Transfer function filter utilizes the transfer function and the Convolution theorem to produce a filter. In this article, an example of such a filter using finite impulse response is discussed and an application of the filter into real world data is shown.

In digital signal processing, an FIR filter is a time-continuous filter that is invariant with time. This means that the filter does not depend on the specific point of time, but rather depends on the time duration. The specification of this filter uses a transfer function having a frequency response that will only pass the desired frequencies of the input. This type of filter is non-recursive, which means that the output can be completely derived from a combination of the input without any recursive values of the output. This means that there is no feedback loop that feeds the new output the values of previous outputs. This is an advantage over recursive filters such as IIR filter (Infinite Impulse Response) in applications that require a linear phase response because it will pass the input without phase distortion.

Let the output function be ${\displaystyle y(t)}$ and the input is ${\displaystyle x(t)}$. The convolution of the input with a transfer function ${\displaystyle h(t)}$ provides a filtered output. The mathematical model of this type of filter is:

h(${\displaystyle \tau }$) is a transfer function of an impulse response to the input. The convolution allows the filter to only be activated when the input recorded a signal at the same time value. This filter returns the input values (x(t)) if k falls into the support region of function h. This is the reason why this filter is called finite response. If k is outside of the support region, the impulse response is zero which makes output zero. The central idea of this h(${\displaystyle \tau }$) function can be thought of as a quotient of two functions.

According to Huang (1981) Using this mathematical model, there are four methods of designing non-recursive linear filters with various concurrent filter designs:

Define the input signal:

${\displaystyle rand({\begin{bmatrix}1&200\end{bmatrix}})}$ adds a random number from 1 to 200 to the sinusoidal function which serves to distort the data.

Use an exponential function as the impulse response for the support region of positive values.