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SCIENTIFIC PERIODICAL OF THH BAUMAN MSTU
SCIENCE and EDUCATION
EL № FS77 - 48211. N»0421200025. ISSN 1994-0408
electronic scientific and technical journal
Lossless Compression of Telemetry Information using Adaptive
Linear Prediction
# 04, April 2014
DOI: 10.7463/0414.0707364
M. A. Elshafey, I.M. Sidyakin
Bauman Moscow State Technical University, 105005, Moscow, Russian Federation
elshafev.m a gmail.com ivan, sidvakin a gmail. com
1. Introduction
Normal requirement for telemetry data compression algorithms is an ability to recover initial data "as is" without loss of information. This feature is very important in various telemetry processing applications. Precise recovery of the telemetry data as it is acquired from the original source of information is necessary for the analysis of any kind of abnormal events, recovery of bad sites within the telemetry data stream and for other types of post- or real-time data processing [1,2]. The effectiveness of methods of lossless compression is largely determined by the properties of the data under compression [3]. Compression algorithms show better compression ratios if they can adapt to the characteristics of the input data, which are in most cases rapidly change. In this paper we present the results of studies conducted to develop an efficient method of reversible telemetry data compression based on adaptive linear prediction of telemetry data packed according to IRIG-106 format. IRIG-106 is an open standard, developed specifically for aerospace industry, but now used in wide range of telemetry registration applications [4]. Data is packed to frames of fixed length and predefined internal structure. Frame can carry different sources of information: digitized samples of analog signals, as well as pure digital data. For each source a channel of the recording system is provided. The source sample in each channel is introduced by telemetry word. All words in the frame have the same bit width. Telemetry frame contains additional service information in purpose of detecting bit errors, frame synchronization, etc.
Lossless data compression algorithm can be divided into two stages; the first stage -decorrelation stage, which exploits the redundancy between the neighboring samples in the data sequence, the second stage - entropy coding, which takes advantage from decreasing variance and lowering entropy of the data made on the first stage [5,6,7].
Normally, linear prediction is used for decorrelation stage, i.e. forthcoming data samples can be predicted based on known values of previous data samples, as shown by equation (1).
* i = I % i a-j * x i _ j (1)
Where xi - the expected value of the current discrete data sample xi and
a={ai,a2,......,ap} is the estimated values of the coefficients of the finite impulse response
(FIR) filter of order p. At each step i prediction error is calculated by (2).
ei = xi — Round (xi) (2)
Where is the rounding function which produces the nearest integer value to its
non-integer input. Filter coefficients are used by the decoder to reconstruct original data samples from the sequence of prediction errors. These filter coefficients are calculated by minimizing the sum of squares of prediction errors and if they are chosen properly, the entropy of should be less than that of .
Efficiency of the decorrelation algorithm can be referenced by two parameters: filter gain, which is relation between variance of the source data with respect to variance of the prediction error signal , or the entropy (which is the smallest average number of bits needed to rep-
resent the source output) of the prediction error signal. In practice, efficiency of the algorithm for estimating the entropy is easier to use, since this parameter does not depend on the shape of the source signal [5].
To adapt to changing of non-stationary signals , lossless data systems work on fragmenting data samples into blocks and hence, finding process of these cooeffients becomes increasingly computational expensive with large data block size. So, adaptive FIR filters have been proposed and used successfully for solving such matter [8,9,10].
2. NLMS adaptive filters
An adaptive filter has the property of self-modifying its frequency response to change the behavior in time, allowing the filter to adapt to the characteristics change of the input signal [11,12]. Due to this capability, the overall performance and the construction flexibility, the adaptive filters have been employed in many different applications like linear prediction. Adaptive filter encompass many different classic stochastic gradient algorithms [13].
Figure (1). Linear prediction layout using adaptive filter. One common method is the Normalized Least Mean Square (NLMS) algorithm which is suitable for both stationary as well as non-stationary environment and provides a tradeoff between convergence speed and computational complexity [14]. In the case of an NLMS adaptive
FIR filter of prediction order(p) and the set of prediction coefficients a (n)= {a -i(n), a2(n),......,ap(n)} is a set of variables varying with time (n). The error of the prediction e(n) of for each sample x(n) is calculated by the expression:
e(n) = x(n) — round [ aT(n) * IX(n)] (3)
Where = =
and is the transpose operator.
The set of cooeffients computed iteratively as follows:
a (n + 1) = a (n) + i (n) * e (n) * IX (n) (4)
1(n = (5)
a2(n) = ¡3* o2(n —1) + (1 — ¡¡)* e(n — 1)2 (6)
Where, U - the convergence parameter, ¡3 - the smoothing parameter (0 < ¡3 <1) and Op -
the input power estimation. For lossless recovery of data samples, original values of U, ¡¡, a(0 ) and IX(0) are added within the compressed data sequence [8,9,10]. In case of adaptive filtering, it is not necessary to fragment data into blocks as it is the case of linear prediction method.
3. Preparing experimental data
Digitized data samples of analog signals, which represent typical telemetry parameters of automatic control system, such as temperature, pressure and positioning information, are obtained in the laboratory. These data then switched into different telemetry frame structures that conform to IRIG-106 standard. The telemetry stream itself is made by the telemetry simulation software described in [15]. But all telemetry parameters are acquired from physical real sources.
Figure (2) shows a timed portion of the acquired data samples and figure (3) shows the histogram of the values of all examined data samples. This graph shows that the values are distributed over the entire range with several peaks.
400 350 300
T 250
o
D
s
3 200
3
m
£
< 150 100
50
0 - -
2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500
Time ^
Figure (2). A sample of the acquired data samples.
Sensor 1 Sensor 2 Sensor 3 Sensor 4 Sensor 5 Sensor 6 Sensor 7 Sensor 8 Sensor 9
0.09 0.08 0.07
?? 0.06
c o
g 0.05
'EZ
S
2 0.04 2
< 0.03 0.02 0.01 0
0 50 100 150 200 250 300
Data Values (0..255)
Figure (3). Distribution histogram of values of data words in a stream of telemetry data. Figures (4) and (5) depict two different types of telemetry frame structures that were used in the experiments (data word length for each of them equals to 8 bits long). The first type of examined frame structure (Type-A) - is a telemetry frame structure that consists of nine channels which are connected to the sources of information arranged in one minor frame (major telemetry frame does not contain sub-frame which means one stage of commutation). This frame further also includes additional information that provides frame synchronization in the telemetry stream.
SYNC_F Data Word: 1 Data Word: 1 Data Word: il Data Word: 4 Data Word: 5 Data Word: 6 Data Word: 7 Data Word: 8 Data Word: 9
SYNC16 I Sensor 1 I 1 Sensor 2 \ \ Sensor 3 1 1 Sensor 4 1 1 Sensor 5 1 1 Sensor & 1 1 Sensor 7 1 1 Sensor 8 1 1 Sensor 9
Figure (4). Telemetry frame structure without sub-frames (one stage multiplexing), Type-A.
The second type of examined frame structure, Type-B - is a telemetry frame structure that contains a sub-frame, resulting from a two-stage switching circuit. Figure (5) shows a four serial data frame IRIG-106 in which the major frame contains one sub-frame of length four and this sub-frame is attached to the fourth data word. In that fourth channel of the main multiplexer, values are passed to the second multiplexing level. Thus, a complete cycle that reads a complete set of telemetry parameters takes four frames of main multiplexer. During this cycle the main multiplexer channels are sampled four times.
SYNC_F SYNC_SF1 I Data Word: 1 Data Word: 2 Data Word: 3 Data Word: 4 Data Word: 5 Data Word: 6
SYNC16 C1 Sensor 1 Sensor 2 Sensor 3 Sensor 5 Sensor 4 Sensor 7
SYNC1G C2 Sensor 1 Sensor 2 Sensor 3 Sensor 6 Sensor 4 Sensor 7
SYNC16 C3 Sensor 1 Sensor 2 Sensor 3 Sensor 8 Sensor 4 Sensor 7
SYNC16 C4 Sensor 1 Sensor 2 Sensor 3 Sensor 9 Sensor 4 Sensor 7
Figure (5) . Telemetry frame structure with a sub-frame (two stage multiplexing), Type-B.
For both types of telemetry frame, experiments involve analysis and comparison of implementing a single non-adaptive FIR filter for different block sizes and adaptive NLMS filtering with initially zero coefficients. Decorrelation stage performed by adaptive NLMS filtering is implemented through two different strategies. First decorrelation method (Strategy-1) - is by implementing a single linear prediction for the raw stream of the telemetry data and the second decorrelation method (Strategy-2) proposed in this paper, is by implementing a separate linear
prediction for each data channel based on the structure of the telemetry frame. Compression ratio is used as criterion for assessing the compression algorithm. This parameter is computed as the ratio of the incoming data size to the size of data at the output of the compression system.
Figure (6) shows results of experiments with the single non-adaptive FIR used for linear prediction in conjunction with different entropy coders (Huffman coding [16], Arithmetic coding [17,18,19,20,21] and Rice coding [22]) for Type-A and Type-B telemetry frame structures with respect to data block size.
a)
b)
Figure (6). Compression performance of linear prediction and different entropy coders with respect to data block size. a) Type-A telemetry frame structure using FIR filter of order (9), b) Type-B telemetry frame structure using
FIR filter of order (24).
4. Implementing single NLMS FIR filter for the serial data stream(Strategy-l)
Table (1) shows results of applying a single NLMS adaptive FIR on both types of telemetry frame structures (Type-A ,.Type-B) and selecting the best value for smoothing parameter U which leads to best results of filter gain a^ / <r| and the entropy of the prediction error signal.
Decorrelation Single NLMS FIR filter of order (9). (variance of data samples a* = 6085,7736 , entropy of data samples = 7,4595)
It 0.001 0.01 0.1 0.2 0.9
Filter Gain a2 / ae2 3.6135 11.8452 42.9828 50.9835 44.9741
Entropy of prediction errors 6.9212 4.3352 3.6536 3.7631 4.2909
a) b )
Decorrelation Single NLMS FIR filter of order (24). (variance of data samples a* = 5967,1569 , entropy of data samples = 7,2611)
It 0.001 0.01 0.1 0.3 0.9
Filter Gain ст2 / <те2 2.8751 6.7916 18.3531 22.0918 19.0140
Entropy of prediction errors 6.9428 5.4030 4.6006 4.6758 5.0757
Table (1). Performance of single NLMS FIR of order (9). a) Type-A telemetry frame structure, b) Type-B telemetry
frame structure.
The best value of U is; for Type-A telemetry frame structure: U = (0.1) and for Type-B telemetry frame structure U = (0.3).
Table (2) shows results of the complete two stage lossless compression algorithm that uses adaptive NLMS FIR filter (with initially zero coefficients) combined with different entropy coders for both types of telemetry frame structures. Adaptive NLMS filter produces very close results of compression ratio in comparison with that produced by the non-adaptive FIR filter with no need for data fragmenting.
First stage Non-adaptive FIR filter of order (9) and data block size= (6144) A single NLMS FIR filter of order (9) ( 1 = 0.1, p = 0.9 )
Second stage Huffman Arithmetic. Rice Huffman Arithmetic. Rice
Compression ratio 2,47 19 2, 4405 2 ,0008 2,2239 2,2107 1,9415
a)
First stage Non-adaptive FIR filter of order (24) and data block size= (8192) A single NLMS FIR filter of order (24) ( 1= 0.3, p = 0.9)
Second stage Huffman Arithmetic. Rice Huffman Arithmetic. Rice
Compression ratio 1,7 872 1, 7802 1 ,6946 1,6872 1,6919 1,6313
b)
Table (2). Compression performance of a single NLMS FIR filter and different entropy coders. a) Type-A telemetry
frame structure, b) Type-B telemetry frame structure.
5.Implementing separate NLMS FIR filter for each data channel(Strategy-2)
Table (3) shows results of implementing a separate NLMS adaptive FIR filter on each data channel (nine data channels) for both types of telemetry frame structures and selecting the best value for smoothing parameter U for initially zero cooeffients._
Decorrelation Nine separate NLMS FIR filters, each of order (9) for each data channel (variance of data samples = 6085,7736 , entropy of data samples = 7,4595)
U 0.01 0.1 0.2 0.4 0.5 0.9
Filter Gain / a| 8.7294 32.0928 41.6631 48.9751 49.5996 42.5778
Entropy of prediction errors 3.9844 2.9520 2.8228 2.8203 2.8545 3.0519
a)
Decorrelation Nine separate NLMS FIR filters, each of order (9) for each data channel (variance of data samples = 5967,1569 , entropy of data samples = 7,2611)
U 0.01 0.1 0.2 0.4 0.5 0.9
Filter Gain / ae2 7.3237 24.3165 30.1311 33.4632 33.5121 28.5575
Entropy of prediction errors 4.2792 3.4387 3.2682 3.2284 3.2641 3.4413
b)
Table (3). Performance of nine separate NLMS FIR filters each of order (9). a) Type-A telemetry frame structure, b)
Type-B telemetry frame structure.
The best value of U is; for Type-A telemetry frame structure U = (0.4) and for Type-B telemetry frame structure: U = (0.4).
Table (4) shows results of the complete two stage lossless compression that implements single adaptive NLMS FIR filter or separate adaptive NLMS FIR filter for each data channel; with different entropy coders for both types of telemetry frame structures.
First stage Nine separate NLMS FIR filters each of order (9) for each data channel ( = 0.4, = 0.9 ) Single NLMS FIR filter of order (9) ( 1 = 0.1, p = 0.9)
Second stage Huffman Arithmetic. Rice Huffman Arithmetic. Rice
Compression ratio 2,818 2,8039 2,1233 2,2239 2,2107 1,9415
a)
First stage Nine separate NLMS FIR filters each of order (9) for each data channel ( = 0.4, = 0.9 ) Single NLMS FIR filter of order (24) ( 1 = 0.3, p = 0.9)
Second stage Huffman Arithmetic. Rice Huffman Arithmetic. Rice
Compression ratio 2,5472 2,4799 1,9571 1,6872 1,6919 1,6313
b)
Table (4). Compression performance of nine separate NLMS FIR filters; one for each data channel; and different entropy coders. a) Type-A telemetry frame structure, b) Type-B telemetry frame structure.
Conclusion and work outcome
Experiments under real telemetry data, provides numerical results that allow drawing conclusions about the effectiveness of the proposed methods and developing recommendations for lossless compression of telemetry information. Analysis of the experimentally acquired results leads to the following main results:
-- Compression ratios retrieved from experiments with NLMS FIR filter are very close to that of non-adaptive FIR filter, while NLMS FIR filter shows slightly better results than non-adaptive filter for telemetry data.
-- NLMS FIR filter allows improving performance for (Strategy-1), i.e. when telemetry channels are demultiplexed before applying prediction filters. These experiments shows entropy value is about 1.5 times lower at the predictor output in experiments with adaptive algorithm.
-- Huffman method almost in all the experiments shows better compression ratio in comparison with Arithmetic and Rice codes.
-- As expected, results are slightly worse for more complex structure of Type-B frame.
Results enlisted, depict an advantage of the approach to lossless telemetry data compression based on adaptive NLMS FIR filter prediction applied at decorrelation stage. The combination of adaptive linear prediction with Huffman source coding is proved to be an effective method for lossless telemetry data compression and is recommended as core algorithm for lossless telemetry compression system. Frame structures with one and two multiplexing levels are examined, which illustrate advantage of the proposed approach for different data packing formats.
References
1. Lossless Data Compression. Report Concerning Space Data System Standard. Informational Report CCSDS 120.0-G-3 Green Book. April 2013.
2. Staudinger P., Hershey J., Grabb M., Joshi N., Ross F., Nowak T. Lossless Compression for Archiving Satellite Telemetry Data. 2000 IEEE Aerospace Conference Proceedings. Vol. 2. IEEE, 2000, pp. 299-304. DOI: 10.1109/AER0.2000.878236
3. Konecki M., Kudelic R., Lovrencic A. Efficiency of Lossless Data Compression. 2011 Proc. of the 34th International Convention MIPRO, Opatija, 23-27 May, 2011. IEEE, 2011, pp. 810-815.
4. IRIG Standard 106-13. Part 1: Telemetry Standards. Secretariat Range Commanders Council Us Army White Sand Missile Range, New Mexico 88002-5110, 2013. Available at: http://www.irig106.org/docs/106-13/ , accessed 01.03.2014.
5. Mandyam G., Magotra N., Stearns S.D. Lossless Waveform Compression. CiteSeerx, 2004. 32 p. Available at: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.49.2138&rank=1 accessed 01.03.2014.
6. Hans M., Schafer R.W. Lossless Compression of Digital Audio. Technical Report. Hewlett-Packard Company, 1999. 37 p.
7. Robinson T. SHORTEN: Simple Lossless and Near-lossless Waveform Compression. Technical Report Cued / F-Infeng / Tr.156. Cambridge University, 1994. 16 p.
8. McCoy J.W., Magotra N., Stearns S. Lossless predictive coding. IEEE Proceedings of the 37th Midwest Symposium on Circuits and Systems. Vol. 2. IEEE, 1994, pp. 927-930. DOI: 10.1109/MWSCAS.1994.518963
9. Magotra N., McCoy W., Livingston F., Stearns S. Lossless data compression using adaptive filters. 1995 International Conference on Acoustics, Speech, and Signal Processing International Conference (ICASSP). Vol. 2. IEEE, 1995, pp. 1217-1220. DOI: 10.1109/ICASSP.1995.480457
10. Mandyam G, Magotra N, McCoy W. Lossless seismic data compression using adaptive linear prediction. International Geoscience and Remote Sensing Symposium (IGARSS'96). Vol. 2. IEEE, 1996, pp. 1029-1031. DOI: 10.1109/IGARSS.1996.516556
11. Haykin S. Adaptive Filter Theory. 3rd ed. Prentice Hall, Inc., Upper Saddle River, NJ, 1996, pp. 365-440.
12. Poularikas A.D., Ramadan Z.M. Adaptive Filtering Primer with MATLAB. Taylor & Francis Group LLC, 2006, pp. 101-167.
13. Douglas S.C., Losada R. Adaptive filters in Matlab: from novice to expert. Proceedings of 2002 IEEE 10th Digital Signal Processing Workshop and the 2nd Signal Processing Education Workshop. IEEE, 2002, pp. 168-173. DOI: 10.1109/DSPWS.2002.1231097
14. Raj Kumar Thenua, Agarwal S.K. Simulation and performance analysis of adaptive filter in noise cancellation. International Journal of Engineering Science and Technology, India, 2010, vol. 2, no. 9, pp. 4373-4378.
15. El'shafei M.A., Sidyakin I.M. [Simulation of telemetry information transmission over AWGN channel]. Inzhenernyy vestnik MGTU im. N.E. Baumana - Engineering Herald of the Bauman MSTU, 2014, no. 1. Available at: http://engbul.bmstu.ru/doc/697480.html , accessed 01.03.2014. (in Russian).
16. David A. Huffman. A Method for the Construction of Minimum-Redundancy Codes. Proceedings of the I.R.E., 1952, pp. 1098-1101.
17. Rissanen J., Langdon Jr. G.G. Arithmetic Coding. IBM Journal of Research and Development, 1979, vol. 23, no. 2, pp. 149-162.
18. Howard P.G., Vitter J.S. Analysis Of Arithmetic Coding for Data Compression. Data Compression Conference (DDC'91), Snowbird, Utah, 08-11 April, 1991. IEEE, 1991, pp. 3-12. DOI: 10.1109/DCC.1991.213368
19. Howard P.G., Vitter J.S. Arithmetic Coding for Data Compression. Proceedings of the IEEE, 1994, vol. 82, iss. 6, pp. 857-865. DOI: 10.1109/5.286189
20. Stems S.D. Arithmetic Coding in Lossless Waveform Compression. IEEE Transactions on Signal Processing, 1995, vol. 43, no. 8, pp. 1874-1879. DOI: 10.1109/78.403346
21. Moffat A., Neal R.M., Witten I.H. Arithmetic Coding Revisited. ACM Transactions on Information Systems, 1998, vol. 16, no. 3, pp. 256-294. DOI: 10.1145/290159.290162
22. Rice R.F. Some Practical Universal Noiseless Coding Techniques. Technical Report. Jet Propulsion Laboratory Jpl-79-22, 1979. 130 p.
НАУЧНОЕ ИЗДАНИЕ МГТУ ИМ. Н. Э. БАУМАНА
НАУКА и ОБРАЗОВАНИЕ
Эл № ФС77 - 4 8211. Государственная регистрация №042 1200025. ISSN 1994-0408
электронный научно-технический журнал
Применение метода адаптивного линейного предсказания для
сжатия телеметрической информации
# 04, апрель 2014
DOI: 10.7463/0414.0707364
Эльшафеи М. А., Сидякин И. М.
УДК 004.632.4
Россия, МГТУ им.Баумана elshafev.m Д gmail.com i van. si dyakin Ж gm ail. com
Эффективность обратимых алгоритмов сжатия во многом определяется тем, насколько точно подобрана модель предсказывающая поведение обрабатываемых данных. Характерной особенностью телеметрической информации является быстрое изменение статистических свойств контролируемых параметров при переключении режимов работы объекта или возникновения аварийных ситуаций. В этих условиях применение адаптивных методов сжатия позволяет существенно улучшить характеристики системы сжатия за счет повышения точности предсказания. В статье приведены результаты исследований метода обратимого сжатия телеметрической информации построенного на основе адаптивного алгоритма линейного предсказания на этапе декорреляции данных. Приводятся результаты исследований эффективности декоррелятора, использующего метод нормализованного наименьшего среднеквадратического отклонения и экспериментальное сравнение нескольких стратегий декорреляции. Определение эффективности работы алгоритмов на этом промежуточном этапе декорреляции производится на основе экспериментально полученных значений дисперсии и энтропии ошибок предсказания. В экспериментах использовались характерные для систем телеизмерений источники данных: параметры систем автоматического управления, записанные в лабораторных условиях, такие как температура, давление и данные позиционирования. Эти данные формируют поток телеметрической информации в соответствии со стандартом IRIG-106, который широко используется в аэро-космической промышленности. Эксперименты проводились с разными видами телеметрических кадров, составленных на основе одно- и двух- ступенчатой системы коммутации каналов. Исследованы комбинации адаптивного предсказателя с различными методами кодирования источника, включая метод Хаффмана, арифметическое кодирование и коды Райса. Проведено сравнение эффективности нескольких комбинаций алгоритмов декорреляции и энтропийного кодирования для сжатия данных телеизмерений.
Получены количественные показатели, позволяющие оценить эффект применения адаптивных методов декорреляции в процедуре сжатия телеметрической информации. На основе полученных экспериментальных данных выработаны рекомендации по разработке системы обратимого сжатия для этого вида информации.
Публикации с ключевыми словами: коммутация, энтропия, телеметрическая информация, декорреляция, обратимое сжатие, кодирование источника, адаптивные методы, арифметическое кодирование, код Хаффмана, код Райса, ТЯЮ-Юб Публикации со словами: коммутация, энтропия, телеметрическая информация, декор-реляция, обратимое сжатие, кодирование источника, адаптивные методы, арифметическое кодирование, код Хаффмана, код Райса, ТЯЮ-Юб
Список литературы
1. Lossless Data Compression. Report Concerning Space Data System Standard. Informational Report CCSDS 120.0-G-3 Green Book. April 2013.
2. Staudinger P., Hershey J., Grabb M., Joshi N., Ross F., Nowak T. Lossless Compression for Archiving Satellite Telemetry Data // 2000 IEEE Aerospace Conference Proceedings. Vol. 2. IEEE, 2000. P. 299-304. DOI: 10.1109/AER0.2000.878236
3. Konecki M., Kudelic R., Lovrencic A. Efficiency of Lossless Data Compression // 2011 Proc. of the 34th International Convention MIPRO (Opatija, 23-27 May 2011). IEEE, 2011. P. 810815.
4. IRIG Standard 106-13. Part 1: Telemetry Standards. Secretariat Range Commanders Council Us Army White Sand Missile Range, New Mexico 88002-5110, 2013. Available at: http://www.irig106.org/docs/106-13/ , accessed 01.03.2014.
5. Mandyam G., Magotra N., Stearns S.D. Lossless Waveform Compression. CiteSeerx, 2004. 32 p. Available at: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.49.2138&rank=1 accessed 01.03.2014.
6. Hans M., Schafer R.W. Lossless Compression of Digital Audio. Technical Report. Hewlett-Packard Company, 1999. 37 p.
7. Robinson T. SHORTEN: Simple Lossless and Near-lossless Waveform Compression. Technical Report Cued / F-Infeng / Tr.156. Cambridge University, 1994. 16 p.
8. McCoy J.W., Magotra N., Stearns S. Lossless predictive coding // IEEE Proceedings of the 37th Midwest Symposium on Circuits and Systems. Vol. 2. IEEE, 1994. P. 927-930. DOI: 10.1109/MWSCAS.1994.518963
9. Magotra N., McCoy W., Livingston F., Stearns S. Lossless data compression using adaptive filters // 1995 International Conference on Acoustics, Speech, and Signal Processing Interna-
tional Conference (ICASSP). Vol. 2. IEEE, 1995. P. 1217-1220. DOI: 10.1109/ICASSP.1995.480457
10. Mandyam G, Magotra N, McCoy W. Lossless seismic data compression using adaptive linear prediction // International Geoscience and Remote Sensing Symposium (IGARSS'96). Vol. 2. IEEE, 1996. P. 1029-1031. DOI: 10.1109/IGARSS.1996.516556
11. Haykin S. Adaptive Filter Theory. 3rd ed. Prentice Hall, Inc., Upper Saddle River, NJ, 1996. P. 365-440.
12. Poularikas A.D., Ramadan Z.M. Adaptive Filtering Primer with MATLAB. Taylor & Francis Group LLC, 2006. P. 101-167.
13. Douglas S.C., Losada R. Adaptive filters in Matlab: from novice to expert // Proceedings of 2002 IEEE 10th Digital Signal Processing Workshop and the 2nd Signal Processing Education Workshop. IEEE, 2002. P. 168-173. DOI: 10.1109/DSPWS.2002.1231097
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