The document outlines a filter-type fault detection and exclusion (FDE) algorithm for multi-frequency GNSS receivers. It discusses using least squares residuals, subset, and parity space methods for fault detection on dual-frequency GPS. It then describes how to extend these methods to triple-frequency GPS and combined GPS/Galileo receivers by incorporating additional frequency measurements into the linearized measurement equations and parity relationships. The goal is to improve integrity, availability and continuity by leveraging multi-frequency observations for satellite fault detection.

1. Filter-Type Fault Detection
and Exclusion (FDE) on Multi-
Frequency GNSS Receiver
Advisor: Prof. Chang, Fan-Ren
Presenters: Tsai, Yi-Hsueh

2. 2
Outline
1. Introduction
2. GNSS Architecture, Observables, and
Covenantal Fault Detection and Exclusion
3. Using Multi-Frequency Tech. on FDE
4. Using ARMA-Filter on FDE
5. Using Kalman Filter on FDE
6. Conclusions and Future Works

3. 3
Introduction (1/2)
accuracy
FDE algorithm
(stand-alone GPS use)
integrity
availability
continuity
ICAO (International Civil Aviation Organization)
navigation performance requirements

4. 4
Introduction (2/2)
current
data
time
process
Filter-Type FDE Algorithm
(use current & past data)
Snapshot FDE Algorithm
(use current data)
past data

5. 5
GNSS Architecture
System GPS GLONASS Galileo
Developed by United State Russian European
Sat. No. / planes 24 / 6 24 / 3 27 / 3
Inclination 55 64.8 56
Altitude 20200 km 19100 km 23616 km
Period 11 hr 58 min 11 hr 15 min 14 hr 22 min
Modulation CDMA FDMA CDMA
L1*, E1-L1-E2** 1575.42 MHz* 1602+0.5625n MHz* 1575.42 MHz**
L2*, E6** 1227.60 MHz* 1246+0.4375n MHz* 1278.75 MHz**
E5B** 1207.14 MHz**
L5*, E5A** 1176.45 MHz* 1176.45 MHz**

6. 6
Observables
ionosphere
 
 2
frequencycarrier
1

 advancephase
delaygroupcode
Doppler shift
delta range

7. 7
Least-Squares-Residuals (1/3)
              kkkkkkk TT
yRHHRHx 111
ˆ 

     kkk xHy ˆˆ 
The estimate of the state vector is
and the range residual vector is
The estimate of y(k) is
The linearized GPS measurement equation is
     kkk yyy ˆ~ 
       ,kkkk wxHy      kNk R0w ,~

8. 8
Least-Squares-Residuals (2/3)
The test statistic
       kkkks T
yRy ~~ 1

0 5 10 15 20
0
0.05
0.1
0.15
0.2
false alarm rate = 1/100
normalized SSE (m2
)
probabilitydensityfunction
determination of threshold
Td
2(4)Parkinson showed that
the distribution of s(k)
is chi-square distributed
with degrees of freedom
as n4.

9. 9
Least-Squares-Residuals (3/3)
detection threshold Td under FAR = 1/15000
Number of satellites
in view, n
Chi-square
degrees of freedom
Detection Threshold
5 1 15.9032
6 2 19.2316
7 3 21.9546
8 4 24.3914
9 5 26.6521
10 6 28.7899
11 7 30.8356
12 8 32.8089

10. 10
> Te
Subset Method
Subset solutions are formed by removing
one of the visible satellites at a time.
s1(k)
si (k)
sn(k)
snf
(k)  Te

11. 11
satisfying and
Parity Space Method (1/2)
The parity matrix
The parity vector
    0HP kk
     kkk yPp 
        kkkk ni pppP 1
     kkk wxH    nfkb e
0
   kkb nfp         kkkkk wPxHP 
      4 n
T
kkk IPRP
nfth channel vector

12. 12
Parity Space Method (2/2)
ith channel vector
nfth channel vector
(failed satellite)
parity vector














 k
kk
maxarg
i
i
T
n,,i p
pp
1
No of visible satellite = 6

13. Using Multi-Frequency
Technique on Failure
Detection and Exclusion

14. 14
Dual Frequency GPS (1/2)
 
 
 
 
 
 
 
 

























k
k
k
k
fk
fk
k
k
L
L
nL
nL
L
L
2
1
2
2
2
1
2
1
w
w
κ
x
IH
IH
y
y
The estimated positioning result
The linearized measurement equation
          kkkkk L
TT
DF 12
1
ˆ yHHHx


        kfkfffk LLLLLLL 2
2
11
2
2
12
1
2
212 yyy 
where
     kkk DFDFDF wHy

15. 15
Dual Frequency GPS (2/2)
The parity matrix
The parity vector
      kfkfk LLLLDFDF PPP 2
1
2
2
1
2
1
1
1 
 
     
     kkff
kkk
LLLLLDF
DFDFDF
12
2
1
2
2
1
2
1
1
1
yP
yPp




The algorithm to identify the failed satellite
     kkks DF
T
DFDF ppThe test statistic
   
 
   
  















 k
kk
k
kk
L
iDF
L
iDF
T
DF
L
iDF
L
iDF
T
DF
ni
2
,
2
,
1
,
1
,
,,1
,maxmaxarg
p
pp
p
pp


16. 16
GPS Triple Frequency (1/2)
The estimated positioning result
Extend to the triple frequency
          kkkkk L
TT
TF 125
1
ˆ yHHHx


 
 
 
 
 
 
 
 
 
 
 







































k
k
k
k
k
fk
fk
fk
k
k
k
L
L
L
nL
nL
nL
L
L
L
5
2
1
2
5
2
2
2
1
5
2
1
w
w
w
κ
x
IH
IH
IH
y
y
y
       kckckck LLLLLLL 552211125 yyyy where
     kkk TFTFTF wHy

17. 17
GPS Triple Frequency (2/2)
The algorithm to identify the failed satellite
The test statistic      kkks TF
T
TFTF pp
   
 
   
 
   
  















 k
kk
,
k
kk
,
k
kk
maxmaxarg L
i,TF
L
i,TF
T
TF
L
i,TF
L
i,TF
T
TF
L
i,TF
L
i,TF
T
TF
n,,i
5
5
2
2
1
1
1 p
pp
p
pp
p
pp

The parity vector      kkk TFTFTF yPp 
The parity matrix
       
      






 

nLLLLLnLLLLLnLLLLL
LTFLTFLTF
TFTF
ffffff
kckckc
k
III
PPP
P 2
2
2
1
1
5
1
2
1
1
2
1
2
5
1
5
1
2
1
1
2
5
2
2
1
5
1
2
1
1
5211




18. 18
GNSS (GPS + Galileo)
To simultaneously use the measurements
of both Galileo and GPS system
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 





















































k
k
k
k
k
k
k
fk
fk
fk
fk
k
k
k
k
E
L
E
L
mE
nL
mE
nL
E
L
E
L
6
2
1
1
2
6
2
2
2
1
2
1
6
2
1
1
w
w
w
w
κ
κ
x
I0H
0IH
I0H
0IH
y
y
y
y



     kkk DFDFDF wHy


19. 19
Simulation Results:
Positioning (1/2)
Dual freq.
L1/L2
Single
freq. L1
Triple freq.
L1/L2/L5
-20 -15 -10 -5 0 5 10 15 20
-20
-15
-10
-5
0
5
10
15
20
Positioning error (L1)
east error (m)
northerror(m)
-20 -15 -10 -5 0 5 10 15 20
-20
-15
-10
-5
0
5
10
15
20
Positioning error (L1/L2)
east error (m)
northerror(m)
-20 -15 -10 -5 0 5 10 15 20
-20
-15
-10
-5
0
5
10
15
20
Positioning error (L1/L2/L5)
east error (m)
northerror(m)

20. 20
Simulation Results:
Positioning (2/2)
Standard
Derivation
Horizontal error (m)
Vertical error (m)
East error North error
Single Freq. 5.5499 3.9982 21.2627
Double Freq. 2.2234 3.3094 6.1387
Triple Freq. 1.8737 2.8852 5.4481

21. 21
Simulation Results:
Failure Detection (1/2)
sound an
alarm
begin
failure   
11521152
1
timedetectionaverage DTADT
detection time (DT)
24 space 48 time
(RTCA DO-208)

22. 22
Simulation Results:
Failure Detection (2/2)
0.5 1 2 5 10 20
1
2
5
10
20
50
80
Slope (m/s)
AverageDetectionTime(sec)
Average Detection Time
Single frequency
Dual frequency
Triple frequency
 The best improvement
percentage for dual
freq. and triple freq.
are 48.3% and 55.9%,
respectively.
 Application of multi-
frequency algorithms
will shorten the failure
detection time.

23. 23
Simulation Results:
Failure Exclusion (1/2)
%100
exclusiontotalofno.
exclusionincorrectofno.
rate)exclusion(incorrect

IER
failed excluded

24. 24
Simulation Results:
Failure Exclusion (2/2)
15 20 25 30 35 40 45 50
0
5
10
15
20
25
30
35
Bias (m)
IncorrectExclusionRate(%)
Incorrect Exclusion Rate
Single frequency
Dual frequency
Triple frequency
 The IER obtained
through the dual (triple)
freq. is about 5% (12%)
lower than the one
through single freq.
 Application of multi-
frequency algorithms
will reduce the IER.

25. Failure Detection
and Exclusion
via ARMA-Filter

26. 26
Failure Detection
via ARMA-filter
      

M
i
i
N
i
i iksikzkz
11
1
The scheme of the ARMA-filter
(Autoregressive Moving Average filter)
Failure
z(k) > Td
Normal
no
yes
ARMA-filters(k)
z(k)

27. 27
Detection Threshold (1/3)
FAMMean time to false alarm (MTFA):
 
FAM
FAR
1
RateAlarmFalse 
sound a
false alarm
begin
detection
time to false alarm (TFA)

28. 28
Detection Threshold (2/3)
Determining Td via the Markov Chain Approach
L
1 1
L
j
i
k-1
L(k)
i(k)
1(k)
terminating state
transient state k
Tij
z(k)  Td
z(k) > Td

29. 29
Detection Threshold (3/3)
   1 kk Tππ
where T is the transition probability matrix
The transition of (k) can be represented as
   0
1
πTΙl

 L
T
LFAMMTFA:
Threshold Td
FAM
FAR
1


30. 30
MA-filter
     
   




kkz
kskk
T
MMA
θβ
eθΦθ 1,1
The dynamic equation of the MA-filter state space
model
where i is the weight satisfies
The scheme of the MA-filter
   

M
i
i ikskz
1
1
11

M
i i
Markov Chain

31. 31
MA-filter: Window Size 2 (1/2)
MA-filter with window size 2:      121  kskskz 
dT1
1


1S
LS
2
1
0
dT1
2


  dT 2211 
iS
 ii  ,
 
 
 
 
 
     































kkkz
ks
k
k
k
k
2211
2
1
2
1
0
1
1
1
01
00





state space model

32. 32
MA-filter: Window Size 2 (2/2)
dT1
2


1S
LS
1S
LS iSjS
 k2
 k1
dT1
1

dT1
1


 12 k
dT1
2


 11 k
0 0
ijT
  dT 2211 
 ii  ,
    jiij SstateinwaskSstatetogoeskT 1|Pr  θθ

33. 33
PFARMA-filter
The scheme of the PFARMA-filter
(parallel–form ARMA)
     
   




kkz
kskk
T
PFARMA
θη
λθΦθ 1
parallel-form
structure
      

M
i
i
N
i
i iksikzkz
11
1
The scheme of the ARMA-filter
Markov Chain

34. 34
Probability Integral
Transformation (PIT)
0 5 10 15 20
0
0.2
0.4
0.6
0.8
0.9
1
random variable
cdfvalue
Probability Integral Transformation
4.6 10.6
2
(2)

2
(6)
F
-1
F
10 Sat.
2(6)
9 Sat.
2(5)
6 Sat.
2(2)
Visible Satellite
   ZZF 
1log21
2

35. 35
Failure Exclusion
via Multivariate ARMA
The scheme of the multivariate ARMA-filter
      

M
i
i
N
i
i ikikk
11
1pqq 














 k
kk
i
i
T
ni p
pq
,,1
maxarg

Multivariate
ARMA-filterp(k)
q(k)

36. 36
Simulation Results (1/5)
Window Size Threshold
1 19.2316
2 12.0159
3 9.3713
4 7.9669
5 7.0898
MA-filter with equal weights

37. 37
Simulation Results (2/5)
1 2 3 4 5
2
5
10
20
50
100
120
Average Detection Time
Window Size
AverageDetectionTime(sec)
slope = 0.2
slope = 0.5
slope = 1
slope = 5
slope = 10
slope = 15
 Under small ramp-type
failures (slope = 0.2, 0.5
and 1 m/s), the ADT will
decrease as the window
size increase.
 Under large ramp-type
failures (slope = 5, 10
and 15 m/s), the window
size has little influence
on the ADT.

38. 38
Simulation Results (3/5)
 Under small step-type
failures (bias = 20, 25
and 30 m), the ADT will
decrease as the window
size increase.
 Under large step-type
failures (bias = 40 m), the
window size has little
influence on the ADT.
1 2 3 4 5
1
2
5
10
20
50
90
Average Detection Time
Window Size
AverageDetectionTime(sec)
step = 20
step = 25
step = 30
step = 40

39. 39
Simulation Results (4/5)
35 40 45 50 55
0
5
10
15
20
25
30
35
40
45
time (sec)
IncorrectExclusionRate(%)
Incorrect Exclusion Rate
Window Size = 1
Window Size = 2
Window Size = 3
Window Size = 4
Window Size = 5
 The IER will reduce as the
window size increase under
small ramp-type failures.
 The window size has little
influence on the IER under
large ramp-type failures.
3 4 5 6
0
2
4
6
8
10
12
14
16
18
time (sec)
IncorrectExclusionRate(%)
Incorrect Exclusion Rate
Window Size = 1
Window Size = 2
Window Size = 3
Window Size = 4
Window Size = 5
slope=0.5m/s
slope=10m/s

40. 40
Simulation Results (5/5)
 The IER will reduce as the
window size increase under
small step-type failures.
 The window size has little
influence on the IER under
large step-type failures.
5 10 15 20 25 30
5
10
15
20
25
27
time (sec)
IncorrectExclusionRate(%)
Incorrect Exclusion Rate
Window Size = 1
Window Size = 2
Window Size = 3
Window Size = 4
Window Size = 5
2 3 4 5
3
4
5
6
7
8
9
10
11
time (sec)
IncorrectExclusionRate(%)
Incorrect Exclusion Rate
Window Size = 1
Window Size = 2
Window Size = 3
Window Size = 4
Window Size = 5
bias=25m
bias=40m

41. Failure Detection
and Exclusion
via Kalman Filter

42. 42
PVA Model (1/2)
     
       
,
1





kkkk
kkk
PVAPVAPVAPVA
PVAPVAPVAPVA
wxHz
vxΦx
The dynamic and measurement equations









4
44
4
2
2
1
44
I00
II0
III
Φ S
SS
PVA t
tt
 
  



0H0
00H
H
k
k
PVAwhere
       kkkkkk PVAPVAPVAPVAPVA zKxΦx ~1|1ˆ|ˆ 
The updated state estimate
where        kkkkk PVA
T
PVAPVAPVA
1
| 
 RHPK
    
    


kNk
kNk
PVAPVA
PVAPVA
Rw
Qv
,0~
,0~

43. 43
is the innovation vector with the covariance matrix
PVA Model (2/2)
The normalized innovation squared (NIS)
       kkkks PVAPVA
T
PVAPVA zSz ~~ 1

       1|1ˆ~  kkkkk PVAPVAPVAPVAPVA xΦHzz
         kkkkkk PVA
T
PVAPVAPVAPVA RHPHS  1|
Chi-square
distributed

44. 44
FDE via Kalman filter
Kalman
Filter
Kalman
Filter 1
Kalman
Filter n
Pseudorange
Delta range
Doppler shift
Satellite Failure
Exclusion
Satellite Failure
Detection
All Measurements
Exclude Meas. n
Exclude Meas. 1

45. 45
Maneuvering Vehicle
delta range information

46. 46
Multiply L(k)
on both side
(L(k)H(k) = I4)
Delta Range Equation (1/3)
           kkkkkk ωxHxHu  11
The linearized delta range measurement
L(k)H(k-1)
           kkkkkk DRDR vuLxΦx  1
-L(k)(k)
The dynamic equation

47. 47
Delta Range Equation (2/3)
The dynamic and measurement equations
           
       




,
1
kkkk
kkkkkk DRDR
wxHz
vuLxΦx
The updated state estimate
             kkkkkkkkk DRDRDRDR zKuLxΦx ~1|1ˆ|ˆ 
where        kkkkk T
DRDR
1
| 
 RHPK
    
    


kNk
kNk DRDR
Rw
Qv
,0~
,0~

48. 48
Delta Range Equation (3/3)
The normalized innovation squared (NIS)
         kkkkkk T
DRDR RHPHS  1|
       kkkks DRDR
T
DRDR zSz ~~ 1

              kkkkkkkk DRDRDR uLxΦHzz  1|1ˆ~
is the innovation vector with the covariance matrix

49. 49
Simulation Results
50 100 150 200 250 300 350 400 450 500
0
10
20
30
positioning
error(m)
PVA
50 100 150 200 250 300 350 400 450 500
0.0
50.3
80.0
120.0
160.0
Innovation
time (sec)
50 100 150 200 250 300 350 400 450 500
0
10
20
30
positioning
error(m)
DR
50 100 150 200 250 300 350 400 450 500
0.0
34.7
80.0
120.0
160.0
Innovation
time (sec)
detection threshold
detection threshold

50. 50
Multiple Model Approach (1/2)
Kalman
Filter 0
Kalman
Filter 1
Kalman
Filter n
Pseudorange
Delta range
Positioning
Result+
 
   

n
i
iDRi
MM
kkk
kk
0
|ˆ
|ˆ
,x
x

 k0
 k1
 kn
prior prob.

51. 51
Multiple Model Approach (2/2)
        
   


 n
l ll
ii
kii
kk
kk
k
0
1
1
|Pr


 Z
         kzk iDRikki ,11
~Pr,|Pr zZ  
The prior probability
where
i(k): the likelihood function of the ith model

52. 52
0 25 50 75 100 125 150 175 200
0
20
40
60
detection threshold
Innovation
innovation
0 25 50 75 100 125 150 175 200
0
12.5
25
37.5
50
Delta Range Model
positioningerror(m)
0 25 50 75 100 125 150 175 200
0
12.5
25
37.5
50
Multiple Model Approach
positioningerror(m)
time (sec)
Simulation Results
0 50 100 150 200
0
0.5
1
φ0
(k)
0 50 100 150 200
0
0.5
1
φ1
(k)
0 50 100 150 200
0
0.5
1
φ2
(k)
0 50 100 150 200
0
0.5
1
φ3
(k)
0 50 100 150 200
0
0.5
1φ4
(k)
0 50 100 150 200
0
0.5
1
φ5
(k)
time (sec)
0 50 100 150 200
0
0.5
1
φ6
(k)
time (sec)
0 50 100 150 200
0
0.5
1
φ7
(k)
time (sec)
Ramp-Type Failure (slope = 0.2m/s)

53. 53
0 25 50 75 100 125 150 175 200
0
20
40
60
detection threshold
Innovation
innovation
0 25 50 75 100 125 150 175 200
0
12.5
25
37.5
50
Delta Range Model
positioningerror(m)
0 25 50 75 100 125 150 175 200
0
12.5
25
37.5
50
Multiple Model Approach
positioningerror(m)
time (sec)
Simulation Results (2/2)
0 50 100 150 200
0
0.5
1
φ
0
(k)
0 50 100 150 200
0
0.5
1
φ
1
(k)
0 50 100 150 200
0
0.5
1
φ
2
(k)
0 50 100 150 200
0
0.5
1
φ
3
(k)
0 50 100 150 200
0
0.5
1
φ
4
(k)
0 50 100 150 200
0
0.5
1
φ
5
(k)
0 50 100 150 200
0
0.5
1
φ
6
(k)
time (sec)
0 50 100 150 200
0
0.5
1
φ
7
(k)
time (sec)
Step-Type Failure (bias = 20 m)

54. 54
Conclusions (1/2)
 In Multi-frequency
 propose the dual frequency alogrithm
 extend to the triple frequency algorithm
 to simultaneously use the measurements of both
GPS and Galileo system
 In ARMA-filter
 propose the ARMA filter for fast failure detection
 use the Markov chain approach for calculating the
threshold of ARMA

55. 55
Conclusions (2/2)
 In ARMA-filter (cont.)
 propose the PIT to solve the problem caused by
change of number of visible satellites
 proposed the multivariate ARMA to reduce the IER
 In Kalman filter
 use delta range information to accurately describe
the dynamic behavior of a maneuvering vehicle
 propose multiple model approach to reduce the
positioning error

56. 56
Future Work
 Multi-frequency
 using delta range and/or Doppler shift measurements
 ARMA-filter
 to extend the ARMA detector to multivariate case
 to determine the optimal coefficients for FDE
 Kalman filter
 to reduce the computing burden
 Generalized pseudo-Bayesian (GPB) approach
 Interacting multiple model (IMM) algorithm.