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@@ -42,7 +42,7 @@ with $n$ the normalized version of your signal of length $N$. The time shift $k$
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For example the autocorrelation of an input signal $x=(2,3,-1)$ is $R_{xx}=(c_{-2}, c_{-1}, c_0, c_1, c_2)=(-2, 3, 14, 3, -2)$ and is symmetrical.
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>>>In the following table you can see an illustrative depicition on how the $c_k$'s are calculated.
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>> In the following table you can see an illustrative depicition on how the $c_k$'s are calculated.
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The header contains the input signal x padded with 0's on its sides.
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For autocorrelation, we compute correlation between $x$ and $x$ itself.
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So in visual terms, we slide $x$ from left to right across itself.
@@ -55,11 +55,11 @@ At each step we compute one $c_k$ by first multiplying the numbers that are alig
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| | | 2 | 3 | -1 | | | 4 + 9 + 1 = 14
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| | | | 2 | 3 | -1 | | 6 - 3 + 0 = 3
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| | | | | 2 | 3 | -1 | -2 + 0 + 0 = -2
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>>>As you can see, when reading from top to bottom we get the correct solution $R_{xx}=(-2, 3, 14, 3, -2)$.
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>>As you can see, when reading from top to bottom we get the correct solution $R_{xx}=(-2, 3, 14, 3, -2)$.
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So here are your tasks:
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1. Implement the `auto_corr` function as described above.
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>>>The function expects $x$ to be normalized. That means, that no normalization is done inside the `auto_corr`-function. Instead you normalize the input signal before calling `auto_corr`.
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>> The function expects $x$ to be normalized. That means, that no normalization is done inside the `auto_corr`-function. Instead you normalize the input signal before calling `auto_corr`.
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2. Check your implementation using `nox -s test`.
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If the test passes you can use `np.correlate` for efficiency in the following exercises!
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@@ -89,7 +89,7 @@ Now we want to compare this autocorrelation to the one of a random signal.
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$\pi \in \mathbb{R}$ denotes Pi, $\mu \in \mathbb{R}$ the mean and $\sigma \in \mathbb{R}$ the standard deviation for a random variable $X$.
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$e^x$ denotes the exponential function.
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A random variable $X$ having a gaussian pdf is described as gaussion or normal distribution $\mathcal{N}$.
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>>>Remark: In the notation $\phi_1(x | \mu, \sigma)$, x is the variable that is plugged into the function and $\mu$ and $\sigma$ are parameters which are needed to define the function and that are determined beforehand.
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>> Remark: In the notation $\phi_1(x | \mu, \sigma)$, x is the variable that is plugged into the function and $\mu$ and $\sigma$ are parameters which are needed to define the function and that are determined beforehand.
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2. Explore the behavior of $\mathcal{N}(\mu, \sigma)$ for different values of $\mu$ and $\sigma$.
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