c13-5 (Numerical Recipes in C), страница 3

The function that you constructhas to have its poles only in the upper half-plane, for stability. It should also have theproperty of going into its own complex conjugate if you substitute −w for w, so that thefilter coefficients will be real.For example, here is a function for a notch filter, designed to remove only a narrowfrequency band around some fiducial frequency w = w0, where w0 is a positive number,H(f ) =w − w0w − w0 − iw0w 2 − w02=(w − iw0 )2 − w02w + w0w + w0 − iw0(13.5.14)In (13.5.14) the parameter is a small positive number that is the desired width of the notch, as aSample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited.

To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).w22w + a213.5 Digital Filtering in the Time Domain563(b)Figure 13.5.1. (a) A “chirp,” or signal whose frequency increases continuously with time. (b) Samesignal after it has passed through the notch filter (13.5.15). The parameter is here 0.2.fraction of w0 .

Going through the arithmetic of substituting z for w gives the filter coefficientsc0 =1 + w02(1 + w0)2 + w02c1 = −2c2 =1 − w02(1 + w0)2 + w021 + w02(1 + w0)2 + w02d1 = 2(13.5.15)1 − 2w02 − w02(1 + w0)2 + w02d2 = −(1 − w0)2 + w02(1 + w0)2 + w02Figure 13.5.1 shows the results of using a filter of the form (13.5.15) on a “chirp” inputsignal, one that glides upwards in frequency, crossing the notch frequency along the way.While the bilinear transformation may seem very general, its applications are limitedby some features of the resulting filters.

The method is good at getting the general shapeof the desired filter, and good where “flatness” is a desired goal. However, the nonlinearmapping between w and f makes it difficult to design to a desired shape for a cutoff, andmay move cutoff frequencies (defined by a certain number of dB) from their desired places.Consequently, practitioners of the art of digital filter design reserve the bilinear transformationSample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited.

To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).(a)564Chapter 13.Fourier and Spectral Applicationsfor specific situations, and arm themselves with a variety of other tricks. We suggest thatyou do likewise, as your projects demand.CITED REFERENCES AND FURTHER READING:Hamming, R.W. 1983, Digital Filters, 2nd ed.

(Englewood Cliffs, NJ: Prentice-Hall).Oppenheim, A.V., and Schafer, R.W. 1989, Discrete-Time Signal Processing (Englewood Cliffs,NJ: Prentice-Hall).Rice, J.R. 1964, The Approximation of Functions (Reading, MA: Addison-Wesley); also 1969,op. cit., Vol. 2.Rabiner, L.R., and Gold, B. 1975, Theory and Application of Digital Signal Processing (EnglewoodCliffs, NJ: Prentice-Hall).13.6 Linear Prediction and Linear PredictiveCodingWe begin with a very general formulation that will allow us to make connectionsto various special cases.

Let {yα0 } be a set of measured values for some underlyingset of true values of a quantity y, denoted {yα }, related to these true values bythe addition of random noise,yα0 = yα + nα(13.6.1)(compare equation 13.3.2, with a somewhat different notation). Our use of a Greeksubscript to index the members of the set is meant to indicate that the data pointsare not necessarily equally spaced along a line, or even ordered: they might be“random” points in three-dimensional space, for example. Now, suppose we want toconstruct the “best” estimate of the true value of some particular point y? as a linearcombination of the known, noisy, values. WritingXy? =d?α yα0 + x?(13.6.2)αwe want to find coefficients d?α that minimize, in some way, the discrepancy x? .

Thecoefficients d?α have a “star” subscript to indicate that they depend on the choice ofpoint y? . Later, we might want to let y? be one of the existing yα ’s. In that case,our problem becomes one of optimal filtering or estimation, closely related to thediscussion in §13.3. On the other hand, we might want y? to be a completely newpoint. In that case, our problem will be one of linear prediction.A natural way to minimize the discrepancy x? is in the statistical mean squaresense. If angle brackets denote statistical averages, then we seek d?α ’s that minimize*2 +X 2d?α (yα + nα ) − y?x? =α(13.6.3)XX =(hyα yβ i + hnα nβ i)d?α d?β − 2hy? yα i d?α + y?2αβαSample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use.

Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).Antoniou, A. 1979, Digital Filters: Analysis and Design (New York: McGraw-Hill).Parks, T.W., and Burrus, C.S. 1987, Digital Filter Design (New York: Wiley)..