Pascal's triangle
Since founded Pascal’s triangle has proved to be a very useful applications in mathematics and in other fields. One of these wonderful applications is used to the bilinear z- transformation to convert from s-domain to z-domain and inverse from z-domain to s-domain. By using the Pascal’s triangle, it can make easier to compute and hand-calculated the matrix equation of the analog and digital coefficients. One of the most useful applications of the Pascal’s triangle is to find coefficients and expand the binomial expression and it can be shown in figure below.
The main key is addressed in the next sections considering some specify matrices which are involved the Pascal’s triangle.
The matrix [T] and [Th]
A wonderful application of the Pascal’s triangle is for expansion of a binomial expression (U+L) power of n. Let insert zeros into the Pascal’s triangle to make a matrix T which has a size of (n+1; 2n+1) and matrix Th is a left-half of the matrix TUL having a size of (n+1;n+1) as shown below and they are used for band pass, band stop and narrow band filter.
In the case of low pass and high pass filetr, matrix [Tc] and [Tt] will be used respectively as show
![Diagonal matrix [Tc] and [Tt] for low pass and high pass filter](https://static.wixstatic.com/media/9c282a_4e2bc4733f14472783c4316a2df141d4.png/v1/fill/w_588,h_117,al_c,q_85,usm_0.66_1.00_0.01,enc_avif,quality_auto/9c282a_4e2bc4733f14472783c4316a2df141d4.png)
The matrix [P]
The matrix [P] contains the positive and negative binomial coefficients of the Pascal’s triangle in the first, last row and the first, last column corresponding to the edge size and the nth row of the Pascal’s triangle and another element in the matrix [P] can be calculated from its left, diagonal and above element. It has a size of (N+1, N+1). There are two different matrices respectively for low pass filter PLP and PHBS for high pass, band pass, band stop and they can be found as:
Inverting of matrix [Tc] and [Tt]
The matrix [Tc] and [Tt] are the diagonal matrix, so the inverse of them can be obtained by replacements each element in diagonal with its reciprocal as illustrated below
Inverting of matrix [P] and [Th]
There are some features of the matrix [P] and they can be used to find the inverse matrix. For the matrix low pass [PLP], if multiply the matrix by itself will give a diagonal matrix with all the numbers in the diagonal are equal to 2 power of n, and from this the inverse matrix [PLP] can be found as:
The inverse of the matrix [PHBS] can be found similar way and can be written as
The matrix [T], [Tc], [Tt]. [P] and inverting of matrix [Th], [Tc], [Tt], [P] are easy for hand-calculated and programming, also they are used to make the Pascal matrix equations which will be introduced in the sections "Analog to digital filter" and "Digital to digital filter".