In SciPy 'firwin' method is used to design fir filters using window method.

FIR Filter design

A digital filter is basically a computer algorithm which process discrete information to extract or eliminate unwanted data. There are two types of digital filters

- FIR ( Finite Impulse Response ) Filter
- IIR ( Infinite Impulse Response ) Filter

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- In SciPy a signal is considered as an array of real or complex numbers or as a NumPy array.

- In signal processing a  system that modifies or changes the content of an input signal is called a filter.

- The function of linear filters are described using convolution operation. 


Filtering

https://docs.scipy.org/doc/scipy/tutorial/index.html

SciPy Documentation

Convolution explains the connection between input signal ( $x[n]$ ) ,impulse response ( $h[n]$) and the output signal ( $y[n] $).

              $y[n] = x[n]*h[n]$

Convolution of 1-D signal can also be expressed as

             $ y[n] = \sum_{k=-\infty}^{+\infty} x[k] h[n-k]$

Discrete convolution can be expressed as 

           $y [n] = \sum_{k=max(n-M,0)}^{min(n,K)} x[k] h[n-k]$

In the above expression $x[n]$ and $h[n]$ are signals having finite length of  K+1 and M+1 respectively resulting in a finite signal of length K+M+1.



Convolution

Correlation produces a third signal from the input two signal and the produced signal is called the cross-correlation of the two input  signals.

  $ y[n] = \sum_{k=-\infty}^{+\infty} x[k] h[n+k]$

Correlation of finite length signal can be expressed as
 
   $y [n] = \sum_{k=max(0,-n)}^{min(K,M-n)} x[k] h[n+k]$




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