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Finite Impulse Response Filters

Overview

Given a finite duration of nonzero input values, an FIR filter will always have a finite duration of nonzero output values. No previous filter output value is used to determine a current output value; only input values are used to calculate output values.

Designing FIR Filters

Define \(H(m)\) over the frequency span \(-f_s / 2\) to \(f_s / 2\).

Algebraic
Coefficient Determination
  1. Develop an expression for the discrete frequency response \(H(m)\)
  2. Apply that expression to the inverse DFT equation to get the time domain \(h(k)\), where \(k\) is the time-domain index
  3. Evaluate that \(h(k)\) expression as a function of time index \(k\)
Inverse DFT \(\begin{align*}h(k) &= \frac{1}{N} \sum^{N / 2}_{m = -(N / 2) + 1} H(m) \exp \left\{j 2 \pi m k / N \right\} \\ &= \frac{1}{N} \frac{\sin\left( \pi k K / N \right)}{\sin \left(\pi k / N \right)}\end{align*}\)