Digital-to-digital filter transformation

Using both Pascal matrix equations and Inverse Pascal matrix equations in making a conversion and inversion between the coefficients of an analog low pass filter transfer function H(s) and a digital transfer function H(z) and from them another matrix equations can be derived, called digital-to-digital filter Pascal matrix equations to transform a digital filter to another digital filter. The use of the Pascal's triangle in these matrix equations is very helpful, make them easier to transform between analog filter and digital filter and also for transforming between digital-to-digital filter. Therefore, the procedure of design a digital filter from another digital filter is presented in this article.

Pascal’s triangle is used in the bilinear transform with pre-warping frequency to convert from the transfer function H(s) in s-domain to the transfer function H(z) in z-domain and inverse from H(z) to H(s). Using Pascal’s triangle, the matrix equations are derived, called Pascal matrix equations and Inverse Pascal matrix equations, which are used to find the relationship between the coefficients of the transfer functions in s-domain and z-domain. From both these matrix equations, the relationship between the coefficients of a designed digital filter and the coefficients of a desired digital filter can be found, called Digital to Digital Pascal matrix equations and also they are used to transform a digital to another digital filter. To sum up, to design a digital filter, it can be done by transforming from an analog low pass filter or from a digital filter by using Pascal matrix equations and Inverse Pascal matrix equations.

Transforming between an analog low pass filter and a digital filter

The relationship between the coefficients of an analog filter [A],[B] and the coefficients of a digital filter [a], [b] are described on the Pascal matrix equations and the Inverse matrix equations.

Pascal matrix equations

The general formula of the Pascal matrix equations can be written as below and these equations are used to convert an analog lowpass filter to a digital filter:

[a] = [P]([A][T])

[b] = [P]([B][T])

Inverse Pascal matrix equations

The Inverse Pascal matrix equations are used to convert a digital filter back to an analog low pass filter as show below

[A] = ([P]⁻¹[a])[T]⁻¹

[B] = ([P]⁻¹[b])[T]⁻¹

Transforming a digital filter to another digital filter

In this section introduces a new method for designing a digital filter from another digital filter. A digital filter can be inverted to an analog low pass filter using the Inverse Pascal matrix equations. From an analog low pass filter, it is converted to a desired digital filter using the Pascal matrix equations. Figure 1 illustrates the transformation from one digital filter to another digital filter.

Let the transfer function of a given digital filter is H_g(z)

$$H_{g}(z) = \frac{\sum_{i=0}^{N} (a_{g})_{i} z^{-i}}{\sum_{i=0}^{N} (b_{g})_{i} z^{-i}}=\frac{{(a_{g})_{0}+(a_{g})_{1}z^{-1}+(a_{g})_{2}z^{-2} + \cdots + (a_{g})_{N}z^{-N}}}{{(b_{g})_{0}+(b_{g})_{1}z^{-1}+(b_{g})_{2}z^{-2} + \cdots +(b_{g})_{N}z^{-N}}}$$ And the transfer function of a desired digital filter is H_d(z)

$$H_{d}(z) = \frac{\sum_{i=0}^{N} (a_{d})_{i} z^{-i}}{\sum_{i=0}^{N} (b_{d})_{i} z^{-i}}=\frac{{(a_{d})_{0}+(a_{d})_{1}z^{-1}+(a_{d})_{2}z^{-2} + \cdots + (a_{d})_{N}z^{-N}}}{{(b_{d})_{0}+(b_{d})_{1}z^{-1}+(b_{d})_{2}z^{-2} + \cdots +(b_{d})_{N}z^{-N}}}$$ From Pascal matrix equations and Inverse matrix equations, the new equations, digital filter to digital filter Pascal matrix equations, are found. These equations are used to transform a digital filter to another digital filter:

[a_d]_{N + 1; 1} = [P_d]_{N + 1; N + 1}((([P_g]_{N + 1; N + 1}⁻¹[a_g]_{N + 1; 1})_{n + 1; 1}[T_g]_{n + 1; n + 1}⁻¹)[T_d]_{n + 1; N + 1})_{N + 1; 1}

[b_d]_{N + 1; 1} = [P_d]_{N + 1; N + 1}((([P_g]_{N + 1; N + 1}⁻¹[b_g]_{N + 1; 1})_{n + 1; 1}[T_g]_{n + 1; n + 1}⁻¹)[T_d]_{n + 1; N + 1})_{N + 1; 1}

Designing a digital filter using Pascal matrix equations and digital filter-to-digital filter Pascal matrix equations

There are two methods to design a digital filter based on a given analogue low-pass filter or a given digital filter . Figure 2 outlines a new algorithm for designing a digital filter below.

According to Figure 2, if given coefficients [A], [B] of an n^th-ordered analogue low-pass filter, apply Pascal matrix equations to convert them to coefficients [a], [b] of the desired N^th-ordered digital filter. If the given coefficients of a digital filter are [a_g], [b_g], then apply the digital filter-to-digital filter Pascal matrix equations to obtain the coefficients of the desired digital filter.

References

• 1. Phuoc Si Nguyen, Bilinear z-transformation and Pascal's triangle International Journal of Scientific & Engineering Research, Volume 4, Issue 12, December 2013, ISSN 2229-5518
• 2. Phuoc Si Nguyen, Convert An Analog Low Pass Filter To A Digital Filter Using Pascal’s Triangle International Journal of Advancements in Research & Technology, Volume 2, Issue 12,December-2013 ISSN 2278-7763
• 3. Phuoc Si Nguyen, Transform a Digital Filter from another Digital Filter Using Pascal’s Triangle International Journal of Scientific & Engineering Research, Volume 4, Issue 10, December 2013, ISSN 2278-7763
• 4. Phuoc Si Nguyen, Design of Digital Filters Using Pascal's triangle

See also

• Pascal's triangle
• Pascal matrix
• Filter transformation