Optical transmission characteristic estimation device, optical transmission characteristic estimation method, and program

Abstract

Provided is an optical transmission characteristic estimation device comprising: a transmission waveform reconstruction unit that reconstructs a transmission signal from a reception signal acquired by reception of an optical signal via a coherent detection method; and an estimation unit that, on the basis of the reconstructed transmission signal and the reception signal, estimates a nonlinear coefficient in a light wave propagation equation via a linear least-squares method, thereby estimating light power distribution in a transmission path.　

Description

[Technical Field]

[0001] The present invention relates to an optical transmission characteristics estimation device, an optical transmission characteristics estimation method, and a program. [Background technology]

[0002] When operating an optical transmission system, the basic characteristics of the optical fiber that makes up the optical transmission path have a significant impact on transmission performance. Here, the basic characteristics of optical fiber include optical power, distribution of loss and dispersion, and location of fault points. For example, if the optical power is too high, the influence of nonlinear optical effects in the optical fiber increases, reducing the signal-to-noise ratio (hereinafter referred to as "SNR"). If the loss is too high, the optical power attenuates accordingly, reducing the SNR.

[0003] Therefore, it is important to know the characteristics of optical fibers in the operation, maintenance, and monitoring of optical transmission systems. Optical transmission paths are composed of various devices other than optical fibers, such as optical amplifiers and optical filters. Knowing the characteristics of these devices is also important in the operation, maintenance, and monitoring of optical transmission systems.

[0004] The characteristics of devices such as optical fibers, optical amplifiers, and optical filters can generally be measured using analog measuring instruments such as OTDRs (Optical Time Domain Reflectometers) and optical spectrum analyzers. However, measurements using analog measuring instruments require direct measurement of each optical node and optical fiber, which poses the problem of high equipment and operating costs.

[0005] To solve this problem, in recent years, digital longitudinal monitoring (DLM) has been proposed as a technology that detects the characteristics of various devices in an optical transmission system by digital signal processing on the receiving side of the system, instead of measurements using analog measuring instruments (see, for example, Non-Patent Documents 1 and 2). DLM is based on a digital coherent optical transmission system, and monitors the optical power and other characteristics of the optical transmission line by performing digital signal processing on the received signal obtained by coherently detecting the optical signal transmitted by the optical transmission line.

[0006] Non-Patent Document 1 uses a method using correlation, which will be referred to as a correlation method here. Non-Patent Document 2 uses a method called a channel reconstruction method that uses a gradient method. [Prior art documents] [Non-patent literature]

[0007] [Non-Patent Document 1] T. Tanimura, et al., “Fiber-Longitudinal Anomaly Position Identification Over Multi-Span Transmission Link Out of Receiver-end Signals”, JLT, 38(9), 2020. [Non-patent document 2] T. Sasai, et al., “Digital longitudinal monitoring of Optical Fiber Communication Link”, JLT, 40(8), 2022. Summary of the Invention [Problem to be solved by the invention]

[0008] However, the correlation method described in Non-Patent Document 1 is theoretically limited in spatial resolution and can only estimate relative optical power. Therefore, the correlation method described in Non-Patent Document 1 cannot achieve sufficient estimation accuracy. The channel reconstruction method described in Non-Patent Document 2 does not have the spatial resolution limitation of the correlation method, but since it is a non-linear least-squares method using a gradient method, it is necessary to appropriately set hyperparameters (e.g., learning rate, number of learnings, initial values, etc.). Furthermore, the channel reconstruction method described in Non-Patent Document 2 imposes a large computational load. As such, conventional methods have had the problem of being unable to estimate optical transmission characteristics with high accuracy while suppressing the computational load with a small number of parameter settings.

[0009] In view of the above circumstances, an object of the present invention is to provide a technique that can estimate optical transmission characteristics with high accuracy while reducing the calculation load by setting a small number of parameters. [Means for solving the problem]

[0010] One aspect of the present invention is an optical transmission characteristics estimation device including: a transmission waveform restoration unit that restores a transmission signal from a reception signal obtained by receiving an optical signal using a coherent detection method; and an estimation unit that estimates an optical power distribution in a transmission path by estimating a nonlinear coefficient in a light wave propagation equation using a linear least squares method based on the restored transmission signal and the reception signal.

[0011] One aspect of the present invention is a method for estimating optical transmission characteristics, which estimates an optical power distribution in a transmission path by restoring a transmission signal from a reception signal obtained by receiving an optical signal using a coherent detection method, and estimating a nonlinear coefficient in a light wave propagation equation based on the restored transmission signal and the reception signal using a linear least squares method.

[0012] One aspect of the present invention is a program for causing a computer to restore a transmission signal from a received signal obtained by receiving an optical signal using a coherent detection method, and to estimate the optical power distribution in a transmission path by estimating a nonlinear coefficient in a light wave propagation equation using the linear least squares method based on the restored transmission signal and the received signal. [Effects of the Invention]

[0013] According to the present invention, it is possible to estimate optical transmission characteristics with high accuracy while suppressing the calculation load with a small number of parameter settings. [Brief explanation of the drawings]

[0014] [Figure 1] FIG. 1 is a diagram (part 1) for explaining an outline of the present invention. [Figure 2] FIG. 2 is a diagram (part 2) for explaining the outline of the present invention. [Figure 3] 1 is a diagram illustrating an example of the configuration of an optical receiving device according to an embodiment of the present invention. [Figure 4] 10 is an example of a flowchart showing a flow of processing performed by the optical receiving device according to the present embodiment. [Figure 5] FIG. 10 is a diagram showing the results of a simulation for comparing a conventional method (correlation method) with the method of the present invention. DETAILED DESCRIPTION OF THE INVENTION

[0015] Hereinafter, an embodiment of the present invention will be described with reference to the drawings. First, an outline of the present invention will be described. In order to obtain the optical power distribution P(z) in the optical transmission line (optical fiber) of an optical transmission system including an optical transmitter, an optical receiver, and an optical transmission line connecting the optical transmitter and the optical receiver, it is sufficient to obtain γ'(z) in the nonlinear Schrodinger equation shown in the following equation (1), which is an equation describing the propagation of light waves in the optical transmission line. Note that γ'(z) in equation (1) is expressed by the following equation (2).

[0016]

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[0017]

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[0018] In equation (1), z represents the distance (km) of the optical transmission line, t represents time (s), and A is ∫|A(z,t)| 2 dt represents the optical field normalized to 1, and β2 is the group velocity dispersion coefficient (ps 2 / km). In equation (2), γ represents the nonlinear constant (1 / W / km), and P(z) represents the power (W) in the optical transmission line.

[0019] Here, as a method for estimating γ'(z) = γP(z) in an actual optical transmission path, there is a method for preparing a virtual transmission path (a digital twin of an actual optical transmission path, a simulation) by a first-order regular perturbation method, as shown in FIG. 1. FIG. 1 is a diagram (part 1) for explaining the outline of the present invention. The output (virtual received signal) from the virtual transmission path prepared by the first-order regular perturbation method is expressed as A d (L). Received signal A of the virtual transmission line d The parameter γ in the virtual transmission path that (L) is closest to the actual received signal A(L) (the square error is minimized). k This problem can be formulated as shown in the following equation (3).

[0020]

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[0021] The electric field waveform A(L) at the position z=L after transmission through the optical transmission line can be expressed as the following equation (4) using the first-order regular perturbation method.

[0022]

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[0023] A0(L) and A1(L) in equation (4) are calculated based on the following equations (5) and (6), respectively.

[0024]

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[0025]

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[0026] By substituting the equation for the linear regular perturbation shown in the above equation (4) into equation (3), it can be seen that this problem can be reduced to a linear least squares problem as shown in Figure 2. Figure 2 is a diagram (part 2) for explaining the outline of the present invention.

[0027] 3 is a diagram showing an example of the configuration of an optical receiving device 10 according to this embodiment. The optical receiving device 10 receives a transmission signal transmitted from an optical transmitting device connected via an optical transmission path. The optical receiving device 10 includes a coherent receiver 11, a chromatic dispersion compensator 12, an adaptive equalizer 13, a frequency offset compensator 14, a carrier phase noise compensator 15, and a transmission characteristics estimator 16.

[0028] The coherent receiver 11 is connected to the optical transmission line and receives and coherently detects the optical signal transmitted through the optical transmission line. The coherent receiver 11 separates the received optical signal into X-polarized and Y-polarized waves. The coherent receiver 11 detects the I- and Q-components of each of the X- and Y-polarized waves by causing interference between each of the X- and Y-polarized optical signals after the separation and a laser beam emitted from a local oscillator light source provided inside the coherent receiver 11. The coherent receiver 11 converts each of the I- and Q-component optical signals of each of the X- and Y-polarized waves into four analog electrical signals, and then converts the four analog signals into four digital signals using four internal analog-to-digital converters and outputs them. Hereinafter, the four digital signals output by the coherent receiver 11 will be referred to as received signals.

[0029] The chromatic dispersion compensator 12 estimates the chromatic dispersion experienced in the optical transmission line, compensates for the estimated chromatic dispersion in the received signal output from the coherent receiver 11 , and outputs the result to the adaptive equalizer 13 .

[0030] The adaptive equalizer 13 is a functional unit that compensates for distortions that occur in the waveform of the optical signal in the optical transmission path using the received signal output from the chromatic dispersion compensator 12. In other words, the adaptive equalizer 13 is a functional unit that corrects code errors that occur in the optical signal due to inter-symbol interference in the optical transmission path. The adaptive equalizer 13 performs adaptive equalization processing using an FIR (Finite Impulse Response) filter (Finite Impulse Response filter) in accordance with set tap coefficients.

[0031] The frequency offset compensator 14 performs processing to compensate for the frequency offset of the received signal that has been subjected to adaptive equalization processing.

[0032] The carrier phase noise compensator 15 performs a process of compensating for the phase offset of the received signal, whose frequency offset has been compensated for.

[0033] The transmission characteristic estimation unit 16 estimates the optical power distribution (optical transmission characteristics) of the optical transmission path. The transmission characteristic estimation unit 16 includes a chromatic dispersion adding unit 161, a decoding unit 162, a transmission waveform restoration unit 163, a linear solution estimation unit 164, a perturbation term estimation unit 165, a matrix calculation unit 166, and an estimation unit 167. The transmission characteristic estimation unit 16 is one aspect of an optical transmission characteristic estimation device.

[0034] The chromatic dispersion adding unit 161 estimates the chromatic dispersion experienced in the optical transmission path and adds the estimated chromatic dispersion to the received signal output from the carrier phase noise compensating unit 15. As a result, the chromatic dispersion adding unit 161 generates a signal from which polarization separation, frequency offset, phase noise, etc. have been removed from the received signal received by the coherent receiver 11. In other words, the chromatic dispersion adding unit 161 generates a signal from which polarization separation, frequency offset, phase noise, etc. have been removed from the received signal obtained by receiving an optical signal using coherent detection. The received signal generated by the chromatic dispersion adding unit 161 is a pre-chromatic dispersion compensated signal from which polarization separation, frequency offset, phase noise, etc. have been removed. Hereinafter, the received signal generated by the chromatic dispersion adding unit 161 will be referred to as received signal A[L].

[0035] The decoding unit 162 decodes the received signal output from the carrier phase noise compensation unit 15 .

[0036] The transmission waveform restoration unit 163 restores the waveform of the transmission signal transmitted by the optical transmitting device based on the received signal decoded by the decoding unit 162. The waveform of the transmission signal restored by the transmission waveform restoration unit 163 is the waveform of the transmission signal input to a virtual transmission path represented by a digital twin (linear regular perturbation method) of the optical transmission path shown in Fig. 1. Hereinafter, the waveform of the transmission signal restored by the transmission waveform restoration unit 163 will be referred to as transmission signal A[0].

[0037] The linear solution estimation unit 164 estimates a linear solution that is affected only by chromatic dispersion (linear phenomenon) from the transmission signal A[0] restored by the transmission waveform restoration unit 163. Hereinafter, the linear solution estimated by the linear solution estimation unit 164 will be referred to as a linear solution A0[L].

[0038] The perturbation term estimation unit 165 receives as input the received signal A[L] generated by the chromatic dispersion adding unit 161 and the linear solution A0[L] estimated by the linear solution estimation unit 164. The perturbation term estimation unit 165 estimates the perturbation term A1[L] by removing the component of the linear solution A0[L] from the received signal A[L]. In this way, the perturbation term estimation unit 165 estimates the perturbation term A1[L] by subtracting the linear solution A0[L] of the light wave propagation equation from the received signal A[L].

[0039] The matrix calculation unit 166 calculates a matrix G that represents the characteristics of the virtual transmission channel based on the transmission signal A[0] restored by the transmission waveform restoration unit 163.

[0040] The estimation unit 167 estimates the nonlinear coefficient γ' of the optical wave propagation equation based on the perturbation term A1[L] estimated by the perturbation term estimation unit 165 and the matrix G calculated by the matrix calculation unit 166. The estimation unit 167 estimates the optical power distribution in the optical transmission path using the nonlinear coefficient γ' in the estimated optical wave propagation equation.

[0041] FIG. 4 is an example of a flowchart showing the flow of processing performed by the optical receiving device 10 in this embodiment. The coherent receiver 11 of the optical receiving device 10 receives an optical signal transmitted from the optical transmitting device (step S101). The coherent receiver 11 outputs the received signal to the chromatic dispersion compensator 12. The chromatic dispersion compensator 12 performs chromatic dispersion compensation on the received signal output from the coherent receiver 11 (step S102). The chromatic dispersion compensator 12 outputs the received signal after wavelength separation processing to the adaptive equalizer 13.

[0042] The adaptive equalization unit 13 performs adaptive equalization processing to compensate for distortion that occurs in the waveform of the received signal after wavelength division processing output from the chromatic dispersion compensator 12 (step S103). The adaptive equalization unit 13 outputs the received signal after adaptive equalization processing to the frequency offset compensator 14. The frequency offset compensator 14 performs frequency offset compensation processing to compensate for a frequency offset on the received signal after adaptive equalization processing output from the adaptive equalizer 13 (step S104). The frequency offset compensator 14 outputs the received signal after frequency offset compensation processing to the carrier phase noise compensator 15.

[0043] The carrier phase noise compensator 15 performs carrier phase compensation processing to compensate for the phase offset on the received signal after the frequency offset compensation processing output from the frequency offset compensator 14 (step S105). The carrier phase noise compensator 15 outputs the received signal after the carrier phase compensation processing to the chromatic dispersion adding unit 161 and the decoding unit 162.

[0044] The chromatic dispersion adding unit 161 adds chromatic dispersion to the received signal after carrier phase compensation processing output from the carrier phase noise compensating unit 15 (step S106). As a result, the chromatic dispersion adding unit 161 generates a received signal A[L]. The chromatic dispersion adding unit 161 outputs the generated received signal A[L] to the perturbation term estimating unit 165. The decoding unit 162 decodes the received signal after carrier phase compensation processing output from the carrier phase noise compensating unit 15 (step S107). The decoding unit 162 outputs the decoded received signal to the transmission waveform restoring unit 163.

[0045] The transmission waveform restoration unit 163 restores the waveform of the transmission signal transmitted by the optical transmitting device based on the reception signal decoded by the decoding unit 162 (step S108). The transmission waveform restoration unit 163 outputs the transmission signal A[0] indicated by the restored waveform to the linear solution estimation unit 164 and the matrix calculation unit 166. The linear solution estimation unit 164 uses the transmission signal A[0] restored by the transmission waveform restoration unit 163 to estimate a linear solution A0[L] that has been affected only by chromatic dispersion (linear phenomenon) based on the following equation (7) (step S109). The linear solution A0[L] obtained by the linear solution estimation unit 164 is a reception waveform that has been affected only by chromatic dispersion (linear phenomenon) without being affected by nonlinearity.

[0046]

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[0047] The linear solution estimation unit 164 outputs the estimated linear solution A0[L] to the perturbation term estimation unit 165. The perturbation term estimation unit 165 estimates the perturbation term A1[L] based on the following equation (8) using the received signal A[L] output from the chromatic dispersion adding unit 161 and the linear solution A0[L] output from the linear solution estimation unit 164 (step S110). The perturbation term estimation unit 165 performs a process of removing the component of the linear solution A0[L] from the received signal A[L]. The purpose of this is to remove E[||A-cA0|| 2 ] where c represents a complex number.

[0048]

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[0049] The perturbation term estimation unit 165 outputs the estimated perturbation term A1[L] to the estimation unit 167. The matrix calculation unit 166 uses the transmission signal A[0] restored by the transmission waveform restoration unit 163 to calculate a matrix G representing the characteristics of the virtual transmission channel based on the following equation (9) (step S111).

[0050]

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[0051] The matrix calculation unit 166 outputs the calculated matrix G to the estimation unit 167. The estimation unit 167 estimates a nonlinear coefficient γ' of the optical wave propagation equation based on the perturbation term A1[L] output from the perturbation term estimation unit 165 and the matrix G output from the matrix calculation unit 166. The estimation unit 167 uses the estimated nonlinear coefficient γ' to estimate the optical power distribution P(z) in the optical transmission line based on the above equation (2) (step S112). That is, the estimation unit 167 estimates the optical power distribution in the transmission line by estimating the nonlinear coefficient γ' of the optical wave propagation equation by the linear least squares method.

[0052] (Simulation results) A simulation was carried out under the following conditions to compare the conventional method (correlation method) with the method of the present invention. Modulation: Probabilistically shaped 64QAM 130GBd Root raised cosine: rolloff 0.1 Transmission line model: Split-step Fourier method Fiber loss: a=0.20(dB / km) Fiber dispersion: β2 = -21.7 (ps 2 / km) Fiber nonlinearity coefficient: g = 1.30 (W -1 km -1 ) Optical amplifier noise figure: NF=5.0(dB) Measurement particle size: 0.5(km) 50km x 3 spans

[0053] Figure 5 shows the results of a simulation comparing the conventional method (correlation method) with the method of the present invention. In the example shown in Figure 5, optical power was attenuated at the 75 km point to simulate anomalous loss in the optical transmission line. Referring to Figure 5, it can be seen that the method of the present invention is relatively consistent with the theory. As such, the method of the present invention can detect anomalous loss with high spatial resolution. On the other hand, it can be seen that the conventional method can only detect it imprecisely.

[0054] The optical receiving device 10 configured as described above includes a transmission waveform restoration unit that restores a transmission signal from a received signal obtained by receiving an optical signal using a coherent detection method, and an estimation unit that estimates the optical power distribution in the transmission path by estimating the nonlinear coefficient in the optical wave propagation equation using the linear least squares method based on the restored transmission signal and the received signal. This makes it possible to estimate optical transmission characteristics with high accuracy while reducing the calculation load and setting fewer parameters.

[0055] (Application example of the present invention) This invention can be applied to the estimation of various optical transmission path characteristics. By performing this power distribution estimation on optical signals of various wavelengths, it becomes possible to estimate the gain spectrum of an optical amplifier and the power spectrum at any position on an optical fiber. Furthermore, by acquiring the optical power distribution for both X polarization and Y polarization, it becomes possible to estimate the amount and position of PDL (Polarization Dependent Loss).

[0056] (Variation 1) In the above-described embodiments, the propagation of light waves in an optical transmission line may be calculated using a different model instead of the light wave propagation equation. For example, in the above-described embodiments, a model based on the nonlinear Schrodinger equation is used, but the present invention is not limited to this. Any model that can represent propagation in an optical transmission line may be used. For example, the Manakov PMD (Polarization Mode Dispersion) equation may be used as a model for calculating the propagation of light waves in an optical transmission line.

[0057] (Variation 2) In the above-described embodiment, Δz k In order to improve the spatial resolution at a specific location, Δz k can be made finer.

[0058] (Variation 3) In the above-described embodiment, γ'(z) is estimated so as to minimize the squared error between the received signal obtained by coherent detection and the virtual received signal obtained by propagating the transmitted signal through the virtual transmission path. Conversely, γ'(z) may be estimated so as to minimize the squared error between the transmitted signal and the signal obtained by backpropagating the received signal obtained by coherently detecting it through the virtual transmission path.

[0059] (Variation 4) The transmission characteristics estimation unit 16 does not have to be provided in the optical receiving device 10. In this case, the transmission characteristics estimation unit 16 is configured as one transmission characteristics estimation device. The transmission characteristics estimation device receives the received signal after carrier phase compensation processing from the optical receiving device 10. The transmission characteristics estimation device outputs the received signal after carrier phase compensation processing to the chromatic dispersion adding unit 161 and the decoding unit 162. Subsequent processing is the same as the processing shown in the above-mentioned embodiment (for example, processing after step S106).

[0060] (Variation 5) The transmission characteristic estimation unit 16 estimates the received signal A of the virtual transmission path as explained with reference to FIG. d The parameter γ in the virtual transmission path that (L) is closest to the actual received signal A(L) (the square error is minimized). k In this configuration, the transmission characteristic estimator 16 may estimate the optical power distribution in the transmission path by calculating the pseudo received signal (A d The optical power distribution in the transmission path is estimated by estimating the nonlinear coefficients in the light wave propagation equation using the linear least squares method using the received signal (A(L)) and the received signal (A(L)).

[0061] Some or all of the functional units of the optical receiving device 10 described above are realized as software by a processor such as a CPU (Central Processing Unit) executing a program stored in a storage device having a non-volatile storage medium (non-transitory storage medium) and a storage unit. The program may be recorded on a computer-readable non-transitory storage medium. Examples of computer-readable non-transitory storage media include portable media such as flexible disks, magneto-optical disks, ROMs (Read Only Memory), and CD-ROMs (Compact Disc Read Only Memory), and storage devices such as hard disks built into computer systems.

[0062] Some or all of the functional units of the optical receiving device 10 described above may be realized using hardware including an electronic circuit (electronic circuit or circuitry) using, for example, an LSI (Large Scale Integrated circuit), an ASIC (Application Specific Integrated Circuit), a PLD (Programmable Logic Device), or an FPGA (Field Programmable Gate Array).

[0063] Although an embodiment of the present invention has been described above in detail with reference to the drawings, the specific configuration is not limited to this embodiment, and includes designs within the scope of the gist of the present invention. [Industrial Applicability]

[0064] The present invention can be applied to a technique for estimating transmission characteristics in a digital coherent optical transmission system. [Explanation of symbols]

[0065] 10...optical receiving device, 11...coherent receiver, 12...chromatic dispersion compensation unit, 13...adaptive equalization unit, 14...frequency offset compensation unit, 15...carrier phase noise compensation unit, 16...transmission characteristic estimation unit, 161...chromatic dispersion addition unit, 162...decoding unit, 163...transmission waveform restoration unit, 164...linear solution estimation unit, 165...perturbation term estimation unit, 166...matrix calculation unit, 167...estimation unit

Claims

1. a transmission waveform restoration unit that restores a transmission signal from a received signal obtained by receiving an optical signal using a coherent detection method; an estimation unit that estimates an optical power distribution in a transmission path by estimating a nonlinear coefficient in a light wave propagation equation by a linear least squares method based on the restored transmission signal and the received signal; Equipped with a linear solution estimation unit that estimates a linear solution affected by chromatic dispersion using the restored transmission signal; a perturbation term estimation unit that estimates a perturbation term by subtracting the linear solution from a signal based on the received signal; a matrix calculation unit that calculates a matrix representing characteristics of a virtual transmission channel using the restored transmission signal, the estimation unit estimates a nonlinear coefficient in the propagation equation of the light wave by a linear least squares method using the perturbation term estimated by the perturbation term estimation unit and the matrix calculated by the matrix calculation unit, thereby estimating an optical power distribution in the transmission path.

2. the estimation unit uses an approximate solution using a perturbation method as a numerical solution of the light wave propagation equation. The optical transmission characteristics estimation device according to claim 1 .

3. a chromatic dispersion adding unit that estimates chromatic dispersion received in an optical transmission line and adds the estimated chromatic dispersion to the received signal to generate the signal; 3. The optical transmission characteristics estimation device according to claim 1 or 2.

4. receiving an optical signal by a coherent detection method and restoring the transmitted signal from the received signal; Estimating a linear solution subjected to chromatic dispersion using the restored transmitted signal; estimating a perturbation term by subtracting the linear solution from a signal based on the received signal; Calculating a matrix representing characteristics of a virtual transmission channel using the restored transmission signal; An optical transmission characteristic estimation method for estimating an optical power distribution in a transmission path by estimating a nonlinear coefficient in a light wave propagation equation by a linear least squares method using the estimated perturbation term and the calculated matrix.

5. On the computer, receiving an optical signal by a coherent detection method and restoring a transmission signal from the received signal; estimating a linear solution subjected to chromatic dispersion using the restored transmitted signal; estimating a perturbation term by subtracting the linear solution from a signal based on the received signal; Calculating a matrix representing characteristics of a virtual transmission channel using the restored transmission signal; A program for estimating the optical power distribution in a transmission path by estimating the nonlinear coefficient in the optical wave propagation equation by the linear least squares method using the estimated perturbation term and the calculated matrix.

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