O'Reilly logo

Digital Filters Design for Signal and Image Processing by Mohamed Najim

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

4.3. Butterworth filters and the maximally flat approximation

4.3.1. Maximally flat functions (MFM)

Here we look at a low-pass normalized transfer function whose squared amplitude is shown in equation (4.3). We try to find a filter with the flattest possible frequency response in the passband when x is close to 0. To come as close as possible to the specification, the synthesized filter must have an amplitude diagram as flat as possible when x = 0. For that, we find the conditions that allow us to cancel the successive derivations of the function |H(jx)|2.

We know that the squared amplitude function |H(jx)|2, being analytic when x=0, can be developed by using the McLaurin series as follows:

images

This development introduces the successive derivatives of |H(jx)|2 used for x=0, or:

images

Moreover, |H(jx)|2 being a rational fraction (see equation (4.3)), we can approach it by a polynomial by proceeding to the division following increasing powers of polynomials that constitute the numerator and the denominator of |H(jx)|2. In this way we obtain the following development:

images

Since the McLaurin development is unique in the convergence region, we can then identify the coefficients of equations (4.7) and ...

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required