Filtering in the Time and Frequency Domains
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Long regarded as a classic of filter theory and design, this book stands as the most comprehensive treatment of filtering techniques, devices and concepts as well as pertinent mathematical relationships. Analysis and theory are supplemented by detailed design curves, fully explained examples and problem and answer sections. Discussed are the derivation of filtering functions, Fourier, Laplace, Hilbert and z transforms, lowpass responses, the transformation of lowpass into other filter types, the allpass function, the effect of losses on theoretical responses, matched filtering, methods of timedomain synthesis, and digital filtering. This book is invaluable for engineers other than those who are filter design specialists who need to know about the possibilities and limits of the filtering process in order to use filters competently and confidently in their system designs.
Inspec keywords: bandpass filters; matched filters; digital filters; Hilbert transforms; convolution; linear systems; allpass filters; Fourier transforms; transfer functions; Z transforms; Laplace transforms; step response; transient response; lowpass filters; distortion; timefrequency analysis; differential equations; delays
Other keywords: step response; frequencydomain system; timedomain synthesis; transfer function; finiteQ elements; Laplace transform; time delay; impulse response; narrowband filter; allpass function; Z transform; group delay response; linear systems analysis; matched filter; bandpass filter; Hilbert transform; magnitude response; signal predistortion; convolution integral; lowpass response; differential equation; quasistationary approach; phase response; Fourier transform; phase delay equalization; group delay equalization; digital filter; weighting function
Subjects: Signal processing theory; Differential equations (numerical analysis); Integral transforms; General electrical engineering topics; Filtering methods in signal processing; General and management topics; Integral transforms; Differential equations (numerical analysis)
 Book DOI: 10.1049/SBEW008E
 Chapter DOI: 10.1049/SBEW008E
 ISBN: 9781884932175
 eISBN: 9781613530719
 Format: PDF

Front Matter
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1 TimeDomain Analysis
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In this chapter we have consolidated the mathematical developments necessary to relate the physical system and the differential equation describing it. This included the derivation of the impulse response, step response, and convolution integral. These quantities, which are important to the theory of filtering, were shown to be a function of the L.I. solutions of the homogeneous equation. It is hoped that this timedomain approach has clarified the transition from the basic differential equation concepts to the everyday tools of the linear system analyst.

2 FrequencyDomain Analysis
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Although the differential equation is a basic system description, obtaining this equation can be tedious and timeconsuming. Consequently for a timeinvariant system this approach is avoided in practice, except in special cases. The Fourier and Laplace transforms offer an alternative approach for characterizing and analyzing these systems. Insight into system behavior is often obtained by the transform method. These transforms change a function of one variable into a function of another variable, and, when applied to problems in the physical sciences, the transform pairs and variables may correspond to physical quantities. We assign time and frequency as the transform variables because these are the variables associated with the filtering devices considered in this text.

3 Linear System Responses
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The filtering system response plays a major role in determining the overall system response. For this reason familiarity with the fundamental aspects of filter responses is not only a must for the filter designer, but it is advantageous for the system designer and those who specify and use filters. Appreciation of the factors affecting the various responses results in more realistic system and filter requirements. Consequently design time, cost, and problems are reduced.

4 Frequency Transformations
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In this chapter we discuss the realization of lowpass (LP), highpass (HP), bandpass (BP), and bandstop (BS) filters by applying suitable frequency transformations to the normalized LP filter. These transformations generally preserve the LP magnitude response (attenuation); other LP characteristics are often retained, however, especially in the case of the narrowband BP filter. We also give the necessary equations (where possible) for denormalizing the LP responses, allowing them to characterize the appropriate filter type. When the transformation preserves the important attributes of the LP response, the tedious and time consuming approximation step in the design sequence is eliminated. This is the big advantage of the frequencytransformation approach. Furthermore, if the transformation function is of the same form as a reactance function, the filter element values are easily determined. Unfortunately certain LP characteristics are distorted by the transformation, and to avoid these distortions we either resolve the approximation problem or use a different transformation. We present an example of each approach. The first results in the wideband constantdelay filter and the second results in the low transient HP filter. This chapter also includes narrowband BP filter design, which is applicable not only to lumpedconstant filters but also to transmission line and waveguide filters.

5 AllPass Functions
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The magnitude response of an allpass (AP) filter is unity for all frequencies, thus all frequencies are passed without attenuation. The associated phase response, however, is useful for approximating a specified phase characteristic, and the AP group delay function is useful for approximating a specified delay characteristic. Moreover, if we indicate that a parameter α can be adjusted for a specific phase response, it is understood that α can likewise be adjusted for a specific group delay response. Approximating a linear phase by the AP phase corresponds to approximating a constant delay by the AP delay. Theoretically these approximations are not necessarily the same, but in practice the difference is often negligible. The phase (delay) properties of AP filters are so important for achieving a specified phase (delay) response of the electronic system that we devote an entire chapter to them, and we hope to consolidate much of the scattered information on this subject.

6 FiniteQ Elements and Predistortion
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In addition to increased insertion loss, other undesirable features of predistorted networks are a wide range of element values, increased sensitivity to small elementvalue changes, increased mismatch between the filter's input impedance and the source resistance, and the appearance of passband ripples when the element q is small and differs from the design value. Consequently predistortion should only be used when q is relatively high and the exact response is desired. Otherwise, a more practical solution is the adjustment of the filter's bandwidth to accommodate losses as explained. In most cases the exact response is not a necessity; the lossy response is satisfactory, and the disadvantages associated with the predistortion method are then absent.

7 Optimum Linear Filtering
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The majority of this chapter is devoted to the matched filter because of its importance in modern communication and radar systems. It is the optimum filter under a wide variety of criteria when the input noise is additive, white, and Gaussian. Even if this optimum filter cannot be exactly realized, it remains as the standard against which other realizations can be compared. The study of the matched filter here includes its derivation, its time and frequencydomain characteristics, its synthesis, its sensitivity to waveform changes, its relationship to crosscorrelation, and its characterization when the input noise spectrum is nonconstant. Also discussed is pulse compression, a topic of both theoretic and practical importance. For large timebandwidth products, the pulse compression system is essentially a matched filter system, thus an optimum detection system. The two methods of pulse compression considered here use the linear FM signal and the Barker sequences. The compressed waveforms include undesirable sidelobes, and we discuss methods of reducing them by appropriate filtering. Again, these filters are not included in the usual treatments of electronic filtering.

8 TimeDomain Operations
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We have obtained a value of system delay by minimizing the integrated squared error between the input signal and the appropriately delayed output signal. Most important, this theoretically obtained value is consistent with previous results and our physical reasoning. The various examples bear this out. The average delay time can also be denormalized for the BP equivalents of the previously discussed signals applied to a BP filter by multiplying by 1/πΔf, where Δf is the BP 3dB bandwidth in hertz and the LP prototype has unityradian cutoff frequency. The LP response then corresponds to the envelope of the BP response.

9 Digital Filtering
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This chapter introduces and reviews the important properties of the ztransform and shows how this transform is used to solve difference equations. From this development we obtain the system function and the discretetime convolution, both useful for design, analysis, and synthesis. Three design methods are discussed  two for recursive filters and one for nonrecursive filters. This chapter then examines sources of error in filter responses, introduces the fast Fourier transform, and discusses its application to digital filtering.

Back Matter
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