# filter out an expected but not regular signal

I am stuck with a case where I need to get rid of a portion of sinusoidal signal that suddenly appears in my main signal (sine wave).

my total signal will look like:

$s(n) = \begin{cases} A\sin(\frac{2\pi}{8000}*n + \theta) + b\sin(\omega*n) & \text{if } 4500 \leq n \leq 5500 \\ A\sin(\frac{2\pi}{8000}*n + \theta) & \text{elsewhere} \end{cases}$

in the given example, I have $8000$ samples which represents one period of the main sine signal, but at a certain range I have the interference of another(weaker) sine signal, that occurs and disappears abruptly (for almost 1000 samples out of total 8000).

My previous approach:

The strange signal used to appear for a very short range (around 50 samples out of 8000), and also a less amplitude (experimentally), so I used to pass all my signal through a Least-squares estimator to come out with the closest $A\sin(\alpha)+B\sin(\beta)$. It used to give a satisfying results. But for the current conditions, least squares is really helpless in front of the big impact of this [noise].

How I am trying to solve

Now, I am thinking of correlation, I am trying to gather as more information as possible about the (noise) signal through experiments, once I can expect the exact shape (freq, length, ...). I am thinking of applying a correlation between $s(n)$ and the designed $noise$. [still in process, but I would love to hear guidance and advise].

Here I attach an illustrative image of the case • Do you know anything about $\omega$, the frequency of the interference? Couldn't a low pass filter do the job? – Matt L. Feb 9 '15 at 13:02
• @MattL. as you can see in the example, $\omega$ is roughly 8 times bigger than the main frequency, so hard to split them (nicely) with LPF. In addition to that, the $\omega$ signal appears only in a specific range, which makes it harder to apply any differentiation, it risks of getting unstable sharp edges. – chouaib Feb 9 '15 at 23:50
• You could try using a least squares estimator for state $[A,\theta, b, \omega, n_1,n_2]$ where $[n_1, n_2]$ is the range of samples with the interference signal. I am also a bit worried about the phase of the interference, maybe that should also be in the state. – Conrad Turner Feb 10 '15 at 11:08
• If $\omega$ is about 8 times the frequency of the desired sine, I don't see why a low pass filter wouldn't help. – Matt L. Feb 10 '15 at 12:30
• @MattL. I would appreciate if you can help designing an LPF for this example, it'd be nice if you add it as an answer – chouaib Feb 11 '15 at 0:47