Can someone show the details of how to apply AIC for sinusoidal models to specific data?
The PDF (likelihood function) of the multi-tone estimation problem is:
$$
{\cal L}({\bf x}; {\bf a}, {\bf f},{\bf \phi}, {\sigma}^2) = \frac{1}{(2\pi \sigma^2)^{\frac{N}{2}}} 
\exp\left( - \frac{1}{2\sigma^2} \sum_{n=0}^{N-1} ( x(n) - \sum_{p=1}^{P} {a}_p \cos(2\pi {f}_p + {\phi}_p) )^2   \right)
$$

The paper by Djuric quotes the AIC and MDL values as:



The code at the end attempts to:


Generate a 5-tone noisy signal.
Do the ML estimates of the amplitudes, frequencies and phases for a given value of $p$.
Find the (log) likelihood values for each value of $p$.
Calculate the AIC and MDL values from Djuric.


The plot for $p = 1 \ldots 60$ is:



Usually, the AIC over-estimates the model order. In this particular case, AIC gets 7 and MDL gets 5 (the correct value). The minima of each curve are plotted (as a o on the AIC curve and as a + on the MDL curve).

Djuric, P.M., "A model selection rule for sinusoids in white Gaussian noise," Signal Processing, IEEE Transactions on  (Volume:44 ,  Issue: 7 ), pp. 1744 - 1751.



scilab CODE ONLY BELOW

P = 5;
f = [ 0.1 0.12 0.231 0.333 0.478634];
phi = [0.329847 5.90324 3.09834 4.907983 1.984];
a = [ 1 1 2 2 3];
sigma = 1;

N = 100;

x = cos(2*%pi*[0:N-1]'*f + ones(N,1)* phi )*a';

xn = x + sigma*rand(N,1,'normal');


function [L,LL] = likelihood(data,ahat,fhat,phihat,sigmahat)
    N = length(data);
    datahat = cos(2*%pi*[0:N-1]'*fhat + ones(N,1)* phihat )*ahat';
    L = 1/(2*%pi*sigmahat*sigmahat)^(N/2) * exp(-1/(2*sigmahat*sigmahat)*sum((data - datahat).^2));
    LL = - log(2*%pi*sigmahat*sigmahat)*N/2 - -1/(2*sigmahat*sigmahat)*sum((data - datahat).^2);
endfunction

function [a,m] = aic(p,L,N)
    a = 3*p - log(L);
    m = 3*p/2*log(N) - log(L);
endfunction

function [ahat,fhat,phihat,regr] = sin_estimate(data)
    PAD = 10;
    N = length(data);
    DATA = fft([data; zeros((PAD-1)*N,1)],-1);
    [mx,ix] = max(abs(DATA(1:N*PAD/2,1)));
    ahat = mx/(N/2);
    fhat = (ix-1)/N/PAD;
    phihat = atan(imag(DATA(ix)),real(DATA(ix)));

    regr = data - cos(2*%pi*[0:N-1]'*fhat + ones(N,1)* phihat )*ahat
endfunction

pValues = [1:60];
likeValues = [];
loglikeValues = [];
aicValues = [];
mdlValues = [];
for p=pValues,
    signal = xn;
    a_est = zeros(1,p);
    f_est = zeros(1,p);
    phi_est = zeros(1,p);
    for pp=1:p
        [a_est(pp),f_est(pp),phi_est(pp),signal] =  sin_estimate(signal);   
    end

    [l,ll] = likelihood(xn,a_est,f_est,phi_est,sigma);

    likeValues = [likeValues l];
    loglikeValues = [loglikeValues ll];

    [aicv,mdlv] = aic(p,likeValues(p));
    aicValues = [aicValues; aicv]
    mdlValues = [mdlValues; mdlv]
end

clf;
plot(aicValues);
plot(mdlValues,'r');
title('AIC and MDL plots');