Passive, active, and digital filters (3ed., CRC, 2009) - tiera.ru

1-22 Passive, Active, and Digital Filterswith arbitrary amplitude and phase response characteristics (see Ref. [1, Chapter 14]) for the application ofthese methods for the design of digital filters). Some classical closed-form solutions are the so-calledButterworth, Chebyshev, and elliptic* approximations to be described in Chapter 2 by A.M. Davis.In general, the designer is interested in simple and reliable approximation methods that yield precisedesigns with the minimum amount of computation.1.8.2 The Realization StepThe synthesis of a filter is the process of converting some characterization of the filter into a network. Theprocess of converting the transfer function into a network is said to be the realization step and thenetwork obtained is sometimes called the realization.The realization of a transfer function can be accomplished by expressing it in some form that allowsthe identification of an interconnection of elemental filter subnetworks and=or elements. Many realizationmethods have been proposed in the past that lead to structures of varying complexity and properties.In general, the designer is interested in realizations that are economical in terms of the numberof elements, do not require expensive components, and are not seriously affected by variations inthe element values such as may be caused by variations in temperature and humidity, and drift due toelement aging.1.8.3 Study of ImperfectionsDuring the approximation step, the coefficients of the transfer function are determined to a high degreeof precision and the realization is obtained on the assumption that elements are ideal, i.e., capacitors arelossless, inductors are free of winding capacitances, amplifiers have infinite bandwidths, and so on.In practice, however, the filter is implemented with nonideal elements that have finite tolerances and areoften nonlinear. Consequently, once a realization is obtained, sometimes referred to as a paper design,the designer must embark on the study of the effects of element imperfections. Several types of analysisare usually called for ranging from tolerance analysis, study of parasitics, time-domain analysis, sensitivityanalysis, noise analysis, etc. Tight tolerances result in high-precision filters but the cost per unitwould be high. Hence the designer is obliged to determine the highest tolerance that can be toleratedwithout violating the specifications of the filter throughout its working life. Sensitivity analysis is a relatedstudy that will ascertain the degree of dependence of a filter parameter, e.g., the dependence of theamplitude response on a specific element. If the loss characteristic of a filter is not very sensitive to certaincapacitance, then the designer would be able to use a less precise and cheaper capacitor, which would, ofcourse, decrease the cost of the unit.1.8.4 ImplementationOnce the filter is thoroughly analyzed and found to meet the desired specifications under idealconditions, a prototype is constructed and tested. Decisions to be made involve the type of componentsand packaging, and the methods are to be used for the manufacture, testing, and tuning of the filter.Problems may often surface at the implementation stage that may call for one or more modifications inthe paper design. Then the realization and possibly the approximation may have to be redone.* To be precise, the elliptic approximation is not a closed-form method, since the transfer function coefficients are given interms of certain infinite series. However, these series converge very rapidly and can be treated as closed-form formulas formost practical purposes (see Ref. [1, Chapter 5]).