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Dec 17th, 2017
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  1. % Test Cramer Rao
  2.  
  3. close all, clc, clear all
  4.  
  5. runs = 10;
  6. MSE  = 0;
  7. CRLB = 0;
  8. interpFactor = 500;
  9. for r = 1:runs
  10. Iterations = 500;
  11. N = 512;
  12. Delay = 0;
  13.  
  14. % 1 : Generate a Random White Gaussian Signal
  15. % ---
  16. signal = randn(1,N);
  17. signal = repmat(signal,[Iterations,1]);
  18.  
  19. % 2 : Generate the noise required to obtain a given SNR.
  20. % ---
  21. SNR = 5;
  22. noise_variance = 1/SNR* var(signal(1,:));
  23. noise  = sqrt(noise_variance)*randn(Iterations,N);
  24.  
  25. % 3 : Calculate the actual SNR for both the zero-mean noise and zero-mean
  26. %     signal
  27. % ---
  28. SNR = (rms(signal(1,:))/rms(noise(1,:)))^2;
  29.  
  30. % 4 : Corrupt signal with generated noise
  31. % ---
  32. signal_w_noise = signal + noise;
  33.  
  34. % 5 : Process cross correlations to calculate lags,
  35. %     interpolate as defined
  36. % ---
  37. x1 = interpft(signal_w_noise(1,:),N*interpFactor);
  38. lags = [];
  39. for seg = 1:Iterations
  40.     x2 = interpft(signal_w_noise(seg,:),N*interpFactor);
  41.     [Correlation, lag_array] = xcov(x1,x2);
  42.     [~,peak_index] = max(Correlation);
  43.  
  44.     coarse = lag_array(peak_index);
  45. %     fine = -(1/2)*(Correlation(peak_index+1) - Correlation(peak_index-1)) / (Correlation(peak_index+1) + Correlation(peak_index-1) - 2*Correlation(peak_index));
  46.  
  47.     lag = coarse/interpFactor;
  48.     lags = [lags lag];
  49. end
  50.  
  51. MSE  = MSE + (mean((lags - Delay).^2))/runs;
  52. CRLB =  CRLB + ((3/((pi.^2)*(N))) * ((1 + 2 .* SNR)./(SNR.^2)))/runs;
  53.  
  54. disp(CRLB/MSE)
  55. end
  56. % Calculate the mean square error and compare it to the CRLB
  57. % ---
  58.  
  59. disp(CRLB/MSE)
  60.  
  61. % CRLB is LARGER than the supposed limit.
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