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@@ -145,7 +145,7 @@ We should now have all the tools we need to digitally implement any linear analo |
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#### IIR filters |
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An infinite impulse response (IIR) filter is a digital filter that implements all possible rational transfer functions. |
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By multiplying the denominator of the rational $H(z)$ definition above on both sides and applying it to an input $x_k$ and output $y_k$, we obtain |
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By multiplying the denominator of the rational $H(z)$ definition above on both sides and applying it to an input $x_k$ and output $y_k$, we obtain the difference relation |
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$$\sum_{m=0}^M a_m y_{k-m} = \sum_{n=0}^N b_n x_{k-n}$$ |
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Usually $a_0$ is normalized to 1, and $y_k$ can be written explicitly. |
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$$y_k = \sum_{n=0}^N b_n x_{k-n} - \sum_{m=1}^M a_m y_{k-m}$$ |
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@@ -157,7 +157,7 @@ $$H(z) = \frac{b_0 + b_1 z^{-1} + b_2 z^{-2}}{1 + a_1 z^{-1} + a_2 z^{-2}}$$ |
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#### FIR filters |
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A finite impulse response (FIR) filter is a specific case of an IIR filter with $M = 0$ (a transfer function denominator of 1). For an input and output signal, |
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A finite impulse response (FIR) filter is a specific case of an IIR filter with $M = 0$ (a transfer function denominator of 1). For an input and output signal, the difference relation is |
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$$y_k = \sum_{n=0}^N b_n x_{k-n}$$ |
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They are computationally straightforward and always stable since they have no poles. |
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@@ -185,7 +185,7 @@ The signal $h(t)$ is the result of processing a [delta function](https://en.wiki |
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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. |
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Repeating this process in the digital realm gives us the discrete convolution. |
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$$ y_k = \sum_{n=-\infty}^\infty h_n x_k $$ |
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$$ y_k = \sum_{n=-\infty}^\infty h_n x_{k-n} $$ |
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Note that $h_n$ is both non-causal (nonzero for negative $t$ or $n$) and infinitely long, which is addressed later. |
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@@ -205,6 +205,17 @@ where $\operatorname{sinc}(x) = \sin(\pi x) / (\pi x)$ is the [normalized sinc f |
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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. |
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### Window functions |
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*Coming soon* |
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### To-do |
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- digital filters |
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- windows |
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- oscillators |
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- minimum phase filters |
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- minBLEP/polyBLEP |
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- analog circuit modeling |
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- integration techniques |
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- optimization (will wait for https://github.com/VCVRack/manual/issues/3 to be completed) |
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- profiling |
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- mathematical optimization |
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- vector instructions |
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- compiler optimization |
Andrew Belt