Combining and Filtering Telemetry Data

by Michael J. Wilson

ARL-TR-3143 March 2004

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ARL-TR-3143 March 2004

Combining and Filtering Telemetry Data

Michael J. Wilson

Weapons and Materials Research Directorate, ARL

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14. ABSTRACT

Multiple receiver telemetry systems are common in the aeroballistics test and evaluation community. These systems typically 

record telemetry data independently, requiring post-flight data processing to produce the most accurate combination of the 

available data. This report addresses the issues of time synchronization between multiple data sources and determination of the 

best choice for each data word. Additional filtering is also developed for the case when all available data are corrupted. The 

performance of the proposed algorithms is presented.

15. SUBJECT TERMS

 flight data processing; telemetry

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Standard Form 298 (Rev. 8/98)

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iii

Contents

List of Figures iv

1. Introduction 1

2. Timing 2

3. Creating the Master Word Table 4

4. "De-noising" With Bayesian Estimation 5

5. Performance 8

6. Conclusion and Impact 9

7. References 10

Distribution List 11
iv

List of Figures

Figure 1. Frame counter example: (a) cframe, (b) b(n), (c) e(n), and (d) bf(n)..................................2

Figure 2. f(b|a) for different a when the number of bits per word is 12 .........................................7

Figure 3. Bayesian estimation example of a sine wave corrupted by noise ...................................8

Figure 4. Word value plot for source merging example: three sources.........................................9
1. Introduction

The United States military is a major employer of telemetry systems for test and

evaluation purposes. Such systems typically format data from various sensors, such as

accelerometers, magnetometers, and global positioning system (GPS), into frames which

are then transmitted from the projectile or moving body to a ground station. The sensor

readings may be sampled and quantized for digital transmission. In practice, the test and

evaluation community uses legacy pulse code modulation/frequency modulation

(PCM/FM) systems (1,2) without any channel coding. While this communications system

may not be optimal, it is a proven telemetry standard. As a result of such a system,

telemetry data are prone to bit errors that must be addressed.

To add redundancy, multiple receivers are used, which connect to antennas that cover

different portions of the projectile flight. This helps to compensate for problems that an

individual receiver may experience. Multiple receivers result in multiple sources of the

telemetry data. It is therefore desirable to combine the multiple data files into one master

file that represents the best possible combination of the data. This results in many

challenging signal processing problems which this report addresses.

The first problem associated with combining telemetry data from multiple sources is

identifying the same data across different sources. A receiver will use a time source to

stamp each frame with the time when it was received. Because of the ambiguity associated

with these times, a periodic counter is included within a frame to help identify that frame

across multiple sources. Section 2 discusses how this identification is made.

Once multiple sources are identified as having the same frame of data, it must be decided

which source of data to use for each word in the master. Since oversampling is largely

employed, the correlation between successive samples is used in determining which source

contains the best data. This method is explained in section 3.

Even with multiple sources for a word, the final choice for the master word may still be

corrupted by noise. Bit errors introduced by wireless channels result in unusual discrete

noise densities that depend on the number of bit differences between the transmitted and

received symbol. An error-correcting methodology that uses a Bayesian estimate for this

type of noise in conjunction with an oversampling assumption is derived in section 4. The

Bayesian estimate allows a priori information to improve the estimate.

1
2. Timing

Frame counters are included within telemetry word tables and can be used to identify the

same frame across different receivers. Let cframe be an Nc bit counter (usually the size of a

word) that wraps to zero when it overflows. At a sampling frequency of Fs, the counter

repeats every 2Nc (1/Fs) seconds. Typical sampling frequencies are ∼3 kHz with flight

times of ∼15 seconds. Therefore, the frame counter is expected to overflow and wrap many

times. Each set of consecutive frames with cframe ∈ [0, 2Nc − 1] is called a major frame.

Because several major frames are transmitted and “dropouts” occur, absolute time values

cannot be determined with cframe. A typical plot of cframe is shown in figure 1(a). Notice

that the received frame counter includes dropouts and erroneous values.

0 1 2 3

x 104

0

1000

2000

3000

4000

5000

(a)

Frame Counter

0 1 2 3

x 104

 3

 2. 5

 2

 1. 5

 1

 0. 5

0

0.5 x 104 (b)

b(n)

0 1 2 3

x 104

 3

 2. 5

 2

 1. 5

 1

 0. 5

0

0.5 x 104 (d)

bf(n)

0 1 2 3

x 104

 4000

 2000

0

2000

4000

6000

(c)

e(n)

Figure 1. Frame counter example: (a) cframe, (b) b(n), (c), e(n), and (d) bf (n).

The first steps in using the frame counter to determine the time associated with a frame of

data are to identify and eliminate errors with the frame counter. To begin, assume that the

data are received in the order in which they were transmitted. In other words, no frames

are out of order. Therefore, the correct form of the frame counter is assumed to be

monotonically increasing except when it wraps. The increase between successive samples

may be greater than one if a dropout occurs. Note that dropouts are capable of producing

any cframe time series as well as the type assumed here. However, high signal-to-noise ratio

(SNR) implies a low probability of deviation from the assumed form of cframe. Also,

received data that do not reflect the assumptions made here are fundamentally useless. In

order to detect errors in cframe, a simple transformation of the data and a median

smoother are employed.

2
Let n be a discrete time index for the received frames with 0 corresponding to the first

frame, and let b(n) equal the difference between cframe(n) and n, therefore

b(n) = cframe(n) − n. (1)

The structure of b can help identify errors. To see how, consider a plot of b (figure 1(b)) for

the frame counter in figure 1(a). Notice that single errors in the frame counter are

transformed into impulses in b. Dropouts produce a positive step whereas wraps produce a

negative step. Because of the local monotonicity of the steps, b is invariant to a median

smoother with a properly chosen window length. To see this, consider the median

smoother defined as

y(n) = median[x(n − N1), ..., x(n), ..., x(n + N1)]. (2)

The median operation sorts the samples by magnitude in the window of length Nm, where

Nm is any positive odd integer, and selects the middle sample. The selection property is

desirable since the only possible output of the smoother are unsigned integer values. This

eliminates the need for any requantization of the output. If the samples in the window are

monotonically increasing or decreasing, the center sample in the window is the output of

the smoother. Therefore, when a window is selected that is larger than an error run and

smaller than time between local steps, a median smoother will change the error points

without changing the correct points.

The following algorithm is used to detect errors in the frame counter:

1. Compute b(n), using equation 1.

2. Smooth b(n) with a median smoother with a properly chosen window length.

3. Repeat step 2 multiple times, depending on the noise level.

4. Let bf (n) be the output of the last smoother (see figure 1(d)) and compute the

difference e(n) = b(n) − bf (n).

5. An error is expected to occur in cframe(n0) if e(n0) 6= 0.

The output after each smoother stage will be invariant to additional smoothing if it

consists only of step functions that change slowly with respect to the window length of the

smoother; e(n) could also conceivably correct errors in cframe. However, the missing frames

are interpolated instead since the data are typically oversampled. Figure 1(c) shows the

error e(n) in the example.

Because of the ambiguity attributable to multiple wraps in the frame counter, additional

information must be used in order to identify the same frame in different sources.

3
Fortunately, most telemetry receivers will use a reference time base such as GPS to stamp

each frame with the time when it was received, tRX. Now consider how tRX relates to the

actual time the data in the frame were acquired, tT X. Define the time difference,

Ds(n) = tRX,s(n) − tT X,s(n), (3)

for each source, s, which is, in general, a function of the discrete time index n. Ds(n)

depends on the relative positions of the receiver and the projectile. Therefore, different

times are assigned to the same frame across multiple receivers. However, if the time errors

that result satisfy

Ds(n) ¿

2Nc (1/Fs)

2 (4)

for all s, TRX can be used in conjunction with the frame counter to identify the same frame

across multiple sources. Reference times for all frames created in the master are then

generated with two parameters, the time of the first frame, t(0) and the sampling period,

Ts, as

t(n) = t(0) + nTs. (5)

The parameters are computed as

Ts = 1

N − 1

X

n∈S1

[tRX(n + 1) − tRX(n)], (6)

in which S1 is the set of all indices so that TRX(n + 1) and TRX(n) exist, and

t(0) = 1

N

X

n∈S2

[tRX(n) − nTs]. (7)

in which S2 is the set of all indices so that TRX(n) exists.

3. Creating the Master Word Table

After a master time base is created and the frames from each source that correspond to the

time points are identified, the master word table assembly can begin. When a frame of

data is transmitted, it is possible that all or most of the receivers receive the frame. With

the timing discussed in the previous section, specific data from each source as representing

the same frame can be identified.

Suppose that there are NS sources with word tables Φs1 , Φs2 , ..., ΦsNS , and let Φs(n, w)

denote the wth word in frame n from source s. An empty master word table, Φm, is now

created with frame times that span all the sources’ times. The elements in Φm must be

generated for each word in each frame as a function of all the data available:

Φm(n, w) = f({Φs1 , Φs2 , ..., ΦsNS }) (8)

4
First, note that is it highly improbable for the errors to produce the same corrupted word

in different sources. Therefore, the occurrences of the same word across different sources

can be exploited. Let Q(n, w) denote the mode (element that occurs the most often) or set

of modes (if a unique mode does not exist) defined as

Q(n, w) = mode(Φs1 (n, w), Φs2 (n, w), ..., ΦsNS (n, w)) (9)

If there is only one mode Φm(n, w) = Q(n, w); otherwise, an additional criterion must be

used to choose from the set.

It will be shown in section 4 that the transmission noise is impulsive. As such, erroneous

data tend to be far away in magnitude from their true value. Also, sensor output is

generally oversampled, which produces highly correlated data. Thus, an erroneous datum

tends to be far away in magnitude from the words in neighboring frames. Therefore, the

distance between the word of interest and its neighbors in different frames can be exploited

as a criterion for choosing the master word.

Let dq be a distance metric for each q ∈ Q(n, w); Φm(n, w) is then defined as

Φm(n, w) = arg min

q {dq : q ∈ Q(n, k)}. (10)

The square differences in magnitudes of neighboring points are used to calculate each

distance as

dq = X

s∈Sq

n

X

+Np

k=n−Np

1

|n − k|

[Φs(n, w) − Φs(k,w)]2 (11)

in which Sq is the set of sources in which q occurs and Np is the number of neighbors to each

side of the current word. Np = 1 provides good results in high SNR. This step completes

the building of the master word table with timing information. Next, the individual words

are filtered to eliminate any remaining errors.

4. “De-noising” With Bayesian Estimation

Even with multiple sources of word values, errors can still be present in the master.

Therefore, we derive a filtering approach to de-noise the word values by analyzing the

channel and methods that generated the errors. This method is applicable when additional

information about the data is known a priori. The example considered here is when the

signals are oversampled.

To begin, consider a binary symmetric channel (BSC) so that the output z(n) is defined as

z(n) = x(n) ⊕ y(n), (12)

5
in which x(n) ∈ {0, 1} is the input sequence to the BSC, and y(n) ∈ {0, 1} is an

independent and identically distributed sequence of Bernoulli random variables with

parameter p = pe, the bit error rate. The BSC models the effect of the wireless channel.

The BSC is used to transmit unsigned integer words or symbols that are M bits long. The

words, a(n), are converted directly into their binary representation, sent through the BSC,

and the output of the BSC is converted back to create the new output sequence, b(n). The

combined system is therefore a discrete memory-less channel (DMC) (3) that is completely

described by a set of conditional probability functions. The bit error rate of the DMC

causes the symbols to change. The probability of receiving any b when a symbol a is

transmitted depends on the hamming distance of a and b defined as

d(a, b) = X

M

i=1

αi ⊕ βi (13)

αi and βi are the ith bits of a and b, respectively, and ⊕ is the Boolean exclusive OR or

modulo-2 addition. It is possible to minimize errors using error-correcting codes; however,

many telemetry systems do not employ coding because of hardware complexity. Bit errors

produced from this model result in a very unusual conditional output probability density,

which is now derived.

Given that the input symbol to the previously described channel is a, the probability

density function (pdf) of the output, B, is

f(b|a) ≡ P(B = b|A = a) = pd(a,b) e (1 − pe)

M−d(a,b) (14)

in which d(a, b) is the hamming distance between a and b. Since each of the bits received is

assumed to be independent, equation 14 follows from the fact that d(a, b) bits change, and

M − d(a, b) bits remain the same in order for the channel to change a into b. Equation 14

is similar to a binomial distribution, but no counting term is needed since only one possible

combination is accounted for. Figure 2 shows equation 14 for an a of 2047, 2048, and 2730,

pe = 0.3, and the number of bits per word equal to 12. Notice that the densities are not

symmetric or semi-monotonic. The peaks or high points correspond to errors in only a few

bits. That error may cause a large change in the output if the bit is significant. Likewise,

small changes in the output may require more bit errors and are therefore not as probable.

If the data source was assumed to be memory-less (i.e., each symbol generated is

independent of all the other symbols), a maximum likelihood estimate would be optimal

(3). However, such an estimate would not change the received data since no a priori

information is used. Therefore, a Bayesian maximum a posteriori (map) estimate approach

is considered. In order to compute the joint pdf of a and b, the a priori pdf of a must be

determined. If the data are oversampled, the current sample value should be very close to

the preceding and succeeding sample values. Therefore, the pdf of the input sample can be

6
0 500 1000 1500 2000 2500 3000 3500 4000 4500 0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

f(b|a), a = 2047

0 500 1000 1500 2000 2500 3000 3500 4000 4500 0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

f(b|a), a = 2048

0 500 1000 1500 2000 2500 3000 3500 4000 4500 0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

f(b|a), a = 2730

b

Figure 2. f(b|a) for different a when the number of bits per word is 12.

estimated as a Gaussian distribution with mean,

μˆ = median[b(n − Np), ..., b(n − 1), b(n), b(n + 1), b(n + 2), ..., b(n + Np)] (15)

in which Np is the number of points to the left and right, and variance, ˆσ2 equal to a

constant that depends on the amount of oversampling. The map estimate is defined as

aˆ = arg max

a f(a|b). (16)

In other words, since the output, b, is known, the best estimate of a is the one that is most

probable to occur given b. Since

f(a|b) = f(a, b)

f(b) , (17)

and f(b) is not a function of a, maximizing the numerator of equation 17 is equivalent to

equation 16. Now,

f(a, b) = f(b|a)f(a). (18)

7
Substituting equation 14 into equation 17 and setting f(a) equal to a Gaussian density

with parameters (ˆμ, σˆ2) yields

f(a, b) = pd(a,b) e (1 − pe)

M−d(a,b) 1

√

2πσˆ2 exp ∙

(a − μˆ)2

σˆ2

 ̧

. (19)

The estimate is therefore

aˆ = arg max

a pd(a,b) e (1 − pe)

M−d(a,b) exp ∙

(a − μˆ)2

σˆ2

 ̧

. (20)

For implementation purposes, the first two factors in equation 20 are computed and stored

for all possible values while the exponential factor is computed at run time. The estimate

is computed for each Φm(n, w). Figure 3 shows the performance of the estimate for a sine

wave contaminated by the previously described channel with pe = 0.01. Notice that the

median smoother is successful in removing the impulsive noise but does not correct to the

most probable values as the Bayesian estimate does.

410 420 430 440 450 460 470 480

0

500

1000

1500

2000

2500

3000

3500

4000

sample

Input

Median Smoother

Bayesian Estimate

Figure 3. Bayesian estimation example of a sine wave corrupted by noise.

5. Performance

To demonstrate the effectiveness of the algorithms presented, a merger of three source files

is shown. Figure 4 plots an individual word generated by a sensor for each source and the

master. Notice that source 1 has clean data but contains large gaps in the beginning and

the middle. Source 2 does not have any large gaps but the noise increases with time. The

8
7.7105 7.7106 7.7107 7.7108 7.7109 7.711 7.7111 7.7112 7.7113 7.7114

x 104

0

2000

4000

Source 1

7.7105 7.7106 7.7107 7.7108 7.7109 7.711 7.7111 7.7112 7.7113 7.7114

x 104

0

2000

4000

Source 2

7.7105 7.7106 7.7107 7.7108 7.7109 7.711 7.7111 7.7112 7.7113 7.7114

x 104

0

2000

4000

Source 3

7.7105 7.7106 7.7107 7.7108 7.7109 7.711 7.7111 7.7112 7.7113 7.7114

x 104

0

2000

4000

Time (seconds from midnight)

Master

Figure 4. Word value plot for source merging example: three sources.

data from source 3 have many small gaps and a substantial amount of noise, but they

contain some frames not present in sources 1 or 2. The algorithm successfully merges all

three sources to produce a continuous set of data with no gaps and practically no impulse

noise.

6. Conclusion and Impact

The problems associated with combining multiple telemetry data sources have been

presented. The previous algorithms are currently employed in DFuze (4) flight tests. The

proven methods result in virtually error-free master files, despite substantial noise present

in the received data. The algorithms presented here are completely general and can be

applied to other applications with multiple telemetry receivers.

9
References

[1] Secretariat, Range Commanders Council, Telemetry Standards, White Sands Missile

Range, NM: RCC, IRIG 106-00.

[2] F. Carden, R. Jedlicka, R. Henry, Telemetry Systems Engineering, Artech House, 2002.

[3] J. G. Proakis, Digital Communications, 4th Ed., McGraw Hill, 2001.

[4] D. J. Hepner et al., “Aeroballistic diagnostic fuze,” U.S. Patent No. 6349652.

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ATTN STERT TE F TD R EPPS

CSTE DTC RT F TD (B 7855) 

 S HAATAJA

REDSTONE ARSENAL AL 35898-8052

1 CDR US ARMY 

YUMA PROVING GROUND

ATTN CSTE DTC YP YT ED M LAUSS

YPG AZ 85365-9498

4 CDR US ARMY AMCOM

ATTN AMSAM RD MG NC P RUFFIN

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AMSAM RD MG IP G HUTCHESON

REDSTONE ARSENAL AL 35898-5247

6 CDR US ARMY AMCOM

ATTN AMSAM RD WS DP TD S BURGETT

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AMSAM RD WS ID P ASHLEY

 T HUDSON

REDSTONE ARSENAL AL 35898-5247

1 TEST ARTICLE PREP DEP

NAWCAD CODE 540000A 

R FAULSTICH

BLDG 1492 UNIT 1

PATUXENT MD 20670-1456

1 CDR US ARMY

ATTN STEYP TD ATO A HART 

YPG AZ 85365-9106

2 IDA SCIENCE AND TECH DIV

ATTN H LAST K WALZL

4850 MARK CENTER DRIVE

ALEXANDRIA VA 22311-1882
13

NO. OF

COPIES ORGANIZATION

1 ARROW TECH ASSOCIATES

ATTN W HATHAWAY

1233 SHELBURNE RD STE 8

SOUTH BURLINGTON VT 05403

1 ALLIANT DEFENSE

ATTN A GAUZENS

PO BOX 4648

CLEARWATER FL 33758-4648

1 LOCKHEED/MARTIN-SANDERS

ATTN M CARLSON

NCA1-2078 95 CANAL ST

NASHUA NH 03061-0868 

1 CAMBER CORP

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200 VALLEY RD SUITE 403

MOUNT ARLINGTON NJ 07856

2 ALLIANT TECHSYSTEMS

ATTN C CANDLAND R DOHRN

600 SECOND ST NE

HOPKINS MN 55343-8384

2 SAIC

ATTN J DISHON G PHILLIPS

16701 W BERNARDO DR

SAN DIEGO CA 92127

1 SAIC

ATTN J GLISH

8500 NORMANDALE LAKE BLVD

SUITE 1610

BLOOMINGTON MN 55437-3828

1 SAIC

ATTN M PALMER

1410 SPRING HILL RD STE 400

MCLEAN VA 22102

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1429 QUINCY ST

ARLINGTON VA 22207

2 ROCKWELL COLLINS

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350 COLLINS RD NE

CEDAR RAPIDS IA 52498

NO. OF

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ATTN Q LAM

5050 POWDER MILL RD

BELTSVILLE MD 20705

2 JOHNS HOPKINS UNIV

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ATTN W D’AMICO K FOWLER

1110 JOHNS HOPKINS RD

LAUREL MD 20723-6099

5 CHLS STARK DRAPER LAB

ATTN J CONNELLY J SITOMER

R POLUTCHKO T EASTERLY

A KOUREPENIS F HARRISON

555 TECHNOLOGY SQUARE

CAMBRIDGE MA 02139-3563

1 UC BERKELEY

ATTN A PISANO

DEPT OF MECH ENG

5101-B ETCHEVERRY HALL

BERKELEY CA 94720-1740

2 GEORGIA TECH RSCH INST

COBB CO RSCH FACILITY

ATTN J MCMICHAEL A LOVAS

7220 RICHARDSON RD

SMYRNA GA 30080

1 GEORGIA INST OF TECH

SCOOL OF AEROSPACE ENG

ATTN A CALISE

270 FERST DR

ATLANTA GA 30332-0150

1 AEROVIRONMENT

ATTN C MIRALLES

4685-3H INDUSTRIAL ST

SIMI VALLEY CA 93063

1 CDR NAVSEA

ATTN M SIMMS

CODE 6024 BLDG 2940W

CRANE IN 47522

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ATTN S BISHOP

5600 SAND LAKE RD

MP-562

ORLANDO FL 32819
14

NO. OF

COPIES ORGANIZATION 

ABERDEEN PROVING GROUND

1 DIRECTOR

US ARMY RESEARCH LABORATORY

ATTN AMSRL CI LP (TECH LIB)

BLDG 4600

2 DIR USARL

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AMSRL WM B A HORST

AMSRL WM SG B RINGERS

BLDG 4600 

19 DIR USARL 

ATTN AMSRL WM BA D LYON

J CONDON B DAVIS 

T HARKINS D HEPNER

G KATULKA D LYON

P MULLER P PEREGINO

A THOMPSON T BROWN

R HALL B PATTON

M WILSON (5) W CLAY

BLDG 4600 

2 DIR USARL

ATTN AMSRL WM BC P PLOSTINS

B GUIDOS

BLDG 390

2 DIR USARL

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ATTN AMSRL WM TC R COATES

BLDG 309

4 CDR US ARMAMENT RD&E CTR

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F MIRABELLE

BLDG 120

1 CMDR USA DTC

ATTN CSTE DTC TTM J SCHNELL

RYAN BLDG

1 CDR ATC

ATTN K MCMULLEN 

BLDG 359