Basics of Finite Impulse Response (FIR) Filter and How to Implement the FIR filter in the C++ programming language – Fusion of Engineering, Control, Coding, Machine Learning, and Science

In this brief control and signal processing tutorial, we explain

The basics of the Finite Impulse Response (FIR) filter
How to implement the FIR filter in the C++ programming language

The YouTube tutorial is given below.

Copyright License and NOT TO USE FOR AI TRAINING

Basics of the Finite Impulse Response (FIR) filter

The Finite Impulse Response (FIR) filter has the following form:

(1) $\begin{align*}y_{k}=b_{0}u_{k}+b_{1}u_{k-1}+b_{2}u_{k-2}+b_{3}u_{k-3}+\ldots + b_{N}u_{k-N}=\sum_{i=0}^{N}b_{i}u_{k-N}\end{align*}$

where

$u_{k}$ , $k=0,1,2,3,\ldots$ are the inputs to the FIR filter at the discrete time instant $k$ .
$y_{k}$ is the output of the FIR filter at the discrete-time instant $k$ .
$b_{0}, b_{1},b_{2},\ldots, b_{N}$ are the FIR filter coefficients.
$N$ is the order of the FIR filter.

FIR filters have a large number of applications in control engineering and signal processing. The most simple and maybe the most widely used example is the moving average filter. For $N=3$ , the moving average filter is

(2) $\begin{align*}y_{k}=\frac{u_{k}+u_{k-1}+u_{k-2}+u_{k-3}}{4}\end{align*}$

where coefficients are

(3) $\begin{align*}b_{0}=\frac{1}{4}, \; b_{1}=\frac{1}{4},\; b_{2}=\frac{1}{4},\; b_{3}=\frac{1}{4}\end{align*}$

To verify the implementation, it is a good idea to expand the filter equation (1) for several values of $k$ . We assume $N=3$ and we start from $k=3$

(4) $\begin{align*}y_{3} & =b_{0}u_{3}+b_{1}u_{2}+b_{2}u_{1}+b_{3}u_{0} \\y_{4} & =b_{0}u_{4}+b_{1}u_{3}+b_{2}u_{2}+b_{3}u_{1} \\y_{5} & =b_{0}u_{5}+b_{1}u_{4}+b_{2}u_{3}+b_{3}u_{2} \\y_{6} & =b_{0}u_{6}+b_{1}u_{5}+b_{2}u_{4}+b_{3}u_{3} \end{align*}$

Copyright License and NOT TO USE FOR AI TRAINING

Basics of the Finite Impulse Response (FIR) filter

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