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I am trying to estimate the clean form of a time series, $u(t)$ that is corrupted by additive White Gaussian noise $w(t)$ at a particular SNR. The received signal is:

$$y(t) = u(t) + w(t)$$

  1. My first choice was to apply MLE or OLS since the gaussianity would yield an optimal estimate. But I don't know which terms would go into the pseodiinverse component: $u_{est} = y*pinv(\cdot)$ ?

Is OLS the correct approach or is there any other technique?

  1. I thought of applying state-space estimators such as Kalman filtering but I do not understand how the design matrices would be applicable here. Is there a simpler technique?
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  • $\begingroup$ Start with a simple OLS estimator. Since the noise is white, you have a nice closed-form expression to compute $\hat{u}(t)$ $\endgroup$
    – Maxtron
    Aug 23 at 22:45
  • $\begingroup$ Thank you for your comment. I tried writing out the OLS equation but since there is no deterministic parameter ,I don't know what term goes inside the pseudo inverse. I have mentioned this as a question. Could you please help with the OLS formulation? $\endgroup$
    – Sm1
    Aug 24 at 19:33
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I think you may do one of the following:

  1. Given a Parametric Model of the Signal
    You may use least squares. In case the model is Linear you may use linear least squares (For instance, polynomial regression). If the model is not linear, then a non linear least squares.
  2. Given a Dynamic Model of the Signal
    If you have a model which connect the signal u[t] to u[t - 1] then you may use the Kalman Filter framework. There also linear and non linear flavors here.
  3. No Model for Signal
    You may use Kernel Smoothing which is a non parametric method and will reduce the noise.
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  • $\begingroup$ Thank you for your answer. I tried writing out the OLS equation but since there is no deterministic parameter ,I don't know what term goes inside the pseudo inverse. I have mentioned this as a question. Could you please help with the OLS formulation? Also what do you mean by "model is linear"? I just have an observed noisy time series/signal and would want to find out the clean signal. In the OLS formulation what goes into the pine terms since I do not have any deterministic term ie: $u_{hat}(t) = y(t)*pinv(?)$ Could you please help how to apply OLS $\endgroup$
    – Sm1
    Aug 24 at 19:37
  • $\begingroup$ If you don't have a model (Like polynomial model) you can't use OLS. $\endgroup$
    – David
    Aug 26 at 10:18
  • $\begingroup$ Can I reformulate my problem of estimating the time series by using polynomial? Something like $y(t) = \theta(t)* u(1) + ....+\theta(t)*u(t) + w(t)$? How many terms should I have? Or do I use some time series model such as ARMA? $\endgroup$
    – Sm1
    Aug 26 at 14:27
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Hi: The random walk + noise kalman filter formulation is

$y_{t} = u_{t} + w_{t}$ # observation equation

$u_{t} = u_{t-1} + \epsilon_{t}$ # state equation

But, the way you wrote it, it seems like $u_{t}$ is constant and there is no state equation which means that it's just ols and the $\hat{u}$ estimate is $\bar{y}$.

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