CN109525984B - Method for improving safety rate of untrusted relay energy-carrying communication system - Google Patents

Method for improving safety rate of untrusted relay energy-carrying communication system Download PDF

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CN109525984B
CN109525984B CN201811203398.3A CN201811203398A CN109525984B CN 109525984 B CN109525984 B CN 109525984B CN 201811203398 A CN201811203398 A CN 201811203398A CN 109525984 B CN109525984 B CN 109525984B
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CN109525984A (en
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张沉思
贾凡
葛建华
岳安军
宫丰奎
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Xidian University
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    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04W—WIRELESS COMMUNICATION NETWORKS
    • H04W12/00—Security arrangements; Authentication; Protecting privacy or anonymity
    • H04W12/02—Protecting privacy or anonymity, e.g. protecting personally identifiable information [PII]
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04B—TRANSMISSION
    • H04B17/00—Monitoring; Testing
    • H04B17/30—Monitoring; Testing of propagation channels
    • H04B17/309—Measuring or estimating channel quality parameters
    • H04B17/336—Signal-to-interference ratio [SIR] or carrier-to-interference ratio [CIR]
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04B—TRANSMISSION
    • H04B17/00—Monitoring; Testing
    • H04B17/30—Monitoring; Testing of propagation channels
    • H04B17/382—Monitoring; Testing of propagation channels for resource allocation, admission control or handover
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04B—TRANSMISSION
    • H04B17/00—Monitoring; Testing
    • H04B17/30—Monitoring; Testing of propagation channels
    • H04B17/391—Modelling the propagation channel
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04W—WIRELESS COMMUNICATION NETWORKS
    • H04W40/00—Communication routing or communication path finding
    • H04W40/02—Communication route or path selection, e.g. power-based or shortest path routing
    • H04W40/04—Communication route or path selection, e.g. power-based or shortest path routing based on wireless node resources
    • H04W40/10—Communication route or path selection, e.g. power-based or shortest path routing based on wireless node resources based on available power or energy
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04W—WIRELESS COMMUNICATION NETWORKS
    • H04W40/00—Communication routing or communication path finding
    • H04W40/02—Communication route or path selection, e.g. power-based or shortest path routing
    • H04W40/22—Communication route or path selection, e.g. power-based or shortest path routing using selective relaying for reaching a BTS [Base Transceiver Station] or an access point

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Abstract

The invention belongs to the technical field of information security, and discloses a method for improving the security rate of an untrusted relay energy-carrying communication system, which comprises the following steps: constructing a model of a communication process of a two-hop bidirectional relay network; calculating the signal relayed by the first time slot relay and the energy collected by the first time slot relay; calculating the signals received by the two sources in the second time slot and their signal-to-noise ratios and channel capacities; calculating instantaneous signal-to-noise ratios and corresponding channel capacities obtained by the two sources when the relay is intercepted; calculating the safe rates of the channels from the two sources to the relay; and optimizing the safe rate, and finding out the optimal transmitting power of the A source and the B source and the proportional coefficient rho of the energy collected by the relay. The invention is superior to equal power distribution, can optimize the reliability of the safe rate, and is suitable for the untrusted selfish relay.

Description

Method for improving safety rate of untrusted relay energy-carrying communication system
Technical Field
The invention belongs to the technical field of information security, and particularly relates to a method for improving the security rate of an untrusted relay energy-carrying communication system.
Background
Currently, the current state of the art commonly used in the industry is such that: relay-assisted cooperative transmission has become a research hotspot in recent years due to the ability to achieve graded gain and extend the coverage of wireless networks. However, in general studies it is assumed that relays are authentic. In practice, due to the complexity of the network structure and the additive interference in the transmission, the relay is less secure than the signal source. The relay may actually be an eavesdropper or a malicious relay. Thus, certain hidden trouble is brought to the safety communication. It is necessary to improve the physical layer security in eavesdropping relay. Under the condition of a certain total power, how to find an optimal power distribution scheme makes the safety rate and the performance of the system optimal becomes the current research direction. In the prior art, the boundary of the security rate is provided in the documents of "X.He and A.Yener", "Two-hop security communication using an untrusted relay", "EURASIP J.Wireless communication, Netw., vol.2009, pp.1-13,2009". In addition, the authors derive an upper bound on the privacy rate when the relay is not trusted. And illustrates that an amplify-and-forward (AF) relay can achieve a non-zero secret rate but higher than a decode-and-forward relay (DF). But the energy generated by the proposed relay is generated by itself. In fact, although a relay is willing to communicate information to a signal source, it may be unwilling to use its own power for the relay. In other words, the relay may be selfish except for the potential eavesdropper identity. It would like to use the collected energy of both source signals for relaying. Thus, harvesting Energy (EH) can extend usage time and ensure signal sustainability. In the prior art two documents of z.chen, l.x.cai, y.cheng, h.shan, "stable cooperative communication in wireless power networks with energy harnessing relay", IEEE trans.wireless communication, vol.16, No.12, pp.8175-8189, oct.2017, "authors study unidirectional relay systems and propose an optimized joint time scheduling and power allocation problem to maximize the throughput of the system. And researches the secret security rate problem of the cognitive internet of things network of the untrusted relay by harvesting Energy (EH). However, the research results are only applicable to the one-way relay system, and how to improve the security rate of the two-way relay system of the untrusted relay and the selfish relay through the adaptive power control is still not well solved.
In summary, the problems of the prior art are as follows:
(1) in the first prior art, the energy generated by the relay is generated by the relay, but the relay is not willing to use the power of the relay, except for the identity of a potential eavesdropper, the relay is self-owned, and only the collected energy of two source signals can be used for relaying; therefore, the original power allocation method is no longer applicable, and it must be considered that a part of power is used for providing energy for the relay and a part of power is used for signal reception. The power allocation factor of one relay needs to be considered.
(2) In the second prior art, a one-way relay system is considered, but a two-way system has a more complex structure than a one-way system and has more complex channel capacity and safety capacity expression, so that the technology is not suitable for the two-way system. For the bidirectional relay system, the power allocation optimization problem needs to be reconstructed.
The difficulty and significance for solving the technical problems are as follows:
since the relay uses the collected energy of the two source signals for signal forwarding, unlike the conventional relay system, the channel capacity and the safety capacity expression need to be updated, an additional power factor is added, and the power of the same source node needs to be jointly optimized.
Different from a one-way relay system, in a two-way relay system, two one-way communications are simultaneously carried out in one channel, the expression of channel capacity and safety capacity is complex, and the difficulty in solving an optimization problem is increased. The bidirectional relay can reduce 4 transmission time slots to 2 transmission time slots, and the frequency spectrum utilization rate of the system is improved.
Disclosure of Invention
Aiming at the problems in the prior art, the invention provides a method for improving the safety rate of an untrusted relay energy-carrying communication system.
The method for improving the safety rate of the untrusted relay energy-carrying communication system is used for establishing a mathematical model of a two-way relay system; calculating the signal received by the relay in the first time slot and the energy collected from the source, and obtaining the signal through the signal source of the second time slot; obtaining the signal-to-noise ratio and channel capacity of a channel relayed between two sources; and finally, calculating the signal-to-noise ratio and the channel capacity if the relay is used as an eavesdropper, and obtaining the optimal solution after comparison.
Further, the calculating the signal received by the first time slot relay and the collected energy specifically includes: the channel gains between the relay r and the sources a and b are h, respectivelyAAnd hBDenoted by the transmit powers of a and b, respectively, being PAAnd PBRepresents; in one time slot, a and b simultaneously transmit signals to the relay r, and the relay receives the signals:
Figure BDA0001830591030000031
wherein n isRIntroducing additive white Gaussian noise with the mean value of zero and the variance of N0 into the receiving relay at the r position; the signal received by the relay receiver is
Figure BDA0001830591030000032
The collected energy is ER＝ηρ(PAgA+PBgB) (ii) a Where η is the attenuation factor of the energy, ρ is the power distribution factor of the relay, gA＝|hA|2,gB＝|hB|2。
Further, the calculating signals received by the two sources in the second time slot specifically includes: after the second time slot, the relay becomes after energy collection and amplification
Figure BDA0001830591030000033
The unit power signal of the relay transmission is
Figure BDA0001830591030000034
Wherein n isBRWhite gaussian noise with mean zero and variance N0; then the signals received by the two sources a and b in the second time slot are obtained as
Figure BDA0001830591030000035
And
Figure BDA0001830591030000036
further, the calculating the signal-to-noise ratio and the channel capacity from the source to the relay specifically includes: a and b obtain their own original signals, and solve the signal-to-noise ratio of the relay r to a and b:
Figure BDA0001830591030000037
Figure BDA0001830591030000038
and (3) obtaining the channel capacity from the relay r to the a and the b:
Figure BDA0001830591030000039
Figure BDA0001830591030000041
further, the calculating the signal-to-noise ratio and the channel capacity when the relay is intercepted specifically includes: when r is an eavesdropper, it will attempt to intercept information from the information receiver. The signal-to-noise ratio of r to a and b is calculated as:
Figure BDA0001830591030000042
Figure BDA0001830591030000043
obtaining the channel capacity from r to a and b when the relay is intercepted:
Figure BDA0001830591030000044
subtracting the channel capacity when the relay is intercepted from the channel capacity when the relay is safely relayed to obtain the confidentiality rate of the one-way channel at the moment as follows:
Figure BDA0001830591030000045
Figure BDA0001830591030000046
further, the calculating the security rate of the unidirectional channel specifically includes:
Figure BDA0001830591030000047
Figure BDA0001830591030000048
finding the optimal solution to make the sum of the channel capacities of the two channels reach the maximum, and obtaining:
Figure BDA0001830591030000049
simplifying the objective function by using a high signal-to-noise ratio yields:
Figure BDA00018305910300000410
further, the secret keeping rate is split into: a function of power and a function of ρ;
(1) calculating an optimal solution of power distribution;
the objective function and constraints are:
Figure BDA0001830591030000051
s.t.0≤PA+PB≤PT；
the lagrange function for the objective function is:
Figure BDA0001830591030000052
the only optimal solution of the objective function can be obtained by the Lagrange multiplier method:
Figure BDA0001830591030000053
(2) optimal solution of relay power allocation factor:
Figure BDA0001830591030000054
s.t.0＜ρ＜1；
setting the derivative of the objective function to zero to obtain the optimal solution of Rootof (w)1ρ4+w2ρ3+w3ρ2+w4ρ+w5) And Rootof (·) represents the root of the polynomial. w is a1＝2η2gAgB-ηgA-ηgB，w2＝-4η2gAgB+5ηgA+5ηgB-2，w3＝-6ηgA-6ηgB+6，w4＝2ηgA+2ηgB-6，w5＝2。
Another object of the present invention is to provide an information security control system applying the method for increasing the security rate of an untrusted relay portable communication system.
Another object of the present invention is to provide a relay-assisted cooperative transmission system applying the method for increasing the security rate of an untrusted relay portable communication system.
Another object of the present invention is to provide an information data processing terminal to which the method for increasing the security rate of an untrusted relay portable communication system is applied.
In summary, the advantages and positive effects of the invention are: the invention considers the energy collection of the relay, and when the relay passes the energy collection, the sustainability of the signal is ensured and the realization is easy; the privacy-rate optimization problem is divided into two independent problems: transmit power allocation problems and power splitter optimization problems; the two problems are respectively solved, a closed form solution is realized, and the computer simulation is adopted to provide how to determine the position of the relay r so as to optimize the secrecy rate. Meanwhile, factors of relay unreliability and selfish are considered, and optimization of the security rate of the communication system is provided when the relay is an eavesdropper.
The invention provides a problem how to carry out power distribution and energy collection of relays to improve the safety rate of a bidirectional relay system when the relays need to acquire energy from a signal source under the condition that an untrusted relay and a selfish relay exist. The invention improves the safety rate of the bidirectional relay system under the condition of EH relay under the condition that the untrusted and selfish relay exists.
Drawings
Fig. 1 is a flowchart of a method for increasing a security rate of an untrusted relay energy-carrying communication system according to an embodiment of the present invention.
Fig. 2 is a schematic system model diagram of a method for improving the security rate of an untrusted relay energy-carrying communication system according to an embodiment of the present invention.
Fig. 3 is a flowchart of an implementation of a method for increasing a security rate of an untrusted relay energy-carrying communication system according to an embodiment of the present invention.
Fig. 4 is a schematic diagram of a hypothetical two-way relay system provided by an embodiment of the present invention;
in the figure: a and b are sources, respectively, and r is a relay.
Fig. 5 is a schematic diagram of a relationship between a relay position and a safety rate under the condition that the signal-to-noise ratio is 15dB according to the embodiment of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is further described in detail with reference to the following embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
Only the collected energy of two source signals can be used for relaying aiming at the prior art; the relay can supplement energy by using a battery or charging and the like to prolong the service life, but the cost is high and the relay is very inconvenient; the problem of energy loss of the signal is not considered; the invention provides an algorithm for improving the security rate of a physical layer under the condition of improving an untrusted selfish relay network; it can also be applied in two-way relay, so that the instantaneous safe rate and energy can reach pareto optimum.
The following detailed description of the principles of the invention is provided in connection with the accompanying drawings.
As shown in fig. 1, a method for increasing a security rate of an untrusted relay energy-carrying communication system according to an embodiment of the present invention includes the following steps:
s101: establishing a two-hop bidirectional relay network communication process model;
s102: calculating the signal relayed by the first time slot relay and the energy collected by the first time slot relay;
s103: calculating the signals received by the two sources in the second time slot and their signal-to-noise ratios and channel capacities;
s104: calculating instantaneous signal-to-noise ratio and corresponding channel capacity obtained by eavesdropping two relay sources;
s105: calculating the secrecy rate of the channels from the two sources to the relay;
s106: and optimizing the secrecy rate, and finding out the optimal transmitting power of the A source and the B source and the proportional coefficient of the energy collected by the relay.
The application of the principles of the present invention will now be described in further detail with reference to the accompanying drawings.
The communication system model established by the invention is a two-hop half-duplex relay network with three nodes. The system consists of two sources a and b and one bi-directional relay r. All nodes have a single antenna and operate in half duplex mode. R relays signals that are actively transmitted using an amplify-and-forward protocol. In all relay protocols, the amplify-and-forward (AF) is the simplest compared to decode-and-forward, i.e. the relay only needs to amplify and forward the received signal to the receiving end. The specific system diagram is shown in fig. 4.
Step one, calculating signals received by a relay in a first time slot and collected energy
Let the channel gains between the relay r and the sources a and b be h, respectivelyAAnd hBIndicating that the transmit power of a and b is denoted PA and PB, respectively. Then in one time slot, a and b send signals to relay r simultaneously, relaying the received signal:
Figure BDA0001830591030000081
wherein n isRAnd introducing additive white Gaussian noise with the mean value of zero and the variance of N0 into the receiving relay at r. Considering that the relay collects the signal from the source, the received signal is divided into two parts, the signal received by the relay receiver is
Figure BDA0001830591030000082
The energy collected is ER＝ηρ(PAgA+PBgB). Where η is the attenuation factor of the energy and ρ is the power allocation factor of the relay. gA＝|hA|2,gB＝|hB|2。
Step two, calculating signals received by two sources in a second time slot;
after the second time slot, the relay becomes after energy collection and amplification
Figure BDA0001830591030000083
Then the unit power signal of the relay transmission is
Figure BDA0001830591030000084
Wherein n isBRIs white gaussian noise with mean zero and variance N0. Then the signals received by the two sources a and b in the second time slot are obtained as
Figure BDA0001830591030000085
And
Figure BDA0001830591030000086
step three, calculating the signal-to-noise ratio and the channel capacity from the source to the relay;
assuming that both a and b are able to obtain their own original signals, the signal-to-noise ratio of the relay r to a and b can be found:
Figure BDA0001830591030000087
Figure BDA0001830591030000088
and then the capacity of the relay r to a and b channels can be obtained:
Figure BDA0001830591030000089
Figure BDA00018305910300000810
step four, calculating the signal-to-noise ratio and the channel capacity when the relay is intercepted:
when r is an eavesdropper, it will attempt to intercept information from the information receiver. The signal-to-noise ratio of r to a and b at this time is calculated as:
Figure BDA0001830591030000091
Figure BDA0001830591030000092
the channel capacity r to a and b at the time of eavesdropping can be obtained:
Figure BDA0001830591030000093
at this time, the channel capacity when the relay is intercepted is subtracted from the channel capacity when the relay is safely relayed, and the privacy ratio of the one-way channel at this time can be obtained as follows:
Figure BDA0001830591030000094
Figure BDA0001830591030000095
step five: calculating the safe rate of the one-way channel:
Figure BDA0001830591030000096
Figure BDA0001830591030000097
finding the optimal solution to make the sum of the channel capacities of the two channels reach the maximum, and obtaining:
Figure BDA0001830591030000098
due to the complexity of the objective function, by simplifying the objective function using a high signal-to-noise ratio, we obtain:
Figure BDA0001830591030000099
at this time, the secret keeping rate can be divided into two parts, one part is a function of power, namely the problem of transmitting power distribution, and the other part is a function of rho, namely the problem of optimizing the energy collected by the relay.
A. Computing a power allocation optimal solution
The objective function and constraints are:
Figure BDA0001830591030000101
s.t.0≤PA+PB≤PT；
the lagrangian function that can be used to obtain the objective function is:
Figure BDA0001830591030000102
the only optimal solution of the objective function can be obtained by the Lagrange multiplier method:
Figure BDA0001830591030000103
B. optimal solution of relay power allocation factor:
Figure BDA0001830591030000104
s.t.0＜ρ＜1；
setting the derivative of the objective function to zero, the optimal solution can be obtained as Rootof (w)1ρ4+w2ρ3+w3ρ2+w4ρ+w5) Here, Rootof (-) denotes the root of the polynomial. w is a1＝2η2gAgB-ηgA-ηgB，w2＝-4η2gAgB+5ηgA+5ηgB-2，w3＝-6ηgA-6ηgB+6，w4＝2ηgA+2ηgB-6，w5＝2。
The application effect of the present invention will be described in detail with reference to the simulation.
In an embodiment, the proposed power distribution image is realized by computer simulation. The invention is provided withThree nodes are positioned on a straight line, and the relay is positioned between two source nodes. The signal loss eta is 0.9, the signal transmission slow fading loss coefficient tau is 3, and the normalized distance between A and B is unified and used as dRThe normalized distance of a to R is indicated.
The relay position d under 15dB signal-to-noise ratio is indicated in FIG. 5RAnd its corresponding safe rate. For comparison, the results of equal power allocation are also given. It can be seen that the proposed power allocation is superior to equal power allocation regardless of relay location. Secret ratio of dRThe best performance is achieved at 0.5, where the proposed power allocation may enable the untrusted relay energy-carrying communication system to achieve the best performance at the group rate.
The above description is only for the purpose of illustrating the preferred embodiments of the present invention and is not to be construed as limiting the invention, and any modifications, equivalents and improvements made within the spirit and principle of the present invention are intended to be included within the scope of the present invention.

Claims (5)

1. A method for improving the safety rate of an untrusted relay energy-carrying communication system is characterized in that the method for improving the safety rate of the untrusted relay energy-carrying communication system establishes a mathematical model of a two-way relay system; calculating the signal received by the relay in the first time slot and the energy collected from the source, and obtaining the signal-to-noise ratio and the channel capacity of a channel between the relay and the two sources through the signal obtained by the signal source of the second time slot; finally, calculating the signal-to-noise ratio and the channel capacity if the relay is used as an eavesdropper, and obtaining an optimal solution after comparison;
the calculating of the signal-to-noise ratio and the channel capacity during the eavesdropping of the relay specifically includes: when r is an eavesdropper, the information from the information receiver is attempted to be intercepted; the signal-to-noise ratio of r to a and b is calculated as:
Figure FDA0003469448300000011
Figure FDA0003469448300000012
wherein the channel gains between the relay r and the sources a and b are h respectivelyAAnd hBSimultaneously order gA＝|hA|2,gB＝|hB|2ρ is the power allocation factor of the relay, PAAnd PBRespectively representing the transmitting power of a source a and the transmitting power of a source b, and obtaining the channel capacity from r to a and b when the relay is intercepted:
Figure FDA0003469448300000013
Figure FDA0003469448300000014
wherein the eavesdropping SNR of the relay r to the source a and the source b is respectively
Figure FDA0003469448300000015
And
Figure FDA0003469448300000016
the eavesdropping channel capacity of the relay r to the source a and the source b is respectively
Figure FDA0003469448300000017
And
Figure FDA0003469448300000018
subtracting the channel capacity when the relay is intercepted from the channel capacity when the relay is safely relayed to obtain the confidentiality rate of the one-way channel at the moment as follows:
Figure FDA0003469448300000019
Figure FDA00034694483000000110
the secret keeping rate is split into: a function of power and a function of ρ;
(1) calculating an optimal solution of power distribution;
the objective function and constraints are:
Figure FDA0003469448300000021
s.t.0≤PA+PB≤PT；
the lagrange function for the objective function is:
Figure FDA0003469448300000022
the only optimal solution of the objective function can be obtained by the Lagrange multiplier method:
Figure FDA0003469448300000023
(2) optimal solution of relay power allocation factor:
Figure FDA0003469448300000024
wherein eta is an energy attenuation factor, and the derivative of the objective function is set to zero to obtain the optimal solution of Rootif (w)1ρ4+w2ρ3+w3ρ2+w4ρ+w5) Rootof (·) represents the root of the polynomial; w is a1＝2η2gAgB-ηgA-ηgB，w2＝-4η2gAgB+5ηgA+5ηgB-2，w3＝-6ηgA-6ηgB+6，w4＝2ηgA+2ηgB-6，w5＝2。
2. The method of claim 1, wherein calculating the received signal and the collected energy for the first time slot relay comprises: the channel gains between the relay r and the sources a and b are h, respectivelyAAnd hBDenoted by the transmit powers of a and b, respectively, being PAAnd PBThe transmitted signals of a and b are respectively represented by xAAnd xBRepresents; in the first time slot, a and b simultaneously send signals to the relay r, relaying the received signals:
Figure FDA0003469448300000025
wherein n isRIntroducing zero mean and N variance into r receiving relays0Additive white gaussian noise of (1); the signal received by the relay receiver is
Figure FDA0003469448300000031
The collected energy is ER＝ηρ(PAgA+PBgB) (ii) a Where η is the attenuation factor of the energy, ρ is the power distribution factor of the relay, gA＝|hA|2,gB＝|hB|2。
3. The method of claim 1, wherein calculating the signals received by the two sources in the second time slot comprises: after the second time slot, the relay becomes after energy collection and amplification
Figure FDA0003469448300000032
The unit power signal of the relay transmission is
Figure FDA0003469448300000033
Wherein n isBRMean value is zero and variance is N0White Gaussian noise of (y)RRelaying the received signal for a first time slot; then the signals received by the two sources a and b in the second time slot are obtained as
Figure FDA0003469448300000034
And
Figure FDA0003469448300000035
where n isAAnd nBRepresenting additive white gaussian noise relayed to source a and source b channels, respectively.
4. The method of claim 1, wherein computing the source-to-relay signal-to-noise ratio and channel capacity specifically comprises: a and b obtain their own original signals, and solve the signal-to-noise ratio of the relay r to a and b:
Figure FDA0003469448300000036
Figure FDA0003469448300000037
N0represents the variance of additive white gaussian noise in the system, thus, the relay r to a and b channel capacity is obtained:
Figure FDA0003469448300000038
Figure FDA0003469448300000039
5. the method of claim 1, wherein computing the security rate of the unidirectional channel comprises:
Figure FDA0003469448300000041
Figure FDA0003469448300000042
finding the optimal solution to make the sum of the channel capacities of the two channels reach the maximum, and obtaining:
Figure FDA0003469448300000043
wherein
Figure FDA0003469448300000044
In order to provide an eavesdropping signal-to-noise ratio of relay r to source a when relay r is an eavesdropper,
Figure FDA0003469448300000045
and
Figure FDA0003469448300000046
respectively representing the signal-to-noise ratio of the relay r to the source a and the source b during the safe relay; simplifying the objective function by using a high signal-to-noise ratio yields:
Figure FDA0003469448300000047
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