Filter

A medium through which a Signal propogates that alters its amplitude, frequency, or phase characteristics, in order to pass desired characteristics while attenuating undesired ones.

Basic Filter Types

Low-Pass Filter

Low-Pass Filter

A Filter that only passes low frequencies.

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High-Pass Filter

High-Pass Filter

A Filter that only passes highest frequencies.

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Band-Pass Filter

Band-Pass Filter

A Filter that only passes a specific Frequency Band

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Band-Stop Filter

Band-Stop Filter

A Filter that passes every frequency except for a specific Frequency Band

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Impulse Response Filters

Impulse Response

Impulse Response

In Digital Signal Processing, Impulse Response is the output of a system when it is subjected to an Impulse.

For example, the impulse response of a reverb filter would continuously fade the volume from the impulse over time. An echo filter would periodically repeat the impulse, while fading the volume over time. A gain filter would have no temporal effect like the aforementioned.

$h[n]$ is used to denote the impulse response of a system where

$$h[n]=T\{\delta [n]\}$$

Impulse response is important in DSP because if a system in a Linear Time-Invariant System, then knowing $h[n]$ means you know everything about the system, where $T$ is the system.

$$T\{x[n]\}=x[n]\ast h[n]$$Link to original

Finite Impulse Response Filter

Finite Impulse Response Filter

A Finite Impulse Response Filter is a Linear Time-Invariant System Filter whose Impulse Response eventually becomes zero, or in other words, the duration of response to the Impulse Signal is finite. This property is due to such filters being Feed-Forward.

$$\begin{array}{c}\mathcal{H}:{\mathbb{R}}^{\mathbb{Z}}\times \mathbb{Z}\to \mathbb{R}\\ \mathcal{H}(x,n)=y[n]\\ y[n]={\sum }_{k=0}^M{b}_k\cdot x[n-k]\end{array}$$

• Because FIR Filters are Feed-Forward, they are massively more parallel and belong in CUDA algorithms on the GPU, whereas IIR Filter is inherently iterative and thus is better suited for the CPU

Z Transform

$$\begin{array}{rl}H(z)& =\sum _{k=0}^M{b}_k\cdot z^{-k}\\ & =b_0\prod _{k=1}^M(1-c_k{z}^{-1})\\ & \end{array}$$

• See Z Transform

Impulse Response

$$h[n]=\{\begin{array}{ll}{b}_n& 0\le n\le M\\ 0& \text{else}\end{array}$$

• See Impulse Response

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Infinite Impulse Response Filter

Infinite Impulse Response Filter

An Infinite Impulse Response Filter is a Linear Time-Invariant System Filter whose Impulse Response never becomes zero, or in other words, the duration of response to the Impulse Signal is infinite. This property is due to such filters being recursively defined, i.e. having Feedback.

$$\begin{array}{c}\mathcal{H}:{\mathbb{R}}^{\mathbb{Z}}\times {\mathbb{R}}^{\mathbb{Z}}\times \mathbb{Z}\to \mathbb{R}\\ \mathcal{H}(x,y,n)=y[n]\\ y[n]={\sum }_{k=0}^M{b}_{k}x[n-k]-{\sum }_{k=1}^N{a}_k\cdot y[n-k]\\ \end{array}$$

• Because IIR Filters are Feed-Back
  • They are is inherently iterative and thus better suited for the CPU, rather than GPU which is where FIR belongs
  • Caching becomes even more important
  • They can be Divergent, which can lead to Overflow. Thus, Stability needs to be considered

Z Transform

$$\begin{array}{rl}H(z)& =\frac{{\sum }_{k=0}^M{b}_k{z}^{-k}}{{\sum }_{k=0}^N{a}_k{z}^{-k}},a_0=1\\ & =\frac{b_0{\prod }_{k=1}^M(1-c_k{z}^{-1})}{{\prod }_{k=1}^N(1-d_k{z}^{-1})}\\ & \end{array}$$

• See Z Transform

Divergence

For a system with poles $d_k$

• Asymptotically Stable $⟺\forall k,|d_k|<1$
• Marginally Stable $⟺\forall k,|d_k|\le 1$ and all poles with $|d_k|=1$ are Algebraic Multiplicity of 1
• Unstable $⟺\exists k,|d_k|>1$ or $\exists k,|d_k|=1$ with Algebraic Multiplicity greater than 1

Impulse Response

$$h[n]=\sum _{k=1}^N{A}_k(d_k{)}^{n}u[n]$$

• See Impulse Response

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