1 Enhancing the SNR of the Fiber Optic Rotation Sensor using the LMS Algorithm Hani Mehrpouyan, Student Member, IEEE, Department of Electrical and Computer Engineering Queen s University, Kingston, Ontario, Canada Abstract This paper investigates the application of forward linear prediction based on the least mean square (LMS) algorithm in the design of a finite impulse response (FIR) filter for the purpose of improving the signal to noise ratio (SNR) of the measurements acquired from a fiber optic gyroscope (FOG) The proposed approach determines the optimum filter tap weights and eliminates the noise and other high frequency disturbances without the need for a training sequence, specific model, or state-space formulation The designed traversal tap-delay line filter is validated by processing raw sensor measurements of the KVH fiber optic gyroscope acquired using a National Instrument 12 bit data acquisition system Simulation results demonstrate significant SNR gain when the LMS filter is applied to the measurements results Moreover, the convergence rate of the LMS algorithm for small, large, and variable step sizes and the effect of the filter order on the cut off frequency and SNR gain is analyzed and compared Index Terms fiber optic gyroscope, measurement signal to noise ratio, forward linear prediction, least mean square, and transversal tap delay line filter I INTRODUCTION FIBER optic gyroscopes (FOGs) have become a well established navigational and guidance tool due to their long life, ruggedness, small size, low cost, and environmental insensitivity [1] However, the performance characteristics of FOGs are highly affected by the bias drift and angle random walk (ARW) Bias drift is defined as the deviation in rotation measurements due to temperature (increases with the temperature), affecting the long term performance of the FOGs [2], [3] On the other hand, ARW affects the short term performance of the system and is the broad noise component of the FOG output [3] ARW can be modeled as a random process which is the result of the combined effect of the noise introduced by the photodetector and the light source intensity noise [4], [5] During the alignment process, FOGs are used to monitor the components of Earth rotation rate along the sensitive axis to determine the initial movements of the platform [6] The accuracy by which Earth s rotation can be determined is significantly affected by the signal to noise ratio (SNR) and the magnitude of the ARW, where the higher the SNR the lower the variance of the rotational measurements Therefore, many different schemes based on hardware modifications or signal processing implementations have been proposed to try to reduce the ARW level and increase the SNR In this paper we focus on the latter Forward linear prediction (FLP) techniques have been used successfully to reduce the noise level in many applications In [7] the FLP algorithm is applied as a channel equalizer, compensating the negative effect of the channel fluctuations and significantly improving the average bit error rate and performance of the communication system FLP uses a set of past samples from a stationary process to predict future sample values [8] and subsequently reduce the noise The most common and practical predictor is the single time-unit predictor which is implemented using a tap-delay line filter with a predetermined order To simplify the filter implementation a finite impulse response (FIR) filter structure as presented in Fig 1 is used for the design of the filter The LMS algorithm is used to determine the optimum tap weights for the FIR filter based on fixed and variable step sizes [9] Fig 1 The structure of the tap-delay FIR filter that uses the past samples {d(n 1), d(n), d(n M)} to provide and estimate of the current sample value, ˆd(n) In this paper we investigate the use of FLP, based on the LMS algorithm to improve the SNR for the measurements data collected using FOGs As one of its main advantages the proposed strategy does not require any assumption on the distribution of the noise or a specific state-space model to perform the prediction and reduce the noise level Moreover, the optimum tap weights of the FIR filter are determined without the need for a training sequence, achieving the proposed performance gains with significantly lower complexity To validate the algorithm
2 outlined here, the designed filter is used to process the raw sensor measurements of a KVH fiber optic gyroscope at 128 Hz and the simulation results demonstrate significant SNR gain Moreover, we have investigated the convergence rate and the SNR for the LMS algorithm based on fixed and variable step sizes Finally, the effect of filter order on the cut off frequency and the achievable SNR gain for the overall system are reported and simulation results are presented to support the findings of the paper This paper is organized as follows: Section II outlines the system model and establishes the algorithms under consideration to determine the optimum filter coefficients with respect to a minimum mean square () design criteria for the LMS algorithm Section III discusses the extensive simulation results and examines the effect of filter order and the step size on the performance and convergence rate of the proposed algorithm, respectively This following notation is used throughout this report: italic letters (x) represent scalar quantities, bold lower case letters (x) represent vectors, bold upper case letters (X) represent matrices, and () T denotes transpose II SYSTEM AND FILTER MODEL In this section we define the system model for the proposed project Eq (1) defines the relationship between the raw sensor measurements r(n) and the desired signal d(n) r(n) = d(n) + ν(n), (1) where ν(n) is the additive noise representing ARW, with mean zero and variance σ 2 n The LMS algorithm is used to to extract the desired signal and improve the signal to noise ratio Fig 2 represents the block diagram for the FLP system setup Fig 2 The block diagram representing the two filter designs used to remove the disturbances from the received signal An FIR filter, illustrated in Fig 1, is used to estimate the desired signal, d(n) Based on the design criteria the input and output relationship for the filter can be illustrated as M ˆd(n) = w k r(n k), (2) k= where {w, w 1,, w M } represent the filter coefficients and M is the order of the FIR filter Eq (2) in vector form is represented as ˆd(n) = w T r, (3) where w T is transpose of the M 1 vector of the filter coefficients and r is the M 1 vector of the input parameters (r = {r(n), r(n 1),, r(n M)} T Based on the above system model the mean square error () for the above estimation problem can be defined as j(n) = E [(d(n) ˆd(n)) 2] = E[(d(n) w T r)(d(n) r T w)], (4) where j(n) is the cost function and can be rewritten as j(n) = E [ (d 2 (n)] 2w T E[rd(n)] + w T E[rr T ]w ] (5) Assuming that the input and the desired sequence are stationary zero-mean random processes, Eq (5) can be modified as follows j(n) = σ 2 d 2w T p + w T Rw, (6) where σd 2 is the variance of d(n), p is the cross correlation vector between the input sequence and the desired sequence and is expressed as E[r(n)d(n)] E[r(n 1)d(n)] p = E[rd(n)] = E[r(n 2)d(n)], (7) E[r(n M)d(n)] and matrix R is the autocorrelation matrix of the input sequence and is defined in Eq (8) on the next page The objective of this design is to determine the filter coefficients, w, such that the cost function expressed in Eq (6) is minimized j(n) which has been derived based on the in its quadratic form can be presented as j(n) = j min + (w w o ) T R(w w o ), (9) where j min represents the minimum mean square error (M) corresponding to the optimal filter weights, w o By taking the gradient of j(n) in Eq (9) with respect to the filter weights and moving in small steps in the opposite direction of the gradient vector, the following relationship between the filter coefficients can be found as w n+1 = w n µ 2 j(n) w(n), (1) where the negative sign guarantees that the movement is in the negative direction of the gradient and the parameter µ is the step size The choice of µ dictates the convergence speed of the algorithm and also the value of the M The smaller the value of µ the lower the M, however the slower the algorithm converges to the optimum filter
3 R = E [ rr T ] = E[r 2 (n)] E[r(n)r(n 1)] E[r(n)r(n M)] E[r(n 1)r(n)] E[r 2 (n 1)] E[r(n 1)r(n M)] E[r(n M)r(n)] E[r(n M)r(n 1)] E[r 2 (n M)] (8) weights After further algebraic manipulation Eq (1) can be represented as [9] w n+1 = w n + µ e(n)r(n), (11) 2 where e(n) is the error function, defined as d(n) ˆd(n) and can be further expressed as e(n) = d(n) w T (n)r(n) (12) w(n) 35 x 1 3 3 25 2 15 Filter impulse response III SIMULATION RESULTS In this section we investigate the performance of the FLP algorithm based on LMS, in terms of improving the SNR for the measurement data obtained using the KVH FOG The LMS algorithm presented in the previous section is used to find the optimum tap-weights for the FIR filter and subsequently the filter is applied to the measurement data acquired using a National Instrument 12 bit data acquisition system at a 128 Hz The effect of filter order and the choice of the step size on the performance of the filter are also examined and the simulation results are presented Fig 3 represents the impulse response of the LMS filter Based on the results presented in Fig 3 the designed FIR filter is a moving average filter that removes the effect of the additive white noise by averaging over the samples This is an interesting development and demonstrates that the application of a simple averaging filter could significantly improve the SNR ratio for the measurements results performed using a FOG Fig 4 represents the magnitude and phase response of the LMS FIR filter One desired property of the filter is its linear phase characteristics Thus, the output of the filter does not suffer from variable group delay and is not distorted Based on the magnitude response we can deduce that the filter is a low pass filter with an approximate cut off frequency of 1 Hz as illustrated more clearly in Fig 5 The low cut off frequency of the filter removes the effect of high frequency disturbances, thus increasing the overall SNR Fig 6 represents the mean square error () for the FIR filter derived in Eq (12) with the step size µ = 1 As noted in Fig 6 approximately 1 samples are required for the filter to reach the minimum min square error (M) with µ = 1 This is a reasonably fast response considering that on a average it takes 1 to 15 minutes for the gyroscope to perform its measurements and demonstrates that the proposed algorithm does not suffer from significant delays and is applicable to real 1 5 5 1 15 2 25 3 n Fig 3 Filter impulse response with filter order set to 3 and the step size µ = 1 world scenarios The following results examine the effect of variable step sizes on the achievable M Fig 7 represents the input output relationship for the LMS filter As noted in Fig 7, the digital input to the filter suffers from considerable distortion caused by ARW additive noise However, the output of the LMS filter is capable of removing a significant portion of the distortions and improve the SNR Quantitatively the SNR for the input to the system is measured to be 3397dB and after applying the FIR filter the SNR improves to 544dB The step size parameter plays an important role in the overall M for the LMS algorithm and how quickly the M is reached The maximum value for the step size, µ is calculated as 1 µ max =, (13) λ min + λ max where λ min and λ max are the minimum and maximum eigen values for autocorrelation matrix of the input signal, defined in Eq (8) For the data used throughout this paper and a filter order of, M = 3, λ min = 6274 1 5 and λ max = 7819 making µ max = 1278 based on Eq (13) One of the main goals of this project is to investigate the effect of the step size, µ, on the M of the LMS algorithm As stated previously a larger step size results in a faster convergence rate, however it also results in a larger M Fig 8 represents the derived in Eq (12) for different values of µ and also provides a comparison for the case of
4 1 x 14 5 5 4 6 Fig 4 The magnitude and phase response of the FIR filter with M = 3 and µ = 1 1 1 1 1 1 1 1 3 1 4 1 5 1 2 3 4 5 6 Fig 6 The for the the FIR filter with M = 3 and µ = 1 1 5 5 1 15 5 3 35 4 45 5 55 6 1 2 3 4 5 6 7 8 9 1 Fig 5 The magnitude of the FIR filter with M = 3 and µ = 1 variable µ, ranging from the upper limit to the chosen lower limit for the step size At µ = 1278 the LMS algorithm converges very quickly and the M is reached At µ = 4, even though a lower overall M can be reached compared to µ max, close to 2 samples are required to for the LMS to converge, making µ = 4 an unattractive choice for the step size due to this significant delay Fig 8 also represents the plot for the case of variable step size, when 4 µ 1278 It is interesting to note that the by varying the step size from one iteration to another, one can achieve both fast convergence and low M Table I quantifies the SNR gain associated with different step sizes and demonstrates by choosing a variable step size, the LMS algorithm both converges very quickly and also results in the best SNR gain Fig 9 compares the linear phase characteristics of FIR filters with different step sizes It is important to note that as the step size increases the FIR filter loses its linear phase d(t) 18 16 14 12 1 8 6 4 2 The filtered measurments Raw meas Filtered meas t (sec) Fig 7 The input/output relationship for the FIR filter with M = 3 and µ = 1 characteristics which negatively affects the overall system performance Thus, when choosing the step size it is also important to analyze the both the magnitude and phase response of the filter to ensure that designed filter does not distort the signal negatively Although, in this specific application, since the signal from the FOGs only consists of low frequency components the nonlinear phase property of the FIR filter does not affect the system performance The filter order is another important design parameter that affects the performance of the system in terms of SNR gain and cut off frequency Figs 1 and 11 represent the frequency response of the FIR filter when M = 1 and M = 2, respectively Comparing the results in Figs 1, 11, and 5, one can conclude that as the filter order is increased the cut off frequency for the FIR filter decreases, eliminating a larger portion of the high frequency
5 1 1 1 µ=12788 µ=4 µ=variable x 14 1 Max step size 1 1 3 1 4 1 2 3 4 5 6 Fig 8 The for the the FIR filter with M = 3 and µ = 1278, µ = 4, and 4 µ 1278 TABLE I A COMPARISON OF THE EFFECT OF THE STEP SIZE, µ AND THE FILTER ORDER, M, ON THE SNR GAIN OF THE FLP ALGORITHM SNR gain Normalized Cut off frequency (f s = 2kHz) M = 3 µ = 1278 1646dB 225 1 3 Hz µ = 4 165dB 225 1 3 Hz 4 µ 1278 1652dB 225 1 3 Hz M = 2 µ = 191 1259dB 18 1 3 Hz µ = 4 1261dB 18 1 3 Hz 4 µ 191 1261dB 18 1 3 Hz M = 1 µ = 3835 963dB 675 1 4 Hz µ = 4 969dB 675 1 4 Hz 4 µ 3835 969dB 675 1 4 Hz disturbances caused by ARW noise Table I quantitatively represents the normalized cut off frequency for the FIR filters of order 1, 2, and 3 Moreover, Table I also demonstrates that the filter order plays a more significant role in noise reduction compared to that of the step size Therefore, the choice of filter order is an important design parameter since it affects, the cut off frequency, the SNR gain, and the operating delay of the filter, because the higher the filter order, the larger the overall delay Figs 12 and 13 represents the curves for the FIR filters of order M = 1, and M = 2, respectively The following observations can be made based on the simulation results: 1) The maximum step size, µ max, is different for different filter orders as pointed out in Figs 12, 13, 8, and Table I 3 4 x 14 6 1 3 x 14 Min step size Variable step size 4 Fig 9 The Phase response for the FIR filter with M = 3 and µ = 1278, µ = 4, and 4 µ 1278 2) The minimum step size, µ min, needs to be adjusted according to the filter order, since as shown in Fig 12, µ min = 4 is too small for a filter order of 1, where close to 5 samples are required before the LMS algorithm converges, resulting in considerable delay 3) The M is not affected by the filter order since for filters of order M = 1, M = 2, and M = 3, the M 6 1 4 4) The variable step size approach, where µ max is used in the first iteration and is then replaced by µ min in the following iterations can be applied to any filter order and is even more effective for smaller filter orders IV CONCLUSION In this paper noise reduction for the fiber optic gyroscopes using the forward linear prediction based on the least mean square error algorithm was investigated and developed The signal received from a FOG is affected by many different sources of noise, which greatly affects the accuracy of rotational measurements performed by such
6 1 1 1 1 1 µ=3835 µ=4 µ=variable 3 4 5 1 1 1 1 3 6 2 4 6 8 1 1 4 1 2 3 4 5 6 7 8 Fig 1 The magnitude of the FIR filter with M = 1 and µ = 1 Fig 12 The for the the FIR filter with M = 1 and µ = 3835, µ = 4, and 4 µ 3835 1 1 1 1 1 µ=191 µ=4 µ=variable 3 4 5 6 1 1 1 1 3 7 2 4 6 8 1 Fig 11 The magnitude of the FIR filter with M = 2 and µ = 1 devices Using forward linear prediction we have demonstrated that the previous samples received by the system can be used to estimate the current samples and subsequently reduce the amount of noise and improve signal to noise ratio The LMS algorithm is affected by the filter order and also the step size parameter The effect of filter order on the cut off frequency of the filter was investigated and it was demonstrated that the higher the filter order the lower the cut off frequency and the higher the overall SNR gain for the system Moreover, we investigated the use of a variable step size strategy to reduce the convergence delay associated with the LMS algorithm when keeping the M the same By applying the largest possible step size in the first iteration and subsequently applying the lower limit for the step size, we were able to reach the M 1 4 1 2 3 4 5 6 7 Fig 13 The for the the FIR filter with M = 2 and µ = 191, µ = 4, and 4 µ 191 as fast as when the maximum value for the step size is applied Finally, it is important to note that the approach outlined in this paper can be applied to other applications to significantly reduce the effect of the noise REFERENCES [1] W K Burns, Optical Fiber Rotation Sensing Academic Press, Boston, 1994 [2] M Bowser M J Hammond, M Perlmutter, and R Christopher, Broad fiber optic gyroscopes for a broad range of applications, in IEEE Position Location and Navigation Symp, 1996, pp 553 558 [3] H Lefevre, The Fiber Optic Gyroscope Artech-House, Norwood, MA, 1993
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