The convolution of a signal \((\htmlClass{sdt-0000000041}{x}[\htmlClass{sdt-0000000117}{n}])\) and a filter \((\htmlClass{sdt-0000000058}{h}[\htmlClass{sdt-0000000117}{n}])\), represents the response of a system for a given signal.
| \( M \) | This is the symbol for the order of a difference equation. It refers to the maximum number of points back in a filter (or lags) that are used. In the case of a digital filter, it usually refers to the number of elements in the filter. |
| \( k \) | This symbol represents any given integer, \( k \in \htmlClass{sdt-0000000122}{\mathbb{Z}}\). |
| \( x \) | This symbol represents a function that represents a signal. |
| \( h \) | This is the symbol for a Finite Impulse Response (FIR), the unit Impulse Response (\( \htmlClass{sdt-0000000113}{h} \)) of a FIR filter. Because the result of this happens to be equal to the coefficients of the FIR filter, it is commonly also used to represent the FIR filter. |
| \( x \) | This symbol describes a discrete function. Discrete meaning that it only has a valid output for inputs from the set of integers \( \htmlClass{sdt-0000000122}{\mathbb{Z}} \). |
| \( \ast \) | This symbol represents convolution, a mathematical operation on two functions resulting in a third function. The convolution is obtained by taking the integral of the product of the two functions after reflecting one of the two function about the y-axis and shifting it. |
| \( \sum \) | This is the summation symbol in mathematics, it represents the sum of a sequence of numbers. |
| \( n \) | This symbol represents any given whole number, \( n \in \htmlClass{sdt-0000000014}{\mathbb{W}}\). |
The symbol \(x\) represents a function that represents a signal. It can either be continuous, in which case it will look like \(x(\htmlClass{sdt-0000000118}{t})\), or discrete, in which case it will look like \(x[\htmlClass{sdt-0000000117}{n}]\), where n is a whole number \(\htmlClass{sdt-0000000117}{n} \in \htmlClass{sdt-0000000014}{\mathbb{W}}\), representing the sample number by the analog to digital converter.
A convolution, denoted by the symbol \(*\), is a mathematical operation on two functions resulting in a third function. The convolution is obtained by taking the integral of the product of the two functions after reflecting one of the two function about the y-axis and shifting it. A convolution is commutative, meaning that it does not matter which of the two function is reflected and shifted. A convolution plays a big role in the definition and implementation of FIR filters. (General form of a Finite Impulse Response Filter)
Consider a FIR filter \(\htmlClass{sdt-0000000058}{h}[\htmlClass{sdt-0000000117}{n}]\) with coefficient values \(\htmlClass{sdt-0000000033}{b}_{\htmlClass{sdt-0000000015}{k}} = [1, 3, 5]\) and a signal \(\htmlClass{sdt-0000000061}{x}[\htmlClass{sdt-0000000117}{n}]\) with values \(\htmlClass{sdt-0000000041}{x}_{\htmlClass{sdt-0000000018}{i}}\) = [1, 2, 3, 4].
To get the convolution of the signal and the filter, we arbitrarily reflect one of the two about the y-axis and shift it over.
Flip the signal to get: [4, 3, 2, 1].
We shift the signal and add the terms after multiplying them to get [1, 5, 14, 23, 27, 20]