Exeley Inc.

Horn, T.

Hamilton, J.

Kosanke, J.

Hare, V.W.

Kluver, W.

Beres, W.

Nance, S.

Keller, M.A.

Assessment of common red blood cell pretreatments to yield an accurate serologic antigen phenotype compared with genotype-predicted phenotype

SZTUBECKA, Małgorzata

SKIBA, Marta

MRÓWCZYŃSKA, Maria

BAZAN-KRZYWOSZAŃSKA, Anna

VARIOUS PRESENTATION OF NOISE PERCEPTION IN BYDGOSZCZ GREEN AREAS

CHU, CHISHIH

WONG, MIN YI

CHIU, CHENG-HSUN

TSENG, YUAN-HSI

CHEN, CHYI-LIANG

HUANG, YAO-KUANG

Salmonella-Infected Aortic Aneurysm: Investigating Pathogenesis Using Salmonella Serotypes

Wang, Tao

Wang, Xiaodong

English

Research on IPv6 Address Space and Super IP Upgrade

Xu, Jiangtao

Liu, Pingping

International Journal of Advanced Network, Monitoring and Controls          Volume 04, No.02, 2019 
28 
Application of Wavelet Analysis in The Prediction of 
Telemetry Data 
Xu Jiangtao 
School of Computer Science and Engineering 
Xi’an Technological University 
Xi’an, 710032, China 
e-mail: 1400693814@qq.com 
 
Liu Pingping 
School of Computer Science and Engineering 
Xi’an Technological University 
Xi’an, 710032, China 
e-mail: 134369601@qq.com 
 
 
Abstract—With the rapid development of space technology, the 
increasing number of spacecraft, in-orbit risk also increases, 
how to ensure that the spacecraft safety and reliability is 
particularly important. Prediction technology can predict the 
failure of the spacecraft in advance, and it has won valuable 
time for the fault of the spacecraft troubleshooting, thereby 
increasing the safety and reliability of spacecraft operation. In 
this paper, based on the non-stationary and periodicity of 
telemetry data. Based on the wavelet analysis, the prediction of 
the data is introduced, the establishment of a short-term 
forecasting model based on Mallat algorithm. The 
experimental results show that the prediction curve is basically 
consistent with the actual curve. 
Keywords-Wavelet Analysis; Fourier Transform; Periodic 
Autoregression; Models; Mallat. 
I.  INTRODUCTION 
The prediction technology of spacecraft fault has 
been a hot research field. After 20 years development 
of prediction theory, until the discrete parameters of 
the linear model of a finite parameter linear model is 
proposed, and it is possible to combine the prediction 
theory with the computer. According to the different 
properties of the forecast, the forecasting methods are 
generally divided into two categories: time series 
forecasting and causal prediction. Time series 
prediction is made by the past predict the future value 
of the prediction, and causal forecasting is through the 
known variables to predict the values of other variables. 
In this paper, the time series forecasting method is used 
to forecast the future development trend of the 
telemetry data. 
II.   WAVELET ANALYSIS THEORY 
A. Wavelet analysiss 
The wavelet analysis method has the characteristics 
of low frequency and high frequency of the 
non-stationary signal that change with the 
low-frequency information signals using a wide time 
window, high frequency information using a narrow 
time window. Wavelet is a small area of the wave, 
waveform with special length, average of 0.Wavelet 
are defined as follows
[1]
. 
Set ( )t  to one square integrable function, namely 
2( ) ( )t L R  , if the Fourier transform to meet the 
conditions: 
 
2
ˆ ( )
R
C d
 


  
 (1) 
(1)formula called ( )t  is a basic wavelet or 
wavelet generating function. When the generating 
function ( )t  is expanding and translating, it can get 
function , ( )a t : 
DOI: 10.21307/ijanmc-2019-044
International Journal of Advanced Network, Monitoring and Controls          Volume 04, No.02, 2019 
29 
 
,
1
( ) ( )a
t
t
aa


 


, among them , ; 0a R a    (2) 
In (2) formula, a is the scaling factor, t is the 
translation factor, Because the value of scale factor and 
translation factor is continuously changing, and 
depends on the parameters, it is a set of sequence of 
functions which are obtained by the expansion and 
translation of the generating function, also called 
sub-wavelet. 
B. Mallat algorithm 
The basic idea of the Mallat algorithm is as follows: 
Let  jH f  as the approximation of the energy limited 
signal 
 
2( )f L R  in the resolution  2 j ,Then the  jH f  
is further decomposed into the approximation of  1jH f  
under the  f  resolution  12 j , and the details of 
 
1jD f  between  12 j  and  2 j . 
1) Mallat algorithm based on wavelet decomposition 
From Multi-resolution analysis: 
2
( ) j Z jL R W  , To 
arbitrary function  2( ) ( )f t L R ,get 
 
 
,
,
( ) ( )jk j k
j k Z
f t c t

 
 (3) 
Take the inner product in the side of the equation 
with 
 
,j k , because 
  , ,( )j k j k Zt   is the 
orthonormal basis of 
 2 ( )L R , get 
 
,,
j
k j kc f  , 
thus to be 
 
 
, ,
,
( ) , ( )j k j k
j k Z
f t f t 

 
 (4) 
From multi-resolution analysis, we can know that 
any function  jf  of  jV ,can be expressed as the 
following form  2 ( )L R , 
 1 1
2 2 1
1 1
   
   
   
j j j
j j j
M M M j
f f d
f d d
f d d d
 
  
 
 
  

    


 
Among 
 
 1 1
, 1, 1,( ) ( ) ( ) ( )
j j j
j k j k k j k k j k
k k k
f t c t c t d t        
 (5) 
 
jf  represents the low frequency components of 
 ( )Mf t , while  ( ), , , 1ld t l M j   indicates the high 
frequency components of  jf  at different resolutions. 
Because of 
 1 1
1, 1, 1,( ) ( ) ( ) ( )
j j j
j k j k k j k k j k
k k k
f t c t c t d t         
 and  ,  and binary translation and scalability of 
orthogonality, Can be obtained 
 
 1
, 1, 2,
j j j
k n j n j k n n k
n n
c c c h       
 (6) 
 
 1
, 1, 2,
j j j
k n j n j k n n k
n n
d c c g       
 (7) 
The formula (6) and (7) called Wavelet 
decomposition algorithm of Mallat algorithm, among 
wherein   k k Zh   is a filter coefficient sequence by a 
two-scale equation corresponding orthogonal scaling 
functions. 
2) Reconstruction algorithm of mallat algorithm  
The reconstruction algorithm of mallat algorithm is 
the inverse process of its decomposition algorithm. the 
convolution of mallat algorithm is represented: 
International Journal of Advanced Network, Monitoring and Controls          Volume 04, No.02, 2019 
30 
 
 
 1
1
1 1
( )
( )
( ) ( )
j j
j j
j j j
c D c h
d D c g
c Uc h Ud g
 
 
 
  

 
     (8) 
Among , h  is represented conjugate inversion of 
filter h;  jc h  represent conjugate of  jc  and  h  ; 
 ( )jD c h   represent Under the dual sampling of 
conjugate  jc h  . 
III.  THE RESEARCH OF TELEMETRY DATA TIME 
SERIES PREDICTION BASED ON MALLAT ALGORITHM 
A. The characteristics of telemetry data 
Telemetry data has the characteristics of 
non-stationary variation, commonly used statistics of 
the telemetry data (such as the mean and 
autocorrelation function, etc.) often varies with time 
changing, it bring very great difficulty to the telemetry 
data forecast. Through the telemetry data 1 and 2 (table 
1, 2) statistics, difference is very big, every stage of the 
statistical parameters show that the sequence of 
non-stationary time series. Wavelet analysis to deal 
with this kind of data has a great advantage. 
TABLE I. THE TEST RESULTS OF A REMOTE SENSING DATA 1 
STATIONARITY 
Time 24 60 120 240 
Mean 
Value 
608.618 620.9572 622.6287 622.6287 
Variance 198.2080 604.1601 611.2902 674.6010 
TABLE II. THE TEST RESULTS OF A REMOTE SENSING DATA 2 
STATIONARITY 
Time 24 60 120 240 
Mean 
Value 
29.2479 35.413 37.1234 39.2378 
Variance 10.3366 30.9019 37.4902 42.5010 
Figures 1, 2 is telemetry data 1 and 2 for four hours 
of change curve, it can be seen that the output power 
sequence is periodicity, as well as randomness. The 
coexistence of periodicity and randomness, the result 
can be seen as the superposition of different frequency 
components, these frequency components 
superimposed on each other in the interior have similar 
frequency characteristics and the same variation. If the 
subsequence to establish a prediction model for single 
change, due to the change of the data characteristics of 
a single, reduce the difficulty of forecasting model 
selection. 
 
Figure 1. Data of 1 consecutive 4 hours curve graph 
Figure 2. Data of 2 consecutive 4 hours curve graph 
B. The choice of wavelet function 
There are some mutations in the trend of spacecraft 
telemetry data, and these mutations reflect the actual 
state of the satellite. In order to accurately capture the 
point of the mutation, the wavelet function is usually 
required to have a fast convergence, which can quickly 
attenuate to zero 
[5]
. In this paper, db3 wavelet is 
chosen as the wavelet base for the different scale of a 
certain remote data sequence. DbN wavelet in N=1, 3, 
4, 1, the results of 2 scale decomposition of a telemetry 
data (Figure 3). 
International Journal of Advanced Network, Monitoring and Controls          Volume 04, No.02, 2019 
31 
 
Figure 3. Comparison results of dbN wavelet based N=1,2,3,4 
The choice of wavelet function can satisfy the 
following 3 conditions except that the condition and 
the regularity condition
[2]
. 
1)Good compact support； 
2) ( )t has vanishing moments; vanishing moments 
can make the wavelet function has a good locality in 
the frequency domain; 
3)Satisfy orthogonality. 
C. Wavelet decomposition scale study 
Due to the telemetry data is not stable, its change 
cycle is difficult to see, and its change is slow and fast 
change together, that is, the change of telemetry data 
cycle is the size of the cycle of nested together. 
Therefore, Separating different frequency component 
can make its change rule is more intuitive, and can also 
improve data non-stationary
[6]
.The figure 4 Show the 
results that comparison db3 wavelet approximation 
part aN decomposed at different scales and a0 of the 
original sequence.  
 
Figure 4. Comparison of different decomposition scale approximation 
section 
It can be seen that the decomposition scale is 2, the 
curve of the approximate part a2 has smooth enough, 
and basically maintained the shape of the original 
curve, while A3, A4, because of the increase in the 
number of points, the sampling point is reduced, the 
approximate partial curve is too smooth, the sequence 
of changes in the trend has been distorted, so this paper 
chose the decomposition of 2. 
D. Forecasting model of time series 
Time series forecasting
[3][7]
 is one of the methods of 
statistical analysis. Its modeling idea is the basic 
assumption that the change of the past of the telemetry 
data will continue into the future, that is, the future is a 
continuation of the past. In this paper, the use of the 
periodic autoregressive model (PAR model)
[8][9]
 is as 
follows: 
If there is a time series X, the expression is 
 
 
0 1, 1 2, 2 ,t t t t t t p t t p tX a a X a X a x          (9) 
Meet the following conditions: 
1) t  is independent sequence,Expected Value is 
 0tE  , variance is  
2 2
t tE  ; 
2)For any  0,1, ,i p  ,  it it Ta a  ,  
2 2
t it T   , 
 0, 1,t    , among T is a positive number,and the 
model is model of  PAR, T is the Cycle length of PAR 
model,t is phase of PAR model. 
The forecast model of the telemetry data is: 
 
International Journal of Advanced Network, Monitoring and Controls          Volume 04, No.02, 2019 
32 
Set  1 2, , , nX X X  is telemetry data samples of per 
minute, the value of  ( )tX h  in the future h is the 
 
t hX   in the condition of t, and thus as its predictive 
value, denoted as  ˆ (1)nX , according to the definition 
of: 
Decomposition for telemetry data sequence 
prediction model is established for an hour, the 
selection of the length of the cycle is 60, namely, 
p=T=60. This load sequence PAR prediction model 
can be represented by the expression of the following: 
 
E. Predicted results analysis 
The predicted values of the reconstructed sub 
sequences of the scales are compared with the original 
output power trends(Figure 5). 
Figure 5. Comparison results between predicted values and actual results 
By analyzing the comparison between an actual 
telemetry data and the predicted value, it can be seen 
that the boundary of the predicted results and the trend 
of mutation is not very ideal. In this paper, the 
extension of the periodic continuation to the boundary 
of the sequence is extended. The idea of periodic 
continuation is that the signal is considered as a 
periodic signal, and the extension process is as 
follows
[10][11]
:  
 ,        0
, 1
n n M
n n M
x x n
x x n M


 

    
Among, M is the length of sequence. The 
comparison result of the prediction curve and the 
actual curve after eliminating the boundary error is 
shown in Figure 6. 
Figure 6.  The comparison results between the predicted values and the 
actual values of the modified boundary 
From Figure 6, we can see that the results of the 
prediction of the sequence are better than the results 
obtained by the wavelet transform. 
IV.  THE PERFORMANCE EVALUATION OF THE 
TELEMETRY DATA FORECASTING MODEL 
Thought of optimal decision method is using linear 
transform to normalize the attribute value, and at the 
same time using the ideal point and negative ideal 
point, compared with the traditional method has more 
rationality and reliability. The ideal point is the best 
solution, and its target value is the best, the worst is the 
worst. The worst is the worst.The optimal solution 
algorithm is as follows: 
 
 
International Journal of Advanced Network, Monitoring and Controls          Volume 04, No.02, 2019 
33 
A. Set decision matrix of A is 
 
 
1
1 1 1 2 11
2 2 2 22
1 2
( ) ( ) ( )( )
( ) ( ) ( )( )
( ) ( ) ( ) ( )
n
n
p P P P n
f x f x f xf x
f x f x f xf x
A
f x f x f x f x
 
 
 
 
 
  


    

 (10) 
to 
 
 
1 2min( ( ), ( ), , ( )}, 1,2, ,i i i i nU f x f x f x i k    (11) 
 
1 2max( ( ), ( ), , ( )}, 1, 2, ,i i i i nU f x f x f x i k k P      
 
1 2max{ ( ), ( ), , ( )}, 1,2, ,i i i i nV f x f x f x i k    
 
1 2min{ ( ), ( ), , ( )}, 1, 2, ,i i i i nV f x f x f x i k k P      
B. To determine the ideal point and negative ideal 
point 
The ideal point: 
 
 *
1 2 1 2( , , , , , , , )
T
k k k Px U U U U U U     (12) 
Negative ideal point: 
 
1 2 , 1,( , , , , )
T
k k Px V V V V V    
C. Calculate the close degree 
The proximity of the ideal point  
 
 
1 1
( )1
[ ], 1,2, ,
( )
pk
j j i
i
j j kj i j
U f x
R i n
P f x U  
    
 (13) 
The proximity of the negative ideal point 
 
 
1 1
( )1
[ ], 1,2, ,
( )
pk
j i j
i
j j kj j i
f x V
r i n
P V f x  
    
 (14) 
D. Calculate the relative close degree 
 
 
,0 1, 1,2, ,ii i
i i
R
i n
r R
    


 (15) 
By calculating the formula (1) (11) (12) is as 
follows: 
The ideal point: 
 *
1 2 1 2 1.23,0.978,4.219,1.49( , , , , , , , ) 1
T
k k k Px U U U U U U    （ ） 
Negative ideal point: 
 
1 2 , 1,( , , , , ) 3.76,3.679,7.896,4.118
T
k k Px V V V V V    （ ） 
Calculate the relative close degree by the formula 
(13) (14) (15): =0.34 
From the relative closeness of view, the time series 
forecasting model program has a higher rationality and 
reliability. 
V.  CONCLUSION 
This paper studies the telemetry data forecasting 
method based on wavelet analysis. Through the 
analysis of the characteristics of the telemetry data, the 
characteristics of the non - stationary and certain 
periodicity of the telemetry data are established, and 
the prediction algorithm based on wavelet analysis is 
established. By choosing different wavelet bases and 
the decomposition scale, the decomposition results 
show that 2, db3 wavelet decomposition scale is the 
best. Based on the Mallat algorithm, the time series 
forecasting model is established according to the 
characteristics of detail data and approximate data. The 
experimental results show that the predicted values are 
in good agreement with the actual values. Finally, 
through the analysis of the performance of the 
forecasting model, the forecasting model is reasonable 
and reliable. 
 
International Journal of Advanced Network, Monitoring and Controls          Volume 04, No.02, 2019 
34 
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