All analog prototype filters, regardless of their type, have scaled frequency range so that the passband cut-off frequency amounts to Ω = 1. For this reason, it is necessary to scale filter during the design process, so that passband and stopband cut-off frequencies have the appropriate, predetermined values.

Reference analog prototype filter is also a low-pass filter so it requires to be transformed into the appropriate type of filter, i.e. high-pass, band-pass or band-stop filter, if needed.

Transformation from an analog prototype filter to appropriate analog filter is performed before transforming it in digital filter.

Frequency scaling depends on the type of analog filter being designed. Scaling is explained for low-pass, high-pass, band-pass and band-stop filters.

All the results obtained in this chapter are tested in the Filter designer tool program.

3.4.1 Low-pass filter

The transformation from a reference analog prototype filter to a low-pass analog filter is the simplest type of transformation. Analog prototype filter is a low-pass filter with the cut-off frequency of Ωp=1. In this case, the transformation comes to a simple frequency scaling. In the transform function, sΩc is used instead of s, where Ωc is a desirable cut-off frequency in the passband.

Generally, the transform function of the reference analog prototype filter can be expressed as follows:
where
H0 is a costant;
zk is the k-th zero of the transfer function of the reference analog prototype filter;
M is a number of zeros of the transfer function of the reference analog prototype filter;
pk is the k-th pole of the transfer function of the reference analog prototype filter; and
N is a number of poles of the transfer function of the reference analog prototype filter and the filter order as well.

By performing the transformation:
each expression within brackets in the transfer function numerator is transformed into:
and the whole numerator is transformed into:
Each bracket in denominator is transformed similarly:
and the entire denominator is transformed into:
By replacing numerator and denominator by their transformed expressions, the transform function of the analog filter is obtained:
Example:

The transform function of the Butterworth reference analog prototype filter of the 3rd order is expressed as follows:
The transformation in a band-pass analog filter with the cut-off frequency Ωc = 0.2929 in the passband is obtained via expression:
Example:

The transform function of inverse Chebyshev reference analog prototype filter of the 3rd order is expressed as follows:
The transformation to a band-pass analog filter with the cut-off frequency Ωc=0.3719 in the passband is obtained via expression:  

3.4.2 High-pass filter

Analog prototype filter is a low-pass filter with the cut-off frequency Ωp = 1. Its transformation to a high-pass analog filter can be split into two steps. The first step refers to the transformation to a high-pass analog filter, whereas the second one refers to frequency scaling. The final objective is that passband cut-off frequency of the resulting high-pass analog filter amounts to Ωc.

The transformation to a high-pass analog filter:
Scaling of frequency axis is performed by transformation:
These two transformations can be represented by one transformation:
Generally, the transform function of the reference analog prototype filter can be expressed as follows:
where:
H0 is a constant;
zk is the k-th zero of the transfer function of the reference analog prototype filter;
M is a number of zeros of the transfer function of the reference analog prototype filter;
pk is the k-th pole of the transfer function of the reference analog prototype filter; and
N is a number of poles of the transfer function of the reference analog prototype filter and the filter order as well.

By performing the following transformation:
each bracket in the numerator of the transfer function is transformed into
and the entire numerator is transformed into:
Each bracket in denominator is transformed similarly:
and the entire denominator is transformed into:
Substituting these transformed expressions for numerator and denominator, the transform function of the analog filter is obtained:
Example:

The transfer function of the Chebyshev reference analog prototype filter of the 3rd order is expressed as follows:
The transformation in a band-pass analog filter with the cut-off frequency Ωc = 0.3721 in the passband is expressed as:
The result is the transfer function of analog filter:
Example:

The transfer function of inverse Chebyshev reference analog prototype filter of the 2nd order is expressed as follows:
The transformation in a band-pass analog filter with the cut-off frequency Ωc=0.1434 in the passband is expressed as:
The result is the transform function of analog filter:  

3.4.3 Band-pass filter

The transformation in a band-pass analog filter is more complex than the transformation in a low-pass and high-pass analog filters. The filter order is doubled by this transformation. This is why it is not possible to design an odd order band-pass filter.

When designing, the required filter order is divided by two. The resulting filter order is used to design a low-pass reference analog prototype filter. By transforming it into a band-pass analog filter the filter order is doubled. The required filter order is obtained this way.

Example:

Assume that it is necessary to design an 8th order band-pass digital filter.

A low-pass reference analog filter of the 4th order is designed first.

Reference analog filter is further transformed in a bandpass analog filter. This transformation doubles the filter order. The order of the resulting filter is 8.

The transformation into a band-pass analog filter is expressed as:
The value of the constant Ω0 can be found via expression:
where:
Ωp1 is a lower cut-off frequency in the passband (refer to figure 3-4-1); and
Ωp2 is a higher cut-off frequency in the passband (refer to figure 3-4-1).  
The value of Ωc, which is necessary for normalization, is found via expression:
The transfer function of an analog prototype filter is transformed first in a band-pass analog filter, and normalized after that with Ωc.

Generally, the transfer function of the reference analog prototype filter can be expressed as:
where:
H0 is a constant;
zk is the k-th zero of the transfer function of the reference analog prototype filter;
M is a number of zeros of the transfer function of the reference analog prototype filter;
pk is the k-th pole of the transfer function of the reference analog prototype filter; and
N is a number of poles of the transfer function of the reference analog prototype filter and the filter order as well.

By performing the transformation:
each bracket in numerator of the transfer function is transformed into:
and the entire numerator is transformed into:
The scaling of frequency axis is performed after transformation:    
Each bracket in denominator is transformed similarly:
and the entire denominator is transformed into:
The scaling of frequency axis is performed after transformation:        
Substituting the transformed expressions for numerator and denominator, the transform function of analog filter is obtained:    
Example:

The transfer function of the Chebyshev reference analog prototype filter of the 2nd order is expressed as:
The transformation in a band-pass analog filter with the cut-off frequency Ωc = 0.1626 in the passband is expressed as:
The result is the transfer function of analog filter:  

3.4.4 Band-stop filter

The transformation in a band-stop analog filter is similar to the transformation in a bandpass analog filter. Similarly, the filter order is doubled and the order of a band-stop digital filter cannot be an odd number.

When designing, the required filter order is divided by two. The resulting filter order is used for designing a low-pass reference analog prototype filter. The transformation into a bandstop analog filter causes the filter order to double. This is how the required filter order is obtained.

Example:

It is necessary to design a band-stop digital filter of the 10th order.

A low-pass reference analog filter of the 5th order is designed first.

The reference analog filter is transformed in a band-stop analog filter. This transformation causes the filter order to double. The result is a 10th order filter.

The transformation in a band-stop analog filter:
The value of the constant Ω0 can be found via expression:
where:
Ωs1 is a lower cut-off frequency in the stopband (refer to figure 3-4-2); and
Ωs2 is a higher cut-off frequency in the stopband (refer to figure 3-4-2).  
The value of Ωc, which is necessary for normalization, is found via expression:
The transfer function of the reference analog prototype filter is transformed first into a band-stop analog filter, and normalized after that with Ωc.

Generally, the transfer function of the reference analog prototype filter can be expressed as follows:
where:
H0 is a constant;
zk is the k-th zero of the transfer function of the reference analog prototype filter;
M is a number of zeros of the transfer function of the reference analog prototype filter;
pk is the k-th pole of the transfer function of the reference analog prototype filter; and
N is a number of poles of the transfer function of the reference analog prototype filter and the filter order as well.

The first step refers to normalization with the frequency Ωc:
The transformation is performed in the second step:
and each bracket in the transfer function numerator is transformed in:
and the entire numerator is transformed in:
Each bracket in denominator is transformed similarly:
and the entire denominator is transformed in:
Substituting the transformed expressions for numerator and denominator, the transform function of analog filter is obtained:
Example:

The transfer function of inverse Chebyshev reference analog prototype filter of the 2nd order is expressed as:
The transformation into a band-stop analog filter with the cut-off frequency Ωc=0.252.5727 in the passband is expressed as:
The result is the transfer function of analog filter: