A method for reducing peak-to-average ratio of OFDM signals based on minimum coverage circle

Abstract

A method for reducing the peak-to-average ratio of an OFDM signal based on the minimum covering circle method. At the signal receiving end, the signal processing process is as follows: after obtaining the original frequency domain signal X, the N subcarriers in the OFDM symbol are divided into reserved subcarriers and data subcarriers, the number of reserved subcarriers is N / Q, and the positions of the reserved subcarriers are distributed at equal intervals; through IFFT transformation, the time domain signal of the frequency domain signal of the data subcarrier is obtained: the time domain signal of the frequency domain signal of the reserved subcarrier is obtained by the minimum covering circle method; the time domain signal corresponding to the frequency domain signal of the data subcarrier is added to the time domain signal corresponding to the frequency domain signal of the frequency domain subcarrier, that is, the time domain transmission signal of the OFDM symbol is obtained. The method of the present invention uses reserved subcarriers with special position distribution characteristics, and adopts the minimum covering circle search method of a finite point set to obtain the time domain signal of the frequency domain signal of the reserved subcarrier, which reduces the complexity of the search calculation process while ensuring the peak-to-average ratio reduction.

Description

Technical Field

[0001] The invention belongs to the technical field of signal processing, and in particular relates to a method for reducing the peak-to-average ratio of an OFDM signal based on a minimum coverage circle. Background Art

[0002] Orthogonal Frequency Division Multiplexing (OFDM) is a multi-carrier transmission technology that can achieve high-speed data transmission with high spectrum utilization and effectively combat multipath fading. Therefore, it is widely used in many digital broadband communication systems such as 4G, 5G, digital audio broadcasting (DAB), digital video broadcasting (DVB), and high-definition television (HDTV). However, the peak-to-average power ratio (PAPR) problem of the transmitted signal caused by the multi-carrier system will greatly reduce the power efficiency of the RF amplifier, and put forward higher requirements on the linear range of the current amplifier, increasing the difficulty and cost of system implementation. Therefore, the peak-to-average power ratio reduction technology has always been one of the key technologies of the OFDM system.

[0003] The reserved subcarrier method (tone reservation, TR for short) belongs to the signal pre-distortion technology. It is a method without distortion and effectively reduces the PAPR of the OFDM system. Its basic principle is that the transmitter reserves a part of special subcarriers to generate peak-suppressing signals that suppress PAPR. The receiver ignores the data on the subcarriers reserved for suppressing PAPR and recovers the useful signal from other normal data subcarriers used to transmit information. The core of the reserved subcarrier method is how to obtain the data on the reserved subcarriers to reduce the peak value of the original OFDM signal. Someone proposed a solution. Among the reserved K subcarriers, if each subcarrier has V values, V corresponding values ​​can be generated. K There are a number of alternative peak clipping signals, but the generation process of each alternative peak clipping signal requires an inverted fast Fourier transform (IFFT), which is very computationally intensive. Another peak clipping method, subcarrier reservation, has been proposed. First, the original OFDM signal is limited, and then the limiting noise generated by the limiting is used to generate a peak cancellation signal. However, this method requires many or even infinite iterations to generate a peak cancellation signal that is good enough to approximate the ideal peak cancellation signal, and the computational complexity is also very high. Summary of the invention

[0004] The object of the present invention is to provide a low-complexity method for reducing the peak-to-average ratio of an OFDM signal based on a minimum covering circle.

[0005] In order to achieve the above object, the present invention adopts the following technical solutions:

[0006] A method for reducing the peak-to-average ratio of OFDM signals based on the minimum coverage circle. At the signal receiving end, the signal processing process is as follows: After obtaining the original frequency domain signal X,

[0007] S1. Divide the N subcarriers in the OFDM symbol into reserved subcarriers and data subcarriers, where the number of reserved subcarriers is N / Q, and the positions of the reserved subcarriers are equally spaced and fixed to the θth, θ+Qth, θ+2Qth, ..., θ+(N / Q-1)Qth subcarriers in the OFDM symbol, where θ and Q are both integers;

[0008] In the original frequency domain signal of the OFDM symbol, the data subcarrier frequency domain signal is X1=[X1(0), X1(1), ..., X1(N-1)], The reserved subcarrier frequency domain signal is V = [V(0), V(1), ..., V(N-1)], X(n) is the nth element in the original frequency domain signal X of the OFDM symbol;

[0009] S2. Obtain the time domain signal x1 of the data subcarrier frequency domain signal X1 through IFFT transformation:

[0010] S3. Using the minimum covering circle method to obtain the time domain signal v of the reserved subcarrier frequency domain signal V, the specific steps are as follows:

[0011] S301, construct the first constant vector and the first signal

[0012] First constant vector

[0013] First signal

[0014] S302: The first signal Convert to a matrix Γ with N / Q rows and Q columns:

[0015]

[0016] S303. For the matrix Γ, find the one with the first signal from the N / Q row vectors The element with the largest amplitude The row vector Γ r , the symbol |·| indicates taking the absolute value, In the formula represents the rth row vector of the matrix Γ, represents the r+N / Qth row vector of the matrix Γ, and so on;

[0017] S304: Constructing the second signal And solve for the complex constant Δ r , so that the second signal Minimize the maximum magnitude of the elements in ;

[0018]

[0019] The minimum covering circle method is used to solve the complex constant Δ r , the center coordinate of the smallest circle found is -Δ r ;

[0020] S305, Order For the remaining N / Q-1 row vectors in the matrix Γ, the following calculation process is performed one by one:

[0021] For a row vector Γ in the matrix Γ q , calculate Γ q The maximum amplitude value max(|Γ q |);

[0022] If max(|Γ q |)≤R, then the complex constant value corresponding to the row vector is Δ q =0, if max(|Γ q |)>R, then the same method as step S304 is used to solve the complex constant Δ q , that is, construct the third signal The minimum covering circle method is used to solve the complex constant Δ q , so that the third signal The maximum magnitude of the elements in is minimized if Γ q The radius of the corresponding minimum circle Use To replace R, the center coordinates of the smallest circle found are -Δ q ;

[0023] S306. After calculating the N / Q row vectors of the matrix Γ through the above steps, N / Q complex constants are obtained, and these complex constants are combined into a vector Δ, Δ=[Δ0, Δ1, ..., Δ N / Q-1 ],make Construct the second constant vector P,

[0024] The time domain signal v corresponding to the reserved subcarrier frequency domain signal V is:

[0025] S4. Add the time domain signal x1 corresponding to the data subcarrier frequency domain signal X1 and the time domain signal v corresponding to the frequency domain subcarrier frequency domain signal V, so as to obtain the time domain transmission signal x of the current OFDM symbol, x=x1+v.

[0026] The method for reducing peak-to-average ratio of OFDM signals based on the minimum covering circle method as described above, further, in step S304, the minimum covering circle method is used to solve the complex constant Δ r When, for the point set Γ r , the corresponding minimum circle search steps are as follows:

[0027] a. From the point set Γ r Find the two points with the longest distance from each other and add them to the point set L, and construct a circle with the line connecting these two points as the diameter

[0028] b. Judgement circle Whether to cover the point set Γ r , if yes, then the circle is the smallest circle and the search process ends; if no, proceed to the next step;

[0029] c. From the point set Γ r Find an out-of-circle The point farthest from the center of the circle is added to the point set L, and the minimum covering circle of the point set L is calculated using the random increment method, and then the circle is replaced by the circle Then return to step b.

[0030] It can be seen from the above technical solutions that the present invention provides a low computational complexity peak-to-average ratio reduction method based on subcarrier reservation technology for the peak-to-average ratio reduction problem of OFDM signals. The method of the present invention first uses reserved subcarriers with special position distribution characteristics to make the calculation process of optimally offsetting time domain signals equivalent to the minimum circle covering problem, and then further improves the search for the minimum circle, using a finite point set minimum covering circle search method to obtain the time domain signal of the reserved subcarrier frequency domain signal, which can reduce the complexity of the search calculation process while ensuring the peak-to-average ratio reduction. BRIEF DESCRIPTION OF THE DRAWINGS

[0031] In order to more clearly illustrate the embodiments of the present invention, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0032] Figure 1 The signal processing flow chart at the signal transmitting end when the reserved subcarrier method is used to reduce the peak-to-average ratio of OFDM signals;

[0033] Figure 2 A schematic diagram of the position distribution of data subcarriers and reserved subcarriers;

[0034] Figure 3 The signal processing flow chart of the method of the present invention at the signal transmitting end;

[0035] Figure 4 It is a comparison diagram of the peak-to-average ratio reduction performance simulation curve using the method of the present invention and the peak-to-average ratio simulation curve not using the method of the present invention. DETAILED DESCRIPTION

[0036] The present invention is described in detail below in conjunction with the accompanying drawings. When describing the embodiments of the present invention in detail, for the convenience of explanation, the drawings representing the device structure will not be partially enlarged according to the general proportion, and the schematic diagram is only an example, which should not limit the scope of protection of the present invention. It should be noted that the drawings are simplified in form and use non-precise proportions, which are only used to facilitate and clearly assist in explaining the purpose of the embodiments of the present invention. At the same time, in the description of the present application, the terms "first", "second", etc. are only used to distinguish the description, and cannot be understood as indicating or implying relative importance or implicitly indicating the number of technical features indicated; the terms "positive", "negative", "bottom", "upper", "lower", etc. indicate the orientation or position relationship based on the orientation or position relationship shown in the drawings, which is only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and therefore cannot be understood as a limitation to the present invention.

[0037] In the description of the present invention, it should be noted that, unless otherwise clearly specified and limited, the terms "connected" and "connection" should be understood in a broad sense, for example, it can be a fixed connection, a detachable connection, or an integral connection; it can be a mechanical connection or an electrical connection; it can be a direct connection, or it can be indirectly connected through an intermediate medium, it can also be the internal connection of two elements, it can be a wireless connection, or it can be a wired connection. For ordinary technicians in this field, the specific meanings of the above terms in the present invention can be understood according to specific circumstances.

[0038] In an OFDM symbol with N subcarriers, the frequency domain signal X = [X(0), X(1), ..., X(N-1)] corresponds to the time domain signal x = [x(0), x(1), ..., x(N-1)], and the relationship between the two is as follows:

[0039] IDFT(X) represents the inverse discrete Fourier transform result.

[0040] Figure 1 The signal processing flow chart at the signal transmitting end when the reserved subcarrier method is used to reduce the peak-to-average ratio of OFDM signals. Figure 1 As shown, in order to reduce the PAPR of OFDM signals, the reserved subcarrier technology divides N subcarriers into two categories, among which N1 subcarriers carry the data symbols to be sent. This type of subcarrier is a data subcarrier, and the frequency domain signal of the data subcarrier is recorded as X1, and the corresponding time domain signal is recorded as x1; the remaining N-N1 reserved subcarriers do not carry data symbols, and are used to generate a peak clipping signal to reduce the peak-to-average ratio of the transmitted signal. This type of subcarrier is a reserved subcarrier, and the frequency domain signal of the reserved subcarrier is recorded as V, and the corresponding time domain signal is recorded as v. The essence of the reserved subcarrier method is to find the frequency domain signal V that matches the time domain signal x1 of the data subcarrier in a certain way, so as to minimize the peak-to-average ratio of the final transmitted signal x=x1+v.

[0041] When the signal is subjected to inverse fast Fourier transform (IFFT), the number of subcarriers N satisfies 2 K (K≥2) frequency domain signal X=[X(0),X(1),...,X(N-1)], which satisfies the following conditions:

[0042] In the formula, θ and Q are both positive integers, and Q = 2 q (q≥2)≤N / 2, 0≤θ≤Q-1;

[0043] The corresponding time domain signal x=[x(0),x(1),...,x(N-1)] has the following properties:

[0044] Where P is a constant vector. is the time domain signal corresponding to the frequency domain signal [X(θ),X(θ+Q),...,X(θ+NQ)], symbol Represents the dot product operation between two vectors of the same length, time domain signal

[0045] An OFDM symbol has N subcarriers, and the subcarriers are numbered 0, 1, 2, ..., N-1. The present invention proposes a method for reducing the peak-to-average ratio of OFDM signals by using a reserved subcarrier technology based on the minimum covering circle theory, such as Figure 3 As shown, the steps of the method of the present invention are as follows:

[0046] At the signal transmitting end, the signal processing process is:

[0047] S1. Divide the subcarriers in the OFDM symbol into reserved subcarriers and data subcarriers, wherein the reserved subcarriers are used to generate peak-cutting signals that reduce the peak-to-average ratio of the transmitted signal, and the data subcarriers are used to carry the data symbols to be sent. The number of reserved subcarriers is N / Q, and the positions of the reserved subcarriers are evenly spaced and fixed to the θth, θ+Qth, θ+2Qth, ..., θ+(N / Q-1)Qth subcarriers in the OFDM symbol, where θ and Q are both integers, 0≤θ≤Q-1, Q=2 q , q≤1, Q≤N / 2;

[0048] For example, assuming that the number of subcarriers in an OFDM symbol is N=16, and q=2, Q=4, and θ=2, the reserved subcarriers are the 2nd, 6th, 10th, and 14th subcarriers, respectively, that is, the 2nd, 6th, 10th, and 14th subcarriers in the OFDM symbol are reserved subcarriers;

[0049] For an OFDM symbol containing N subcarriers, its original frequency domain signal is

[0050] X=[X(0),X(1),...,X(N-1)], in the frequency domain signal of the OFDM symbol, the data subcarrier frequency domain signal is X1, the reserved subcarrier frequency domain signal is V, and the data subcarrier frequency domain signal is

[0051] X1=[X1(0),X1(1),...,X1(N-1)], reserved subcarrier frequency domain signal V=[V(0),V(1),...,V(N-1)], where,

[0052]

[0053] S2. Obtain the time domain signal x1 of the data subcarrier frequency domain signal X1 through IFFT transformation, where x1=[x1(0), x1(1), ..., x1(N-1)];

[0054] S3. Using the minimum covering circle method to obtain the time domain signal v of the reserved subcarrier frequency domain signal V, the specific steps are as follows:

[0055] S301, construct the first constant vector and the first signal

[0056] First constant vector

[0057] First signal

[0058] S302: The first signal Convert to a matrix Γ with N / Q rows and Q columns:

[0059]

[0060] S303. For the matrix Γ, find the one with the first signal from the N / Q row vectors The element with the largest amplitude The row vector Γ r , the symbol |·| indicates taking the absolute value;

[0061] The first signal The element with the largest amplitude The row vector of In the formula represents the rth row vector among the N / Q row vectors of the matrix Γ, represents the r+N / Qth row vector among the N / Q row vectors of the matrix Γ, and so on;

[0062] S304: Constructing the second signal And solve for the complex constant Δ r , so that the second signal Minimize the maximum magnitude of the elements in ;

[0063]

[0064] Vectors of complex numbers represents a finite point set containing Q points on a two-dimensional plane, where the values ​​of the real and imaginary parts represent the horizontal and vertical coordinates respectively, and the complex constant -Δ r It can be regarded as the coordinates of the center of a circle, and the complex constant Δ r The solution process is essentially to solve the point set Γ r Find the corresponding minimum covering circle, the radius of the minimum covering circle is equal to

[0065] The present invention improves the existing minimum covering circle method and r , the corresponding minimum covering circle search steps are as follows:

[0066] a. From the point set Γ r Find the two points with the longest distance from each other and add them to the point set L, and construct a circle with the line connecting these two points as the diameter

[0067] b. Judgement circle Whether to cover the point set Γ r , if yes, then the circle is the smallest circle and the search process ends; if no, proceed to the next step;

[0068] c. From the point set Γ r Find an out-of-circle The point farthest from the center of the circle is added to the point set L, and the minimum covering circle of the point set L is calculated using the random increment method, and then the circle is replaced by the circle Then return to step b;

[0069] S305, Order For the remaining N / Q-1 row vectors in the matrix Γ, the following calculation process is performed one by one:

[0070] For a row vector Γ in the matrix Γ q , calculate Γ q The maximum amplitude value max(Γ q |);

[0071] If max(|Γ q |)≤R, then the complex constant value corresponding to the row vector is Δ q =0, if max(|Γ q |)>R, then the same method as step S304 is used to solve the complex constant Δ q , that is, construct the third signal And use the minimum covering circle method to solve the complex constant Δ q , so that the third signal Minimize the maximum amplitude of the elements in (let max(|Γ q +Δ q |) minimized), if Γ q The radius of the corresponding minimum circle Use To replace R, the center coordinates of the smallest circle found are -Δ q ;

[0072] S306. After calculating the N / Q row vectors of the matrix Γ through the above steps, N / Q complex constants are obtained, and these complex constants are combined into a vector Δ, Δ=[Δ0, Δ1, ..., Δ N / Q-1 ],make Construct the second constant vector P,

[0073] The time domain signal v corresponding to the reserved subcarrier frequency domain signal V = [V(0), V(1), ..., V(N-1)] is

[0074] S4. Add the time domain signal x1 corresponding to the data subcarrier frequency domain signal X1 and the time domain signal v corresponding to the frequency domain subcarrier frequency domain signal V, to obtain the time domain transmission signal x of the OFDM symbol, where x=x1+v. After receiving the signal, the signal receiving end can restore the received signal according to the conventional method.

[0075] The method of the present invention adopts the minimum covering circle method to obtain the time domain signal v of the reserved subcarrier frequency domain signal V. The calculation process of the minimum covering circle is based on the following basic properties:

[0076] 1) The minimum covering circle is unique;

[0077] 2) A circle can be determined by taking the line between two points as the diameter;

[0078] 3) Three non-collinear points can determine a circle, and all three points are on the circumference of the circle;

[0079] 4) There are at least two points in a finite point set that fall on the circumference of its minimum covering circle, and if there are more than two points, then the minimum covering circle is the circumscribed circle of the triangle or polygon formed by these points.

[0080] Based on the above properties, an intuitive solution to the minimum covering circle problem is to traverse the point set S = {s1, s2, ..., s N}, but the calculation process of the conventional minimum covering circle still has a high complexity, up to O(N 4 ). Even though some existing algorithms have made some improvements to the traversal process, such as the random increment method based on recursive thinking, its computational complexity is still at a high level. For example, the specific calculation steps of the random increment method are:

[0081] 1) Randomly select two points from the point set S and construct a circle with the line connecting these two points as the diameter

[0082] 2) Randomly select a new point from the point set S. If the new point is on the circle If inside, the circle is not updated If the new point is on the circle Outside, from the circle Traverse the point set in the circle to select two points and construct a circle with the new point; after completing all combinations, select the points that can cover the circle at the same time. The circle with the smallest radius of all points and new points in it is then used to replace the circle

[0083] 3) Repeat the above steps until the circle If it can cover all points in the point set S, then the circle is the smallest circle.

[0084] The present invention improves the search process of the minimum circle based on the following theory:

[0085] For any finite point set S, the corresponding minimum covering circle is denoted by C, then C∈Ω, where the six circles contained in the circle set Ω={C1,....,C6} are defined as follows:

[0086] 1) Circle C1: The circle determined by the two points L1 and L2 with the longest distance in the point set S;

[0087] 2) Circle C i ,i=2,...,6: in In the case of point set {L1,...,L i+1} corresponds to the minimum covering circle, where point L i+1 is the point set S and the circle C i-1 The point farthest from the center of the circle.

[0088] From the above content, it can be seen that the above method can definitely find the minimum covering circle of the point set S, and the theoretical upper limit of the number of searches is 6 times. Therefore, the method of the present invention adopts the improved minimum circle search method in steps S404 and S405, which can have a very low computational complexity.

[0089] The following simulation experiment verifies the peak-to-average ratio reduction effect of the method of the present invention. In the simulation, the complementary cumulative distribution function (CCDF) in the time domain is used to describe the distribution of the signal peak-to-average ratio (PAPR), and its mathematical calculation formula is Pr(PAPR>z)=1-Pr(PAPR≤z), where z represents the threshold. Figure 4 It is the CCDF curve when N / Q is equal to different values. Figure 4 The TR curve in the figure represents the CCDF curve after the peak-to-average ratio is reduced by the method of the present invention, and the No TR curve represents the CCDF curve without the method of the present invention. Figure 4 It can be seen that the method of the present invention can obtain very good peak-to-average ratio reduction performance. For example, when N / Q=4, that is, 4 (accounting for 4 / 256=1.56% of the overall bandwidth) subcarriers are used as reserved subcarriers, compared with the OFDM signal that does not use the method of the present invention to reduce the peak-to-average ratio, its CCDF is reduced by about 2.3dB at 10-3. Simulation results show that the method of the present invention can obtain very good peak-to-average ratio reduction performance with only a small proportion of reserved subcarriers.

[0090] The above description of the disclosed embodiments enables one skilled in the art to implement or use the present invention. Various modifications to these embodiments will be apparent to one skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the present invention. Therefore, the present invention will not be limited to the embodiments shown herein, but rather to the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A method for reducing peak-to-average ratio of OFDM signals based on the minimum covering circle method, characterized by: At the signal receiving end, the signal processing process is as follows: After obtaining the original frequency domain signal X, S1. Divide the N subcarriers in the OFDM symbol into reserved subcarriers and data subcarriers, where the number of reserved subcarriers is N / Q, and the positions of the reserved subcarriers are equally spaced and fixed to the θth, θ+Qth, θ+2Qth, ..., θ+(N / Q-1)Qth subcarriers in the OFDM symbol, where θ and Q are both integers; In the original frequency domain signal of the OFDM symbol, the data subcarrier frequency domain signal is X1=[X1(0), X1(1), ..., X1(N-1)], The reserved subcarrier frequency domain signal is V = [V(0), V(1), ..., V(N-1)], X(n) is the nth element in the original frequency domain signal X of the OFDM symbol; S2. Obtain the time domain signal x1 of the data subcarrier frequency domain signal X1 through IFFT transformation: S3. Using the minimum covering circle method to obtain the time domain signal v of the reserved subcarrier frequency domain signal V, the specific steps are as follows: S301, construct the first constant vector and the first signal First constant vector First signal S302: The first signal Convert to a matrix Γ with N / Q rows and Q columns: S303. For the matrix Γ, find the one with the first signal from the N / Q row vectors The element with the largest amplitude The row vector Γ r , the symbol |·| indicates taking the absolute value, In the formula represents the rth row vector of the matrix Γ, represents the r+N / Qth row vector of the matrix Γ, and so on; S304: Constructing the second signal And solve for the complex constant Δ r , so that the second signal Minimize the maximum magnitude of the elements in ; The minimum covering circle method is used to solve the complex constant Δ r , the center coordinates of the smallest circle found are -Δ r ; S305, Order For the remaining N / Q-1 row vectors in the matrix Γ, the following calculation process is performed one by one: For a row vector Γ in the matrix Γ q , calculate Γ q The maximum amplitude value max(|Γ q |); If max(|Γ q |)≤R, then the complex constant value corresponding to the row vector is Δ q =0, if max(|Γ q |)>R, then the same method as step S304 is used to solve the complex constant Δ q , that is, construct the third signal The minimum covering circle method is used to solve the complex constant Δ q , so that the third signal The maximum magnitude of the elements in is minimized if Γ q The radius of the corresponding minimum circle Use To replace R, the center coordinates of the smallest circle found are -Δ q ; S306. After calculating the N / Q row vectors of the matrix Γ through the above steps, N / Q complex constants are obtained, and these complex constants are combined into a vector Δ, Δ=[Δ0, Δ1, ..., Δ N / Q-1 ],make Construct the second constant vector P, The time domain signal v corresponding to the reserved subcarrier frequency domain signal V is: S4. Add the time domain signal x1 corresponding to the data subcarrier frequency domain signal X1 and the time domain signal v corresponding to the frequency domain subcarrier frequency domain signal V to obtain the time domain transmission signal x of the current OFDM symbol.

2. The method for reducing peak-to-average ratio of OFDM signals based on the minimum covering circle method according to claim 1, characterized in that: In step S304, the minimum covering circle method is used to solve the complex constant Δ r When, for the point set Γ r , the corresponding minimum circle search steps are as follows: a. From the point set Γ r Find the two points with the longest distance from each other and add them to the point set L, and construct a circle with the line connecting these two points as the diameter b. Judgement circle Whether to cover the point set Γ r , if yes, then the circle is the smallest circle and the search process ends; if no, proceed to the next step; c. From the point set Γ r Find an out-of-circle The point farthest from the center of the circle is added to the point set L, and the minimum covering circle of the point set L is calculated using the random increment method, and then the circle is replaced by the circle Then return to step b.

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