First, I'd like to introduce a problem. I try to solve a set of linear equations in LS (Least Squares) sense, i.e. Ax=y (LS) => A^TA x = A^T y.

x is a vector I'm looking for, but y is a measured signal. This signal contains almost sinusoidal contributions with know frequencies, but as always in the nature, is contaminated with some noise. It's uniformly sampled (1hr). Matrix A contains the sinusoidal contributions that I know add to the signal, but I don't know how much each.

Prior to solving LS, I filter the data with Finite Impulse Response (FIR) lowpass filter, which is causal. Filtering is a convolution in time domain, no DFT is needed. Filter length (1/2) is excluded from the ends of time series.

The next step is the one that puzzles me. To avoid spectral leakage, I'd like to taper that data. Most people recommend that after taking the actual data set for analysis, taper them. For example, I want to investigate 6 months of data. Let's say my data is 5 years. I'd say, that okay, I have longer data set, so I use window taper (e.g. Hann) for this data block, and I select only some data for analysis afterwards. The others would taper 6-month block only.

Which one is correct and why? I actually quite don't understand this while in fact the problem boils down to algebraic set of equations (frequencies are known). Is it related to the noise level in LS setup? I've read tutorials about taper, but vast majority corresponds to DFT properties.

  • $\begingroup$ Okay, after a while I found an answer to this. In Fourier analysis we are given a set of frequencies. The same happens in the method of Least-Squares. The only difference how amplitude and phase are attributed is that the frequency distribution is not the same. In LS it's just given, in DFT it depends on the window length. However, it appears, that the leakage due to sampling a finite window is also the case. And it's window-length dependent. $\endgroup$ May 29, 2021 at 23:21


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