P1

Let $Y=X^2$, and let $X$ be a uniform random variable over $(-1, 1)$

1. Find the linear minimum mean squared estimator of $Y$ in terms of $X$
2. Find its mean squared error

Solution

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P2

Let us consider the estimation of the value of an (unknown) constant $s=s[n]$, $n$ given measurements

that are corrupted (but uncorrelated) with a zero mean white noise $w[n]$ that has variance $\sigma^2_w$

1. Write the estimation problem as a Kalman filter problem and compute the Kalman gain $K[n]$ and the variance of the estimation error $M[n|n]$
   (You are asked to find close forms of $K[n]$ and $M[n|n]$ as functions of $n,\sigma^2_w,M[-1|-1]=e^2_0$
2. Now suppose that we do not have no a priori information about $s$
   (i.e. $\hat s[-1|-1]=0$ and $e^2_0\rightarrow \infty$)
   Show that the Kalman filter simply becomes the sample mean
   $\hat s=\frac{1}{N}\sum^{N-1}_{n=0}x[n]$

Solution

• 다음 문제
  $\hat s[n|n-1]=a\hat s[n-1|n-1]=\hat s[n-1|n-1]\\[0.2cm] e^2_0\rightarrow \infty,\quad K[n]=\frac{1}{n+1},\quad M[n|n]=\frac{\sigma^2_n}{n+1}\\[0.3cm] \hat s[n|n]=\hat s[n-1|n-1]+K[n](x[n]-\hat s[n-1|n-1])\\[0.3cm] =\hat s[n-1|n-1]+\frac{x[n]}{n+1}-\frac{\hat s[n-1|n-1]}{n+1}\\[0.3cm] =\frac{n}{n+1}\hat s[n-1|n-1]+\frac{x[n]}{n+1}\cdots \color{red}(*)$

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P3

For the scalar state-scalar observation Kalman filter assume that $\sigma^2_n=0$ for all $n$ so that we observe $s[n]$ directly. Find the innovation wequence. Is it white?

Solution