Recursive Matrix Pencil Method Olga Ibryaeva South Ural State University Chelyabinsk, Russia
[email protected]
Abstract—The main concern for many applications (for example, for Coriolis Mass Flow Meter signal processing in twophase flow conditions) is to track several domain poles of signals with minimum delay. The Matrix Pencil method (MPM) estimates the signal as a sum of complex exponentials. Creating a computationally efficient moving MPM implementation is of high interest. The most computationally expensive step of the classical MPM is to calculate the singular value decomposition (SVD) of a matrix composed from the signal samples. When a new data point enters the data window, this matrix changes only slightly and it is reasonable to find its SVD not directly but using the SVD of the old matrix and a low-rank SVD modification procedure. In this paper a well-known SVD modification procedure is adapted for MPM and a recursive version of MPM is proposed. The amended method is validated by numerical examples and is faster than the original, suggesting it may be feasible to track signal parameters on-line. The errors accumulating over time due to the recursive calculation require the recursive MPM to be restarted periodically. Keywords—Matrix pencil method, Prony-like methods, recursive estimation, singular value decomposition (SVD), updating and downdating SVD, Coriolis Mass Flow Meter signal processing.
I. INTRODUCTION The Matrix Pencil method (MPM), that was first proposed in [1], is one of the Prony-like methods [2] which estimate the signal as a sum of complex exponentials. Determining the amplitudes, phases and damping factors is a regular task in power system electromechanical oscillation [3], biomedical monitoring [4], radioactive decay [5], sonar [6], radar [7], geophysical sensing, processing of seismic data [8], and Coriolis Mass Flow (CMF) Meter signal processing [9]. CMF Meter signal processing methods for single phase (pure gas or liquid) measurement are well established. Techniques in the recent literature [10], [11], [12] include Fourier Transform, Digital Correlation, Digital Phase Locked Loop, Adaptive Notch Filter, and the Hilbert Transform. CMF Meter signal processing in two-phase (liquid and gas) flow conditions is challenging [11]: all signal parameters are subject to large and fast variations and must be tracked to high accuracy, and with minimum delay, to provide good The work was conducted with the financial support of the Ministry of Education and Science of the Russian Federation of applied scientific research within the framework of the basic part of the State task "Development, research and implementation of data processing algorithms for dynamic measurements of spatially distributed objects", Terms of Reference 8.9692.2017 / BP from 17.02.2017
Fig. 1. Timing diagram of Coriolis signal processing (from [13])
control and precise measurements. For example, Fig. 1 (from [13]) shows a timing diagram of the response of a transmitter to flow tube data. The lag of three flowtube resonant cycles between the raw data and the drive signal response arises from a combination of the delays induced by ADC and DAC operation, filtering, data processing and phase synchronization. Under normal operating conditions, such a delay rarely induces any difficulty. However, in two-phase flow conditions the frequency and the amplitude of the sensor signals change much more rapidly, therefore the ability of tracking algorithms to follow these changes with high precision and minimum delay becomes more and more important. A range of CMF signal processing techniques have been developed [14], [15]. In our earlier paper [16] MPM is shown to be promising new approach to CMF signal analysis. Our intention to use MPM on-line in a moving window manner prompted us to develop the Recursive Matrix Pencil Method (RMPM). In many respects, this work echoes the paper [17] in which the Moving Window Prony (MWP) Method is described. The main difference is that MWP method uses a rank-one SVD modification procedure twice, while RMPM uses a rank-two SVD modification procedure once and is therefore computationally more efficient. The structure of the paper is as follows. In section II, the classical MPM is described. Based on a low-rank SVD modification procedure [18], RMPM is proposed in Section III. This method is evaluated using numerical examples in Section IV. Finally, Section V offers concluding remarks for the paper.
II. CLASSICAL MATRIX PENCIL METHOD The Matrix Pencil Method is a technique for estimating signal parameters ( )=∑
(
)
)
=∑
where ,…, are the largest singular values of the matrix , and are the corresponding singular vectors, ( , … , ). = ( , … , ), = ( , … , ), = It can be seen [1] that the estimates of computing the eigenvalues of the × matrix:
can be found by nonsymmetrical
from its samples
(
(
)=∑
MPM finds estimates for the amplitudes and poles from samples ( ) = , = 0,1, … , − 1, in two steps. First, it finds the poles as the solution of a generalized eigenvalues problem by using the matrix pencil formed from the sampled values . In the second step, it uses these poles to estimate by solving a least squares problem. matrices
,
.
where is the sampling period, = are the complex amplitudes, are the damping factors, =2 are the ) frequencies and = ( are the poles of ( ).
We define two ( − ) ×
=
,
Once and are known, the complex amplitudes are then found by solving the following least squares problem:
1
1
⋮
⋮
=
⋮
⋯ ⋯ ⋱ ⋯
1 ⋮
⋮
We have set out the classical MPM as it was first presented in [1] which can be summarized as follows: Algorithm 1. Classical Matrix Pencil Algorithm
, as follows:
Input: Signal samples
=
=
⋮
⋮
… … ⋱ … … … ⋱ …
⋮
,
⋮
Output: ⋮
⋮
,
where ≤ ≤ − is the pencil parameter. It has been shown [1] that and are the best choices for to ensure the MPM method is least sensitive to noise. In our simulations has been set as floor(N/3). (The function floor in Matlab rounds a number to the next smaller integer.) ) In the noise-free case, the poles = ( can be found [1] as the generalized eigenvalues of the matrix pencil − . This means that the are the eigenvalues of . The superscript † denotes the pseudoinverse or MoorePenrose inverse. For noisy data, Singular Value Decomposition (SVD) is used to reduce the noise and to estimate the number of the signal poles, thus: =
.
Here , are the unitary matrices and is the diagonal matrix containing the singular values of . The superscript notes the transpose. Typically, the singular values beyond are very close to zero, provided that they are arranged from largest to smallest. The order is thus estimated and the pseudoinverse replaced by the rank- truncated pseudoinverse:
=∑
(
=
=
,
is
,
)
,
=
,
= 1, … ,
= 0,1. … ,
− 1.
.
Start ,
, as in (3), (4).
1.
Form matrices
2.
Carry out the SVD of the matrix
3.
Estimate the number
4.
Find the rank-
5.
Estimate
by computing the eigenvalues of
6.
Estimate
by solving the least squares problem (8).
.
of signal poles.
truncated pseudoinverse
(6). (7).
End We should note that several modified versions of Matrix Pencil Method have been proposed since it was first introduced: MPM for undamped sinusoids is thoroughly studied in [19], the improved MPM using low-rank Hankel approximation is suggested in [20], the quantum version of MPM is developed in [21]. In the next section we propose our recursive modification of this method. III. RECURSIVE MATRIX PENCIL METHOD A. SVD Modification in Matrix Pencil Method Denote the matrices , from the last section as ( ) ( ) , , formed using the signal samples , ,…, . Once the new data point is appended, the older sample should be deleted. Therefore, the new matrices ( ) ( ) , become
( )
( )
=
… … ⋱ …
⋮
=
⋮
)×(
=
… … ⋱ …
⋮
,
⋮
⋮
.
⋮
= ( )
,…, in the first column and by deleting the , ( ) last column of . The addition and deletion of columns are denoted as "updating" and "downdating", respectively. We ( ) ( ) will obtain the SVD of based on the SVD of using the computationally attractive route to SVD modifications for column updates and downdates, which is proposed in [18]. By direct substitution, we can sure that that the following equality holds: ( )
( )
0 = 0
+
,
where
(
=
)×
⋮
,
⋮
(
)×
1 0 = 0 0 . ⋮ ⋮ 0 −1 ( )
Recall that having a known -truncated SVD of , we ( ) intend to find an -truncated SVD of . It can be realized by the procedure of low rank SVD modification method [18]. We have adapted this procedure and the result is in Algorithm 2. One step of RMPM ( )
-truncated SVD of
Input: Output:
-truncated SVD of
.
( )
.
0
0
.
Therefore,
It can be seen that the new matrix can be obtained ( ) from the matrix by appending the data
0 0
Perform the SVD on this smaller matrix
4.
∈
Construct a matrix with lower dimensions ) :
3. (
.
Start ( ) 1. Find SVD factors of 0 using the known ( ) truncated SVD of = . One can observe that
-
( )
)
(
Obtain the desired
5.
( )
0 =(
(: , 1:
=
)
)
=
-truncated SVD of (1:
, 1:
(16) ( )
) (1: , 1:
:
) .
End ( )
)×( ) Remark 1. Notice that ∈ ( while ∈ ( )× . Since usually ≪ ≪ − , replacing the SVD ( ) of the matrix by the SVD of the smaller matrix leads to lower computational costs. The computational benefit is more noticeable for large values of and is shown in Section IV.
Remark 2. For simplicity, in (17) we use notation from Matlab. So for example, (: , : ) includes all subscripts in the first dimension but uses the vector : to index in the second dimension. Further we use this SVD modification step in order to propose the Recursive Matrix Pencil Method. B. Recursive Matrix Pencil Algorithm The main concern for many applications (for example, for Coriolis Mass Flow Meter signal processing in two-phase flow conditions) is to track several domain poles of the input signals. Thus, a computationally efficient moving MPM implementation is of high interest. This section describes the operation of the Recursive Matrix Pencil Method for sets of consecutive signal segments. Note that we can't run RMPM "forever" because from time to time we should "restart" it to remove accumulated error. The corresponding Example 1 will be given in Section IV. Algorithm 3. Recursive Matrix Pencil Method
2. ( −
but
0
( )
=
(
=
0) 0
.
0
Find orthogonal bases , of the column spaces ) , ( − ) , respectively, and set =
( −
Notice that = × ,
≠
) , (
=
=
)×( (
)×(
),
( − and ).
) . ≠
(
)×(
),
Input: Signal samples (
= Output: ,…, Start
(
)
,
, ,
=
sets of
,
signal samples)
= 0,1. … ,
−2+
.
= 1, … , , for each set of samples = 1, … , .
1. ,…,
( )
≡
Form matrix .
as in (3) from the samples
2.
Carry out the SVD of the matrix
3.
Estimate the number
4.
Find the rank-
for 5.
=
to
.
of signal poles.
truncated pseudoinverse
do
Form matrix ,…,
as in (4) from the samples .
6.
Estimate
7.
Estimate problem (8).
by
Find rankAlgorithm 2.
truncated SVD
8.
(6).
by computing the eigenvalues of solving
the
(7).
least
squares Fig. 2.. Input signal waveform and FFT [17]
of
( )
using
end 9. Form matrix ,…, .
as in (4) from the samples
10. Estimate
by computing the eigenvalues of
(7).
11. Estimate
by solving the least squares problem (8).
End Let's now evaluate the proposed RMPM algorithm using numerical examples. IV. SIMULATION RESULTS A. Example 1 - Restart strategy First we would like to repeat the example from [17] that the authors used to validate the effectiveness of MWP method. Consider the signal: ( ) = 150
sin
+
+ 160
sin
+
,
= 3, = 5, = 2 ∙ 51.2, = 2 ∙ 48.8, that is sampled at a frequency Fs=1000 Hz. The input signal waveform is presented in Fig. 2. Performing Fast Fourier Transform (FFT) on 1000 samples of thee input signal, only one peak could be detected. As expected,, the resolution of FFT is not sufficient enough to resolve two very close frequencies.
red straight lines),, but with the growth of the moving window step, tep, the error is accumulated owing to the algorithm nature of recursive computation. Fig. 4 presents the logarithm of the absolute error of ( ) and the steady decline in measurement quality. The absolute errors of ( ), Im(( ), Im( ) behave in a similar way. In practice, a ”restart strategy” is recommended to use to secure the calculation precision [17]. B. Example 2 - MPM and RMPM speed comparison on a model example We now compare the performance of MPM and RMPM by applying both to tracking rapid change in the amplitude of the signal ( ) = ( ) sin 2 ∙ 150 +
that is sampled at a frequency Fs= Fs=1000 Hz. The amplitude change is shown in blue in Fig. 5. To make the example more realistic,, aadd white Gaussian noise ( ) with the standard deviation = 3 to the signal (19) so that the signal-to-noise ratio (SNR) is about 30 dB. SNR is calculated by the formula:
Choose (as in [17]) the data window length = 40. SVD ( ) decomposition of indicatess that the effective rank is 4, as there are only 4 non-zero zero diagonal elements in the singular value matrix. It fully corresponds to the fact that the signal was generated as a linear combination of two damped sinusoids. Fig. 3 shows the real cos and imaginary ) sin parts of the signal poles = ( , = 1,2, estimated by the RMPM from moving window implementation in 220 steps. One can see that in the beginning they are very close to the real values (they they are shown by the
Fig 3. Real and imaginary parts of signal poles estimated by RMPM
Fig 4. The logarithm of the absolute error of
∑
= 10 lg ∑
,
( )
Fig. 5. Real and estimated amplitude change
where , are the signal and the noise samples, respectively. The noised input signal is presented in Fig. 6. We select = 100 and apply RMPM and MPM to this frequency tracking. Both methods work in a moving window manner, but RMPM begins with the "old" SVD in order to evaluate the "new" SVD according to Algorithm 3, whereas MPM evaluates it directly every time. Note also that in this example we use RMPM without restarting. RMPM and MPM provide broadly similar results as shown in Figs. 5, 7. As one can see, both methods respond to this amplitude change, giving a correct result for frequency and amplitude values. RMPM works the same way as MPM, but it has a significantly reduced computing requirement. The time of operation of the methods for different data window lengths is given in Table I. (This time was calculated using the Matlab function tic-toc.) You can see that the time of RMPM is less than MPM, and the difference is greater at large . It is expected because large values of correspond to the large size of the matrix ,
Fig. 6. Input signal with noise
and consequently, to the large computational cost for calculating its SVD. We note that the time given in the Table I corresponds to the processing of the entire signal. That is at = 100 we processed 900 segments of the signal, moving each time by 1 sample and this took 3.1 seconds. Since 900 times we did about the same thing, we can roughly consider that the processing of a single segment takes approximately 3.1/900=0.0034 seconds. Thus RMPM is able to work in real time. TABLE I. Window length, N
ELAPSED TIME
MPM
Time, seconds RMPM
50
3
2.3
100
8
3.1
200
24.1
5.5
So how numerical experiments have shown, RMPM gives the same results as MPM in less time, but sometimes requires restart.
Fig. 7. Real and estimated frequency
measurements: Preliminary results using the matrix pencil method,” Contract Report, DRDC-RDDC-2016-C106, 2016. (http://cradpdf.drdcrddc.gc.ca/PDFS/unc223/p803588_A1b.pdf)
V. CONCLUSIONS Recursive Matrix Pencil Method has been proposed in this paper. The most computationally expensive step of MPM of calculating the SVD was optimized in terms of computational complexity. RMPM can be effectively used in a moving window manner since the SVD of the new matrix occurring at each step is found not directly but from the SVD of the old matrix. This leads to a significant reduction in the operation time of the method. The Algorithm of RMPM is presented and validated by numerical examples. It is shown that RMPM is feasible for online tracking: during certain period of time, the estimation precision is satisfactory. Among the shortcomings of RMPM we should note the need for restart. This disadvantage is not significant since even with the restart recursive MPM will still operate faster than classical MPM in a moving window manner. The more significant shortcomings is that once the effective rank (the number of poles) is defined, it is supposed to be constant during the operation of RMPM. However, for non-stationary signals, the effective rank might be varying. From this point of view, the restart necessity is rather an advantage, since during the restart, we can overestimate the rank. Further investigations should be conducted to develop the rules for data window length selection, restart strategy, adaptive rank detection. Our experimental experience has also convinced us that the choice of the signal sampling frequency is of great importance for its subsequent processing by MPM, RMPM, as well as by Prony method [22].
Acknowledgment The author would like to express her gratitude to Professor A.L. Shestakov for his continuous encouragement, support and guidance. The authors would like to thank Professor Manus Henry for his valuable comments which helped to improve the manuscript. Special thanks are due to the staff of the laboratory of technical self-diagnostics and self-control for instruments and systems (SUSU), especially to A.S. Semenov for providing valuable advices and recommendations.
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