信号处理视角下的差分麦克风阵列设计与性能分析

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《研究与设计不同阵列》一书旨在从信号处理的角度系统地探讨差分麦克风阵列的设计、实现和性能分析。本书的主要目标是建立一个严谨但易懂的理论框架，涵盖第一、二、三及N阶差分麦克风阵列的设计。作者提供了详尽的实例，如双极子、卡迪奥伊德、超卡迪奥伊德、次卡迪奥伊德和四极子等标准方向性图案的形成方法。通过这些示例，读者能够深入了解不同阶数阵列在指向性图谱、方向增益因子、白噪声增益以及点源处的增益方面的性能。书中深入讨论了差分处理与自适应波束形成之间的内在联系，这有助于增强对差分麦克风阵列高方向性增益的理解。通过这种方式，读者能够学习如何设计抵抗白噪声放大效应的阵列，确保在实际应用中的稳定性。此外，本书属于Springer Topics in Signal Processing系列的第六卷，由Jacob Benesty教授编写，他来自加拿大魁北克大学INRS-EMT。版权方面，所有内容受Springer-Verlag Berlin Heidelberg于2013年出版，强调保留所有权利，包括翻译、重印、再利用图像、朗诵、广播、微缩胶片复制等任何形式的复制，以及电子适应、计算机软件或未来开发出的类似或不同方法。然而，对于学术评论或特定用于计算机系统输入执行的短摘除外，仅供购买者个人使用。《研究与设计不同阵列》是一本实用的指南，对于音频工程师、信号处理专家和研究者来说，它提供了一套完整的工具和技术，帮助他们在实际应用中设计和优化具有高性能的差分麦克风阵列系统。通过阅读这本书，读者不仅能掌握理论知识，还能学会如何在嘈杂环境中提高信号的捕捉和处理能力。

10 1 Introduction

acoustic environment and then being corrupted by some unwanted noise. A

DMA consisting of M microphones is used to pick up the signals. To estimate

the source signal, the M noisy signals are partitioned into small overlapping

frames of a few milliseconds. Each frame is transformed into the short-time

Fourier transform (STFT) domain. In each subband, a diﬀerential beam-

former is designed and applied to the multichannel signals, thereby produc-

ing an estimate of the clean signal spectrum in this subband. Finally, the

time-domain clean speech estimate is constructed using the overlap-add or

overlap-save technique with the inverse STFT. Apparently, the most crucial

step in this paradigm is the design of the diﬀerential beamformers, which is

one of the main focuses of this book.

The problem addressed in this work is of great importance from both the

theoretical and practical viewpoints. From the theoretical side, it is essen-

tial to understand the theory and principles underlying DMAs from a signal

processing perspective. On the practical side, DMAs can be used in a wide

range of applications such as teleconferencing, hands-free communications,

3D sound recording, mobile phones, hearing aids, bluetooth headphones, and

navigation systems in cars, just to name a few. Therefore, it is important to

understand the design issues of DMAs and their limitations as well as the

way to circumvent those limitations.

1.3 Organization of the Book

The material in this book is organized into seven chapters, including this

one. In the next six chapters, we attempt to cover the most basic concepts

and fundamental techniques used in the design and implementation of the

diﬀerent orders of DMAs and the associated beamforming algorithms. All

is explained from a signal processing perspective. The material discussed in

these chapters is as follows.

Chapter 2 presents the general formulation of a DMA design problem. It

also introduces several basic concepts that are important and useful in the

design and evaluation of a DMA including the steering vector (for a plane

wave with the conventional anechoic and farﬁeld model), the beampattern,

the gain in SNR, the white noise gain, the directivity factor (the SNR gain in

diﬀuse noise), etc. In the design of a DMA, a particularly structured matrix

called the Vandermonde matrix generally appears (explicitly or implicitly) in

the formulation. This matrix and its inverse is also discussed.

The simplest form of a DMA is the ﬁrst-order one that uses only two

omnidirectional sensors. If the two microphones are properly placed, many

beampatterns, such as the dipole, cardioid, subcardioid, hypercardioid, and

supercardioid, can be formed by processing the two microphones’ outputs.

Chapter 3 is dedicated to the design of ﬁrst-order DMAs. It discusses how

to form diﬀerent beampatterns and studies the performance associated with

1 Introduction

References 11

each beampattern. It also illustrates the impact of the sensor spacing on the

performance.

Chapter 4 is concerned with the design and implementation of second-order

DMAs. It studies how to form the popularly used beampatterns such as the

dipole, two types of cardioid, quadrupole, hypercardioid, and supercardioid.

The performance of these beamformers in terms of the SNR gain in spatially

correlated and uncorrelated noise is also presented.

Chapter 5 investigates a family of third-order DMAs. Their beampatterns

have three distinct nulls, which can be viewed as three fundamental con-

straints. By properly adjusting these constraints, diﬀerent beampatterns can

be formed, leading to diﬀerent performance in suppressing noise impinging

at the array from diﬀerent directions. As examples, we show how to design

the hypercardioid and supercardioid beampatterns and study their SNR gain

in white, diﬀuse, and point source noise. We also generalize this method to

the design of N th order DMAs with N distinct nulls.

The performance of a DMA in reducing noise depends on several factors

such as the order of the DMA and the characteristics of the noise. In general,

the higher the order of the DMA, the more eﬀective is the array in suppressing

noise. However, a high-order DMA leads to huge white noise ampliﬁcation,

which is a signiﬁcant barrier that prevents using them. In Chapter 6, we

show how DMAs are related to the classical adaptive beamforming technique.

From this important and useful relationship, we then derive a minimum-norm

solution for the design of a general Nth-order DMA. Unlike the traditional

way of DMA design where exactly N + 1 sensors are required to design an

Nth order DMA, the proposed approach can use more than N + 1 sensors to

design an Nth order DMA. The advantage of this technique is its robustness

against white noise ampliﬁcation. The more the number of microphones, the

more robust is the DMA.

In Chapter 7, we show that the patterns of DMAs can be obtained from

the general deﬁnition of the beampattern by approximating the exponential

function with its MacLaurin’s series expansion. In other words, a directional

pattern of order N can be obtained from the MacLaurin’s series of order N,

as long as this approximation holds. In this chapter, we also show how to

design diﬀerential arrays based on this approach and their relationship to

adaptive beamforming.

References

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2. M. Branstein and D. B. Ward, Eds., Microphone Arrays: Signal Processing Techniques

and Applications. Berlin, Germany: Springer-Verlag, 2001.

1.3 Organization of the Book

12 1 Introduction

3. J. Benesty and Y. Huang, Eds., Adaptive Signal Processing: Applications to Real-

World Problems. Berlin, Germany: Springer-Verlag, 2003.

4. Y. Huang, J. Benesty, and J. Chen, Acoustic MIMO Signal Processing. Berlin, Ger-

many: Springer-Verlag, 2006.

5. J. Benesty, J. Chen, and Y. Huang, Microphone Array Signal Processing. Berlin, Ger-

many: Springer-Verlag, 2008.

6. D. B. Ward, R. C. Williamson, and R. A. Kennedy, “Broadband microphone arrays

for speech acquisition,” Acoust. Australia, vol. 26, pp. 17-20, Apr. 1998.

7. O. L. Frost, III, “An algorithm for linearly constrained adaptive array processing,”

Proc. IEEE, vol. 60, pp. 926–935, Aug. 1972.

8. J. Benesty, J. Chen, Y. Huang, and J. Dmochowski, “On microphone-array beamform-

ing from a MIMO acoustic signal processing perspective,” IEEE Trans. Audio, Speech,

Language Process., vol. 15, pp. 1053–1065, Mar. 2007.

9. B. Widrow, P. Mantey, L. Griﬃths, and B. Goode, “Adaptive antenna systems,” Proc.

IEEE, vol. 55, pp. 2143–2159, Dec. 1967.

10. J. Capon, “High resolution frequency-wavenumber spectrum analysis,” Proc. IEEE,

vol. 57, pp. 1408–1418, Aug. 1969.

11. W. F. Gabriel, “Adaptive arrays–an introduction,” Proc. IEEE, vol. 64, pp. 239–272,

Feb. 1976.

12. R. A. Monzingo and T. W. Miller, Introduction to Adaptive Arrays. New York, NY:

Wiley, 1980.

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ference,” IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-33, pp. 527–536,

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14. H. Cox, R. M. Zeskind, and M. M. Owen, “Robust adaptive beamforming,” IEEE

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15. H. F. Olson, “A uni-directional ribbon microphone,” J. Acoust. Soc. Am.

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315, 1932.

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17. G. M. Sessler and J. E. West, “Directional transducers,” IEEE Trans. Audio Electroa-

coustic., vol. 19, pp. 19–23, Mar. 1971.

18. P. Graven and M. Gerzon, “Coincident microphone simlation covering three dimen-

sional space and yielding various directional outputs,” U.S. Patent 4,042,779, Aug.

1997.

19. J. Eargle, The Microphone Book. Burlington, MA: Focal Press, 2005.

20. G. W. Elko, “Steerable and variable ﬁrst order diﬀerential microphone array,” U.S.

Patent 6,041,127, Apr. 1997.

21. G. W. Elko and A.-T. N. Pong, “A steerable and variable ﬁrst-order diﬀerential mi-

crophone array,” in Proc. IEEE ICASSP, 1997, pp. 223–226.

22. G. W. Elko, “Superdirectional microphone arrays,” in Acoustic Signal Processing for

Telecommunication, S. L. Gay and J. Benesty, Eds. Boston, MA: Kluwer Academic

Publishers, 2000, Chapter 10, pp. 181–237.

23. G. W. Elko and J. Meyer, “Microphone arrays,” in Springer Handbook of Speech Pro-

cessing, J. Benesty, M. M. Sondhi, and Y. Huang, Eds. Berlin: Germany, Springer-

Verlag, 2008, Chapter 50, Part I, pp. 1021–1041.

24. M. Williamsen, “Design and measurement of a dipole microphone,” AudioXpress, vol.

7, pp. 1–8, 2009.

25. H. Teutsch and G. W. Elko, “First- and second-order adaptive diﬀerential microphone

arrays,” in Proc. IWAENC, 2001.

26. J. Pekonen, “Microphone techniques for spatial sound,” in Proc. Acoustics Seminar

on Spatial Sound Modeling, 2008.

27. T. D. Abhayapala and A. Gupta, “Higher order diﬀerential-integral microphone ar-

rays,” J.Acoust.Soc.Am., vol. 127, pp. EL227–EL233, May 2010.

1 Introduction

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