t Observation Equations : z t Ht xt + Ct vt (i) (ii) (1) 4 In (1) Digital Filters zt is the m dimensional vector of observed variables for t 1, 2,, N ; dimensional stochastic state vector; xt is an n... simultaneously, where the local means are modeled by the dynamics implied by the state equations, i.e I I 0 w1t xt 1 0 I xt I w 2t zt signal noise I 0xt vt 2 w1 Q 2 2 w1 w2 2 2 w w 1 2 w 2 v2 v1v2 2 ; R 1 v v v2 2 12 (2) In model (2) all the system matrices are time invariant: I is a two dimensional identity matrix;.. .Digital Filters for Maintenance Management 1 1 X Digital Filters for Maintenance Management Fausto Pedro García Márquez and Diego José Pedregal Tercero Ingenium Research Group, University of Castilla-La Mancha Spain 1 Abstract Faults in mechanisms must be detected quickly and reliably in order to avoid important... two disturbances; is the correlation coefficient between the two noise signals in the state equation and 8 Digital Filters By comparing systems (2) and (1) it is easy to see the system matrices values in this particular case, i.e w I I 0 ; Et ; wt 1t ; Ht I 0; Ct 1 Φt 0 I I w2t The unknown hyper-parameters to be estimated by ML in this model are Q... in this chapter are cast, is the so called State Space systems, that have experienced a remarkable attention during the last decades, as the extended literature about it reveals [3], [7], [13 ], [15 ], [16 ], [17 ], [ 21] , [24], [26] and [27] A stochastic discrete-time State Space system (SS) is a model composed of two sets of equations, the Observation Equations, and State Equations The former relates the... the estimate tends to increase as more information becomes available 1 0.8 Rho 0.6 0.4 0.2 0 1 2 3 4 2 3 4 5 6 7 8 5 6 7 8 1 0.8 Rho 0.6 0.4 0.2 0 1 Fig 5 Recursive estimation of Time (s) (stars) and 95% confidence bands (solid) for one “as commissioned” curve (top) and one “faulty” curve (bottom) 5 Random Walks and smoothing 5 .1 Device and data Following successful implementation on a level crossing... compared to other infrastructure elements, about 3.4 million UKP (United Kingdom Pound) per year for about 10 00 km of railway TC-TCR trade circuits, for example, cost 2 .1 million UKP per year for the same area Of the points expenditure, 1. 2 million UKP is for clamp lock type (hydraulic) turnout and 1. 4 UPK million for electrically operated turnouts (data provided by a British asset manager) Turnouts can... and/or complementary Fixed Interval Smoother (FIS) algorithms, exposed in general terms in the following section 2 Digital Filters Based on this common framework, the following subsections in this introduction show the particular applications shown in later sections of the chapter 2 .1 Filtering with Integrated Random Walks (IRW) One possible way to analyze faults on line is to work with a reference... provides excellent results for the data collected during this series of experiments [10 ] 2.3 Advance Dynamic Harmonic Regression (DHR) A different case study was based on data collected from point mechanisms at Abbotswood Junction (UK) Three electro-mechanical and four electro-hydraulic point machines were Digital Filters for Maintenance Management 3 monitored by a RCM system Processed information . T ttt T T ttttt T tttttt tttt T ttt tt t T ttttt T tttt T tttt EQEHPΦKΦPΦP zKxHKΦx FHPΦK HPHCRCF 1| 111 |1| 1 1| 1 1 1 1| 1 1| ˆˆˆ ˆˆ ˆ ˆ The backward FIS recursions are: tt T tttttt Ntt T ttt T tt Nt T tttttt T tt tttttttNt ttttt Nt HFHPΦΦΦ 0SΦSΦHFHS 0ssΦxHzFHs PSPPP sPxx 1 1| 1 1 1| 1 1 1| 11| 1|| 11 |1| ˆ . T ttt T T ttttt T tttttt tttt T ttt tt t T ttttt T tttt T tttt EQEHPΦKΦPΦP zKxHKΦx FHPΦK HPHCRCF 1| 111 |1| 1 1| 1 1 1 1| 1 1| ˆˆˆ ˆˆ ˆ ˆ The backward FIS recursions are: tt T tttttt Ntt T ttt T tt Nt T tttttt T tt tttttttNt ttttt Nt HFHPΦΦΦ 0SΦSΦHFHS 0ssΦxHzFHs PSPPP sPxx 1 1| 1 1 1| 1 1 1| 11| 1|| 11 |1| ˆ . tt T tttttt Ntt T ttt T tt Nt T tttttt T tt tttttttNt ttttt Nt HFHPΦΦΦ 0SΦSΦHFHS 0ssΦxHzFHs PSPPP sPxx 1 1| 1 1 1| 1 1 1| 11| 1|| 11 |1| ˆ with with ˆ ˆˆˆˆ ˆ ˆˆ This general