## PSN-L Email List Message

Subject: more on transfer functions
From: Randall Peters PETERS_RD@..........
Date: Wed, 06 Feb 2008 16:11:33 -0500

```Brett,
Probably the most important thing about the differences among the transfer
functions for the different state variables is their differing
functional dependence of SNR.  The multiplication by angular frequency, when
working
with the derivative ('velocity sensor')--causes
the power spectral density in that case to approach the electronics noise level
more
rapidly at low frequencies than is true for the
position sensor case.  In other words the signal goes below noise more rapidly for
the
velocity sensor than for the position sensor, as
the frequency decreases.  A sensor that is 'flat to velocity' is not immune to
this
limitation; since acceleration, being fundamental (and not velocity) is what
regulates
the frequency dependence of the signal to noise ratio.
A way to understand the importance of the electronic noise in this matter is
as
follows.  Nobody should question the fact
that the only thing that allows any sensor to function is the transfer of power to

it.  In the case of a seismometer, the specific power
(power divided by the magnitude of the inertial mass) is given by the product of
velocity and acceleration.  Students of physics should
remember the expression for mechanical power as the dot product of force and
velocity.  In terms of acceleration, the specific power is given by the square of
the
peak acceleration divided by the angular frequency--having units of meters squared
per
second cubed.  When the spectral density of the power is graphed Log-Log (or
dB-Log),
the logarithmic linear 'compression' in frequency of the FFT values (which are
equispaced for a linear scale) causes the reciprocal omega term to disappear.  In
other words, for a position sensor, the mechanical specific power, in a spectral
density sense, is constant for frequencies below the corner frequency.  Those
familiar
with Jon Berger's well known paper on earth noise will remember that the ordinate
of
his power spectral density (PSD) graphs is specified in terms of meters squared
per
second cubed per one-seventh decade (expressed in dB).  (Note: his graphs do not
specifically mention the bin-width of one-seventh decade; this must be understood
from
written descriptions in the paper.)  His units are consistent with what I have
just
indicated, but the common (erroneous) meters squared per second to the fourth per
Hz
are not!  In fact, these common units cannot be a proper power spectral density
statement, because they are dimensionally unacceptable.
Whereas the PSD is flat below the corner when calculated with data from a
position sensor, the same is not true in the case of a velocity sensor.  In the
velocity case, the power is given by omega times the square of the peak value of
the
velocity.  The PSD is in turn (because of the compression mentioned in the usual
Log-Log representation) given by omega squared times the square of the peak
velocity.
Thus, as omega (two pi times the frequency) decreases below the corner value, the
PSD
expression decreases with the square of the frequency--falling off 20 dB per
Just from the electronics noise alone, we see that with the velocity
sensor--as
frequency decreases--a point is reached where the mechanical PSD falls below the
power
spectral density of electronics noise.  Thereafter, unless some noise reduction
scheme
is employed the signal responsible for mechanical motion below those
frequencies--cannot be seen with the velocity sensor.  They can, however, still be

seen with the position sensor.

Randall

```