40.3 Microphone Array

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40.3.1 Principle of Microphone Array

A microphone array [11] consists of several microphones distributed spatially to measure an acoustic

field. The time signals from each microphone are added, accounting for the time delay between sound

sources and microphones, and a directional output signal can be obtained as a result. The algorithm is

called “beamforming.” Now, consider M omnidirectional microphones distributed in a far field of noise

sources. The output signal of the array focused to a particular location in the source region, r; and zðr; tÞ;

is calculated as a sum of delayed and weighted signals of each microphone:

zðr; tÞ ¼

XM

m¼1

wmpmðt 2 DmÞ ð40:12Þ

Here, pmðtÞ is the signal from the mth microphone, wm is a weighting factor, and Dm is a time delay

applied to signal of the mth microphone, as given by

Dm ¼

ro 2 rm

c0 ð40:13Þ

where ro and rm are the distances from the focus point to the reference point o and the mth microphone,

respectively. When the focus location coincides with the source location, a strong signal is obtained (see

Figure 40.8). If this process is repeated for various focus locations, r, on the source surface, then a

noise-source map can be obtained.

FIGURE 40.7 Time history of the A-weighted one-third octave band ( f0 ¼ 8 kHz) sound pressure level measured

with a parabolic mirror – microphone system (D ¼ 1 m, train speed ¼ 274 km/h).

40-6 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

40.3.2 Array’s Directivity Pattern

The performance of a microphone array is characterized by the spatial resolution and signal-to-noise

ratio. For simplicity, consider a linear array of M ¼ 2N þ 1 microphones spaced equally by d: When a

harmonic plane wave is propagating with an incident angle u; and weighting factors all equal 1=M; the

ratio of the output signal of the array to that of the center microphone is computed using

W ðuÞ ¼

1

M

sinððM=2Þkd sin uÞ

sinðð1=2Þkd sin uÞ ð40:14Þ

where k is the wave number. Figure 40.9 shows the directivity patterns for different values of the product

kd based on Equation 40.14. The highest peak appears at u ¼ 0; which we call a “main lobe,” and lower

peaks also appear at some locations that are separate from a true source direction, which we call “side

lobes.” The width of the main lobe decides the performance of the array to separate two closely lying

sources (which we call spatial resolution), and the ratio of main lobe to side lobe decides the signal-tonoise

ratio of the array. The spatial resolution improves as kd increases, that is, in proportion to the ratio

of the array length to the wavelength. However, when kd ¼ 2p; a peak of the same strength as the true

source appears due to a spatial aliasing at u ¼ 908; which occurs when d . l=2; where l is the

wavelength. Thus, the acoustic frequency, f ; is restricted by f , c0=2d; to avoid aliasing.

θ

P– N (t)

PN (t)

P0 (t)

P1 (t)

P–1 (t)

P–2 (t)

P2 (t)

d

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 30 60 90

W(q)

Incident angle q (degree)

kd=2π

kd=π

kd=π/2

FIGURE 40.9 Directivity patterns of a linear array for different values of the product kd ðM ¼ 2N þ 1 ¼ 9Þ:

Delay

Δ1

Delay

Δ2

Delay

ΔM

Σ p2(t)

p1(t)

pM(t)

z(t)

w1

w2

wM

FIGURE 40.8 Principle of a microphone array. Individual time delays are chosen such that signals arriving from a

given point will be added up coherently.

Instrumentation 40-7

© 2005 by Taylor & Francis Group, LLC

In the above case of the linear array, the directivity exists only in the direction of the array (onedimensional).

If microphones are arranged in a two-dimensional plane, a two-dimensional directivity

can be obtained. Recently, many microphone arrangements have been proposed that obtain better spatial

resolution and to reduce side lobes [12 – 15].

40.3.3 Applications

Microphone arrays have been used for identification of the noise source in various situations, for

example, in wind tunnel tests. Many actual examples can be found in published literature [10,16 – 18].

The measurement with a microphone array has the advantage of much shorter measuring time than that

of a mirror– microphone system. Furthermore, the lower frequency limit is not so serious because the

size of the apparatus can be easily extended.

Another fundamental example is given now. Nine microphones are arranged equally spaced by

d ¼ l=2 for each one-third octave band, and their signals are summed without time delay. In this case, the

array is focused to a fixed direction, perpendicular to the array axis. Figure 40.10 shows a time history of

noise generated by a high-speed train, measured with a linear microphone array located at a point 25 m

away from the track. It is found that pantographs, the leading car, and gaps between cars are the main

noise sources in this example.

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