SVD Based Noise Filter on the Samples in Toeplitz Form

Question

Given a sequence of length $2N$ $$ x[1], x[2], \dots, x[N], \dots x[2N] $$ We can construct the following matrix along with its SVD. $$ M = \begin{pmatrix} x[1] & x[2] & \cdots & x[N] \\ x[2] & x[3] & \cdots & x[N+1] \\ \vdots & \vdots & \ddots & \vdots \\ x[N+1] & x[N+2] & \cdots & x[2N] \end{pmatrix} = U S V^T $$ Small singular values can be removed and matrix recomputed. Input signal can be replaced with e.g. 1st and last cols of $M$.

Unfortunately, I do not know the name and motivation behind this procedure. Where can I find a reference with a detailed description of such a filter? What is matrix $M$ called?

Example:

Test code in Mathematica:

(* define test signal and noise *)
length = 2^10 ;
amplitude = 1.0 ;
signal = amplitude*Sin[2*Pi*0.2741*Range[2*length]] ;
SeedRandom[1] ;
level = 0.1 ;
noise = RandomVariate[NormalDistribution[0.0,level*amplitude],2*length] ;
input = signal + noise ;
(* filter *)
matrix = Most[Partition[input, length, 1]];
{u,s,v} =SingularValueDecomposition[matrix,Tolerance -> 10^-15 ] ;
keep = 2 ;
s = Take[s,keep,keep] ;
u = Take[u,All,keep] ;
v= Take[v,All, keep] ;
matrix = u.s.Transpose[v] ;
result = {First[matrix],Rest[Last[matrix]]} //Flatten;
(* compare *)
Grid[{
{
ListPlot[signal,ImageSize ->300],
ListPlot[input,ImageSize ->300],
ListPlot[result,ImageSize ->300]
},
{
Periodogram[signal,Automatic,Automatic,KaiserWindow[#,Pi/2]&,PlotRange -> All, ImageSize ->300],
Periodogram[input,Automatic,Automatic,KaiserWindow[#,Pi/2]&,PlotRange -> All, ImageSize ->300],
Periodogram[result,Automatic,Automatic,KaiserWindow[#,Pi/2]&,PlotRange -> All, ImageSize ->300]
}
}]

Answer 1

The matrix M is called a Toeplitz matrix and they have a certain type of decomposition (but X should be the ACF and not the signal itself), so the resulting U and V would be a Vandermonde matrix which are very similar to Discrete Fourier transform.

So the whole process is similar to find spectrum of your signal and putting zero in bins which their power are below a threshold.