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@conference{schmidt2010digital, title = {Digital Equalization Filter: New Solution to the Presented at the 128th Convention Frequency Response Near Nyquist and Evaluation Listening 2010 May 22–25 byLondon, UK Tests}, author = {Schmidt, Thorsten and Bitzer, Joerg}, booktitle Engineering Society The papers=at{Audio this Convention have been selected Convention on the basis of128}, a submitted abstract and extended precis that have month = {May}, been peer reviewed by at least two qualified anonymous reviewers. This convention paper has been reproduced from the author’s advance manuscript, without editing, corrections, or consideration by the Review Board. The AES takes year = {2010}, no =responsibility for the contents. Additional papers may be obtained by sending request and remittance to Audio url {http://www.aes.org/e-lib/browse.cfm?elib=15261} nd } Engineering Society, 60 East 42 Street, New York, New York 10165-2520, USA; also see www.aes.org. All rights reserved. Reproduction of this paper, or any portion thereof, is not permitted without direct permission from the Journal of the Audio Engineering Society.

Digital equalization filter: New solution to the frequency response near Nyquist and evaluation by listening tests Thorsten Schmidt1 , Joerg Bitzer2 1

Cube-Tec International, Anne-Conway-Str. 1, 28359 Bremen, Germany (http://www.cube-tec.com)

2

Institute of Hearing Technology and Audiology (IHA), Jade-University of Applied Sciences, Ofener Str. 16, 26121 Oldenburg, Germany (www.hoertechnik-audiologie.de)

Correspondence should be addressed to Joerg Bitzer ([email protected]) ABSTRACT Current design methods for digital equalization filter face the problem of a frequency response increasingly deviating from their analog equivalent close to the Nyquist frequency. This paper deals with a new way to design equalization filters, which improve this behavior over the entire frequency range between 0 Hz (DC) and Nyquist. The theoretical approach is shown and examples of lowpass, peak- and shelving-filters are compared to state-of-the-art techniques. Listening tests were made to verify the audible differences and rate the quality of the different design methods.

1. INTRODUCTION Equalization is one of the most important tasks during the mixing and mastering process. In this process the sound is modified depending on its spectral characteristics. One tool often used for this task is the parametric equalizer which allows to spectrally balance the sound by filtering the signal with a fixed set of adjustable filter parameters. Apart from the conventional filter types like low- and high-

pass, peak- and shelving filters are applied to the signal. Usually, the filter itself is implemented as a second order infinite impulse response filter (IIR). In order to compute the digital filter coefficients from the design parameters like frequency (fc ) , gain g and bandwidth (or quality factor Q) many methods exist e.g. [9, 7, 10, 4]. Most methods start with the design in the analog domain and utilize the bilinear transform (BLT) to get the digital counterpart

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In the next section we will briefly introduce different algorithms for the design, show the effects of the BLT for the different filter types, and introduce some known and some new countermeasures for it. In section 3 we will answer the question if the differences between the different methods are audible by using two different listening test methodologies. In the final section conclusions for this contribution will be drawn. 2. USED ALGORITHMS In the following subsections we will discuss different design methods for the most common filter types in parametric equalizers, which are cut-filters (lowpass only), peak-filters and shelving-filters. 2.1. Lowpass filter In the analog domain an all-pole second order lowpass filter is defined by the s-domain transfer function (TF) ω2 (1) H(s) = 2 ωcc s + Q s + ωc2 where ωc is the cutoff frequency (−3dB) and Q is the so-called quality parameter. Usually Q is defined for a maximally flat transfer function (ButterworthDesign Q = √12 ) at ω = 0. The digital equivalent can be obtained by different methods. All of them have advantages and disadvantages. In this contribution the main goal is to mimic the analog transfer function as well as possible, without introducing any side-effects. However, we will discuss the main features for the given methods. 2.1.1. Bilinear Transform (BLT) The standard design method for lowpass filter design is to transform the s-domain TF to the digital z-domain TF by using the bilinear transform (The

usual 2/T is not necessary if eq. 3 is used.): s=

z−1 z+1

(2)

As mentioned in the introduction, this transformation maps the analog frequency axis from zero to infinity to the digital frequency axis from zero to half of the sampling rate (Nyquist frequency). Since we know exactly what happens we can compensate the deviation for the cutoff frequency by using the so-called tangens pre-warping. fc (3) ωc = tan π fs However, all other frequencies are changed, which leads to a deviation of the TF. In figure 1 a comparison between analog and digital designs are given. Note that the deviation is only significant if fc exceeds fs /8 and that the −3dB points are matched exactly. 0

|H(f)|2 in dB

[8]. Unfortunately, this transformation maps the frequency infinity in the analog domain to the Nyquist frequency (half of the sampling frequency fs ) in the digital domain forcing the digital filter at DC and Nyquist to have the same gain as the analog filter at these frequency points respectively . Due to this compression of the frequency axis the frequency response of the analog and digital filter deviate from each other, especially at frequencies close to Nyquist.

−10 −20 −30 −40 300 500

analog digital 1k 2k 3k 5k 10k Frequency in Hz →

20k

Fig. 1: Comparison of the analog and digital transfer functions for a second order lowpass filter with √ Q = 1/ 2 (Butterworth-Design) for different cutofffrequencies. 2.1.2. Oversampled bilinear Transform One universal workaround for all filter types is to apply oversampling to the signal, which can be found in many professional software and hardware products on the market. This method resamples and filters the signal at a higher sampling rate, reducing the effect of the nonlinear mapping of the BLT

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x 10

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at the frequencies of interest. The drawback of this method are the additional steps needed for upsampling, downsampling and the anti-aliasing filter, which will introduce an artificial delay. In our contribution we use a 128th order FIR filter with half-band least-squares design. We chose this method because it can be implemented easily and is very efficient, since every second coefficient is zero due to the halfband design, and the impulse response is symmetric because of its linear phase. Therefore, we have to store only 33 coefficients and the filtering process can be simplified significantly. In figure 2 we can see the TF of the designed filter. It will only reduce the signal above 20kHz and the aliasing artefacts are reduced down to -100dB below 20kHz. Therefore this filter should be transparent for stationary audio signals. For transient signals the impulse response should be short enough so no pre-echo can be perceived. We tested our oversampling algorithm with several audio files and the remaining differences were below the quantization level for 24Bit signals. For uniformly (range -1 to 1) distributed white noise signals the error signal has only the power of noise at the least significant bit. Therefore, we can state that the oversampling algorithm is transparent.

−10 −20 −30 −40 300 500

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20 30 Frequency in kHz →

20k

2.1.3. Minimum Phase Matching (MiMa) In our approach we designed filters using the frequency domain least squares algorithm (FDLS) which was explained in 2007 by Greg Berchin [1] combined with a phase-reconstruction via HilbertTransformation. The FDLS method works entirely in the digital domain which lets us design filters out of magnitude and phase data only: As is known the output y(k) of a digital filter to an input signal x(k) is determined by a set of nonrecursive and recursive filter coefficients:

−120 10

1k 2k 3k 5k 10k Frequency in Hz →

Fig. 3: Comparison of the analog and digital transfer functions for a second order lowpass filter with √ Q = 1/ 2 (Butterworth-Design) for different cutofffrequencies implemented with an oversampling factor of two.

−100 0

analog digital

40

Fig. 2: Transfer function of the FIR filter used for the upsampling process. The passband is shown in the zoomed section.

The comparison between the analog and digital solution with oversampling is shown in figure 3. The deviation is reduced significantly, however, the complexity of the algorithm, the needed computer power, and an additional latency are the drawbacks of this approach.

y(k) =

N X

n=0

bn x(k − n) −

D X d=1

ad y(k − d)

(4)

where N-1 specifies the number of past input and D the number of past output values. For simplified usage we can write the same expression in matrix notation where x is the input x = x(k) ... x(k − N ) −y(k − 1)... − y(k − D) (5) and c the coefficient vector: c = b0 ... bN a1 ... aD (6) Combining both, the equivalent to equation 4 is: y(k) = xcT


(7)

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0 −0.5

Φ(f)

−1 −1.5 −2 −2.5 −3 300 500


20k

Fig. 4: Comparison of the analog and digital phase function for a second order lowpass filter with √ Q = 1/ 2 (Butterworth-Design) for different cutofffrequencies.

The corresponding transfer function in the frequency domain is given by:

H(ejΩ ) =

N P

n=0 D P

bn e−jΩn (8) ad e−jΩd

Using these basic relationships we can set up the input matrix x with the desired magnitude and phase values for a set of frequencies. The last step is to use eq. 7 and solve the resulting system of linear equations for the coefficients vector c. Due to the fact that the number of designing triplets (frequency, magnitude, phase) is probably higher than M + B our system will have more equations than unknowns. These kind of equation systems are called overdetermined and usually have no single solution. Instead it is possible to calculate a solution that comes close in the sense of least squares utilizing numerical methods like the Q-R-decomposition. In order to create a lowpass filter whose magnitude response matches its analog equivalent we have to calculate the magnitude response out of the analog TF (equation 1) either for equidistant or a fixed set of salient frequencies and take them as our input for the FDLS method. Unfortunately, we cannot use the phase from the analog design in the same manner, because of high deviation near Nyquist as a result of a different magnitude-phase-relationship in both domains. In contrast to the digital domain the phase for analog filter does not have to be a multiple of π at Nyquist. As a consequence the error in the magnitude response increases for non appropriate

d=1

and the phase response: Φ(Ω) = arg H(ejΩ )

(10)

0

|H(f)|2 in dB

which can be further separated into the magnitude response: Hm (Ω) = |H(ejΩ )| (9)

−10 −20

with Ω = 2πfc /fs .

−30

Assuming that the input signal at any discrete time k consists of a single frequency Ω1 with an amplitude A1 x1 (k) = A1 cos(Ω1 k) (11)

−40 300 500

the filtered output is the same signal but shifted in phase and multiplied in amplitude with the phase, or magnitude function, respectively, at that frequency point. y1 (k) = A1 |Hm (Ω1 )| cos(Ω1 k + Φ(Ω1 ))


20k

Fig. 5: Comparison of the transfer functions for a second order analog and √ a third order digital lowpass filter with Q = 1/ 2 (Butterworth-Design) for different cutoff-frequencies implemented with the MiMa method.

(12)


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Φ(Ω) = −H {log(Hm (Ω))}

(13)

where H is the Hilbert-Transformation. A solution proposed in [5]. This allows us to create one possible phase response with a valid magnitude-phase relation while simultaneously guaranteeing stability and a minimized overall group delay. One aspect we have not discussed yet is the selection of the filter order for the MiMa method. Usually, the filter order relies on the complexity of the filter shape, more precisely the number of turning points and the steepness of the magnitude response. For conventional lowpass, peak and shelving second order filters are sufficient. In the MiMa method the number of coefficients, hence the filter order, can be viewed as dimensions in which the FDLS algorithm has to minimize the mean squared error that makes it difficult to predict the required filter order for a certain level of accuracy. Berchin suggests to find the order experimentally. We found out that for our field of application a second order filter produces good results but an additional dimension nearly nullifies the difference between analog and digital shape, without heavy impacts on filter performance, stability and delay but at the cost of additional design performance. Therefore, we use third order filters for all filter types designed with the method presented in this section. Figure 5 shows the example of different magnitude responses for third order lowpass filters designed with the MiMa method. 2.2. Peak filter The most used equalizer filter is the peak filter with the parameter gain G and frequency ωm plus a parameter for the bandwidth, which is defined differently in papers and products, but can be transformed to all other forms as shown in [4]. In this contribution we will use the definition of [4], where the frequencies defining the bandwidth are given at half of the gain in dB.

2.2.1. Standard (Bilinear Transform) The analog peak filter transfer function can be defined as √ 2 s2 + ωmQ G s + ωm (14) H(s) = 2 2 s + Qω√mG s + ωm where Q denotes the relative bandwidth ωm /B and B is the absolute bandwidth at G/2 in dB. Applying the BLT leads to the digital coefficients given in the well-known EQ-cookbook [3]. Figure 6 shows some design examples with this solution, if the parameters are G = 12dB, Q = 1 and the frequency varies from 440Hz to 20kHz in half-octave steps. The working frequency fm is met precisely due to the tangens pre-warping. However, the bandwidth at higher frequencies has nothing to do with the original analog design, due to the bandwidth compression by the BLT. One countermeasure for 12 10 |H(f)|2 in dB

phase functions when the algorithm tries to minimize the mean squared error between both quantities. Figure 4 shows an example for the deviation of the analog and digital phase functions. The solution to this problem is to use minimum-phase systems which have a unique relationship between magnitude and phase which is defined via:

8 6 4 2 0 300 500


20k

Fig. 6: Peak filter designs with standard BLT (Gain = 12dB, Q = 1 and variable frequency). the compression is to adapt the Q factor for higher working frequencies. This was proposed in [4]. For the definition given in eq. 14, the adjustment would be sin 2π ffms Qa = Q (15) 2π ffms If we compare figure 6 with the adjusted solution given in figure 7, we can see that the bandwidth is much closer to the original, but still deviates at


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higher frequency fm . Furthermore, the TF between the peak and the Nyquist frequency is distorted heavily, because of the BLT.

10 |H(f)|2 in dB

12 10 8

0 300 500

6 4

0 300 500

4 2

8

2

6


20k

Fig. 7: Peak filter designs with BLT and adjusted bandwidth Qa (Gain = 12dB, Q = 1 and variable frequency).

2.2.2. Oversampled bilinear Transform As described in section 2.1.2 oversampling is an universal strategy and will work as well for peak filter designs. In figure 8 the comparison to the desired analog version is shown. We can see that this method is much better compared to the former designs especially at very high peak frequencies fm . 2.2.3. Orfanidis with Modification The main drawback of the modified BLT design is the distortion of the TF between the peak and the Nyquist frequency. An alternative solution with a prescribed gain at Nyquist was presented in [7], which is used in many products on the market. The solution was widely adopted, mainly because the Matlab code is freely available. However, the code given in [7] will not work with our bandwidth definition, if the upper frequency is above fs /2. Therefore, we compute the bandwidth between fs /2 and the corresponding lower frequency at the same gain level based on the analog design, and we use this as the design parameter for [7]. Figure 9 shows the resulting TFs with the same parameter as in the preceding examples. We can see that the resulting


20k

Fig. 8: Peak filter designs with oversampled BLT and adjusted bandwidth Qa (Gain = 12dB, Q = 1 and variable frequency).

designs are close to the analog counterparts. However, a deviation especially at the rising curve from 0Hz to fm exists. 12 10 |H(f)|2 in dB

|H(f)|2 in dB

12

8 6 4 2 0 300 500


20k

Fig. 9: Peak filter design after Orfanidis [7].

In order to reduce this effect, we optimized the Q factor until the best match between the analog TF and the designed digital TF in the frequency range from 0Hz to fm is reached. For the optimization we used the simplex method proposed by Nelder and Mead implemented in the fminsearch-command in Matlab. Figure 10 shows the resulting TFs which


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12

10 8 6 4 2 0 300 500


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Fig. 11: Peak filter designed by the MiMa method (Gain = 12dB, Q = 1 and variable frequency, third order).

10 |H(f)|2 in dB

12

|H(f)|2 in dB

are much better adjusted at the left side of the peak, but the deviation on the right side increases slightly. However, because of the decreasing sensibility of the human ear, this is more tolerable, if the design goal is to come as close as possible to the analog designs. The main drawback of this method is the complexity involved in the optimization process. For the final evaluation we pre-computed the correction factor for a matrix of 127 frequencies and 10 Q-factors. Fortunately, the gain-parameter has no influence on the overall form of the TF, but on the height and therefore the optimized Q-factor is valid for all gain values.

8 6

√ where A = G and ωc is defined at half of the √ Gain G in dB. Usually, the Q-factor is set to 1/ 2 for a maximally flat response.

4 2 0 300 500


20k

Fig. 10: Peak filter design after Orfanidis [7] with optimized Q-Factor.

2.2.4. MiMA In the same way we designed our lowpass filter in section 2.1.3 we can use the magnitude response calculated from equation 14 as filter prototype. Figure 11 shows some design examples for different frequencies. 2.3. Shelving filter Shelving filters are alternative solutions to cut filters (low- and highpass) at both ends of the spectrum. In this work we will concentrate on high-shelv-filters, since we are interested in the influence near the Nyquist frequency. The analog transfer function is given by H(s) = A

√ ωc A 2 Q s + ωc √ ωc A 2 Q s + Aωc

As2 + s2 +

(16)

2.3.1. Standard (Bilinear Transform) In order to get the digital filter coefficients from the analog design the BLT is the standard solution in literature and a solution which is easy to apply is given in the EQ-cookbook [3]. However, the influence of the BLT on the shelving filters cannot be neglected. Figure 12 shows the TFs for different working frequencies fc , and the strong deviation at high frequencies is visible. 2.3.2. Oversampled bilinear Transform Again the oversampling method reduces the effect of the compression of the frequency axis. Figure 13 shows the final TFs when an oversampling of factor two is applied. 2.3.3. Matched z-transformation (MZT) The MZT is another method to obtain a digital filter from an analog prototype. In contrast to the BLT this method maps every pole and zero from the s-plane to a corresponding position in the z-plane. Factorizing the TF into its product form with the zeros zd and the poles pd : QN (s − zd ) (17) H(s) = Qn=1 D d=1 (s − pd )


EQing near Nyqust

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12

10

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|H(f)|2 in dB

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8 6 4 2 0 300 500

analog digital

8 6 4 2


20k

0 300 500


20k

Fig. 12: High-Shelv-filters√designed by the standard BLT method (Q = 1/ 2,G = 12dB, variable frequencies).

Fig. 13: High-Shelv filter √ designed by oversampling BLT method(Q = 1/ 2,G = 12dB, variable frequencies, and an oversampling factor of two).

each extreme value term of the form (s − v) is replaced by (s − v) ⇒ evT z −1 (18)

methodology (ITU-R recommendation BS.1534-1) to determine if differences are audible and to what extent. The test persons were asked to judge the differences between a reference signal (which was the oversampled BLT algorithm) and all other algorithms. Furthermore, an anchor signal (lowpass filtered with a cutoff frequency of 3.5kHz) and a hidden reference was added. The test persons had to scale the impairments and one of the test files has to be rated as 100%. Figure 16 shows the graphical user interface.

with T = 1/fs [8]. Stanciu and Banu ([9]) compared this method to the BLT and came to the conclusion that the magnitude responses for shelving filters are superior to the BLT approach. Figure 14 shows some design examples. 2.3.4. MiMA Analog to the design of lowpass and peak filters, equation 16 can be used as our design input. The resulting examples can be viewed in figure 15. 3. EVALUATION After having introduced all the different methods to design equalizer filters the remaining question is if people are able to differentiate between these designs. The resulting quality cannot be measured easily, since this has a strong artistic component, or you have to establish some references a number of audioprofessionals agree on. Therefore, we concentrate on the question if differences are audible and hope that the analog designs are the main goal for high-quality audio equalizer. 3.1. Methodology 3.1.1. MUSHRA-Methodology For our first investigation we used a MUSHRA-

The test files were all pre-computed in matlab with the following parameters which are all appropriate for the given audio material.

Frequency Gain Q

Lowpass 7000Hz n/a √ 1/ 2

Peak 10000 12dB 1

Shelv 10000 12dB √ 1/ 2

3.1.2. 2 AFC-Method The second method we used was a two-alternative forced choice test. In this test method the user can switch between two unknown alternatives while listening to the audio material. Finally, the preferred alternative is saved. This procedure is repeated until all possible combinations between two algorithms are tested. Therefore, the number of comparisons


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analog digital

8 6 4 2


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Fig. 14: High-Shelv filters designed by the MZT √ method(Q = 1/ 2,G = 12dB, variable frequencies).

depends on the number of algorithms NA in the test and is given by NA (NA −1)/2. In order to be as close as possible to real equalizers in a studio environment, we implemented all the design in a VST plugin1 and used a digital audio workstation as the host for the plugin. The 2AFC test was implemented in the plugin. Each test person could play with the plugin before starting the test to get used to the parameters and the different design algorithms and was asked afterwards to conduct the test. After some preliminary tests we decided to use one drum record with some electric piano recorded in our own studio as test material. The drums are very clear and, therefore, the impact at high frequencies can be perceived easily. Figures 17 and 18 show the final graphical user interface (GUI) of our test plugin in the training and test mode. For the test we asked 12 normal hearing persons, most of them experienced in listening tests and with a general understanding of equalizers. We asked them to rate the algorithms with regard to their quality. Without any reference this question is tricky. However, it was necessary that each test person developed their own taste in order to judge consistently during the test. 1 Which is available at www.hoertechnik-audiologie.de/software

0 300 500


20k

Fig. 15: High-Shelv filters designed by the MiMa √ method (Q = 1/ 2,G = 12dB, variable frequencies, third order).

Fig. 16: GUI for the MUSHRA-like listening test.

3.2. Results Figures 19 to 21 show the results for the MUSHRA test with 12 test persons. The results are given with boxplots showing the median (line in the box), the 25% and 75% percentiles (the box), the range of the data values considered not to be outliers (the whiskers), and the outliers (cross marker). The results show that only for the lowpass signals the test persons were able to find differences between the reference and the BLT standard algorithm. All other results are without any significance. One reason for this result could be that our MUSHRA test is Flash-based and this web-technology needs material that is coded with lossy audio codecs (we chose mp3 with 320kBit/s).


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100

Rating (%)

80 60 40 20 0

Fig. 17: Plugin GUI in training mode with the 5 different design methods.

anchor

MiMA

BLT

reference

Fig. 19: Results for the lowpass-filter listening test.

100

Rating (%)

80 60 40 20 0

Fig. 18: Plugin in test mode with the two unknown choices.

The results for the 2AFC test are shown in table 1. For the lowpass design the consistency is one, which means that all test persons were able to discriminate the three algorithms. The test persons had much more difficulties with the peak and shelvingfilters. Interestingly, the accordance is only significant for the lowpass, which means that most of the test persons decided for or against the same algorithms. This was not true for the peak and shelving designs. For lowpass filters the MiMa algorithm was rated best. However, the oversampling BLT method is very close to it in the ranking scale, computed with the Bradley-Terry-Luce (BTL) model [2, 6]. Since the accordance is not significant and the consistency is low, the ranking table for the other designs should not be taken too seriously. Some test persons were consistent over all tests, though, and could repeat their performance. This is a strong indication that

anchor

OrfanMod

Orfan

MiMA

BLT

reference

Fig. 20: Results for the peak-equalizer listening test.

professional sound engineers would be able to distinguish the different methods, and based on their experience with high-class equalizers they might prefer the designs that are closer to the analog counterpart. 4. CONCLUSIONS In this contribution we presented a new way to create digital equalization filters, we suggested minor improvements for existing peak filter designs and compared state-of-the-art methods in terms of audible differences and quality. The magnitude response for filters designed with the MiMa and Oversampling method are remarkably close to the analog filter shape. The differences between the design methods are audible. The decision which design to choose depends on the different advantages and disadvantages that heavily depend on the field of application. A real quality test was not possible due to the lack of a professional test


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Table 1: Results of the 2AFC listening test for the different filter types. Lowpass Peak Shelv Consistency 1 0.64 0.53 Accordance yes no no Position 1 MiMa (0.00) OrfanMod (0.00) OS BLT (0.00) Position 2 OS BLT (0.3) BLT (0.19) MiMA (8.48) Position 3 BLT (1.78) MiMa (0.26) MZT (9.08) Position 4 Orfan (0.46) BLT (18.47) Position 5 OS BLT (0.47) Processing Magazine, 24(1):137 –139, jan. 2007.

100

[2] Ralph A. Bradley and Milton E. Terry. Rank Analysis of Incomplete Block Designs – I. The Method of Paired Comparisons. Biometrika, 39(3–4):324–345, 1952.

Rating (%)

80 60 40

[3] R. Bristow-Johnson. Cookbook formulae for audio eq biquad filter coefficients. Internet file. http://www.musicdsp.org/files/Audio-EQCookbook.txt, visited last on March 2010.

20 0

anchor MiMA

MZT

BLT reference

Fig. 21: Results for the shelv-equalizer listening test.

set-up, such as reference audio material, high quality equipment and more experienced listeners. However, if analog filters are the design goal, MiMA and OS BLT will deliver very good results. The main difference between the two designs is the moment, when computational power is needed. For MiMa that would be during the design, and for OS BLT it is during the processing. Furthermore, OS BLT introduces some latency, which may not be tolerable in applications such as equalization during live performances. 5. ACKNOWLEDGEMENT We would like to thank all our students and colleagues for their efforts during the listening tests. Special thanks to Ute Bitzer for her language advice. All remaining mistakes are courtesy of the authors. 6.

[4] R. Bristow-Johnson. The equivalence of various methods of computing biquad coefficients for audio parametric equalizers. Nov 1994. preprint No. 3906. [5] K.B. Christensen. A generalization of the biquadratic parametric equalizer. Oct 2003. preprint No. 5916. [6] Robert D. Luce. Individual Choice Behaviour: A Theoretical Analysis. Wiley, 1959. [7] S. Orfanidis. Digital parametric equalizer design with prescribed nyquist-frequency gain. Journal of the Audio Engeneering Society (JAES), 45(6):444–455, June 1997. [8] John G. Proakis and Dimitris K Manolakis. Digital Signal Processing. Prentice Hall, 4th edition, April 2006. [9] L. Stanciu and L. Banu. Digital parametric equalizers for high quality audio signals. volume 2, pages 573 – 576 Vol. 2, july 2005. [10] U. Zoelzer. DAFX - Digital Audio Effects. John Wiley and Son, 2002.

REFERENCES

[1] G. Berchin. Precise filter design. IEEE Signal
