Find the optimal value to maximize a function

Hi everyone.....could anyone help......I need to find th optimal value of x0 and x1 to maximize R, where x1=d-x0..........the function MATLAB code:

close all; clear all; clc;

% System parameters
F = 10000;    % Number of files
S = 100;     % SBS cache capacity (set to 100)
M = 50;      % Cash capacity fraction
alpha = 2; % Path loss exponent for LOS link
c = 3e8;     % Light speed
fr = 1e12;  % Operating frequency
B = 10e6;    % System bandwidth
epsilon = 0.8; % Skewness factor
K = 0.0016;  % Molecular absorption coefficient
P_M = 10^(64/10); % Transmit power MBS
P_S = 10^(30/10); % Transmit power SBS
sigma = 10^(-90/10); % Noise power
N_L = 512;
N_M = 16;
eta = 1;
x2 =30;      % RIS-MBS distance


d_range = 1:1:40;

R = zeros(size(d_range));

for i = 1:length(d_range)
    d = d_range(i);
    
    sum1 = 0;
    for f = 1:F
        sum1 = sum1 + f^(-epsilon);
    end

    sum2 = 0;  
    for f = 1:M
        sum2 = sum2 + f^(-epsilon)/sum1;
    end

    sum3 = 0;
    for f = (M + 1):(M + (S - M) / eta)
        sum3 = sum3 + (f^(-epsilon)) / sum1;
    end

    sum3 = sum3 * eta;
    
    beta = (c/(4*pi*fr))^2;  % Spreading loss index
    Rs = B * log2(1 + (P_S * beta * (d-x1)^(-alpha) * exp(-K * (d-x1))) / sigma);
    Rm = B * log2(1 + (P_M * (N_L^2) * N_M * (beta * (d-x0)^(-alpha) * exp(-K * (d-x0))) * (beta * x2^(-alpha) * exp(-K * x2))) / sigma);
    Rt = Rs * (sum2 + sum3) + Rm * (1 - (sum2 + sum3));
    R(i) = Rt;
    
  
end

figure
plot(d_range, R, 'b^-')
xlabel('SBS-RIS Distance, d')
ylabel('Achievable Rate')                                         

Answer 1

[Mathieu NOE]

Hi

seems to me that R curves goes up as x0 goes closer to zero (but not rqual to zero otherwise you get Inf values)

also all the small for loops to create the sum1, sum2,sum3 variables can be replaced with sum function directly

close all; clear all; clc;

x0_range  = [0.01 0.25 0.5 1];

% System parameters
F = 10000;    % Number of files
S = 100;     % SBS cache capacity (set to 100)
M = 50;      % Cash capacity fraction
alpha = 2; % Path loss exponent for LOS link
c = 3e8;     % Light speed
fr = 1e12;  % Operating frequency
B = 10e6;    % System bandwidth
epsilon = 0.8; % Skewness factor
K = 0.0016;  % Molecular absorption coefficient
P_M = 10^(64/10); % Transmit power MBS
P_S = 10^(30/10); % Transmit power SBS
sigma = 10^(-90/10); % Noise power
N_L = 512;
N_M = 16;
eta = 1;
x2 =30;      % RIS-MBS distance
d_range = 1:1:40;

R = zeros(numel(d_range),numel(x0_range));

for k = 1:numel(x0_range)
    x0 = x0_range(k);
    
    for i = 1:length(d_range)
        d = d_range(i);
        x1 = d-x0; % added line 

%         sum1 = 0;
%         for f = 1:F
%             sum1 = sum1 + f^(-epsilon);
%         end
        f = 1:F;
        sum1 = sum(f.^(-epsilon));
        
%         sum2 = 0;  
%         for f = 1:M
%             sum2 = sum2 + f^(-epsilon)/sum1;
%         end

        f = 1:M;
        sum2 = sum(f.^(-epsilon))/sum1;        

%         sum3 = 0;
%         for f = (M + 1):(M + (S - M) / eta)
%             sum3 = sum3 + (f^(-epsilon)) / sum1;
%         end
%         sum3 = sum3 * eta;
        
        f = (M + 1):(M + (S - M) / eta);
        sum3 = eta*sum(f.^(-epsilon))/sum1;        
        

        beta = (c/(4*pi*fr))^2;  % Spreading loss index
        Rs = B * log2(1 + (P_S * beta * (d-x1)^(-alpha) * exp(-K * (d-x1))) / sigma);
        Rm = B * log2(1 + (P_M * (N_L^2) * N_M * (beta * (d-x0)^(-alpha) * exp(-K * (d-x0))) * (beta * x2^(-alpha) * exp(-K * x2))) / sigma);
        Rt = Rs * (sum2 + sum3) + Rm * (1 - (sum2 + sum3));
        R(i,k) = Rt;
        leg{k} = ['x0 = ' num2str(x0)];

    end

end

    plot(d_range, R, '^-')
    legend(leg)
    xlabel('SBS-RIS Distance, d')
    ylabel('Achievable Rate')  

[Hadeel Obaid]

Thanks a lot @Mathieu NOE......do you think that I should use x0_range instead of d_range?.....Also how can I use the exaustive search to find the optimal value?

[Mathieu NOE]

I am not sure to understand fully your problem... how many independant parameters do we have ?

only d and x0 ?

what is the range for each one ?

can you confirm that x1=d-x0 so x1 is not an independant parameter (I added this in the corrected code)

then why are you still using x1 in this line :

Rs = B * log2(1 + (P_S * beta * (d-x1)^(-alpha) * exp(-K * (d-x1))) / sigma);

[Hadeel Obaid]

Sorry for the confussion @Mathieu NOE...Let me elaborate more here:

So, I have a fixed distant (d=40) between the RIS (reflector) and the SBS station and I have a user between the RIS and the SBS. The distance between the RIS and the user is x1, and the distance between the user and the SBS is x0. We have (x0 + x1 = d), so I need to find the optimal value of x0 that maximize the rate R..........

Then, if we find the optimal value of x0, x1 will be found already as we have (x1=d-x0).

[Mathieu NOE]

This is what I get now

still I wonder what x2 is doing here

x2 =30; % RIS-MBS distance (what is MBS ? )

as x2 is used in the Rm computaion

Rm = B * log2(1 + (P_M * (N_L^2) * N_M * (beta * (d-x0)^(-alpha) * exp(-K * (d-x0))) * (beta * x2^(-alpha) * exp(-K * x2))) / sigma);

is x2 fix or dependant of x0 ?

so far the new code is showing that maximum rate is obtained if the user is close either to the RIS or the SBS

% System parameters
F = 10000;    % Number of files
S = 100;     % SBS cache capacity (set to 100)
M = 50;      % Cash capacity fraction
alpha = 2; % Path loss exponent for LOS link
c = 3e8;     % Light speed
fr = 1e12;  % Operating frequency
B = 10e6;    % System bandwidth
epsilon = 0.8; % Skewness factor
K = 0.0016;  % Molecular absorption coefficient
P_M = 10^(64/10); % Transmit power MBS
P_S = 10^(30/10); % Transmit power SBS
sigma = 10^(-90/10); % Noise power
N_L = 512;
N_M = 16;
eta = 1;
x2 =30;      % RIS-MBS distance
d = 40;
x0_range  = linspace(0.01*d,0.99*d,100);

R = zeros(1,numel(x0_range));

for k = 1:numel(x0_range)
    
        x0 = x0_range(k);
    
        x1 = d-x0; % added line 
    
        %sum1 computation
        f = 1:F;
        sum1 = sum(f.^(-epsilon));
        
        %sum2 computation
        f = 1:M;
        sum2 = sum(f.^(-epsilon))/sum1;        

        %sum  computation
        f = (M + 1):(M + (S - M) / eta);
        sum3 = eta*sum(f.^(-epsilon))/sum1;        
        
        beta = (c/(4*pi*fr))^2;  % Spreading loss index
        Rs = B * log2(1 + (P_S * beta * (d-x1)^(-alpha) * exp(-K * (d-x1))) / sigma);
        Rm = B * log2(1 + (P_M * (N_L^2) * N_M * (beta * (d-x0)^(-alpha) * exp(-K * (d-x0))) * (beta * x2^(-alpha) * exp(-K * x2))) / sigma);
        R(k) = Rs * (sum2 + sum3) + Rm * (1 - (sum2 + sum3));
    
end

plot(x0_range, R)
xlabel('SBS-user Distance, x0')
ylabel('Achievable Rate')  

[Hadeel Obaid]

@Mathieu NOE.....x2 is fixed and independent of x0......MBS is another basestation in the system model.

Okey, intersteing results!.

But, I am a bit confused here if I should show the results with x0_range or d_range?

[Mathieu NOE]

Imaybe you want the same plot for multiple d values so like this

NB that now the x axis is a ratio x0/d (normalized distance) so the curves can overlay in a meaningfull way

close all; clear all; clc;

% System parameters
F = 10000;    % Number of files
S = 100;     % SBS cache capacity (set to 100)
M = 50;      % Cash capacity fraction
alpha = 2; % Path loss exponent for LOS link
c = 3e8;     % Light speed
fr = 1e12;  % Operating frequency
B = 10e6;    % System bandwidth
epsilon = 0.8; % Skewness factor
K = 0.0016;  % Molecular absorption coefficient
P_M = 10^(64/10); % Transmit power MBS
P_S = 10^(30/10); % Transmit power SBS
sigma = 10^(-90/10); % Noise power
N_L = 512;
N_M = 16;
eta = 1;
x2 =30;      % RIS-MBS distance
d_range  = (20:10:60);


for i = 1:numel(d_range)
    
    d = d_range(i);
    x0_range  = linspace(0.01*d,0.99*d,100);
    R = zeros(1,numel(x0_range));
    
    for k = 1:numel(x0_range)
    
        x0 = x0_range(k);
    
        x1 = d-x0; % added line 
    
        %sum1 computation
        f = 1:F;
        sum1 = sum(f.^(-epsilon));
        
        %sum2 computation
        f = 1:M;
        sum2 = sum(f.^(-epsilon))/sum1;        

        %sum  computation
        f = (M + 1):(M + (S - M) / eta);
        sum3 = eta*sum(f.^(-epsilon))/sum1;        
        
        beta = (c/(4*pi*fr))^2;  % Spreading loss index
        Rs = B * log2(1 + (P_S * beta * (d-x1)^(-alpha) * exp(-K * (d-x1))) / sigma);
        Rm = B * log2(1 + (P_M * (N_L^2) * N_M * (beta * (d-x0)^(-alpha) * exp(-K * (d-x0))) * (beta * x2^(-alpha) * exp(-K * x2))) / sigma);
        R(k) = Rs * (sum2 + sum3) + Rm * (1 - (sum2 + sum3));
    
    end

leg{i} = ['d = ' num2str(d)];
plot(x0_range/d, R)
hold on
  

end

legend(leg);
xlabel('SBS-user normalized Distance, x0/d')
ylabel('Achievable Rate')

[Hadeel Obaid]

Thank you @Mathieu NOE....this is what I want.

[Mathieu NOE]

as always, my pleasure !