# Question on Microphones

David Lubman dlubman at ix.netcom.com
Tue Aug 5 13:33:28 EST 1997

```Jeff:

Parabolic microphones work on the same principle as parabolic (radar)
antennas and are probably impractical for use with hearing aids because
of their great size.

Here's my off-hand intuitive cut on how they work, though I have not
checked this against textbooks. Sorry for it's length, but it was fun to
think about and write! Feel free to correct my errors or oversights!

Parabolic microphones "collect" sound energy "coherently" from a large
"area".

The "collection area is the cross-sectional (c-s) area of the parabolic
dish.

The potential "gain" of the parabolic microphone over a single
omnidirectional microphone depends on frequency. More intuitively, it
depends on wavelength.

The gain of a "large" parabolic microphone is equal to the c-s area
expressed in squared wavelengths. This is found by dividing the c-s area
by the square of wavelength, keeping the same units.

According to my side algebra the potential gain is (Pi/4) * (Df/c)^2,
where Pi = 3.1416 ..., D is the diameter of the parabolic dish, f is the
frequency, and c is the speed of sound. Use consistent units, i.e., D in
meters, c in meters/sec.

This formula may work for a parabolic microphone that is "large" in
wavelengths. For small parabolic microphones (Df/c) <=1 my formula
predicts gains smaller than 1, which is unrealistic.

To find the gain in dB, take 10* log of the number, i.e,
10*log[(Pi/4)*(Df/c)^2].

Parabolic microphones "collect" sound energy at a single point - the
"focal point" - where an omnidirectional microphone is placed.

The reason for the parabololic shape (I think it is actually a truncated
paraboloid) can be understood in terms of ray theory, Snell's law of
reflection, and elementary geometry.

Sound "rays" arriving from a distant (farfield) source exactly on the
paraboloidal axis are parallel to each other and to the axis. At each
point where a ray strikes the paraboloid it is (assumed to be totally)
reflected without any phase change at an angle governed by Snell's law.
Simply, Snell's law states that the angles of incidence and reflection
are equal.

The paraboloidal shape guarantees "coherent" summation of rays: each ray
arriving parallel to the parabolic microphone axis travels the same
total distance. Sound rays striking the parabolic microphone from other
directions sum "incoherently". Thus, preference is given to sources
toward which the parabolic microphone is steered or aimed.

This brings to mind several practical limitations of parabolic
microphones.

They may not work well at low frequencies. First, because the gain is
low due to its limited size in wavelengths. Second, because the
paraboloid dish may not be a good sound reflector at low frequencies. To
be a good sound reflector, the surface density must be very high.
Possibly about 2 pounds per square foot for the low frequencies of music
but perhaps much less for speech or birdsong.

Also, for coherent summation, the phase change upon reflection needs to
be the same at every point on the dish. I expect that is satisfied by
structural homogeneity.

The oncoming signal must have a coherent "wavefront" over the dish. Gain
is lost to the extent that coherency is lost. This is a propagation
problem that limits the potential gain of an otherwise perfect parabolic
microphone. Atmospheric inhomogenieties such as turbulence may be one
cause.

Since the gain of a parabolic microphone increases with the square of
frequency, I expect that an equalizer network is necessary to flatten
the frequency response. If my formula is correct, a simple low pass
filter (RC circuit giving -6 dB/octave) would be appropriate to flatten
the response.

Be sure and let me know what I screwed up!

Jeffrey Sirianni wrote:

A patient from our office asked me the other day about microphones used
in
> hearing aids.  As he has problems with detection of speech in noise, he
> inquired about Audio Zoom from Phonak, which he will be trying.  He asked
> about parabolic microphones, used for surveillance and such, and wanted to
> know why the this technology could not be incorporated into a hearing aid.
> I know this is a silly question, but could someone with technical
> background in microphones explain how a parabolic microphone works and what
> are the essential components.
>
> Jeff
>
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--
David Lubman in Westminster, California

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