Minimize norm of the complex number array with subtraction of a variable

data_info_bit = randi([0,1],N_bits_perfram,1); %Random Bit Generation
data_temp = bi2de(reshape(data_info_bit,N_syms_perfram,M_bits)); %Data
x = qammod(data_temp,M_mod,'gray'); %Symbols
x = reshape(x,N,M); 
s = OTFS_modulation(N,M,x); %Time Signal
s_sub = mat2cell(s.', 1, (M*ones(1,N)));
for ii = 1:N
    .....
        s(ii*M-M+1:M*ii) = s(ii*M-M+1:M*ii) - rdc;
    x_ccdf_rdc(1,:) = s'*sqN; % #Nblc Signal 
    CFx_rdc(ii,ifram)=PAPR(x_ccdf_rdc(1,:)); %Crest Factors
    ik = ik + 1;
end  

I am dividing the time signal into sub-blocks, trying to find the optimum complex number (rdc) that minimizes the norm of the sub-blocks. This rdc needs to be extracted from each sub-block element. I was using CVX optimization as below in dotted line but I am iterating more than 1e4 and this is taking a lot of time. Is there any function in optimization toolbox which can i use ?

cvx_begin;
variable rdc complex;
minimize(max(abs(s_sub{ii} - rdc)));
cvx_end;

Answer 1

[Torsten]

for ii  = 1:N
    rng default
    rdc0 = 1+1i; % You add one complex value to try to minimize the resulting sum of squares
    fun = @(rdc_1)abs(rdc_1 - s(ii*M-M+1:M*ii)); % x expands to the size of M
    [rdc_1,res] = fminimax(fun,rdc0);
    rdc_1s = [rdc_1s rdc_1];
    s_1(ii*M-M+1:M*ii)  = s(ii*M-M+1:M*ii) - rdc_1; 
end

[Ahmet Sacid Sümer]

fminimax did not accept complex number. I try to seperate real and imaginary part and try to optimize but i could not do it.

[Torsten]

for ii  = 1:N
    rng default
    rdc0 = 1+1i; % You add one complex value to try to minimize the resulting sum of squares
    fun = @(rdc_1_r,rdc_1_i) (rdc_1_r-real(s(ii*M-M+1:M*ii))).^2+(rdc_1_i-imag(s(ii*M-M+1:M*ii))).^2;
    %fun = @(rdc_1)abs(rdc_1 - s(ii*M-M+1:M*ii)); % x expands to the size of M
    [rdc,res] = fminimax(@(x)fun(x(1),x(2)),[real(rdc0),imag(rdc0)]);
    %[rdc_1,res] = fminimax(fun,rdc0);
    rdc_1 = rdc(1) + 1i*rdc(2);
    rdc_1s = [rdc_1s rdc_1];
    s_1(ii*M-M+1:M*ii)  = s(ii*M-M+1:M*ii) - rdc_1; 
end

[Ahmet Sacid Sümer]

Thank you very much. It works.

[Torsten]

I just observed that you will have to replace the line

fun = @(rdc_1_r,rdc_1_i) (rdc_1_r-real(s(ii*M-M+1:M*ii))).^2+(rdc_1_i-imag(s(ii*M-M+1:M*ii))).^2;

by

fun = @(rdc_1_r,rdc_1_i) sqrt((rdc_1_r-real(s(ii*M-M+1:M*ii))).^2+(rdc_1_i-imag(s(ii*M-M+1:M*ii))).^2);

If not, you find rc that minimizes max((abs(rc-s))^2), not max(abs(rc-s)).

[Ahmet Sacid Sümer]

Thank you. My goal is minimize the power so, i need to minimize max((abs(rc-s))^2) but is it make a difference ? if i find rc that minimizes the max(abs(rc-s)) that is mean also i find that minimize the max((abs(rc-s))^2) ?

My question is why do i use rng default. Is it not necessary right ?

Because when I used this I got the same symbols even though I wanted full randomness.

data_info_bit = randi([0,1],N_bits_perfram,1); %Random Bit Generation

from this line i think. But when i removed i, got fluctuating BER-SNR graph interestingly

Answer 2

[Alan Weiss]

I am not sure that I understand exactly what you are doing, but if you are willing to accept a least squares solution (rather than minimum absolute value) then you can probably use lsqnonlin to minimize the complex norm. See Complex Numbers in Optimization Toolbox Solvers.

Alan Weiss

MATLAB mathematical toolbox documentation

[Ahmet Sacid Sümer]

I have a complex array lets say 1x8. I want to minimize peak value of this array by substracting constant complex number lets called it rdc. I have not function, i have only an array. My thing is i need to substract rdc from every element of array. So rdc needs to be reduce the peak but also not create another one.

1+1i

-2-2i

in this example my rdc is -0.5-05i

(1+1i)-(-0.5-05i)=1.5+1.5i

(-2-2i)-(-0.5-0.5i)=-1.5-1.5i

I need to find rdc.

Thank you for your interest

[Alan Weiss]

Try this:

rng default
M = 100*randn(10,1) + 100*randn(10,1)*1i; % 10 random complex points
x0 = 1+i; % You add one complex value to try to minimize the resulting sum of squares
fun = @(x)(x-M); % x expands to the size of M
[x,res] = lsqnonlin(fun,x0)

Local minimum possible.
lsqnonlin stopped because the size of the current step is less than
the value of the step size tolerance.

x = 62.4282 + 70.4894i

res = 4.1568e+05

Alan Weiss

MATLAB mathematical toolbox documentation

[Ahmet Sacid Sümer]

N               = 4; 
M               = 16; 
M_mod           = 4;
M_bits          = log2(M_mod);
N_syms_perfram  = N*M;
N_bits_perfram  = N*M*M_bits;        
data_info_bit   = randi([0,1],N_bits_perfram,1); %Random Bit Generation
data_temp       = bi2de(reshape(data_info_bit,N_syms_perfram,M_bits)); %Data
x               = qammod(data_temp,M_mod,'gray'); %Symbols
x               = reshape(x,N,M);
s               = OTFS_modulation(N,M,x); %Time Signal 
rdc_1s          = [];
rdc_2s          = [];
for ii  = 1:N
    rng default
    rdc0 = 1+1i; % You add one complex value to try to minimize the resulting sum of squares
    fun = @(rdc_1)(rdc_1 - s(ii*M-M+1:M*ii)); % x expands to the size of M
    [rdc_1,res] = lsqnonlin(fun,rdc0);
    rdc_1s = [rdc_1s rdc_1];
    s_1(ii*M-M+1:M*ii)  = s(ii*M-M+1:M*ii) - rdc_1; 
end
% CVX Optimization
for ii  = 1:N
    cvx_begin;
    variable rdc_2 complex;
    minimize(max(abs(s(ii*M-M+1:M*ii) - rdc_2)));
    cvx_end;
    rdc_2s = [rdc_2s rdc_2];
    s_2(ii*M-M+1:M*ii)  = s(ii*M-M+1:M*ii) - rdc_2;       
end

It did not work. Convex work properly. s_1 should be almost equal to s_2. Also i get this

Local minimum found.

Optimization completed because the size of the gradient is less than

the value of the optimality tolerance.

<stopping criteria details>