Introduction - Study and Design of Differential Microphone Arrays - page 7

extended to the directional electret (capacitor) microphones [17], [18]. Un-
like the ribbon microphones, electrets are pressure sensors and they have an
omnidirectional response. To form a directional pattern, one would need to
measure the differentials of the acoustic pressure field, which can be achieved
by combining the outputs of a number of omnidirectional sensors. For in-
stance, the first-order differential of the acoustic pressure can be obtained by
subtracting two closely spaced omnidirectional microphones' outputs while a
general N th-order differential of the acoustic pressure can be formed by sub-
tracting two differentials of order N − 1. A directional microphone consists
of two or more omnidirectional microphones and, therefore, it is indeed a
microphone array. (Note that today, a directional microphone may have only
a single transducing element and the desired pattern is achieved by using
calculated front-back delay paths [19].) However, it is named a microphone
instead of a microphone array in its early form primarily because all the sen-
sors inside such a system are physically conjoined in a single housing so the
overall system still looks like a single microphone.
Directional microphones, in their early form, have a prominent limitation,
i.e., once made, their directional response is fixed. If, for a given application,
we find that the microphone we bought does not produce the expected per-
formance, the only option is to try a different type. There is no flexibility
in adapting the directional pattern to fit the application needs. To circum-
vent this limitation, the modern concept of DMAs was developed, in which a
number of pressure microphones are arranged into a particular geometry and
digital signal processing techniques are then used to process the microphones'
outputs to obtain the desired directional response [20]-[35]. Figure 1.4 illus-
trates how first-, second-, and third-order DMAs are constructed with a linear
geometry. Basically, a general N th-order DMA has a response proportional
to a linear combination of signals derived from spatial derivatives from order
0 to (including) order N . Note that an inherent assumption in the construc-
tion of a DMA is that the microphones are placed close enough so that the
true acoustic pressure differentials can be approximated by finite differences
between microphone sensors' outputs. This is one of the major reasons why
a DMA is in general compact in size.
Besides its small size, a DMA has the following number of advantages
in comparison with an additive array. 1) It can form frequency-invariant
beampatten and, therefore, it is more suitable to process broadband speech
signals. 2) It is effective not only for high frequencies, but for low frequencies
as well. 3) For a given number of sensors, differential arrays have the potential
to attain maximum directional gain [15].
Although a DMA has many attractive properties, its design and implemen-
tation is by no means a trivial task. First, the response of an N th order array
has a high-pass filter nature with a slope of 6 N dB/octave, so its frequency
response has to be properly compensated to process broadband speech sig-
nals. Second, the frequency response and level of a DMA are sensitive to
the position and orientation of the arrays relative to the sound source. Con-