Note: Descriptions are shown in the official language in which they were submitted.
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TITLE:
METHOD FOR REDUCING RANDOM WALK IN,
FIBER OPTIC GYROSCOPES
TECHNICAL FIELD
The present invention relates to fiber optic
gyroscopes. More particularly, this invention pertains
to apparatus and a method for lowering random walk
error in the output of a fiber optic gyroscope.
BACKGROUND ART
The Sagnac interferometer is an instrument
for determining rotation by measurement of a
nonreciprocal phase difference generated between a pair
of counterpropagating light beams. It generally
comprises a light source such as a laser, an optical
waveguide consisting of several mirrors or a plurality
of turns of optical fiber, a beamsplitter-combiner, a
detector and a signal processor.
In an interferometer, the waves coming out of
the beamsplitter counterpropagate along a single
optical path. The waveguide is "reciprocal". That is,
any distortion of the optical path affects the
counterpropagating beams similarly, although the
counterpropagating beams will not necessarily
experience such perturbations at the same time or in
the same direction. Time-varying perturbations may be
observed where the time interval is equal to the
propagation time of the light around the optical
waveguide whereas "nonreciprocal" perturbations affect
the counterpropagating beams differently and according
to the direction of propagation. Such nonreciprocal
perturbations are occasioned by physical effects that
disrupt the symmetry of the optical medium through
which the two beams propagate.
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Two of the nonreciprocal effects are quite
well known. The Faraday, or collinear magneto-optic
effect, occurs when a magnetic field creates a
preferential spin orientation of the electrons in an
optical material whereas the Sagnac, or inertial
relativistic effect, occurs when rotation of the
interferometer with respect to an inertial frame breaks
the symmetry of propagation time. The latter effect is
employed as the principle of operation of the ring
gyroscope.
The measured or detected output of a
gyroscope is a "combined" beam (i.e., a composite beam
formed of the two counterpropagating beams) after one
complete traverse of the gyroscope loop. The rotation
rate about the sensitive axis is proportional to the
phase shift that occurs between the couterpropagating
beams. Accordingly, accurate phase shift measurement
is essential.
Figure l is a graph of the well known
relationship between the intensity (or power, a
function of the square of the electric field) of the
detected beam output from the coil of optical fiber and
the phase difference that exists between the two
counterpropagating beams after completion of a loop
transit. (Note: Typically, prior art photodetectors
are arranged to measure output power rather than
intensity.) The figure discloses a fringe pattern that
is proportional to the cosine of the phase difference,
~, between the beams. Such phase difference provides a
measure of the nonreciprocal perturbation due, for
example, to rotation. A DC level is indicated on
Figure l. Such DC level corresponds either to the half
(average) intensity level or the half power level of
the gyro output.
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It is a well known consequence of the shape
of the fringe pattern, that, when a small phase
difference, or a small phase difference +n~ where n is
an integer, is detected (corresponding to a relatively
low rotation rate), the intensity of the output beam
will be relatively insensitive to phase deviation or
error as the measured phase difference will be located
in the region of a maximum or minimum of the output
fringe pattern. This phenomenon is illustrated at
regions 10, 12, 12', 14 and 14' of the fringe pattern
which correspond to phase shifts in the regions of ~ =
0, +2~, +~, -2~ and -~ radians respectively. Further,
mere intensity does not provide an indication of the
sense or direction of the rotation rate.
lS For the foregoing reasons, an artificially
biased phase difference is commonly superimposed upon
each of the counterpropagating beams, periodically
retarding one and advancing the other in phase as the
pair propagates through the sensor coil. The biasing
of the phase shift, also known as "nonreciprocal null-
shift", enhances the sensitivity of the intensity
measurement to phase difference by shifting the
operating point to a region characterized by greater
sensitivity to a phase deviation ~ indicative of the
presence of rotation. In this way, the variation in
light intensity observed at the photodetector, ~I (or
power ~P), is enhanced for a given nonreciprocal phase
perturbation ~.
By enhancing the intensity effect due to the
presence of a given phase perturbation ~, corresponding
increases in photodetector output sensitivity and
accuracy are obtained. These, in turn, may be
translated into a simplification and resulting
economization of the output electronics. Such output
electronics commonly includes a differencing circuit
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for comparing the intensity values of the operating
points between which the electro-optic modulator (often
a multifunction integrated optical chip or "MIOC") is
cycled during a loop transit time r.
Presently, fiber optic gyroscopes are
commonly biased by a periodic modulation waveform,
often either a square wave or a sinusoid. The square
wave is cycled between +~/2 with a period of 2r while
the sinusoid is cycled between maxima and minima of
approximately +1.8 radians. The sinusoidal extremes
correspond to the argument of the maximum of the first
order Bessel function of the first kind, J1(x). The
prior art square wave modulating waveform is
illustrated in Figure 2.
Referring back to Figure 1, the
representative square wave modulation profile of the
prior art square wave modulation corresponds to
alternation of the output intensity curve between the
operating points 16 and 18. Each of the points 16 and
18 lies at an inflection of the intensity fringe
pattern where a small nonreciprocal perturbation ~ of
the phase difference ~ results in a maximum detectable
change, ~ P), in the optical intensity (power)
output. Also, by alternating the bias imposed between
two different operating points, the system can
determine the sign of ~ and, thus, the direction of
rotation.
In addition to phase modulation, "phase-
nulling" is commonly applied to the interferometer
output. This introduces an additional phase shift
through a negative feedback mechanism to compensate for
that due to the nonreciprocal (Sagnac) effect. A phase
ramp (either analog or digital) with slope proportional
to the rate of change of the measured phase difference
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is commonly generated for this purpose. Commonly, a
ramp, varying in height between 0 and 2~ radians,
provides the nulling phase shift since the required
shift cannot be increased indefinitely due to voltage
constraints.
One of the primary uses of inertial systems
is to determine aircraft heading. Such a determination
depends upon the quality of the system sensors,
including the gyros, and is affected by the amount and
type of noise in the gyro outputs.
The noise properties of the outputs of
advanced technology gyros (e.g., those of the laser and
fiber optic type) include a so-called "random walk"
characteristic. This represents a stochastic process
in which each step constitutes a statistically
independent event. When measuring a variable subject
to random walk, such as the output of a fiber optic
gyroscope, a gradual convergence to a so-called "true"
measurement takes place. For example, in measuring the
drift rate of heading angle with a fiber optic
gyroscope known to possess a true drift rate of 0
degrees per hour, one might expect to obtain a rate
measurement of 0.9 degrees per hour over a 100 second
time slice and a measurement of 0.3 degrees per hour
over a 900 second time slice. It is a characteristic
of random walk that the uncertainty of an estimate
diminishes as its length (number of samples) is
increased.
Random walk can include a random, non-
convergent stochastic process known as white noise
(i.e., noise whose power spectral density (PSD) is
"flat"). The presence of white noise is particularly
troublesome when one employs a gyroscope to determine
heading angle. When a noise component of gyro output
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is truly white noise random, the RMS value of the angle
will grow with the square root of time. That is,
o = RW ~ (1)
S where:
RW = random walk coefficient;
T = time; and
a = standard deviation of the heading angle.
The above equation indicates that the random
walk error due to white noise will cause the heading
angle to grow over time. This, of course, is quite
troublesome.
Figure 3 is a graph (not to scale) that
illustrates the relationship that exists between random
walk (curve 20) and light source peak power in a fiber
optic gyroscope. White noise in the output of a fiber
optic gyro can have a number of sources. Electronics
noise (both dark current and Johnson or thermal noise),
shot noise and beat, or synonymously relative intensity
noise, may all contribute. The contributions of
electronic noise and shot noise to gyro random walk
decrease as the peak power is increased, a phenomenon
shown generally in Figure 3. As may also be seen in
that figure, the contribution of synonymously relative
intensity noise (curve 22) is independent of peak power
and thereby limits the extent to which gyro random walk
can be reduced through an increase. In contrast within
a predetermined range, increases, in peak power will
reduce the contributions of electronics noise (curve
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24) and shot noise (curve 26). Beyond such range,
increased power will not lead to better random walk
performance.
The relative importance of white noise
increases with the power of the light source.
Superluminescent diodes provide about 0.5 milliwatts of
peak power whereas rare earth doped sources are
commonly rated in the vicinity of lO milliwatts.
Referring to Figure 3, the contribution of white noise
to random walk is a fraction of that of shot noise
which, in turn, is a fraction of that of electronics
noise when a low power source, such as a
superluminescent diode, is employed. As the power of
the light source is increased, the contribution of
synonymously relative intensity noise eventually
dominates the noise performance of the gyroscope.
A prior art attempt to isolate and remove the
effect of white noise from gyro output has involved
"tapping" the output of the light source, then
differencing such output with that of the gyro. This
relies upon the fact that synomously relative intensity
noise originates with the light source. The
mechanization of such a scheme is complex and fraught
with technical difficulties involving synchronization
of detected outputs and matching and stabilization of
gains with time and temperature as well as a second
detector requirement. In addition to the obvious
costs, including power, incurred, the size of the gyro
is necessarily increased, rendering such approach of
limited feasibility.
DISCLOSURE OF THE INVENTION
The present invention addresses the preceding
and other shortcomings of the prior art by providing,
in a first aspect, an improvement in the method for
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modulating a fiber optic gyroscope of the type that
includes the sensor coil of optical fiber. In such a
gyroscope, the output of a light source is directed
into the coil, then split into a pair of beams. The
beams are input into opposed ends of the coil to
- counterpropagate and are then combined to form a
gyroscope output.
The improvement provided by the invention is
begun by selecting a periodic artificial phase shift
such that the random walk of the output is less than
that of maximum output signal modulation. Such a
periodic artificial phase shift is then applied between
the counterpropagating beams.
A second aspect of the invention provides a
method for modulating a fiber optic gyroscope of the
type that includes a sensor coil of optical fiber.
Such method is begun by directing the output of a light
source into the gyroscope. The light source output is
then split into a pair of beams and such beams are
input into opposite ends of the coil to
counterpropagate. A periodic artificial phase shift in
the form of a s~uare wave ~M(t) is applied between the
counterpropagating beams. Such periodic artificial
phase shift is of the form (4n+1)~/2 < ¦~M(t) < ~2n+1)~
where n is an integer and includes 0. The
counterpropagating beams are then combined to form a
gyroscope output.
In a third aspect, the method described in
the preceding paragraph is modified insofar as the
square wave ~M(t) for applying a periodic artificial
phase shift between the counterpropagating beams is of
the form (2n+1)~ < ¦~M(t)¦ < (4n+3)~/2 where n is an
integer and includes 0.
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In a fourth aspect, the invention provides a
method for modulating a fiber optic gyroscope of the
type that includes a sensor coil of optical fiber. In
such aspect, the output of a light source is directed
into the gyroscope. The light source output is then
split into a pair of beams which are input into
opposite ends of the coil to counterpropagate therein.
A periodic artificial phase shift is applied
between the counterpropagating beams. Such phase shift
comprises a sinusoid ~M(t) where x < ¦~M(t)¦ < y, x
being such that J1(x) is a maximum, Jl(Y) = 0 and Jl( )
being a first order Bessel function of the first kind.
The counterpropagating beams are then combined to form
a gyroscope output.
A fifth aspect of the invention provides a
method for adjusting random walk noise in the output of
a fiber optic gyroscope of the type in which a pair of
light beams counterpropagates within a sensor coil of
optical fiber. Such method is begun by applying a
periodic artificial phase shift between the
counterpropagating beams and varying the magnitude of
the periodic artificial phase shift to responsively
adjust random walk noise.
The foregoing and other features and
advantages of the present invention will become further
apparent from the detailed description that follows.
Such description is accompanied by a set of drawing
figures. Numerals of the drawing figures,
corresponding to those of the written text, refer to
features of the invention. Like numerals refer to like
features throughout both the written description and
drawing figures.
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BRIEF DESCRIPTION OF DRAWINGS
Figure 1 is a graph of the intensity or power
of the output of a fiber output gyroscope as a function
of the phase shift between the counterpropagating
component beams with the effect of square wave
modulation in accordance with the prior art indicated
thereon;
Figure 2 is a graph of a representative
modulation waveform (square wave) for a fiber optic
gyroscope in accordance with the prior art;
Figure 3 is a graph of the relationship(s)
between the random noise components of the output of a
fiber optic gyro and peak power of the light source;
Figure 4 is a graph of gyroscope output
random walk as a function of modulation amplitude;
Figures 5(a) and 5(b) are graphs of a
modulation waveform in accordance with the method of
the invention and the output of a fiber optic gyroscope
subject to such modulation, respectively; and
Figure 6 is a graph of the output of a fiber
optic gyroscope with areas indicated thereon
corresponding to square wave modulation in accordance
with the invention.
BEST MODE FOR CARRYING OUT T~E INVENTION
The present invention overcomes limits to
possible noise reduction in the output of a fiber optic
gyroscope subject to random walk error and including a
white noise component. Referring back to Figure 3,
random walk noise is subject to reduction through
increased light source peak power. However, a limiting
value is approached as power is increased. The
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presence of such a limiting value can be especially
troublesome when measuring heading angle over a
protracted period of time due to the proportionality
between heading error and time in the presence of white
noise.
Superluminescent diodes have commonly been
employed as light sources. However, higher power light
sources of the rare earth doped fiber type may generate
power in the region of ten (10) milliwatts, far in
excess of that provided by a superluminescent diode.
Figure 3 demonstrates that, as the power of the light
source is increased, the relative contribution of a
factor, synonymously relative intensity noise,
insensitive to peak power, becomes the dominant
contributor to output noise.
The inventor has addressed the problem of a
"floor" to noise reduction in gyroscope output by
providing modulation schemes that produce output
signals are characterized by random walk that is less
than is possible when modulation schemes of the prior
art type such as the square wave modulation of Figure
2, are employed. Further, the modulation schemes of
the invention address problems that become more
prominent as the power of the light source is
increased. Thus, while some degree of output
sensitivity to nonreciprocal phase perturbations ~, is
sacrificed, increased light source power compensates to
generate adequate signal output for data processing
purposes.
I. Analysis of Square Wave Modulation
As mentioned earlier, random walk in the
output of a fiber optic gyro is limited by white noise
having a flat PSD. The sources of white noise in fiber
optic gyros include electronics noise, shot noise and
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beat noise or synonomously relative intensity noise.
The random walk due to each of such noise sources for a
gyro modulated in accordance with Figure 2, for
example, is as follows:
RWElex 1 y 2 NEP (2a)
KSSF Po
RWshot KSSF ~ 2e (2b)
RWRIN = RIN/[KSsF 2] (2c)
RW S = ~ RWeleX + RWshot RIN (2d)
where KSsF is the Sagnac Scale Factor of the fiber
gyro,
NEP is the noise equivalent power in units of Watts/~Hz
for the photodetector, e is the charge of an electron,
R is the photodetector responsivity in units of
amps/Watt, RIN is the relative intensity noise
coefficient of the light source in units of l/Hz and PO
is the peak power of the gyro light source.
The Sagnac scale factor, KSsF~ is defined as:
KSSF 2~LD (3)
~c
where.L is the fiber length, D is the fiber diameter, ~
is the mean wavelength of light in the fiber gyro and c
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the free space velocity of light. The detected power
of the fiber gyro is:
P = P ~1 + coS (KSsf n + ~M( )))/
where ~ is the rotation rate to be measured and ~M(t) is
a periodic modulation function. Referring to Figure 2,
~M(t) is chosen to be a square wave of period 2r and
amplitude ~/2 although a sinusoidal waveform of
amplitude varying between maxima of approximately +1.8
radians and having period of 2r (where 1.8 radians is
the argument x of Jl~x)max where Jl(x) is a first order
Bessel function of the first kind) represents the
equivalent prior art sinusoidal modulation scheme. As
mentioned earlier, r is the transit time of light
through a fiber coil of length L. To accommodate the
modulation function as shown in Figure 2 (i.e. square
wave of amplitude +~/2 ), demodulation may be
accomplished via wideband A/D conversion followed by
- digital subtraction.
Referring to equations 2a through 2d in
combination with the graph of Figure 3, it may be noted
that RWeleCt and RWShot is each inversely proportional
to PO while the value of RWRIN, the remaining component
of RWRSs, is independent of PO. Accordingly, it is the
contribution due to synonynously relative intensity
noise that is the source of true white noise in the
output of a fiber optic gyro. This noise component of
gyro random walk limits the noise reduction that can be
achieved through higher power operation.
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The inventor has addressed the preceding
noise limitation with a modulation scheme that reduces
the effect of relative intensity noise. In this way
the noise performance of the output of the gyro
overcomes the limitations that are present when a
"conventional" periodic modulation waveform, such as
the square wave illustrated in Figure 2 or the
corresponding sinusoidal prior art modulation waveform,
is applied.
It is well known that the general equations
for random walk due to electronic noise, shot noise and
beat noise in a gyro subject to square wave modulation
of arbitrary amplitude are as follows:
1 ~ NEP
RW 1 = KSSF P sin ~M (5a)
~ 1 2e~1 + cos ~
RWshot = KSSF I o sin ~M (5b)
1 RIN1 + cos ~
RWRIN = KSSF ~ sin ~M (5c)
As before,
RSS ~ RWeleX + RWshot + RWRIN2 (5d)
The above expressions differ from those of
equations 2a through 2d by the inclusion of terms that
drop out when ~M is set to ~/2. (Corresponding
expressions, well known to those skilled in thee art,
describe gyro output noise subject to sinusoidal
modulation of arbitrary amplitude ~M-) By referring to
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,~
equations 5a through 5c, the inventor has found that
the "limiting" random walk coefficient RWRIN may be
reduced below that given by equation 2c when the
amplitude of the square wave modulation is changed from
+~/2, representative of maximum signal modulation of
the prior art, to a function ~M(t) defined as follows:
(4n+1)~/2 < I~M(t)l < (2n+1)~ (6a)
(2n+1)~ < ¦~M(t)¦ < (4n+3)~/2 (6b)
where n is an integer including 0.
The inventor has found a corresponding
improvement in output random walk in the presence of
sinusoidal modulation ~M(t) when such modulation is
changed from a sinusoid alternating etween maxima and
minima of approximately +1.8 radians to one having
maxima and minima within a range defined as follows:
x < I~M(t)l < Y (6c)
where x is such that J1(x) is a maximum, Jl(Y) = 0 and
J1( ) is a first order Bessel function of the first
kind.
The improvement is gyro random made possible
by modulation in accordance with the invention is
confirmed by the graph of Figure 4 which presents a
plot of random walk (logarithmic scale) versus square
wave modulation amplitude ~M Three curves are
presented, each corresponding to a different peak power
level. One curve presents a plot of the variables at a
peak power of 10 milliwatts, another plots the
variables for a peak power of 100 milliwatts, and the
third curve plots the variables for a peak power level
of 1000 milliwatts. As can be seen, gyro random walk
decreases with increasing peak power. (Note: the
left- most values of the graph indicate prior art
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(+~/2) square wave modulation.)
Comparing the graphs of Figures 2 and 4, one
can see that a dramatic reduction of random walk is
obtained as the square wave modulation amplitude ~M is
increased beyond ~/2 (the same result would be observed
below -~/2). As shown in Figure 4, the random walk is
limited to approximately 0.007 degree/ ~our for all
power levels at ~/2 square wave modulation amplitude.
This value essentially represents the relative
intensity noise term RWRIN of equation 2c. A greater
than ten-fold reduction is obtained when the modulation
amplitude ~M is increased to approximately 31~/32 in a
1000 microwatt peak power gyro. Thereafter, the random
walk of the gyro output increases substantially in the
region of ~ radians, reflecting the relative
insensitivity of output intensity to deviations in
phase difference in the region of a ~ (or n~ where n is
an integer) phase difference.
Table I below lists the random walk that can
¦ 20 be obtained with prior art (i.e. +~/2 square wave
modulation amplitude) operation along with an optimum
modulation amplitude and the random walk value achieved
using such optimum modulation in accordance with the
invention. Further, an improvement factor is indicated
representing the reduction in white noise accomplished
when modulation in accordance with the invention is
employed.
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TABLE I
Peak Power Random Walk, Opt. Mod. Random Walk
Improve
(microwatts) Prior Art Amplitude~ at ~opt
Factor
(deg/ ~ ) ~opt (rad) (deg/ ~ )
0.008 2.6 0.004 2X
100 0.007 3.0 0.001 7X
1000 0.007 3.1 0.0004 17.5X
Figures 5(a) and 5(b) are a pair of graphs
- that illustrate the modulation, operating points and
outputs of a fiber optic gyro modulated in accordance
with the invention (square wave modulation). The
foregoing are compared with the corresponding values
for a lower power fiber optic gyro modulated in
accordance with the prior art (corresponding prior art
values indicated by dashed lines).
Referring to Figure 5(a), one can see that
the modulation waveform 28 comprises a square wave of
period 2r with amplitude varying between the ranges of
~/2 to ~ and -~ to -~/2. (A sinusoidal waveform of
identical period and amplitude x lying between the zero
and maximum value arguments of Jl(x), the first order
Bessel function of the first kind, would be equally
applicable.) The precise value of the modulation
amplitude may be obtained by solving equation 5d for a
minimum. However, the method of the invention is not
addressed solely to attaining a minimum random walk
value but rather to the improvement of random walk
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performance over that possible with prior art
modulation schemes.
Figure 5(b) is a graph of the intensity of
the output of a fiber optic gyroscope as a function of
~. As can be seen, the operating points 30, 32,
corresponding to the n=0 in equation 6a describing
square wave modulation in accordance with the
invention, are located away from the +~/2 inflection
points 30' and 321 of the output curve 34. The output
intensity at such points is characterized by less-than-
maximum sensitivity to a nonreciprocal perturbation ~
indicative of a detected rotation rate. The lessened
signal sensitivity will be partially compensated by
increasing the power of the output curve 34 relative to
that of a curve 36 (indicated by a dashed line)
corresponding to a lower power source, such as a
-- superluminescent diode. Although not essential, the
substitution of a higher-power light source, when
necessary, will overcome the effect of lessened
sensitivity to a phase deviation for purposes of
generating an electronic output. Further, as seen in
Figure 3, the relative benefit of the modulation scheme
of the invention becomes more significant as peak power
increases. By comparing the operating points 30, 32 of
a gyro modulated in accordance with the invention with
the operating points 38' and 40' of one subject to +~/2
square wave modulation, one may observe that, whereas
the prior art modulation maximizes signal, the
modulation scheme of the invention is directed to
maximizing signal-to-noise ratio or, equivalently
minimizing random walk.
Figure 6 is a graph of the output of a fiber
optic gyro as a function of phase shift ~. The areas
is identified as "A'l in this figure correspond to
square wave modulation in accordance with the invention
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as defined in equation 6a while those identified as "B"
correspond to square wave modulation in accordance with
the invention as defined in equation 6b.
II. Sinusoidal Modulation in Accordance With the
Invention
Sinusoidal modulation in accordance with the
invention is defined by equation 6c. Square wave and
sinusoidal modulations in accordance with the invention
share the characteristic that each is arranged to
operate upon the gyro output fringes at points other
than those associated with generating a maximum output
signal ~I (or ~P) for a given nonreciprocal phase shift
~. Differences between the maximum amplitudes of such
prior art square wave and sinusoidal modulations (i.e.,
~/2 as opposed to about 1.8 radians) are related to the
qualititively-different durations of the transition
periods between the maxima and minima of square waves
and sinusoids. Thus Fourier-Bessel analysis, well
known in the art, defines about +1.8 radians as the
sine wave maxima and minima for achieving a maximum
gyro output for a given ~ just as +~/2 defines the
square wave ~ and minima for obtaining a maximum
gyro output for a given c. The modulation defined by
equation 6c, just as those defined by equations 6a and
6b, is directed to operation upon regions of the gyro
output curve other than those associated with maximum
signal output.
Thus it is seen that the present invention
provides a method for modulating a fiber optic gyro
that accomplishes a reduction in the white noise
component of random walk beyond the limitations imposed
by prior art modulation schemes. By reducing such
error, one can significantly improve the accuracy of an
inertial navigation system that employs fiber optic
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2~ 73871
gyros to determine heading angle. The present
invention requires no additional hardware to accomplish
noise reduction and thus increases neither system
complexity nor cost.
While this invention has been described with
reference to its presently preferred embodiment, it is
not limited thereto. Rather, this invention is limited
only insofar as it is defined by the following set of
patent claims and includes within its scope all
equivalents thereof.
