% Low Complexity Magnitude Approximation
% LuminousLogic.com_


% Magnitude computation from complex numbers, mag = sqrt(I^2 + Q^2) can be
% expensive to implement.  The below algorithm is a low-cost alternative.


% The approximation is, mag ~= alpha*max(|I|,|Q|) + beta*min(|I|,|Q|)

% Alpha & Beta can be chosen to be hardware friendly (multiplies become shifts) 
% and optimized for either minimum RMS or minimum peak error.

% How It Works
% Absolute value operations limit I and Q to 0-90 degree range.
% Max,min further limit values to 0-45 degree range.
% In 0-45deg range, a linear combo of I&Q provides a good mag approximation.


% Initialize Workspace
clear all;
close all;
rand ('state',0);
randn('state',0);


% Simulation Parameters
num_iter   = 1E2;
alpha      = 1;
beta       = 1/4;


% Create a set of rectangular data
phase_I   = rand(num_iter,1)*2*pi;
phase_Q   = rand(num_iter,1)*2*pi;
rect_data = cos(phase_I) + 1i*sin(phase_I);


% Introduce random attenuation between 0 and 70 dB
atten_db  = rand(size(rect_data)) * 70;
rect_data = rect_data .* (10.^(-atten_db/20));


% Compute exact magnitudes
mag_exact    = abs(rect_data);
mag_exact_db = 20*log10(mag_exact);



% Approximate magnitudes
mag_approx = zeros(size(mag_exact));
for i=1:num_iter
    mag_approx(i) = alpha*max(abs([real(rect_data(i)) imag(rect_data(i))])) + beta*min(abs([real(rect_data(i)) imag(rect_data(i))]));
end
mag_approx_db = 20*log10(mag_approx);


% Plot Results
figure;
plot(mag_exact_db , 'k'); grid on; hold on;
plot(mag_approx_db, 'r');
legend('Exact','Simple Approx');
xlabel('Carrier #');
ylabel('Magnitude [dB]')