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This will probably be an extremely simple question for some one with any background in signal processing(not my background)

Let say I have signal $$x(t)=A\sin(\omega t)$$ where A is known and $\omega$ is unknown. Let say I only observe a noisy signal of $x(t)$ at discrete time periods $t=1,2,3....$. My signal $y(n)$ has the following form $$y(n)=x(n)+\epsilon$$ where $\epsilon\sim N(0,\sigma^2)$. My question is , what would be the best estimate of $y(n+1)$ given the observations till time time period $n$.

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    $\begingroup$ Assuming you have n evenly spaced samples of the noisy sine wave, why not curve fit a sine wave to the samples and then evaluate that fitted sine wave as desired? $\endgroup$ – Ed V Sep 21 at 22:04
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    $\begingroup$ tracking in this context typically means that the frequency varies over time. does your sine frequency shift with time? $\endgroup$ – Stanley Pawlukiewicz Sep 22 at 1:47
  • $\begingroup$ Hello, fortunately, the frequency does not vary with time?Also, can you guide me resources to read about curve fitting a sine wave? $\endgroup$ – kangkan Dc Sep 22 at 12:00
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    $\begingroup$ @kangkanDc too bad your frequency is fixed. There is a lot of material on using an extended Kalman filter for the slowly varying frequency problem. The EKF includes a prediction step. why do you need to predict the future sample? $\endgroup$ – Stanley Pawlukiewicz Sep 22 at 14:54
  • $\begingroup$ Imagine that one has an asset whose value varies periodically with fixed frequency(our x_t). Suppose one has a stock whose price depends noisily on the value of the asset (our y_t). In this situation, one would want to have the best prediction of y_t $\endgroup$ – kangkan Dc Sep 23 at 8:49
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That's actually not a trivial problem and the best approach really depends on your specific requirements.

My first approach would be a phase-locked loop and simply use the local oscillator as the predictor (with proper amplitude scaling). This can give you extreme accuracy if you are willing to pay with time.

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  • $\begingroup$ Thanks. is there any online resource that you can guide me to? $\endgroup$ – kangkan Dc Sep 22 at 12:00

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