CN111654315B - Co-existing wireless communication system based on interference elimination - Google Patents

Co-existing wireless communication system based on interference elimination Download PDF

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CN111654315B
CN111654315B CN202010483358.XA CN202010483358A CN111654315B CN 111654315 B CN111654315 B CN 111654315B CN 202010483358 A CN202010483358 A CN 202010483358A CN 111654315 B CN111654315 B CN 111654315B
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CN111654315A (en
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梁应敞
周标
龙睿哲
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University of Electronic Science and Technology of China
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    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04B—TRANSMISSION
    • H04B7/00—Radio transmission systems, i.e. using radiation field
    • H04B7/02—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas
    • H04B7/022—Site diversity; Macro-diversity
    • H04B7/026—Co-operative diversity, e.g. using fixed or mobile stations as relays
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L25/00—Baseband systems
    • H04L25/02—Details ; arrangements for supplying electrical power along data transmission lines
    • H04L25/0202—Channel estimation
    • H04L25/0224—Channel estimation using sounding signals
    • H04L25/0228—Channel estimation using sounding signals with direct estimation from sounding signals
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L25/00—Baseband systems
    • H04L25/02—Details ; arrangements for supplying electrical power along data transmission lines
    • H04L25/03—Shaping networks in transmitter or receiver, e.g. adaptive shaping networks
    • H04L25/03006—Arrangements for removing intersymbol interference
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02D—CLIMATE CHANGE MITIGATION TECHNOLOGIES IN INFORMATION AND COMMUNICATION TECHNOLOGIES [ICT], I.E. INFORMATION AND COMMUNICATION TECHNOLOGIES AIMING AT THE REDUCTION OF THEIR OWN ENERGY USE
    • Y02D30/00—Reducing energy consumption in communication networks
    • Y02D30/70—Reducing energy consumption in communication networks in wireless communication networks

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Abstract

The invention belongs to the technical field of wireless communication, and particularly relates to a symbiotic wireless communication system based on interference elimination. The invention designs the information transmission format of the reflecting equipment, the receiver firstly detects the symbol of the transmitter, then reconstructs the direct link signal, subtracts the reconstructed direct link signal from the received signal by utilizing the interference elimination technology, and then detects the symbol of the reflecting equipment by utilizing the Maximum Ratio Combining (MRC) method. The method has the advantages that the serial interference elimination technology is applied to the symbiotic wireless communication system, the recovery method of the reflection equipment symbols in the symbiotic wireless communication system is effectively simplified, and meanwhile, the demodulation reliability is improved.

Description

Co-existing wireless communication system based on interference elimination
Technical Field
The invention belongs to the technical field of wireless communication, and particularly relates to a symbiotic wireless communication system based on interference elimination.
Background
The technology of the internet of things is one of important technical means for realizing the future intelligent society, various devices can be connected to carry out unified management, and the daily life and work of human beings are facilitated. Because a large number of devices exist in the internet of things, the power consumption and the manufacturing cost of the devices are factors which must be considered, namely, the devices of the internet of things should meet the requirements of low power consumption and low manufacturing cost, so that a large number of devices of the internet of things can be arranged to realize the interconnection of everything.
The symbiotic wireless communication system is a communication system with low energy consumption and low manufacturing cost. In the system, three devices, namely a transmitter, a reflecting device and a receiver, exist, the transmitter transmits a wireless signal, the reflecting device changes the amplitude and the phase of the wireless signal according to self information after receiving the signal, and the receiver demodulates the signal after receiving the signal and recovers symbols transmitted by the transmitter and the reflecting device. The reflection device realizes different reflection coefficients by selecting different impedances to realize the modulation of signals. The characteristics of symbiotic wireless communication enable the symbiotic wireless communication to become an Internet of things technology with wide application prospects, and low-power-consumption communication equipment can be arranged on a large scale in the future Internet of things, so that the transmission of equipment information and the coverage of wide regions are realized.
Serial interference cancellation techniques are proposed in third generation mobile communication technology to reduce or eliminate inter-user interference in multi-user detection. The specific method is that firstly, the symbols of the user with large signal power are demodulated, then the symbols are used for reconstructing the signal of the user, the signal of the user is subtracted from the received signal, and then the symbols of other users are detected until the symbol detection of all the users is completed. By using successive interference cancellation techniques, the reliability of user symbol demodulation can be greatly improved.
Disclosure of Invention
The invention provides an interference elimination-based symbiotic wireless communication system, which is based on that two links, namely a direct link and a reflection link, exist in the symbiotic wireless communication system, and the signal intensity of the direct link is stronger than that of the reflection link, so that by utilizing the thought of a serial interference elimination technology, the direct link is demodulated and reconstructed first, the reconstructed signal is subtracted from a received signal, and when the elimination effect is ideal, the obtained signal can be regarded as a reflection link signal polluted by noise, and then the reflection link is demodulated to obtain a symbol sent by reflection equipment.
The technical scheme adopted by the invention is that an information transmission format of the reflection equipment is designed, a receiver firstly detects a transmitter symbol, then reconstructs a direct link signal, subtracts the reconstructed direct link signal from a received signal by utilizing an interference elimination technology, and then detects the symbol of the reflection equipment by utilizing a Maximum Ratio Combining (MRC) method, wherein the method specifically comprises the following steps:
a symbiotic wireless communication system comprises a transmitter, a receiver and a reflection device, wherein the transmitter, the receiver and the reflection device adopt a single antenna, a transmission signal of the transmitter and a signal of the reflection device are synchronous, namely the starting positions and the ending positions of a transmitter symbol and a reflection device symbol are the same, and 1 reflection device symbol corresponds to 1 OFDM symbol; defining a time-domain complex baseband signal transmitted by a transmitter as s cb [n]The time domain complex baseband signal received by the receiver is r cb [n]The signal generated by the reflecting device is cn]Adopting an on-off keying modulation mode, bit 1 corresponds to the reflection device in a reflection state, and bit 0 corresponds toThe reflecting device being in a non-reflecting state, the channel between the transmitter and the receiver being defined by f [ n ]]Representing a total power of 1, called direct link channel, a channel g between the transmitter and the reflecting device, a channel v n between the reflecting device and the receiver]Expressed that the total power is 1,g and v [ n ]]Together forming a reflected link channel;
signal transmission:
the time domain expression of the ith OFDM symbol generated by the transmitter is:
Figure BDA0002518176550000021
wherein N represents the number of subcarriers, N cp Denotes the cyclic prefix length, s i [k]Is a symbol on the k subcarrier in the ith OFDM symbol, N f The number of OFDM symbols contained in one frame of OFDM signal is also the number of reflecting device symbols contained in one frame of reflecting device signal;
i.e. a frame of OFDM signal is represented as:
Figure BDA0002518176550000022
the OFDM signal transmitted by the transmitter reaches the receiver through a direct link channel, and the direct link signal is:
Figure BDA0002518176550000023
wherein P is d For the average power of the direct link signal received by the receiver:
Figure BDA0002518176550000024
wherein P is t Is the average transmission power of the transmitter, G t Is the antenna gain of the transmitter, G r Is the antenna gain of the receiver, λ is the wavelength of the carrier, D f Is the distance from the transmitter to the receiver，v 1 Is the path loss factor;
the reflection device firstly sends a pilot symbol 0, the receiver estimates the direct link channel according to the pilot symbol, and then the reflection device sends a pilot symbol 1 and a data symbol, and the reflected link signal is:
Figure BDA0002518176550000031
where Δ γ is the relative signal-to-noise ratio of the direct link and each reflected link:
Figure BDA0002518176550000032
wherein alpha is the reflection coefficient, G tag Is the antenna gain of the receiver, D g Is the distance from the transmitter to the reflecting device, v 2 、v 3 Is the path loss factor, F = λ 2 /(4π) 2 ；
Signal receiving:
the receiver receives the direct link signal, the reflected link signal and the noise at the same time, and the signals are as follows:
r cb [n]＝r d [n]+r b [n]+u[n]
u[n]is a circularly symmetric complex Gaussian variable with a mean of 0 and a variance of
Figure BDA0002518176550000033
The receiver first demodulates the direct link signal to recover the transmitter symbols, and then estimates the direct link signal
Figure BDA0002518176550000034
Subtracting the recovered direct link signal from the received signal yields:
Figure BDA0002518176550000035
remove r rem [n]Cyclic prefix of each symbol in the sequence to obtain
Figure BDA0002518176550000036
Performing DFT conversion on the signal to obtain the following expression:
Figure BDA0002518176550000037
wherein, V [ k ]]Is channel v [ n ]]The DFT of (d) is transformed,
Figure BDA0002518176550000038
resulting from the DFT-transform addition of the residual direct link signal and noise, which, in the ideal cancellation case,
Figure BDA0002518176550000039
is a DFT transform of noise; in the above equation, the first term on the right side of the equation, except c i The other variables can be regarded as a channel, and then symbols of each path of subcarriers can be combined by using a maximum ratio combining method, and then a reflection device symbol c is solved i ；
Estimating a composite channel consisting of a direct link channel and a reflected link channel:
Figure BDA0002518176550000041
order to
Figure BDA0002518176550000042
Using the maximum likelihood criterion is:
Figure BDA0002518176550000043
wherein A is c Is a set of values for the symbols of the reflecting device,
Figure BDA0002518176550000044
is composed of
Figure BDA0002518176550000045
The column vector of the component(s) is,
Figure BDA0002518176550000046
is formed by R rem,i [k]A column vector of components.
In the above scheme, the specific method for estimating the direct link signal is as follows:
removing the cyclic prefix of each OFDM symbol, wherein the ith OFDM symbol with the cyclic prefix removed is expressed as:
Figure BDA0002518176550000047
then, DFT conversion is carried out:
Figure BDA0002518176550000048
wherein, F [ k ]]Is channel f n]The N-point DFT transform of (1),
Figure BDA0002518176550000049
is the sum of the noise and the DFT transform of the reflected link signal.
Then using the known pilot symbols s 0 [k]Estimating the direct link channel to obtain the following expression:
Figure BDA00025181765500000410
here for convenience of illustration, division on the right of the expression is
Figure BDA00025181765500000411
This need not be done in practice.
As known from the central limit theorem, the OFDM signal can be regarded as an independent and equally distributed complex gaussian random variable sequence. Thus, using maximum likelihood detection to recover the subcarrier symbols of each OFDM data symbol, the following expression can be obtained:
Figure BDA00025181765500000412
wherein A is s Is a set of values for the sub-carrier symbols.
Reconstructing the ith OFDM symbol transmitted by the transmitter using the recovered subcarrier symbol as:
Figure BDA0002518176550000051
and a transmission signal of the transmitter:
Figure BDA0002518176550000052
for the estimated channel
Figure BDA0002518176550000053
IFFT is carried out to obtain the time domain estimation of the direct link channel as
Figure BDA0002518176550000054
The direct link signal arriving at the receiver can then be estimated as
Figure BDA0002518176550000055
The method has the advantages that the serial interference elimination technology is applied to the symbiotic wireless communication system, the recovery method of the reflection equipment symbols in the symbiotic wireless communication system is effectively simplified, and meanwhile, the demodulation reliability is improved.
Drawings
FIG. 1 shows a system architecture diagram of the present invention;
FIG. 2 shows a design of a reflected link signal transmission format in the present invention;
FIG. 3 shows the bit error rate performance of a transmitter symbol for receivers designed in accordance with the present invention at different relative SNR;
fig. 4 shows the error rate performance of the symbol of the reflecting device under different relative snr conditions for the receiver designed by the present invention.
Detailed Description
The technical scheme of the invention is further described in detail by combining the drawings as follows:
the invention considers the situation that a single reflection device, a transmitter, a receiver and the reflection device adopt a single antenna, and the signal of the transmitter and the signal of the reflection device are synchronous, namely the starting position and the ending position of the symbol of the transmitter and the symbol of the reflection device are the same. The OFDM signal has a plurality of subcarriers corresponding to a plurality of subcarrier channels, each subcarrier corresponds to a reflection device symbol in a symbiotic wireless communication system, each subcarrier corresponding to the same reflection device symbol is regarded as a path of signal at the moment, and the reflection device symbol can be detected by using a maximum ratio combining algorithm. For the case that the signal of the transmitter is a single carrier signal, multiple antennas can be used at the receiving end to transmit one reflective device symbol on multiple channels, and then the maximum ratio combining (IC-MRC) algorithm based on interference cancellation proposed by the present invention is used to detect the transmitter symbol and the reflective device symbol.
Suppose that the transmitter transmits an OFDM signal s cb [n]The time domain signal received by the receiver is r cb [n]The signal generated by the reflecting device is cn]An on-off keying (OOK) modulation scheme is used, where bit 1 corresponds to the reflective state of the reflective device and bit 0 corresponds to the non-reflective state of the reflective device. The reflecting device is located in close proximity to the transmitter to ensure that it can be activated. Channel between transmitter and receiver using f [ n ]]Indicating a total power of 1, called the direct link channel. The channel between the transmitter and the reflecting device is g, the channel between the reflecting device and the receiver is v n]Expressed, the total power is 1,g and v [ n ]]Together forming a reflected link channel. The whole system structure is shown in fig. 1.
Upper channelIs the modeling of small scale fading, followed by a discussion of large scale fading of the channel and the received power of the signal. The large scale fading and signal received power of the direct link will be described first. Suppose the average transmission power of the transmitter is P t The average power of the direct link signal received by the receiver is P d From the free space path loss model, the following expression can be derived:
Figure BDA0002518176550000061
wherein, G t Is the antenna gain of the transmitter, G r Is the antenna gain of the receiver, λ is the wavelength of the carrier, D f Is the distance from the transmitter to the receiver, v 1 Is the path loss factor. To simplify the description of the above equation, let F = λ 2 /(4π) 2 Then the above formula becomes
Figure BDA0002518176550000062
The large scale fading and signal received power of the reflected link are next described. The reflection link consists of two paths, transmitter to reflection device and reflection device to receiver. First, the first path is discussed, assuming that the average power of the signal at the reflecting device is P g Then there is
Figure BDA0002518176550000063
Wherein G is tag Is the antenna gain of the receiver, D g Is the distance from the transmitter to the reflecting device, v 2 Is the path loss factor.
Next, the second path is discussed, assuming that the reflection coefficient is α, the average power of the signal reflected by the reflecting device is
P tag ＝|α| 2 P g (4)
The signal then reaches the receiver via a channel v, at a power of the signal
Figure BDA0002518176550000071
Wherein D is v Is the distance, v, of the reflecting device from the receiver 3 Is the path loss factor.
By combining the formulas (3), (4) and (5), the average power of the reflected link signal can be obtained to be
Figure BDA0002518176550000072
The signal received by the receiver is expressed as
r cb [n]＝r d [n]+r b [n]+u[n] (7)
Wherein r is d [n]For direct link signals to reach the receiver, r b [n]For reflected link signals arriving at the receiver, u n]Is a circularly symmetric complex Gaussian variable with a mean of 0 and a variance of
Figure BDA0002518176550000073
The expression for the direct link signal is
Figure BDA0002518176550000074
The expression for the reflected link signal is
Figure BDA0002518176550000075
Where c n is the signal of the reflecting device.
Setting the noise variance to 1, the direct link signal-to-noise ratio can be defined as
Figure BDA0002518176550000076
Accordingly, define the reflected link signal-to-noise ratio as
Figure BDA0002518176550000077
Then defining the relative signal-to-noise ratio of the direct link and the reflected link as
Figure BDA0002518176550000078
Equation (9) can be written as:
Figure BDA0002518176550000079
because the signal power of the reflection link is weaker than that of the direct link signal, the receiver estimates the direct link channel by using the pilot symbols, then regards the reflection link as interference and detects the transmitter symbols, and then reconstructs the direct link signal to obtain
Figure BDA0002518176550000081
Subtracting this signal from the received signal yields the following expression
Figure BDA0002518176550000082
Under the condition of ideal interference elimination effect, r obtained in the last step rem [n]The signal is a reflected link signal polluted by noise, and at this time, the detection method of maximum ratio combining can be utilized to combine each subcarrier, and then the symbol of the reflection device is recovered.
In order to ensure that the receiver can estimate the direct link channel and the reflected link channel, the transmission format of the reflected link signal needs to be designed. The reflecting device first transmits a pilot symbol 0, the receiver can estimate the direct link channel, and then the reflecting device transmits a pilot symbol 1 and a data symbol. After the interference is eliminated, the channel estimation of the reflection link can be obtained by estimating the channel by using two pilot symbols in the received signal, and then the receiver detects the reflection device symbol by using a maximum ratio combining method. The transmission format of the reflected link signal is shown in fig. 2.
Fig. 3 shows the error rate performance of the transmitter symbols at different relative snrs for the receiver designed by the present invention. Fig. 4 shows the error rate performance of the symbol of the reflecting device under the condition of different relative signal-to-noise ratios of the receiver designed by the invention. In simulation, the number of OFDM subcarriers N =64, and the cyclic prefix length N cp =16, subcarrier symbol is 4QAM symbol. The number of symbols in a frame of the reflecting device is 10 and the number of symbols in an ofdm frame is also 10. Channel f [ n ]]And v [ n ]]The channels are independent 8 paths of equal power channels, the total power is 1, and the fading of each path of channel meets the complex Gaussian distribution. Channel g is a rayleigh fading channel, i.e. satisfies a complex gaussian distribution with a mean of 0 and a variance of 1. Channel realization times of 10 6 The noise power is
Figure BDA0002518176550000083
In fig. 3, the OFDM labeled curve shows the bit error rate curve for the receiver using linear least squares channel estimation and maximum likelihood detection for direct link only. When the reflected link signal power is less than the noise, i.e. gamma d Less than- (. DELTA.. Gamma.) dB The bit error rate curve of the transmitter symbols is very close to the bit error rate curve of the OFDM system. When the reflected link signal power is greater than the noise, i.e. gamma d Greater than- (. DELTA.. Gamma.) dB And when the error rate curve of the transmitter symbol is reduced, the reduction speed gradually approaches to an error platform, and the error rate corresponding to the error platform is reduced by one order of magnitude along with the reduction of the relative signal-to-noise ratio by 10dB. From the analysis, it is known that the error rate performance of the transmitter symbols is acceptable.
In fig. 4, the curve marked ED represents the error rate of the symbol of the reflecting device when the receiver uses energy detection directly on the received signal. When the delta gamma is a fixed value, the performance of ED is poor, the curve is reduced slowly, the performance of the IC-MRC algorithm is better than that of ED,signal to noise ratio gamma on direct link d Greater than- (. DELTA.. Gamma.) dB The curve dropped significantly. It can also be observed that as Δ γ decreases by 10dB, the performance of the IC-MRC algorithm also decreases by 10dB. Simulation results show that the performance of the IC-MRC algorithm is superior to that of ED.

Claims (1)

1. A symbiotic wireless communication system based on interference cancellation is characterized by comprising a transmitter, a receiver and a reflection device, wherein the transmitter, the receiver and the reflection device adopt a single antenna, a transmission signal of the transmitter and a signal of the reflection device are synchronous, namely the starting positions and the ending positions of a transmitter symbol and a reflection device symbol are the same, and 1 reflection device symbol corresponds to 1 OFDM symbol; defining the OFDM signal transmitted by the transmitter as s cb [n]The time domain signal received by the receiver is r cb [n]The signal generated by the reflecting device is cn]Adopting on-off keying modulation mode, bit 1 corresponding to reflection device being in reflection state, bit 0 corresponding to reflection device being in non-reflection state, using f [ n ] for channel between transmitter and receiver]Representing a total power of 1, called direct link channel, a channel between the transmitter and the reflecting device g, and a channel between the reflecting device and the receiver v n]Total power of 1,g and v [ n ]]Together forming a reflected link channel;
signal transmission:
the time domain expression of the ith OFDM symbol generated by the transmitter is:
Figure FDA0003869390590000011
wherein N represents the number of subcarriers, N cp Denotes the cyclic prefix length, s i [k]Is a symbol on the k subcarrier in the ith OFDM symbol, N f The number of OFDM symbols contained in one frame of OFDM signal is also the number of reflecting device symbols contained in one frame of reflecting device signal;
one frame OFDM signal is represented as:
Figure FDA0003869390590000012
the OFDM signal transmitted by the transmitter reaches the receiver through a direct link channel, and the direct link signal is:
Figure FDA0003869390590000013
wherein P is d For the average power of the direct link signal received by the receiver:
Figure FDA0003869390590000014
wherein P is t Is the average transmission power of the transmitter, G t Is the antenna gain of the transmitter, G r Is the antenna gain of the receiver in the direct link, λ is the wavelength of the carrier, D f Is the distance from the transmitter to the receiver, v 1 Is the path loss factor;
the reflecting device firstly sends a pilot symbol 0, the receiver estimates a direct link channel according to the pilot symbol, then the reflecting device sends a pilot symbol 1 and a data symbol, and the reflected link signal is:
Figure FDA0003869390590000021
where Δ γ is the relative signal-to-noise ratio of the direct link and the reflected link:
Figure FDA0003869390590000022
wherein alpha is the reflection coefficient, G tag Is the antenna gain of the receiver in the reflection link, D g Is the distance from the transmitter to the reflecting device, D v Is the distance, v, of the reflecting device from the receiver 2 、v 3 Is a cause of path lossSub, F = λ 2 /(4π) 2 ；
Signal receiving:
the receiver receives the direct link signal, the reflected link signal and the noise at the same time, and the signals are as follows:
r cb [n]＝r d [n]+r b [n]+u[n]
wherein u [ n ]]Is a circularly symmetric complex Gaussian variable with a mean of 0 and a variance of
Figure FDA0003869390590000023
The receiver first demodulates the direct link signal to recover the transmitter symbols, and then estimates the direct link signal
Figure FDA0003869390590000024
The method specifically comprises the following steps:
removing the cyclic prefix of each OFDM symbol, wherein the ith OFDM symbol with the cyclic prefix removed is expressed as:
Figure FDA0003869390590000025
then, DFT conversion is carried out:
Figure FDA0003869390590000026
wherein, F [ k ]]Is channel f n]The N-point DFT of (1) is transformed,
Figure FDA0003869390590000027
is the sum of the noise and the DFT transform of the reflected link signal;
then using the known pilot symbols s 0 [k]Estimating the direct link channel to obtain the following expression:
Figure FDA0003869390590000031
recovering the subcarrier symbols of each OFDM data symbol using maximum likelihood detection, resulting in the following expression:
Figure FDA0003869390590000032
wherein A is s A value set of the subcarrier symbols;
reconstructing the ith OFDM symbol transmitted by the transmitter using the recovered subcarrier symbol as:
Figure FDA0003869390590000033
and a transmission signal of the transmitter:
Figure FDA0003869390590000034
for the estimated channel
Figure FDA0003869390590000035
IFFT is carried out to obtain the time domain estimation of the direct link channel as
Figure FDA0003869390590000036
The direct link signal arriving at the receiver is then estimated as
Figure FDA0003869390590000037
Subtracting the recovered direct link signal from the received signal yields:
Figure FDA0003869390590000038
remove r rem [n]Cyclic prefix of each symbol in the sequence to obtain
Figure FDA0003869390590000039
Performing DFT conversion on the signal to obtain the following expression:
Figure FDA00038693905900000310
wherein, V [ k ]]Is a channel v [ n ]]The DFT of the signal is transformed,
Figure FDA00038693905900000311
is a DFT transform of noise;
estimating a composite channel consisting of a direct link channel and a reflected link channel:
Figure FDA0003869390590000041
order to
Figure FDA0003869390590000042
Using the maximum likelihood criterion is:
Figure FDA0003869390590000043
wherein, A c Is a set of values for the symbols of the reflecting device,
Figure FDA0003869390590000044
is formed by
Figure FDA0003869390590000045
The column vector of the component is composed of,
Figure FDA0003869390590000046
is formed by R rem,i [k]A column vector of components.
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