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CHAPTER 5

IMPROVING GPS SIGNAL ACCURACY USING

KALMAN FILTER AND WIENER FILTER

5.1 INTRODUCTION

An inertial combination of GPS and INS systems provide a secure

resolution for launching such as aircrafts, Unmanned Air Vehicles (UAVs),

Precision Guided Munitions, and missiles. GPS/INS works join together to

provide a safe and reliable resolution for the noise problem as taking

predictive action. GPS can be used to continuously keep informed the

IMU/INS to maintain a minimum bias error. The integration of IMU and INS

provide the position of GPS for improving learning and track times, also to

handle multipath conditions during the launch. The GPS with IMU/INS

systems is linked using Kalman techniques to combine navigation

information and handle short intervals of GPS outage caused by interference

or jamming. The grades of performance achieved are dependent on the IMU /

INS accuracy and GPS satellite visibility and are driven generally by failing

to notice distance requirements. The complication and cost of coupling are

generally acceptable for manned missiles, aircraft and UAVs that carry both

systems. The advantages of integrating GPS/INS, depends on either GPS or

INS alone, can be listed as follows

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A high data rate of absolute navigation solution including

position and velocity

Better-quality short-term and long-term positioning accuracy

Enhanced availability and reliability

Fine tuned trajectories,

Greater data security and integrity requirements

Non significant cost, especially when utilized for military

applications

Figure 5.1 Loosely coupled integration algorithm

GPS

Receiver

Inertial

Navigation

Algorithm

Data Fusion

Algorithm

IMU

Sensor

Readings

GPS position

and velocity

Position

Velocity

Attitude Sensor

Errors

70

Figure 5.2 Tightly coupled integration algorithm

Different kinds of data fusion strategies are available. Based on that

GPS/INS integrated systems can be classified into three types: loosely

coupled GPS/INS, tightly coupled GPS/INS, and ultra-tight coupled

GPS/INS.

i) Loosely-coupled integration: the INS and GPS send either

position or velocity these are differences to form the

measurement input to a Kalman filter (KF) to estimate the INS

errors. Nevertheless, the gain is restricted by the need to

account for time-correlated noise on the GPS navigation

solution. For receiving high quality signal at least four GPS

satellites must be covered so that the GPS navigation solution

can calibrate the INS. A block diagram illustration of a

loosely coupled integration algorithm is presented in

Figure 5.1.

GPS

Receiver

Inertial

Navigation

Algorithm

Data Fusion

Algorithm

IMU Sensor

Readings

Pseudo ranges

and Doppler

measurements

Position

Velocity

Attitude Sensor

Errors

Figure 5.2 Tightly coupled integration algorithm

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ii) Tightly-coupled integration: the GPS navigation filter is

combined into the INS/GPS integration filter which uses the

pseudo range and pseudo range rate measurements from the

GPS antenna to each satellite, measured by the GPS receiver

and updates the same variables calculated through the INS

navigation solution. The INS errors are fed back to correct the

inertial navigation equation function. An independent INS

navigation solution is maintained, which is corrected to form

the integrated navigation solution. A block diagram of a

tightly coupled integration algorithm is presented in

Figure 5.2.

Figure 5.3 Ultra-tightly coupled integration algorithm

iii) Ultra-tightly coupled: This combines the INS/GPS integration

and GPS signal tracking functions into a single estimation

algorithm in which the INS measurements are fed back into

the receiver to decrease GPS signal tracking errors and

GPS

Receiver

Inertial

Navigation

Algorithm

Data Fusion

Algorithm

IMU

Sensor

Readings

Pseudo ranges

and Doppler

measurements

Position

Velocity

Attitude Sensor

Errors

72

enhance GPS positioning performance, in addition to assisting

the receiver tracking loop to recover a signal lock if it is

misplaced due to interference or jamming. A block diagram

illustration of an ultra-tightly coupled integration algorithm is

presented in Figure 5.3.

5.2 KALMAN FILTER

Figure 5.4 Block diagram illustration of Kalman filter

The kalman filter is a recursive predictive filter that is based on the

use of state space techniques and recursive algorithms. Figure 5.4 shows the

block diagram illustration of the kalman filter. It is estimated the state of

dynamic system. This dynamic system can be disturbed by some noise mostly

assumed as white noise. To improve the estimated state kalman filter uses the

measurements that are related to the state but disturbed as well.

Thus the kalman filter consists of two steps:

1. Prediction

2. Correction

𝐾𝑘

𝐵𝑘

𝐷𝑒𝑙𝑎𝑦 𝐴𝑘

𝐶𝑘

+

+ +

+

- y(k)

y(k-1)

x(k) x(k-1)

u(k)

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In the first step the state is predicted with dynamic model. In the

second step it is corrected with observation model, so that the error

covariance of the estimator is minimized. In this sense it is an optimal

estimator. This procedure is repeated for each time step with the state of

previous time step as initial value. Therefore the filter is called a recursive

filter. Malleswari et al (2009) has described about the modeling of errors

affecting the GPS signals as they travel from satellite to user on earth using

kalman filters.

Dah-Jing Jwo et al (2007) have explained about obtaining good

estimation solutions using the EKF approach for both dynamic process and

measurement models. Jaime Gomez-Gil et al (2013) used the low-cost GPS

receivers to provide geodetic positioning information using the NMEA

protocol, usually with eight digits for latitude and nine digits for longitude.

Azizi et al (2001) used the new approach for the mathematical modeling of

the GPS/INS supported photo-triangulation based on the Kalman filtering

process is proposed. Elder M. Hemerly et al (2008) multi-sensor based

navigation system is implemented in this thesis using odometer/INS

integration system. Ravindra Babu (2009) has proposed the Ultra-Tight

GPS/INS/PL Integration for analyzing the kalman Filter Performance.

The basic components of the kalman filter are state vector, the

dynamic model and the observation model. These are explained one by one in

the following.

5.2.1 State Vector

The state vector contains the variables to be measured. It describes

the state of the dynamic system and represents its degrees of freedom. The

variables in the state vectors cannot be measured directly, but they can be

inferred from the values that are measurable. Elements of the state vector can

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be positioned, velocity, orientation angles etc., If x1 and x2 are the two state

variables it is represented as

x = {𝑥1

𝑥2} (5.1)

The state vector has the two values at the same time, that is the

priori value, the predicted value before the update and the posteriori value, the

corrected value after the update and it is marked as 𝑥− and 𝑥+ respectively.

5.2.2 Dynamic Model

The dynamic model describes the transformation of the state vector

with respect of the time. The next state of the system can be represented by a

system of differential equation

𝑥𝑘+1 = 𝐴𝑘 𝑥 𝑘 + 𝑤𝑘 (5.2)

where 𝐴 is the dynamic matrix and constant. xk is the state vector and 𝑤𝑘 is

the dynamic noise which is usually assumed as white noise and has the

covariance matrix Q(t).

5.2.3 Observation Model

The observation model represents the relationship between the state

and measurements. In the linear case the measurements can be described by a

system of linear equations, which depend on the state variables. Usually the

observations are made at time steps ti

𝑧𝑘 = ℎ(𝑥𝑘, 𝑣𝑘) (5.3)

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The vector form of this system is

𝑧𝑘 = 𝐻𝑘𝑥𝑘 + 𝑣𝑘 (5.4)

where 𝑧𝑘 is the observation at epoch k , H is the measurement matrix , 𝑣𝑘 is

the noise of measurement process with covariance matrix R(ti). Like the

dynamic matrix in the discrete system, the measurement matrix H is a

constant as well.

5.2.4 Prediction

The prediction is the first step of kalman filter. The predicted state

or the priori state is calculated by neglecting the dynamic noise and solving

the differential equation that describes the system model.

𝑥𝑘+1− = 𝐴 𝑥𝑘 (5.5)

The state vector at time k can be expressed by a taylor series with

respect to an approximate state 𝑥𝑘 and the Equation 5.5 can be written as

𝑥𝑘− = 𝐴𝑘−1 (𝑥𝑘−1

− ) (5.6)

It shows the predicted state is a discrete combination of the initial

state xk-1

where 𝐴𝑘−1 is the state transition matrix which transform any initial state to

the specified state at time k.

In the more generalized form where also the covariance matrix of

the noise Q is a function of time, the covariance matrix is

𝑃 𝑘− = 𝐴𝑘 𝑃𝑘−1 𝐴 𝑘

𝑇 + 𝑄𝑘 (5.7)

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5.2.5 Correction

In the correction step the predicted state vector x(ti) is improved

with observations made at the epoch ti thus the posteriori state has the form

𝑥𝑘+ = 𝑥𝑘

− + ∆ 𝑥𝑘 (5.8)

With covariance matrix

𝑃𝑘+ = 𝑃𝑘

− + ∆ 𝑃𝑘 (5.9)

As said before the kalman filter is an optimal filter means that the

state variance of the state covariance matrix P+ is minimized. As the P

- is

already known from the prediction step it follows that ∆ 𝑃 minimized.

∆ 𝑃𝑘 = ∈ [∆ 𝑥𝑘 ∙ ∆ 𝑥𝑘𝑇] (5.10)

This condition is compiled with

∆ 𝑥𝑘 = 𝑃𝑘−𝐻𝑘

𝑇(𝐻𝑘𝑃𝑘−𝐻𝑘

𝑇 + 𝑅𝑘 )−1 ∙ (𝑧𝑘 − 𝐻𝑘 𝑥𝑘

−) (5.11)

∆ 𝑥(𝑡𝑖) = 𝐾𝑘 ∙ (𝑧𝑘 − 𝑥𝑘−) (5.12)

With

𝐾𝑘 = 𝑃𝑘−𝐻𝑘

𝑇(𝐻𝑘𝑃𝑘−𝐻𝑘

𝑇 + 𝑅𝑘)−1 (5.13)

Where 𝐾𝑘 is the gain matrix and

𝑧𝑘 − 𝑥𝑘− (5.14)

is the measurement residual. It reflects the difference between the predicted

measurement

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�̂�𝑘− = 𝐻𝑘 𝑥𝑘

− (5.15)

and actual measurement 𝑧𝑘.Finally the corrected state is obtained by

𝑥𝑘 = 𝑥𝑘− + 𝐾𝑘(𝑧𝑘 − �̂�𝑘

−) (5.16)

In this equation the estimated state and the measurements are

weighted and combined to calculate the corrected state that is if measurement

covariance is less than the predicted state, then the measured weight will be

high and the predicted state will be low. So the uncertainty can be reduced.

The covariance matrix of the a posteriori state is given by a law of

error propagation by

𝑃𝑘 = 𝑃𝑘− − 𝐾𝑘𝐻𝑘 𝑃𝑘

− (5.17)

𝑃𝑘 = 𝑃𝑘−(𝐼 − 𝐾𝑘𝐻𝑘 ) (5.18)

5.3 KALMAN FILTER SPECIFICATIONS AND DESIGN

Figure 5.5 Complete illustrations of operation of kalman filter

�̂�𝑘− = 𝐴 �̂�𝑘−1

𝑃 𝑘− = 𝐴𝑘 𝑃𝑘−1 𝐴 𝑘

𝑇 + 𝑄𝑘

1. Determine the next state

2. Determine the next error covariance

𝐾𝑘 = 𝑃𝑘−𝐻𝑘

𝑇(𝐻𝑘𝑃𝑘−𝐻𝑘

𝑇 + 𝑅𝑘)−1

𝑃𝑘 = 𝑃𝑘−(𝐼 − 𝐾𝑘𝐻𝑘 )

1. Calculate the kalman gain

2. Update estimate with measurement

𝑧𝑘 �̂�𝑘 = �̂�𝑘− + 𝐾𝑘(𝑧𝑘 − �̂�𝑘

−)

3. Update the error covariance

Predicting Equations for Time

update

Correcting Equations for

Measurement update

Initial estimates 𝑥𝑘−1

and 𝑃𝑘−1

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Figure 5.5 shows the complete illustration of kalman filter. With

the reference of the Figure 5.5 it is easy to describe the selection of tuning

parameters of kalman filter to improve the GPS signal accuracy.

The GPS receiver measurements include position, attitude,

altitude, velocity and time etc. In this work, position and

velocity are taken as two state variables. Hence state variables

‘n’ is set to 2.

One output variable i.e measurement of next state of the GPS

receiver signal is taken at each time interval. Hence the output

variable ‘m’ is chosen as 1.

To start the filtering operation initial state of the state

variables should be defined as given in Figure 5.5. Therefore

the estimation of initial position as given in equation 5.5 is

obtained by taking the mean value of the first two samples.

Similarly, initial velocity as given in equation (5.7) is obtained

by calculating the difference of the first two samples and

dividing it by the sampling period.

𝑥0− = [

𝑃𝑜𝑠𝑖𝑡𝑖𝑜𝑛𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦

] = [𝑣(1) + 𝑣(2) 2⁄

𝑣(1) + 𝑣(2) 50 𝑚𝑠⁄]

Initially the error covariance matrix 𝑃0−

is equal to

(302 ∙ [ 1 1 0 0

] ) i.e the value of the noise covariance matrix

Q = 30 multiplied with matrix [ 1 1 0 0

] of order n=2. As Q is

not possible to measure directly it is taken as 30 .

With the reference of the equation (5.1) State update matrix is

chosen as A= [1 0.0250 1

]

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With the reference of equation 5.4 the measurement matrix is

chosen as H = [1 0] because only position measurement is

taken at each time duration.

5.4 EXPERIMENTAL KALMAN FILTER RESULT

Figure 5.6 Error signal after applying Kalman filter

The errors obtained after applying Kalman filter can be seen in

Figure 5.6. It is the difference in absolute value with regard to the constant

original signal. Here it is observed that the Kalman filter reduces the error

considerably (mean = 7.47 meters, variance = 4.310 meters). The

disadvantage of the Kalman filter is that it behaves computationally complex,

for a large number of input data. So that the fuzzy system is designed for

handling large number of GPS data.

5.5 WIENER FILTER

In the signal processing, Norbert Wiener proposed the wiener filter

during the 1940s and this concept was published in 1949. The main principle

80

of this filter is to condense the quantity of noise present in a signal by

comparison of desired noiseless signal with an estimated signal. Kolmogorov

derived the discrete-time equivalent of Wiener's work independently and

published in 1941. Therefore the theory is always known as the Wiener-

Kolmogorov theory. Wiener filters are known as optimum linear filters which

one involves linear estimation of a desired signal sequence with respective of

an additional related sequence. In the statistical analysis to obtain the solution

using the linear filtering technique that the accessibility of definite statistical

parameters such that mean and correlation functions of the desired signal and

surplus additive noise are assumed.

Wiener filters have the following characteristics

The assumption is that the input signal and added noise are

stationary linear stochastic processes with knowledge of

spectral characteristics or autocorrelation and cross-

correlation functions

The condition is that the filter must be causal i.e physically

realizable

The Performance criterion considered in evaluating the signal

is minimum mean-square error (MMSE)

5.5.1 Model/Problem Setup

The input to the Wiener filter is assumed to be a signal, s (t),

corrupted by additive noise, n (t). The output is calculated by means of a

filter, g (t), using the following convolution

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�̂�(𝑘) = 𝑤(𝑘) ∗ (𝑠(𝑘) + 𝑛(𝑘)) (5.19)

Where

s (k) is the original signal (to be estimated)

n (k) is the noise

�̂�(k) is the estimated signal

H (k) is the Wiener filter's impulse response

The error is defined as

𝑒(𝑘) = 𝑠(𝑘 + 𝜑) − �̂�(𝑘) (5.20)

Where 𝛿 is the delay of the wiener filter (since it is causal)

The error is the difference of estimated signal and the true signal

shifted by 𝛿. Therefore the squared error is

𝑒2(𝑘) = 𝑠2(𝑘 + 𝜑) − 2𝑠(𝑘 + 𝜑)�̂�(𝑘) + �̂�2(𝑘) (5.21)

Where,

𝑠(𝑘 + 𝜑) is the desired output of the filter

𝑒(𝑘) is the error

Depending on the value of 𝛿 the problem name can be changed:

If 𝜑 > 0 then the problem is that of prediction (error is

reduced when �̂�(𝑘) is similar to a later value of s)

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If 𝜑 = 0 then the problem is that of filtering (error is reduced

when �̂�(𝑘)is similar to �̂�(𝑘))

If 𝜑 < 0 then the problem is that of smoothing (error is

reduced when �̂�(𝑘) is similar to an earlier value of s)

For a fixed inputs, the resultant solution is generally known as the

Wiener filter. The Wiener filter is insufficient for processing the signals of

non stationary property of the signal and/or noise is essential to the problem.

In this kind of situations, the optimum filter has to be considered in a time-

varying form.

At the optimum ∆𝐽(𝑤) = 0, so that, the first equation is called the

normal equation and it leads to the second expression for the wiener filter.

Note that

𝑅𝑥𝑤0 = 𝑟𝑥𝑠 ⇒ 𝑤0 = 𝑅𝑥−1 𝑟𝑥𝑠 (5.22)

The foundation of wiener filter is explained by Wiener (1949) in

the time series of extrapolation, interpolation, and smoothing of stationary

time series. Garcia & Chi Zhou (2010) have explained the technique to

improve GPS signal processing time using wiener filters with reduced length.

Wiener filter preprocessing to analyze both non stationary and stationary

phase noises of OFDM systems was discussed by Zhong et al (2013). System

identification of wiener filter using Bayesian semi parametric was analyzed

by Fredrik Lindsten et al (2013). Error present in the GPS signal’s estimates

can be provided by designing using wiener, and kalman filter smoothening

purpose (Shmaliy et al 2000).

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5.6 WIENER FILTER DESIGN AND SPECIFICATIONS

Wiener filter can be implemented in the system whose real signal

and additional noise are known with spectral characteristics or autocorrelation

and cross-correlation functions. Hence the following procedures are followed

to define the stochastic model of the process as well as to find the wiener

filter coefficients.

Step 1: Determine the stochastic model

The first step of designing the wiener filter is that the computation

of System coefficients that can be calculated mainly by two methods are Burg

method and least squares method. The most commonly used one is the least

squares method which is based upon the Yule-Walker equations that can be

written in matrix form as

[

1 𝑟1 𝑟2𝑟1 1 𝑟

… 𝑟𝑁−1

… 𝑟𝑁−2

𝑟2 𝑟1 1⋮ ⋮ ⋮

𝑟𝑁−1 𝑟𝑁−2 𝑟𝑁−3

… 𝑟𝑁−3

⋮ ⋮… 1 ]

[ 𝑟1𝑟2𝑟3⋮𝑟2]

=

[ 𝑟1𝑟2𝑟3⋮𝑟2]

(5.23)

the diagonal value r0 is = 1.

Auto correlated GPS input signal is represented as in Figure5.7. It

comes under the category of real roots with negative dominant roots. Hence

the system parameter can be chosen as a1=0.1 and a2=-0.8.

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Figure 5.7 Autocorrelation waveform of GPS signal

The Figure 5.8 shows the system model with coefficients a1=0.1

and a2=-0.8.

Figure 5.8 System model

Step 2: Determine noise covariance and measurement variance

Σ Σ

Σ a1=0.1

a2=-0.8

z-1

z-1

v1 (n)

v2 (n)

u(n-1)

u(n-2)

u (n)

x(n)

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The signal u(n) is produced by a white noise v1 (n) of zero mean

and variance of σv2. =0.27. The signal u (n) is corrupted by additive noise v2

(n) of zero mean and variance of σ22. =0.1. Wiener filter which is operated by

receiving the signal x (n) as an input signal to produce such that an estimate

of the desired output i.e optimum value in the sense of mean squared error.

Step 3: Calculate autocorrelation vector R

The Autocorrelation function of signal u (n) is

Ru = [R11 R12

R21 R22] ; Ru = [

1.0 0.50.5 1.0

] (5.24)

Since v2(n) is a white-noise process of zero mean and variance

𝜎22 = 0.1 the 2-by-2 correlation matrix Rv2 of this process results

Rv2 = [0.1 00 0.1

] (5.25)

R = Ru + Rv2

R = [1.10 0.500.50 1.10

] (5.26)

Step 4: Calculate cross correlation vector Q

For the 2-by-1 cross-correlation vector Q, then write

Q = [p(0)p(1)

] = [R12 + a2R11

R11 + a2R12] = [

0.5273−0.4457

] (5.27)

Step 6: Determine the wiener filter coefficients

The 2-by-1 optimum tap-weight vector 𝑊 of the Wiener filter is

defined by (5.39). It is obtained by multiplying the inverse matrix R-1

by the

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cross-correlation vector Q . By inverting the correlation matrix R of the

equation (5.36), get

𝑅−1 = [1.1 0.50.5 1.1

]−1

= [ 1.1458 −0.5207

−0.5207 1.1458] (5.28)

𝑊 = 𝑅−1 𝑄 ⇒ [ 1.1458 −0.5207

−0.5207 1.1458] [

0.5273−0.4457

] = [ 0.8362−0.7855

] (5.29)

5.7 EXPERIMENTAL WIENER FILTER RESULT

The errors after applying Wiener filter are calculated as the

difference (in absolute value) between the output of the Wiener filter and the

constant signal as in Figure 5.9. The Wiener filter achieves the smallest error

in both mean and average (mean = 2.014, variance = 0.994).

Figure 5.9 Error signal after applying wiener filter

5.8 CONCLUSION

In this chapter theoretical background of the kalman filter and

wiener filter have been discussed.