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




Correlation

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

Filter Design

In mathematics, a convolution measures the overlap between two functions when one is flipped and shifted. For continuous functions, such as $$f, g: \mathbb{R}^d 	o \mathbb{R}$$, it is defined by the integral:
$$(f * g)(\mathbf{x}) = \int f(\mathbf{z}) g(\mathbf{x}-\mathbf{z}) d\mathbf{z}$$
For discrete sequences over $$\mathbb{Z}$$, the integral becomes a sum. The one-dimensional discrete convolution of functions $$f$$ and $$g$$ is:
$$(f * g)(i) = \sum_a f(a) g(i-a)$$
In signal processing, this operation connects an input signal $$x[n]$$ and an impulse response $$h[n]$$ to produce an output $$(x * h)[n] = \sum_{k=-\infty}^{+\infty} x[k] h[n-k]$$, with bounded summation for finite-length signals.

Convolution

- 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. 


San Jose State University

SciPy contains signal processing functions for

- B-splines

- Filtering

- Spectral Analysis

- Detrend

Signal Processing using SciPy

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

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