JAES_V58_12_Egalizor_digital

ENGINEERING REPORTS

Improving the Magnitude Responses of DigitalFilters for Loudspeaker Equalization*

MOHAMAD ADNAN AL-ALAOUI([email protected])

Department of Electrical and Computer Engineering, American University of Beirut, Beirut, Lebanon

Digital filters used in the equalization of loudspeakers try in some applications to achieve a

digital approximation of analog filters. The bilinear z transform (BZT) and the matched pole–

zero transform (MPZT or MZT) are often used with less than satisfactory results. Recently a

new MZT transform, called MZTi, was introduced, which yields filters with magnitude and

phase responses that match their analog target filters better. A novel s-to-z transform that

yields often closer approximations to the analog target filters than the MTZi method is

described.

0 INTRODUCTION

Digital filters are used in the equalization of loud-

speakers [1]–[5]. The filters are supposed to approximate

analog target filters. The bilinear z transform (BZT) [6]–

[13] and the matched pole–zero z transform (MPZT or

MZT) [6], [11] are often employed with less than

satisfactory results to approximate the target analog

filters. Better filters have been obtained by using

variations of the BZT [2] and MZT [3] methods. As

pointed out by one of the reviewers, whose comments

are quoted in the remainder of this paragraph, ‘‘the

assumption that digital filters used to equalize loud-

speakers are meant to match analog prototypes is not

always true. There are times when digital filters are used

for loudspeaker equalization to do things that an analog

filter cannot do. Also there are other applications, such

as mixing desks, where accurate transforms of analog

filters are often desired.’’ This engineering report ad-

dresses applications where it is desired that the digital

filter responses mimic their corresponding analog

prototypes.

Recently Gunness and Chauhan proposed minimizing

the error by adding zeros or by shifting the positions of

poles and zeros to all-pole or biquadratic filters, re-

spectively. The resulting filters, called MZTi, were shown

to yield better approximations to the target analog filters

[1].

In this study, we compare the performance of the MZTi

approach proposed in [1] and the novel approach to

analog-to-digital transforms that was proposed by Al-

Alaoui [14]. This publication is divided into five sections,

including the Introduction and the Conclusion. Section 1

introduces Al-Alaoui’s proposed transforms, Section 2

presents an overview of analog bell filters, and Section 3

compares Al-Alaoui transforms to the MTZi approach

and demonstrates the superiority of the Al-Alaoui

transforms.

1 AL-ALAOUI MPZ TRANSFORM

In [14] Al-Alaoui developed new s-to-z transforms to

overcome the deficiencies in the traditional transforms.

The traditional transforms approximate s of the analog

transfer function in some fashion. They end up substitut-

ing approximations of s, in effect a scaled value of s, in

the (s þ a) factors while keeping the exact value of a, thus

introducing additional distortions. However, the new

transform approximates the factors (s þ a) in the transfer

function. In the resulting transforms every pole will also

generate a zero and vice versa. This is similar to the MPZ

transform, except that in the latter case the zeros are

assigned at z¼�1, whereas in the Al-Alaoui transform the

zeros are generated automatically.

Several parallel and cascade forms were proposed in

[14]. In all examples the following variants of the cascade

form algorithm presented in [14] are used.

1) Real poles and zeros are treated using the original

cascade transformation approach, that is, the following

applies:

1

sþ a¼ K

1þ e�aTz�1

1� e�aTz�1; sþ c ¼ K

1� e�cTz�1

1þ e�cTz�1:

ð1Þ

2) Higher degree factors are treated by applying

successive first-degree factors.*Manuscript received 2009 January 21; revised 2010 August

8 and October 8.

1064 J. Audio Eng. Soc., Vol. 58, No. 12, 2010 December

3) Complex poles and zeros are treated using the

recombined partial fraction method since that reduces the

overall order. The following applies:

1

s2þ 2asþ a2 þ b2¼ K

0 z�1

1� 2e�aTcosðbTÞz�1 þ e�2aTz�2

ð2Þ

s2þ 2csþ c

2 þ d2 ¼ K

0 1� 2e�cTcosðdTÞz�1þ e�2cTz�2

z�1:

ð3Þ

4) The constant factor in the transfer function of the

resulting discrete-time system is evaluated to obtain the

same magnitude as the analog transfer function at the

scaling frequency xS. Unless specified otherwise, dc

scaling was used.

A corresponding MATLAB code of this procedure is

included in the Appendix.

2 APPLICATION TO LOUDSPEAKEREQUALIZATION

We consider the application of the Al-Alaoui MPZ

transform and the MZTi approach to loudspeaker

equalization as described in [1]. Hence our main concern

is with audio applications, where one of the most

commonly used filters is the bell filter. The analog

transfer function of the bell filter takes the form

HðsÞ ¼ s2 þ ððg0 � sÞ=QÞ þ 1

s2 þ ðs=QÞ þ 1: ð4Þ

In Eq. (4) the resonant frequency is normalized to 1.

The filter is characterized by the parameters g0 and Q,

with g0 being the gain at the resonant frequency. We will

consider several cases by varying these two parameters.

For each case we present figures comparing the

magnitude, phase, and absolute magnitude error for the

MZTi approach and the Al-Alaoui MPZ transform.

3 SIMULATION RESULTS

The following ten cases were implemented and the

corresponding figures plotted. Different values of Q, g0,

and the resonant frequency x0 are studied. The figures

present in each case the analog filter responses and the

corresponding digital filters, which were obtained by

using the MZTi and the Al-Alaoui transforms.

Case 1: Q ¼ 2; g0 ¼ 15 dB, x0 ¼ 1

Case 2: Q ¼ 2.8; g0 ¼ 15 dB, x0 ¼ 1

Case 3: Q ¼ 3; g0 ¼ 20 dB, x0 ¼ 1

Case 4: Q ¼ 2; g0 ¼ 10 dB, x0 ¼ 1

Case 5: Q ¼ 3; g0 ¼ 15 dB, x0 ¼ 1

Case 6: Q ¼ 3.5; g0 ¼ 15 dB, x0 ¼ 1

Case 7: Q ¼ 1.0; g0 ¼ 5 dB, x0 ¼ 1

Case 8: Q ¼ 10; g0 ¼ 20 dB, x0 ¼ 1

Case 9: Q ¼ 5; g0 ¼�12 dB, x0 ¼ 1

Case 10: Q ¼ 5; g0 ¼ 15 dB, x0 ¼ 0.85p

For each of these cases three graphs are plotted: the

magnitude response, the phase response, and the magni-

tude error of the resulting digital filters with respect to the

corresponding analog filter. Fig. 1 plots the magnitudes,

phases, and magnitude errors for case 1; Fig. 2 plots the

magnitudes, phases, and magnitude errors for case 2; and

so on for cases 3 to 10 (Figs. 1–10).

The figures show that the Al-Alaoui approach outper-

forms the MZTi approach in all cases except for cases 1 and

3. In particular, case 10 shows a better behavior of the Al-

Alaoui approach when the resonant frequency gets close to

the Nyquist frequency. Case 9 shows the superiority of the

Al-Alaoui approach when the gain is negative (in dB). The

other examples show the superiority of the Al-Alaoui

approach for a wide range of gains and Q values.

To explain the performance of the Al-Alaoui method in

cases 1 and 3, where it is outperformed by MZTi, we

resort to the information presented in Tables 1 and 2.

Table 1 shows the order of the different filters in all the

cases studied. Consequently it can be seen from Table 1

that the filters obtained from the Al-Alaoui method have

an order less than or are equal to those of MZTi in all

cases studied. Furthermore the Al-Alaoui method pre-

serves the order of the analog filter in eight out of ten

examples. The only two cases where the Al-Alaoui

approach leads to a higher order are cases 1 and 3, that is,

the cases where the Al-Alaoui approach has a worse

performance. Table 2 shows the poles and zeros of the

analog bell filters for the ten cases investigated. From

Table 2 it can be seen that all poles and zeros are complex

conjugate pairs, except for cases 1 and 3, where the zeros

of the bell filter are real. This explains the performance of

the Al-Alaoui method in the examples studied.

In fact each pole or zero generates a pole and a zero

with the Al-Alaoui method when the poles or zeros are

real, as expressed in Eq. (1). When they are complex

conjugates, this is not the case, as shown by Eq. (2),

where a pair of complex conjugate poles generates an

additional zero at the origin, and a pair of complex

conjugate zeros generates an additional pole at the origin.

Hence when both complex conjugate poles and zeros are

present, the extra poles and zeros at the origin cancel out.

In cases 1 and 3 the two real zeros of the analog filters

introduce two additional real poles when the Al-Alaoui

approach is implemented. These additional real poles

tamper with the bell-shaped response, which is controlled

by the complex conjugate poles. This scenario does not

happen in all the other cases, where the Al-Alaoui method

performs better.

It should be noted that this addition of poles and zeros

is inherent in the Al-Alaoui transform and contributes to

its superior performance in various applications, such as

those presented in [14]. However, it presents a drawback

when applied to loudspeaker equalization. To solve this

J. Audio Eng. Soc., Vol. 58, No. 12, 2010 December 1065

ENGINEERING REPORTS DIGITAL FILTERS FOR LOUDSPEAKER EQUALIZATION

Fig. 1. Case 1: Q ¼ 2; g0¼ 15 dB. (a) Magnitude responses. (b) Phase responses. (c) Magnitude errors.


AL-ALAOUI ENGINEERING REPORTS

Fig. 2. Case 2: Q ¼ 2.8; g0 ¼ 15 dB. (a) Magnitude responses. (b) Phase responses. (c) Magnitude errors.



Fig. 4. Case 4: Q ¼ 2; g0 ¼ 10 dB. (a) Magnitude responses. (b) Phase responses. (c) Magnitude errors.



Fig. 8. Case 8: Q ¼ 10; g0 ¼ 20 dB. (a) Magnitude responses. (b) Phase responses. (c) Magnitude errors.



Fig. 9. Case 9: Q¼ 5; g0 ¼�12 dB. (a) Magnitude responses. (b) Phase responses. (c) Magnitude errors.



Fig. 10. Case 10: Q ¼ 5; g0 ¼ 15 dB, x0¼ 0.85p. (a) Magnitude responses. (b) Phase responses. (c) Magnitude errors.



problem, and in order to preserve the bell-shaped

response in the digital filters obtained from the Al-Alaoui

transform, we modify Eq. (1) by simply removing the

additional pole obtained from a real zero in the analog

filter. Hence we obtain

sþ c ¼ Kð1� e�cT

z�1Þ: ð5Þ

The implementation of Eq. (5) instead of Eq. (1) in the

Al-Alaoui transform is denoted in the figures by

‘‘modified Al-Alaoui.’’ The modified Al-Alaoui approach

clearly leads to a considerably enhanced performance in

cases 1 and 3 and outperforms the MZTi method. It leads

to exactly the same performance as the initial Al-Alaoui

approach in all the other cases (since they do not involve

real zeros in the analog filter) and hence is not shown in

the corresponding figures. Furthermore since the addi-

tional poles are removed, the filters obtained by the

modified Al-Alaoui method are all of order 2 in all the

cases studied, that is, they are of the same order as the

analog bell filters, as shown in Table 1.

To summarize, we can conclude that the Al-Alaoui

approach outperforms MZTi whenever the zeros of the

analog bell filter are complex conjugates, and performs

worse when the zeros are real. However, the modified Al-

Alaoui approach is superior in all cases.

4 CONCLUSION

The proposed novel Al-Alaoui MPZ transform was

compared to the state of the art MZTi approach. In all the

examples shown the Al-Alaoui s-to-z transform was

comparable to or superior than the state-of-the-art MZTi

approach. The filters obtained by the Al-Alaoui approach

are closer to the ideal analog response over the entire

frequency range, and in particular over the low-frequency

range (between zero and the resonant frequency). Hence

they are very suitable for the relevant audio applications.

A modified Al-Alaoui approach was presented, which

approximates the ideal analog response over the entire

frequency range.

5 ACKNOWLEDGMENT

This research was supported in part by the University

Research Board of the American University of Beirut.

The author wishes to acknowledge Dr. Elias Yaacoub and

Jimmy Azar for their invaluable contributions to the

production of this engineering report. He is grateful to the

outstanding reviewers whose comments contributed

significantly to the improvement of the study. The author

would further like to thank Ali H. Sayed for providing the

atmosphere conducive to research through his invitation

to the author to spend part of his research leave in the

Adaptive Systems Laboratory at UCLA, as well as the

graduate students Zaid Towfic, Sheng-Yuan Tu, Federico

Cattivelli, Zhi Quan, Qiyue Zou, and Cassio G. Lopes for

their help during his stay at UCLA.

6 REFERENCES

[1] D. W. Gunness and O. S. Chauhan, ‘‘Optimizing the

Magnitude Response of Matched z-Transform Filters

(MZTi) for Loudspeaker Equalization,’’ in Proc. AES

32nd Int. Conf. ‘‘DSP for Loudspeakers,’’ (Hillerod,

Denmark, 2007 Sept. 21–23), pp. 1–10.

[2] S. Orfanidis, ‘‘Digital Parametric Equalizer Design

with Prescribed Nyquist-Frequency Gain,’’ presented at

the 101st Convention of the Audio Engineering Society,

J. Audio Eng. Soc. (Abstracts), vol. 44, p. 1168 (1996

Dec.), preprint 4361.

Table 1. Filter orders of different methods for examples

studied.

Case 1 2 3 4 5 6 7 8 9 10

Analog 2 2 2 2 2 2 2 2 2 2

MZTi 4 3 4 3 3 3 3 3 4 4

Al-Alaoui 4 2 4 2 2 2 2 2 2 2

Modified Al-Alaoui 2 2 2 2 2 2 2 2 2 2

Table 2. Poles and zeros of analog filters in examples

studied.

Case Poles Zeros

1 �0.2420 þ 0.9703i �2.2834

�0.2420 � 0.9703i �0.4379

2 �0.1728 þ 0.9850i �0.9719 þ 0.2354i

�0.1728 � 0.9850i �0.9719 � 0.2354i

3 �0.1650 þ 0.9863i �2.9623

�0.1650 � 0.9863i �0.3376

4 �0.2236 þ 0.9747i �0.7071 þ 0.7071i

�0.2236 � 0.9747i �0.7071 � 0.7071i

5 �0.1613 þ 0.9869i �0.9071 þ 0.4209i

�0.1613 � 0.9869i �0.9071 � 0.4209i

6 �0.1383 þ 0.9904i �0.7775 þ 0.6289i

�0.1383 � 0.9904i �0.7775 � 0.6289i

7 �0.3749 þ 0.9270i �0.6668 þ 0.7453i

�0.3749 � 0.9270i �0.6668 � 0.7453i

8 �0.0495 þ 0.9988i �0.4950 þ 0.8689i

�0.0495 � 0.9988i �0.4950 � 0.8689i

9 �0.3721 þ 0.9282i �0.0935 þ 0.9956i

�0.3721 � 0.9282i �0.0935 � 0.9956i

10 �0.2585 þ 2.6578i �1.4534 þ 2.2402i

�0.2585 � 2.6578i �1.4534 � 2.2402i



[3] R. J. Clark, E. C. Ifeachor, G. M. Rogers, and P. W.

J. Van Eetvelt, ‘‘Techniques for Generating Digital

Equalizer Coefficients,’’ J. Audio Eng. Soc., vol. 48, pp.

281–298 (2000 Apr.).

[4] R. J. Clark, ‘‘Investigation into Digital Audio

Equaliser Systems and the Effects of Arithmetic and

Transform Error on Performance,’’ Department of Com-

munication and Electronic Engineering, Faculty of

Technology, Ph.D. thesis, University of Plymouth, UK

(2001 Apr.).

[5] K. B. Christensen, ‘‘A Generalization of the

Biquadratic Parametric Equalizer,’’ presented at the

115th Convention of the Audio Engineering Society, J.

Audio Eng. Soc. (Abstracts), vol. 51, p. 1233 (2003 Dec.),

convention paper 5916.

[6] E. C. Ifeachor and B. W. Jervis, Digital Signal

Processing: A Practical Approach, 2nd ed. (Pearson

Education, Essex, UK, 2002), pp. 468–471.

[7] F. G. Franklin, J. D. Powell, and A. Emami-Naeini,

Feedback Control of Dynamic Systems (Addison-Wesley,

Reading, MA, 1994).

[8] S. K. Mitra, Digital Signal Processing (McGraw-

Hill, New York, 1998).

[9] A. V. Oppenheim and R. W. Schafer, Discrete-Time

Signal Processing (Prentice-Hall, Englewood Cliffs, NJ,

1989).

[10] C. L. Philips and H. T. Nagle, Digital Control

System Analysis and Design, 3rd ed. (Prentice-Hall,

Englewood Cliffs, NJ, 1995), chap. 11.

[11] J. G. Proakis and D. G. Manolakis, Introduction to

Digital Signal Processing, 3rd ed. (Prentice-Hall, Engle-

wood Cliffs, NJ, 1996).

[12] L. R. Rabiner and B. Gold, Theory and

Applications of Digital Signal Processing (Prentice-Hall,

Englewood Cliffs, NJ, 1975).

[13] K. Steiglitz, A Digital Signal Processing Primer:

With Applications to Computer Music (Addison-Wesley,

Reading, MA, 1994).

[14] M. A. Al-Alaoui, ‘‘Novel Approach to Analog to

Digital Transforms,’’ IEEE Trans. Circuits Sys. I:

Fundamental Theory and Applications, vol. 54, pp.

338–350 (2007 Feb.).

APPENDIXThe MATLAB code for the procedure described in Section 1 is as follows:



APPENDIX continued



APPENDIX continued

The function bell_filter_Wc implements the analog Bell filter that corresponds to [1, eqs. (1)–(4)].

The function mzt_lp_ord2 implements the MZTi second-order low-pass filter that corresponds to [1, eqs. (11)–(15)].



APPENDIX continued

The function fir_curve_fit implements the FIR curve fit transform presented in [1, sec. 2].

The function alalaoui used to implement the proposed approach using the Al-Alaoui transform derived in [14] is as

follows:


APPENDIX continued



APPENDIX continued

THE AUTHOR

M. A. Al-Alaoui

Mohamad Adnan Al-Alaoui received a B.S. degree in

mathematics from Eastern Michigan University, Ypsilan-

ti, MI, in 1963, a B.S.E.E. degree from Wayne State

University, Detroit, MI, in 1965, and M.S.E.E. and Ph.D.

degrees in electrical engineering from the Georgia

Institute of Technology, Atlanta, in 1968 and 1974,

respectively.

After receiving a Ph.D. degree he joined the Electrical

Engineering Department of the Royal Scientific Society,

Amman, Jordan, where he was responsible for the

communications area. From 1977 to 1985 he served as

an assistant professor or associate professor in electrical

engineering at the American University of Beirut (AUB),

Beirut, Lebanon, the University of Connecticut, Storrs,

and the Hartford Graduate Center, Hartford, CT. He was

Chair of the Automatic Control Department at the Higher

Institute for Applied Science and Technology, Damascus,

Syria, and in 1988 he rejoined AUB, where he is currently

a professor and where he also served as chair of the

Department of Electrical and Computer Engineering. His

research interests are in neural networks and in analog

and digital signal and image processing and their

applications in biomedical engineering, communications,

controls, and instrumentation. He was a visiting scholar

with Stanford University, the University of Southern

California, the University of California at Santa Barbara,

and the University of California at Los Angeles.

Dr. Al-Alaoui was the recipient of the First Research

Award in Engineering for 1989–1990 by AUB. He is a

senior member of the IEEE.



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