Reduce signal length while maintaining properties?

Question

I am new to DSP.

So I have a noisy signal with some high-frequency components. I was able to characterize the statistical properties of this signal through a gaussian distribution. My aim is to run a Singular Spectrum Analysis in order to remove some components at certain frequencies and see if they impact the Gaussian distribution (I assume they will not but have to prove this). However, the signal is at a considerably high sampling rate (256 Hz) and for a larger period of time (256 s) and I would have to do this for more than one signal. Hence, this process requires a large computational time, which I can't really afford right now.

I attempted to just use a smaller segment of the signal but noticed that the FFT properties change considerably. So is there a way to reduce the length of the signal while maintaining the FFT properties and the statistical distribution for faster analysis (Keeping in mind that I am new to DSP)?

Thank You.

I have attached the SSA code below.

Tp = 1/freq_intrest;
tendIn=size(y,2)/256;
tendIn=tendIn-mod(tendIn,Tp);
tendindex=(tendFFT*256)-mod(tendFFT*256,2);
M = round(tendFFTindex/2 - mod(tendFFTindex/2,Tp*256)); %Window Length

N = length(y);
t = (1:N)';

Y=zeros(N-M+1,M);
for m=1:M
    Y(:,m) = y((1:N-M+1)+m-1);
end
Cemb=Y'*Y / (N-M+1);
C=Cemb;

[RHO,LAMBDA] = eig(C);
LAMBDA = diag(LAMBDA);               % extract the diagonal elements
[LAMBDA,ind]=sort(LAMBDA,'descend'); % sort eigenvalues
RHO = RHO(:,ind);                    % and eigenvectors
PC = Y*RHO;

RC=zeros(N,M);
for m=1:30                  % m upto 30 to save comptuational time. Already noticed only PC(1:8) are important.
    buf=PC(:,m)*RHO(:,m)';  % invert projection
    buf=buf(end:-1:1,:);
    for n=1:N               % anti-diagonal averaging
        RC(n,m)=mean( diag(buf,-(N-M+1)+n) );
    end
end

Answer 1

So, after reading our discussion, and understanding your performance problems a bit better, it'd seem to me that indeed, there's a way that can do something very similar to SSA on the full correlation matrix.

However it is not SSA in the strict sense of the word, but works by assuming a finite-sized noise subspace, and a finite-sized signal subspace, which you find through the eigenvalue decomposition from a smaller autocorrelation matrix estimate $\mathbf C$, and projecting "test functions" (in this cases, sinusoidal sample vectors) into that to check for presence of a frequency component.

Therefore, this can greatly reduce the time spent on decomposing the matrix, but it also doesn't give you the same result as SSA – unless you're very clear about presence of discrete tones in your signal.

Anyway, the method I describe above is called MUSIC, and it would work for your purpose:

My aim is to run a Singular Spectrum Analysis in order to remove some components at certain frequencies and see if they impact the Gaussian distribution

I assume you don't "know" certain frequencies beforehand (otherwise, you wouldn't need any spectral estimation to remove them), but need to estimate them precisely. You could do that using MUSIC / Root-MUSIC or ESPRIT (which is kind of similar in many ways), then synthesize the component (incl. correct phase and amplitude), subtract it and continue your statistical analysis.