Frequency Transformation

Frequency transformation is a process of synthesizing of filter (high pass, band pass, band stop) from a low pass prototype in which frequency variable of the transfer function is replaced by a function of frequency. The transformation of a network with low pass magnitude response into a low pass, high pass, band pass, band stop network can be directly onto the elements of the network. Such process is known as network transformation.

Frequency transformation is important because 

• the prototype low pass filter can be easily converted into high pass, band pass, band stop filter within same characteristics.

• it promised to utilize the method to approximate low pass filter to approximate the behavior of other filters.

The effects of frequency transformations are on

• magnitude response 

• network function 

• location of poles and zeros 

• network elements

Let \( s' = \sigma' + j \omega' \) be the complex frequency variable for low-pass functions, and  
let \( s = \sigma + j \omega \) be the complex frequency variable.  

Then, the frequency transformation is a function:  

\[
s' = f(s)
\]

Which maps one or several frequency ranges of the pass band of low pass characteristics then the remainder of the frequency will be mapped to stop band of low pass response. From analyzing Pass band and stop band of low pass filter we can obtain different types of transformation such as 

• Low pass to low pass

• Low pass to high pass

• Low pass to band pass

• Low pass to band stop