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| kb:wiener_filtering [2021-05-24 23:50] – jaeyoung | kb:wiener_filtering [2024-04-30 04:03] (current) – external edit 127.0.0.1 | ||
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| A causal Wiener filter allows us to predict future values of a random process $y[\cdot]$ given past values of a related process $x[\cdot]$. | A causal Wiener filter allows us to predict future values of a random process $y[\cdot]$ given past values of a related process $x[\cdot]$. | ||
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| - | That is, given $x[n], x[n - 1], \dots $, we can estimate $x[n+1]$. | ||
| To do this, we can create a model for $x[\cdot]$ that states that it is a filtered version of a white random process: | To do this, we can create a model for $x[\cdot]$ that states that it is a filtered version of a white random process: | ||
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| $$ x[n] = (f \ast w)[n] $$ | $$ x[n] = (f \ast w)[n] $$ | ||
| - | Here, $w[\cdot]$ is a white random process with unit intensity. | + | Here, $w[\cdot]$ is a white random process with unit intensity, and $f[\cdot]$ is the unit sample response of a stable, causal system whose inverse is also stable and causal. |
| Given this model, we know that: | Given this model, we know that: | ||