Separating waves of very close wavelengths

Question

I'm new to wave problems, so excuse my simplified words. I have a signal consists of high range of frequencies . I applied a Band-pass filter to this signal and I have been able to separate it to bands of specific frequencies.

The problem that I want to solve is as follows: Each band have a lot of individual waves of VERY, VERY close wavelengths. For instance 4.08 Hz and 4.09 Hz. even we may have 4.001 Hz and 4.002 Hz as well. The signal data is limited and I can't push the filter to any better resolution.

The question now, How I can separate the individual waves in each band? And if possible please refer me to any books which deal with such problem.

Thanks in advance,

Answer 1

You have an incorrect approach here. Instead of trying to design small range filters, you should just do frequency-domain analysis.

Say you have a signal $x$ consisting of $N$ samples, sampled at the rate of $f_s$ Hz. For simplicity, let's take an example with $N = 1000$ and $f_s$ = 1000 Hz. The Nyquist frequency is half of the sampling frequency, which is $500$ Hz in this case. This means that the signal you are sampling should not have frequencies above that (otherwise there would be aliasing).

Now, if you want to know what frequencies you have in your signal, compute the discrete Fourier transform (DFT). DFT is defined as

$ X(k) = \sum_{n=0}^{N-1} x(n)e^{-2\pi k\frac{n}{N}} \quad 0 < k < N-1 $

From the DFT, you get two things. First, $|X|$ is called the magnitude spectrum, which is of your interest. The second thing would be the phase spectrum, given by $arg(X)$. DFT is symmetric from the middle for real-valued inputs, so the frequency resolution you get in our example is $1$ Hz. Now, say that you would also want to know the half frequencies. Looking at the definition of DFT, you can see it is $N$-point. If you want, you can do it likewise for $M$-points. Thus, to get the half frequencies as well, take $M=2000$.

In the case of your question, what you need is a really long DFT.

Other things to note: do not compute DFT from its definition. Use a fast Fourier transform (FFT). It's faster, and more precise.

Answer 2

You probably have answers since May, but here are a couple of comments anyway.

There are several related questions:
FFT-for-a-specific-frequency-range mentions
$\quad$ZoomFFT — a clear 9-page tutorial, with nice pictures
$\quad$chirp-DFT like a magnifying glass for a certain frequency range
how-can-i-compute-a-log-spaced-power-spectrum

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A quite different method is to make a matrix with columns sin-cos on a fine grid of frequencies $$\quad\quad A = [ \cos\ (2 \pi\, f_0), \sin\ (2 \pi\, f_0), \cos\ (2 \pi\, f_1), \sin\ (2 \pi\, f_1) ... ]$$ and solve $\quad A x = \text{data}\quad $ by least squares.
Some advantages of this approach:
a) you can choose any frequency grid you like, and any basis you like: sin-cos, wavelets ...
b) simple to program, a tiny wrapper around SVD
c) SVD is robust
d) it's easy to add smoothing constraints, i.e. regularize $x$ .

Disadvantages ? don't know — experts please comment. It does seem to be well-known to those who know it well: Least-squares spectral analysis has links to 20-odd papers and books, sigh.