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3

Without more knowledge on $X$, we cannot say a lot. However, if it is IID with mean $\mu$ and variance $\sigma^2$, the sample mean $\hat{\mu} = \frac{1}{N}\left(\sum_{n=0}^{N-1}x_n\right)$ is unbiased, since: $$E[\hat{\mu}] = E\left(\frac{1}{N}\sum_{n=0}^{N-1}x_n\right) = \frac{1}{N}\sum_{n=0}^{N-1}E\left(x_n\right) = N\mu/N=\mu\,.$$ And its variance goes to ...

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A sine wave is cyclo-stationary so is not the best waveform to use for estimating delay between transmit and received signals. It will correlate whenever the delay causes the signal to be an additional $2\pi$ in phase offset. Consider using waveforms with better autocorrelation properties for this purpose such as pseudo-random noise sequences or other ...

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You basically have 4 models here: Accelerating to constant speed. Moving at constant speed. Decelerating to zero speed. Standing. So the basic solution is building the 4 models and switching using Hard Switch between them. Yet there is a smoother framework to handle smooth transition between them called Interacting Multiple Model (IMM) Kalman Filter. Using ...

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My question is, is there a more appropriate model of a Kalman filter for the type of car that I'm trying to predict? Probably yes, because you're generating a command for the car that (I presume) you know, but you're not using that knowledge in the filter. The model you're using in the Kalman filter is \$\mathbf{x}_k = \mathbf{x}_{k-1} + \mathbf{w_k},\ \...

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My question is, is there a more appropriate model of a Kalman filter for the type of car that I'm trying to predict? No. (But also please see below). Although the model works well for constant velocity, there's a trailing when the velocity goes from V to zero as Fig. shown. Is there a good solution to that? Yes. The Kalman filter includes a term for "...

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