US007831645B1

(12) United States Patent (10) Patent No.: US 7,831,645 B1 Berners et al. (45) Date of Patent: Nov. 9, 2010

(54) DIGITAL RESONANT SHELF FILTER OTHER PUBLICATIONS (75)75 Inventors: E.id P. s "E. S. R. SEGs) Azizi.Parametric Seyed-Ali. and Graphic “A New Equalizer Concept Banks'.of Interference Audio CompensationEngineering Soc., for Onanan S. ADel, Menlo Park, Convention Paper 5482, Sep. 21-24, 2001, New York, NY. pp. 1-8. (73) Assignee: Kind of Loud Technologies, LLC, Azizi,E.R.E. Seyed-Ali, “A New C t of Interf C tion f Santa Cruz, CA (US) Convention Paper 5629, May 10-13, 2002, Munich, Germany, pp. 1-8. (*) Notice: Subject to any disclaimer, the term of this Berners, David P. et al., “Discrete-Time Shelf Filter Design for patent is extended or adjusted under 35 Analog Modeling”. Audio Engineering Soc., Convention Paper, Oct. U.S.C. 154(b) by 1274 days. 11-13, 2003, New York, NY. pp. 1-5. Blesser, Barry A., et al., “Audio Dynamic Range Compression for (21) Appl. No.: 11/249,162 Minimum Perceived Distortion', IEEE Transactions on Audio and Electroacoustics, vol. AU-17, No. 1, Mar. 1969, pp. 22-32. (22) Filed: Oct. 11, 2005 (Continued) Primary Examiner Tan V Mai Related U.S. Application Data (74) Attorney, Agent, or Firm Pillsbury Winthrop Shaw (60) Provisional application No. 60/617,343, filed on Oct. Pittman LLP 8, 2004. s (57) ABSTRACT (51) Int. Cl. A method and system for designing a discrete-time filter G06 L/00 (2006.01) having a which approximates that of an G06F 7/10 (2006.01) - 0 52) U.S. C 708/3: 708/300 analog shelf filter is disclosed. Prior art methods include (52) U.Oa ------O- - A------O ------s apolV1ngpplying the bilib11 near tranSIOrmf to thehe analog filter,filter. Whichwhich hhas (58) Field of Classification Search ...... 708/8O1 3OO 708/3, 322 the drawback of warpingrping high-frequencynig quency features of the desired transfer function. In an embodiment of the present See application file for complete search history. invention, an analog filter is designed which anticipates the (56) References Cited warping imposed by the bilinear transform. For filters whose features approach the Nyquist limit, the inventive method U.S. PATENT DOCUMENTS provides a closer approximation to the analog response than 5,287,511 A * 2/1994 Robinson et al...... 717/106 direct application of the bilinear transform. 6,581,080 B1* 6/2003 Richards ...... TO8,320 2004/0042625 A1 3f2004 Brown ...... 381 (103 20 Claims, 8 Drawing Sheets

301 Sampling rate, f,

3OO analog filter, Ho(s)

prewarping bilinear transformation

input, r(t)

filtering US 7831645 B1 Page 2

OTHER PUBLICATIONS Mapes-Riordan, Dan, “A Worst-Case Analysis for Analog Quality (Alias Free) Digital Dynamics Processing'. J. Audio Eng. Soc., vol. Bohn, Dennis A., “Constant-Q Graphic Equalizers”. Audio Engi 47, No. 11, Nov. 1999, pp. 948-952. neering Soc., Convention Paper 2265-D-15. Oct. 12-16, 1985, New McGrath, David, et al., “Raised Cosine Equalization Utilizing Log York, NY. pp. 1-42. Scale Filter Synthesis'. Audio Engineering Soc., Convention Paper 6257. Oct. 28-31, 2004, San Francisco, CA, pp. 1-16. Deczky, Andrew G., “Unispherical Windows'. IEEE, 2001, pp. Olveira, A.J., “A Feed forward Side Chain Limiter/Compressor/De II-85-II-88. esser with Improved Flexibility”. J. Audio Eng. Soc., vol. 37, No. 4. Floru, Fred, “Attack and Release Time Constraints in RMS-Based Apr. 1999, pp. 226-240. Feedback Compressors”. J. Audio Eng. Soc., vol. 47, No. 10, Oct. Orfanidis, S. J., “Digital Parametric Equalizer Design with Pre 1999, pp. 788-804. scribed Gain'. J. Audio Eng. Soc., vol. 45, No. 6, Kraght, Paul, “ in Digital Clippers and Compressors'. J. Jun. 1997, pp. 444-455. Audio Eng. Soc., vol. 48, No. 11, Nov. 2000, pp. 1060-1065. Tarczynski, A., et al., “A WISE Method for Designing IIR Filters'. Lang, Mathias C., "Least-Squares Design of IIR Filters with Pre IEEE Trans. On Signal Proc., 2001, vol. 49, No. 7, Jul. 2001, pp. scribed Magnitude and Phase Responses and a Pole Radius Con 1421-1432. straint', IEEE, 2000, pp. 3109-3121. * cited by examiner U.S. Patent Nov. 9, 2010 Sheet 1 of 8 US 7831645 B1

101 Sampling rate,f 105

analog filter,o H(s) 1 OO

bilinear transformation

input, x(t) 107 H(z) 109 M filtering FG. 1 (PRIOR ART)

upSampledO p Samplingpling rate, nfsn: 2O5

2OO analog filter, H(s)

bilinear transformation

input, x(t) 212 Output, y(t) We 21 O 213 -W- filtering downsampling FIG. 2 (PRIOR ART) U.S. Patent Nov. 9, 2010 Sheet 2 of 8 US 7831645 B1

3O1 Sampling rate, f, 3O2 305

3OO analog filter, Ho(s) H(s) prewarping bilinear transformation

input, x(t) 308 Output, y(t)

N A? 307 M filtering FIG. 3 U.S. Patent Nov. 9, 2010 Sheet 3 of 8 US 7,831,645 B1

4. O 1 accept analog filter H0(s), Sampling ratef

4. O 2 select preserved filter features

4 O 3 determine bilinear warping Constant Td

404 fix shelf gain Yi

405 Compute Q1, Op I

406 apply bilinear transform to produce H(z)

FIG. 4 U.S. Patent Nov. 9, 2010 Sheet 4 of 8 US 7831645 B1

O III II

504 HH 24. IILM|||||1|| II II -1H III || || || || ||

|| ||108 ||1 frequency - Hz

FIG. 5 502 U.S. Patent Nov. 9, 2010 Sheet 5 of 8 US 7831645 B1

HillIIASU -

HW1

IAWHE H-1B|ISSM IIIIII

III || || ||O

FIG. 6 U.S. Patent Nov. 9, 2010 Sheet 6 of 8 US 7831645 B1

*NC)

frequency HZ

FIG. 7 U.S. Patent Nov. 9, 2010 Sheet 7 of 8 US 7831645 B1

10 frequency - Hz

FIG. 8 U.S. Patent Nov. 9, 2010 Sheet 8 of 8 US 7831645 B1

nvention - - - - - Prior Art Analog -

10 1 o' 10 Frequency, Hz

FIG. 9 US 7,831,645 B1 1. 2 DIGITAL RESONANT SHIELF FILTER Because of its increased warping at higher frequencies, the bilinear transform is a poor choice for discrete-time modeling CROSS-REFERENCE TO RELATED of filters with features near the Nyquist limit. APPLICATIONS Another prior art approach is to process the input signal at a high sampling rate, as shown in FIG. 2. By using a suffi The present application is based on, and claims priority ciently large factor, the Nyquist limit may be from, U.S. Provisional Applin. No. 60/617,343, filed Oct. 8, moved much higher than the relevant filter features, and the 2004, commonly owned by the present assignee, the contents warping of those features by the bilinear transform will be of which are incorporated herein by reference. This applica minimized. However, this approach has the drawback of addi tion is also related to commonly-owned and concurrently 10 tional computational cost, and also introduces artifacts due to filed U.S. application Ser. No. 1 1/249,159, the contents the upsampling process. thereof also being incorporated herein by reference. In a prior art method applicable to parametric sections, and described in S. J. Orfanidis, “Digital Parametric Equalizer FIELD OF THE INVENTION Design with Prescribed Nyquist-Frequency Gain”,Journal of 15 the Audio Engineering Society, Vol. 45, no. 6, pp. 444, June This invention pertains to the field of digital signal process 1997, Orfanidis developed formulas which translate filter ing, and in particular to equalization and filtering, especially center frequency, gain and bandwidth into the coefficients of of audio signals. a second-order having a transfer function which interpolated the levels of the analog parametric filter at DC, BACKGROUND OF THE INVENTION the center frequency, and the Nyquist limit. This approach does not suffer from warping of the frequency axis as do the In the recording, production and playback of audio, one prior art methods based on the bilinear transform, and has the important and widely used tool is equalization, the manipu benefit that it is order preserving, and as Such, the resulting lation of signal level and phase as a function of frequency. filter is efficient to implement. Equalization may be used to correct problems in a recorded 25 No such method is available for analog shelf filters. There signal, for instance to eliminate unwanted resonances in a remains a need in the art, therefore, to develop an efficient, drum track by Suppressing energy in selected frequency order-preserving method for designing a digital filter having bands, or to reduce a singer's lisp by enhancing certain high a transfer function which approximates that of an analog shelf frequencies. Equalization is also often used for artistic pur filter. poses to give each instrument its own space in the audio band, 30 or to create a certain feel—different genres of music, for instance, typically have different characteristic power spec SUMMARY OF THE INVENTION tra. A method and system for designing a discrete-time filter In mixing and mastering, probably the most commonly having a transfer function which approximates that of an used equalizers are parametric sections and shelf filters. Para 35 analog shelf filter is disclosed. According to one aspect of the metric sections enhance or suppress a selected band of fre present invention, an intermediate analog filter is designed quencies, whereas shelf filters apply again to all frequencies which anticipates the warping imposed by the bilinear trans either above or below a prescribed frequency. Both first-order form. When the bilinear transform is applied to the interme and second-order shelf filters are used, with the second-order diate analog filter, a discrete-time filter is obtained that more or so-called resonant shelf filters sometimes having a reso 40 nant peak on one side or the other of the transition between its closely matches the desired characteristics of the analog filter low-frequency and high-frequency gains. For filters whose features approach the Nyquist limit, the Analog equalizers, such as the Pultec EQP-1A from Pulse inventive method provides a closer approximation to the ana Techniques, Inc. of Englewood, N.J., are highly prized for log response than direct application of the bilinear transform. their Sonic characteristics and user controls. It is desired to 45 make digital emulations of these analog equalizers for use BRIEF DESCRIPTION OF THE DRAWINGS with digital audio workstations so as to provide the advan tages of a Software environment. Such as automation, along These and other aspects and features of the present inven with the sought-after Sonic characteristics of vintage equal tion will become apparent to those ordinarily skilled in the art izers. 50 upon review of the following description of specific embodi One of the most important characteristics of an analog ments of the invention in conjunction with the accompanying equalizer is its transfer function, which results from a set of figures, wherein: s-plane poles and Zeros. To emulate Such a filter in discrete FIG. 1 illustrates a prior art system for digital equalization time, two prior art approaches are used. In the first, the analog of a signal according to the characteristics of a desired analog prototype filter is converted to the Z-plane via the bilinear 55 filter, in which the desired analog filter is taken to the discrete transform, as shown in FIG. 1. time domain via the bilinear transform; As is known, bilinear transforms map the entire continu FIG. 2 illustrates a system for digital equalization of a ous-time frequency axis onto the unit circle in the Z-plane. signal according to the characteristics of a desired analog See, e.g., A.V. Oppenheim and R. W. Schafer, Discrete-Time filter, in which the signal is upsampled and processed at the , Prentice Hall, 1989. Using a bilinear 60 upsampled rate; transform, analog prototype filters can be carried into the FIG. 3 illustrates an embodiment of the inventive system discrete-time domain while preserving filter order and stabil for digital equalization of a signal according to the features of ity. Through the use of the bilinear warping constant, a single a desired analog filter, in which the features are prewarped; frequency in the continuous-time domain can be positioned FIG. 4 illustrates an example digital shelf filter design anywhere on the unit circle in the Z-plane. All other frequen 65 method according to the invention; cies will be displaced by various amounts, with the amount of FIG. 5 is a graph illustrating a first-order high shelf filter warping becoming more severe near the Nyquist limit. transfer function magnitude; US 7,831,645 B1 3 4 FIG. 6 is a graph illustrating a family of resonant shelf criteria are given for the selection of features to be fixed in the filters; filter design, and methods are disclosed for determining filter FIG. 7 is a graph illustrating an aspect of an iteration coefficients based on the desired features. establishing parameters of the prewarped analog filter, The overall approach of the present invention, therefore, is FIG. 8 is a graph illustrating example desired and pre to design an analog prototype filter in Such away that when it warped analog shelf filters in accordance with the invention; is warped to the digital domain via a bilinear transform, and important transfer function features are mapped to the desired FIG. 9 illustrates an example shelf filter design in accor frequencies. Specifically, the solution of the present invention dance with the invention, along with that of a prior art method. is to design a prewarped analog prototype filter which is then 10 DETAILED DESCRIPTION OF THE PREFERRED converted to a digital filter via abilinear transform. EMBODIMENTS An embodiment of the inventive system is shown in FIG.3. As shown in FIG. 3, given an analog filter Ho(s) and sampling The present invention will now be described in detail with rate f, a pre-warping process 302 is performed to produce an reference to the drawings, which are provided as illustrative intermediate pre-warped analog filter H(s) and a bilinear examples of the invention so as to enable those skilled in the 15 warping constant T. Upon application of the bilinear trans art to practice the invention. Notably, the figures and form 305 using these pre-warped values, the result is a dis examples below are not meant to limit the scope of the present crete-time filter H (Z) having the desired specifications. invention to a single embodiment, but other embodiments are In one example, the inventive system in FIG. 3 is imple possible by way of interchange of some or all of the described mented as Software operating on a computer system Such as a or illustrated elements. Moreover, where certain elements of the present invention can be partially or fully implemented Mac or Windows compatible system. In such a system, the using known components, only those portions of Such known inventive system can be included in a plug-in application for components that are necessary for an understanding of the an audio production platform such as Pro Tools TDM HD present invention will be described, and detailed descriptions from Digidesign of San Francisco, Calif. Such applications of other portions of such known components will be omitted 25 and platforms can include many other features and function So as not to obscure the invention. In the present specification, alities that are not necessary for an understanding of the an embodiment showing a singular component should not be present invention, Such as receiving and formatting an audio signal input and analog filter characteristics, interacting with considered limiting; rather, the invention is intended to a user or other software components to select and/or change encompass other embodiments including a plurality of the processing parameters, and providing an audio signal output same component, and Vice-versa, unless explicitly stated oth 30 erwise herein. Moreover, applicants do not intend for any to further applications. Those skilled in the art will be able to term in the specification or claims to be ascribed an uncom understand how to implement the inventive compressor in mon or special meaning unless explicitly set forth as such. Such plug-in applications and platforms after being taught by Further, the present invention encompasses present and future the present specification. However, the present invention is known equivalents to the known components referred to not limited to such applications. herein by way of illustration. 35 The approach of the present invention will now be The inventive methods described herein will be discussed described in more detail beginning with reference to a typical with respect to digital filters configured as second-order fil high shelf filter as shown in FIG. 5. In particular, consider that ters. It is understood, however, that the techniques discussed a first-order analog high shelf filter has a transfer function herein apply, with modifications that should be clear to those given by skilled in the art, to other filter structures, such as lattice and 40 2 Vy (1) ladder filters, other filter orders, and to any number of imple Y 2 -- S -- 1 s (s mentation platforms, such as signal processing microproces H(s) = 1 2 sors and other discrete time systems. s? + is + 1 Generally, the present invention recognizes that the sec (to (s ond-order shelf filter typically makes use of two complex 45 conjugate poles and two complex-conjugate Zeros to accom As illustrated in FIG. 5, the high shelf filter may be param plish shelving. If the poles lie at a higher frequency than the etrized by the shelf frequency (), 502 and the high-frequency Zeros, the shelf is either a low-cut shelf or a high-boost shelf; gain Y 504. Standard analog second-order shelf-filter design otherwise, it is a low-boost or high-cut shelf. The poles and for a high-shelf filter at frequency (), with asymptotic high Zeros can have independently varying amounts of resonance. 50 Often, one set of features (either the poles or zeros) will be frequency gain Y leads to a transfer function of either critically damped or slightly underdamped, and the 2 (2) other set of features will have an adjustable amount of reso Y-s? -- V. 1 co (s aCC. H(s) = - 2 The present invention further recognizes that in the design 55 s? + S + 1 of the discrete-time filter, five features of the transfer function (to (s can be fixed, as afforded by the five independent coefficients available for use in the biquad (Eq. O). For a high-shelf cut-filter, the transfer function becomes bo + bi z' + b2: (0) 1 2, 2 (3) H(z) = 1 + 1 + as 2. 60 is + -s + 1 H(s) = - y 2 Vy - S1.2 -- S + 1 Salient features of the shelf include: low- and high-fre 2 quency asymptotic gains, resonant frequencies of the poles (to (s and Zeros, and response magnitudes at the resonant frequen cies. Depending upon the locations and amounts of damping 65 where 1/y becomes the high-frequency asymptotic gain. The of the shelf resonances, each of these six features may shelf transfer function can be generalized for a resonant shelf become more or less important. In the present invention, filter to: US 7,831,645 B1 6

Vy (4) (6) S + 1 Q. Cos 2's . S + 1 1 1 H(s) = a -- (2 is + S + 1 to, "I (to Qplwl

This transfer function is shown in FIG. 6. Note that a resonant The bilinear transform, given by: feature 602 may appear on either side or both sides of the 10 transition. According to the invention, therefore, to carry the shelf 2 (1 -: (7) filter from the continuous-time to the discrete-time domain, S the steps outlined in FIG. 4 may be taken. The first step 402 T, 1 + 1 after identifying the analog filter in 401 is to determine which 15 features of the filter's transfer function are desirable to pre SWC. is applied to the pre-warped analog filter lids) with a bilinear According to one aspect of the invention, the analog filter warping constant given by (as shown in step 403 in FIG. 4) can be fully described by specification of the following fea tures: Asymptotic Gain at DC: Asymptotic Gain at High Frequency; Q Associated With Poles; Q Associated With (8) Zeros; Shelving Frequency where, for a high-shelf filter, the shelving frequency is defined as the frequency of the poles for a boost filter, or the frequency of the Zeros for a cut filter. Those skilled will understand how to derive these features 25 Since the resonant frequency of the shelf filter is (), this from a specified transfer function Such as (4). choice of Td will prevent the resonant frequency from being In further accordance with this example of the invention, in displaced by the bilinear transform. Thus, we can choose the discrete-time domain, there is a different set of features that may be more relevant for describing the transfer function: () ()o. Frequency Locations of Resonant Peaks; Gains at Resonant 30 The DC gain of the shelf filter will be preserved across the Peaks: Asymptotic Gain at DC: Gain at Nyquist Limit. These bilinear transform, so nothing needs to be done in order to features will be known to those skilled in the art, and can be preserve this feature. However, in order to preserve the filter easily derived from a specified transfer function such as (O). gain at the Nyquist limit, it is preferable to alter the transfer In general, second-order filters have five degrees of free function in the analog domain. dom. For filters having only one resonant feature, there are 35 The gain of the discrete-time shelf filter at the Nyquist limit only four characteristics to match between the domains—the will be equal to the asymptotic high-frequency gain of the DC and band-edge gains, and the location and gain of the analog prototype. Accordingly, in the next step 404 of the resonant peak. Discrete filter design can thus be carried out design process, we need to choose with the addition of a fifth constraint. In one example of the invention, the filter gain at the frequency of the non-resonant 40 feature is used to complete the set of constraints. y1 = Ho (f, f2), or (9) For filters without any resonant features, the constrained characteristics may be the DC and band-edge gains, the gains (10) at the frequencies of the two non-resonant features, and the location in frequency of one of the non-resonant features. 45 For shelf filters with two resonant peaks, the system is overdetermined: The digital filter is characterized by six fea tures. For accurate modeling of such filters in the discrete time domain, higher order filters could be used. However, it is Suggested that the methods disclosed herein be employed to 50 where (), 27tf/2. design a filter which matches all the features, save the fre At this point, all parameters have been determined except quency of the broadest resonant feature. Q and Q. Accordingly, in step 405 of FIG.4, these param Consider the following analog prototype resonant shelf eters are determined next. filter. 55 According to one example of the invention, these two parameters may be determined iteratively, in Such a way that w (5) the resulting filter has the desired gains at () and (i)/sqrt(Y), 2s2 S + 1 the characteristic frequencies of the filter poles and Zeros. Ho(s) = () 0 OuO. . In order to carry out the iteration, we first need to obtain the s? + S + 1 60 desired gains Hi Desired (c)) and |Hipse (()1/sqrt(Y)) . At (O6 Qp00 co, this is straightforward. Since we have chosen T in Such away that () is not displaced by the bilinear transform, we can As discussed above in connection with an embodiment of simply set H(c))=|H(c)). However, in order to the inventive system shown in FIG. 3, an aspect of the inven determine the desired gain for H at (i)/sqrt(Y), we need to tion is to design a pre-warped analog filter H(s) which, upon 65 evaluate Ho at the frequency to which (i)/sqrt(Y) will be application of the bilinear transform, will result in a discrete warped by the bilinear transform. This frequency is readily time filter meeting the specifications: found as US 7,831,645 B1 8 This exhausts the five degrees of freedom afforded by the biquad structure. As can be seen in FIG. 10, the inventive (11) method provides a very close match to the desired analog ()) = 2A. at tal 2 Vy | S filter characteristic, even at high frequencies, where the prior art method performs poorly. Before concluding, note that when the Zeros rather than the leading to H,(c)1/sqrt(Y)-H(c)-hat). poles are resonant, the design method is similar, but with the We now are able to set up an iterative adjustment to Q, and bilinear warping constant given by Q. Q will be adjusted based upon IH (co-)l, while Q will 10 be adjusted according to H(c)/sqrt(Y)). Frequencies other (13) than () and ()/sqrt(Y) could be used as the basis for deter T - 2 mining Q, and Q. However, as illustrated in FIG. 7 which d coo/ Wyo are: shows the pole and Zero transfer functions for a set of Q values, since () and cofsqrt(Y) are the natural frequencies of 15 This is because the biggest resonance of the filter is now at the poles and Zeros of the filter, this choice provides the most (D/sqrt(Yo). In order to preserve this resonance position when robust and quickly converging iteration: a change in will adjusting the gain at the Nyquist limit, for this filter () and affect the gain of H at () more than at any other frequency, sqrt(Y) must be adjusted proportionally, rather than simply and a change in Q will affect the gain of H at ()/sqrt(Y) adjusting () as before. All other design steps remain more than at any other frequency. unchanged. The iteration works as follows: Beginning with Q-Qo Although the present invention has been particularly and Q-i-Qo. described with reference to the preferred embodiments 1. QQ1|Hose (())/H1 (CD1) thereof, it should be readily apparent to those of ordinary skill in the art that changes and modifications in the form and 3. repeat 25 details may be made without departing from the spirit and After a handful of iterations, Q, and Q will have con Scope of the invention. It is intended that the appended claims Verged to stable values. A very conservative nine iterations encompass such changes and modifications. may be used to good effect for a wide variety of shelf filters. What is claimed is: The simple update expressions within the iteration result 1. A computer-implemented method for equalizing a sig 30 nal, comprising: from the fact that, at the natural frequencies, the zeroth and accepting, at the computer, a desired analog shelf filter; second order terms of a resonator combine to Zero, leaving the determining a set of desired features describing the desired entire gain of the resonator proportional to the linear term: analog filter, the set of desired features including at least one feature of poles or Zeroes of the desired analog filter; 35 accepting, at the computer, a sampling rate; 1 (12) forming, by the computer, a bilinear warping constant based on the sampling rate and the set of desired fea tures; and determining, by the computer, a set of prewarped features Consider an example high-shelf analog filter Ho with a DC 40 describing a prewarped analog shelf filter, Such that gain of unity, a high-frequency gain of Yo 2, a shelving fre when warped through the bilinear transform according quency fo Coo/2t of 8 kHz, a pole resonance of Qo sqrt(2), to the bilinear warping constant, the set of prewarped and a zero resonance of Q-sqrt(2)/2. FIG. 8 shows the features align with the set of desired features, including responses of the filters Ho and H, designed in accordance the feature of the poles or zeroes. with the teachings of this invention for a discrete-time Sam 45 2. A method according to claim 1, further comprising: pling rate of 44.1 kHz. The original filter Ho appears as a solid forming a discrete-time shelf filter by applying abilinear line, the pre-warped filter H dashed. Various points on the transform with said warping constant to the prewarped plots show how the effects of the bilinear transform are analog shelf filter. accounted for by the pre-warped filter. The high-frequency 3. A method according to claim 2, further comprising: asymptote of filter H will be warped to point A on Ho, the 50 accepting said signal; and Nyquist limit. Point B shows the natural frequency of the applying said discrete-time filter to said signal. resonant poles. This point is kept fixed through our choice of 4. A method according to claim 1, wherein the set of bilinear warping constant T. Finally, point C on H will be desired features comprises two or more of asymptotic gain at warped to point Don Ho. DC, asymptotic gain at high frequency, Q associated with Finally, as shown in step 405 of FIG.4, a discrete-time filter 55 poles, Q associated with Zeros, and shelving frequency. H (Z) can be obtained by applying the bilinear transform to 5. A method according to claim 1, wherein the set of H(s) with warping constant T. (2/coo)tan(()1/2f ). desired features includes a shelving frequency (), and FIG. 10 shows the desired analog filter transfer function wherein the bilinear warping constant T is formed according along with digital filters designed with the prior art method of tO: applying the bilinear transform, and the inventive method of 60 prewarping followed by application of the bilinear transform. Comparing the inventive discrete-time filter to the analog T = (2 f (too) tan (coof 2f ) filter yields the following results: responses match at DC; responses match at Nyquist limit; responses match at charac teristic frequency of poles; responses match at characteristic 65 where f is the sampling frequency. frequency of Zeros of the discrete-time filter; characteristic 6. A method according to claim 1, wherein the set of frequencies of poles match for the two filters. desired features includes a shelf gain Yo, Qo associated with US 7,831,645 B1 10 poles, Q associated with Zeroes, and shelf frequency (Do, and 15. A system according to claim 12, wherein the corre the set of prewarped features includes shelf gain Y which is sponding set of features of the specified analog filter includes computed by: a shelfgain Yo, Qo associated with poles, Qo associated with Zeroes, and shelf frequency (), and the set of features y1 = Ho (f. (2), or includes shelf gain Y1 which is computed by: y1 = Sqr{ (1-(a), (yo/Loo)+(co, (yo/IQollao)) (1-(co, if cool?) + (a,f Oolool) y1 = Ho (f. (2), or where (), 27tf/2 and f is a sampling frequency. 10 y1 = Sqr{ (1-(a), (yo/Lao)+(co, (yo/IQol (cool) 7. A method according to claim 1, wherein the set of pre (1-(co, if cool?) + (a,f(Qol (cool) warped features include Q associated with poles Q, and Q associated with Zeroes Q, and wherein the determining step includes iteratively determining Qp1 and QZ1 through an where (), 27tf/2 and f is a sampling frequency. iterative process using desired gains at () and (i)/sqrt(Y), the 15 16. A system according to claim 12, wherein the set of characteristic frequencies of the filter poles and Zeroes. features include Qassociated with poles Q, and Qassociated 8. A method according to claim 1, wherein the feature with Zeroes Q, and wherein the function includes a function includes one or both of a frequency location of the poles or that iteratively determines Qp1 and QZ1 through an iterative Zeroes and again associated with the poles or Zeroes. process using desired gains at () and (0/sqrt(Y), the charac 9. A method according to claim 1, wherein the poles or teristic frequencies of the filter poles and Zeroes. Zeroes are associated with a non-resonant feature of the desired analog filter. 17. A system according to claim 12, wherein the feature 10. A method according to claim 1, wherein the poles or includes one or both of a frequency location of the poles or Zeroes are associated with a resonant feature of the desired Zeroes and again associated with the poles or Zeroes. analog filter. 25 18. A system according to claim 12, wherein the poles or 11. A method according to claim 1, wherein the set of Zeroes are associated with a non-resonant feature of the desired features includes at least one feature of poles and at desired analog filter. least one feature of Zeroes of the desired analog filter. 19. A system according to claim 12, wherein the poles or 12. A system for equalizing a signal, the system including Zeroes are associated with a resonant feature of the desired a computer that is adapted to perform a plurality of functions 30 analog filter. comprising: 20. A computer-implemented method for digitally process a function to convert a specified analog shelf filter to a ing a signal, comprising: prewarped analog shelf filter having a set of features, accepting, at the computer, a desired analog shelf filter; which when warped through a bilinear transform, will determining a frequency location of at least one pole or align with the corresponding set of features of the speci 35 Zero of the desired analog shelf filter, a shelf gain and a fied analog shelf filter, the set of features including at shelf frequency of the desired analog filter; least one resonant feature; and abilinear transformation function that produces a digital accepting, at the computer, a sampling rate; filter version of the prewarped analog shelf filter that is forming, by the computer, a bilinear warping constant operable in the discrete-time domain. 40 based on the sampling rate and the shelf frequency; and 13. A system according to claim 12, wherein the set of determining, by the computer, an intermediate pole or Zero features comprises two or more of asymptotic gain at DC, frequency location, an intermediate shelf gain and an asymptotic gain at high frequency, Qassociated with poles, Q intermediate shelf frequency of a prewarped analog associated with Zeros, and shelving frequency. shelf filter, such that when the prewarped analog shelf 14. A system according to claim 12, wherein the set of 45 filter is warped through the bilinear transform according features includes a shelving frequency (), and wherein the to the bilinear warping constant, a resulting digital filter function includes a function to form a bilinear warping con has a resulting pole or Zero frequency location, resulting stant T. according to: shelf frequency and resulting shelf gain that are approxi 50 mately the same as the pole or Zero frequency location, T = (2 f(u0) tan (coof 2f ) shelf frequency and shelf gain. where f is a sampling frequency.