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Add mention of kernel to DSP

pull/8/merge
Andrew Belt 6 years ago
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      DSP.md

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@@ -182,11 +182,10 @@ $$ y(t) = (h \ast x)(t) = \int_{-\infty}^\infty h(\tau) x(t - \tau) d\tau $$
where $h(t)$ is the *impulse response* of our filter. where $h(t)$ is the *impulse response* of our filter.


The signal $h(t)$ is the result of processing a [delta function](https://en.wikipedia.org/wiki/Dirac_delta_function) through our filter, since $\delta(t)$ is the "identity" function of convolution, i.e. $h(t) = (h \ast \delta)(t)$. The signal $h(t)$ is the result of processing a [delta function](https://en.wikipedia.org/wiki/Dirac_delta_function) through our filter, since $\delta(t)$ is the "identity" function of convolution, i.e. $h(t) = (h \ast \delta)(t)$.
Clapping your hands or popping a balloon (both good approximations of $\delta$) in a large cathedral will generate a very sophisticated impulse response, which can be recorded and processed in a FIR filter algorithm to reproduce arbitrary sounds as if they were performed in the cathedral.
Clapping your hands or popping a balloon (both good approximations of $\delta$) in a large cathedral will generate a very sophisticated impulse response, which can be recorded as $h(t)$ and processed in a FIR filter algorithm to reproduce arbitrary sounds as if they were performed in the cathedral.


Repeating this process in the digital realm gives us the discrete convolution. Repeating this process in the digital realm gives us the discrete convolution.
$$ y_k = \sum_{n=-\infty}^\infty h_n x_{k-n} $$ $$ y_k = \sum_{n=-\infty}^\infty h_n x_{k-n} $$
Note that $h_n$ is both non-causal (nonzero for negative $t$ or $n$) and infinitely long, which is addressed later.




#### Brickwall filter #### Brickwall filter
@@ -205,6 +204,13 @@ where $\operatorname{sinc}(x) = \sin(\pi x) / (\pi x)$ is the [normalized sinc f
Although the impulse response is infinitely long, restricting it to a finite range $[-T, T]$ and shifting it forward by $T$ produces a finite causal impulse response that can be solved by a fast FIR algorithm to produce a close approximation of an ideal brickwall filter. Although the impulse response is infinitely long, restricting it to a finite range $[-T, T]$ and shifting it forward by $T$ produces a finite causal impulse response that can be solved by a fast FIR algorithm to produce a close approximation of an ideal brickwall filter.




### Windows

The impulse response $h_n$ is defined for all integers $n$, so it is both non-causal (requires knowledge of future $x(t)$ to compute $y(t)$) and infinitely long.

*TODO*


### To-do ### To-do


- digital filters - digital filters


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