Frequency Transformationi.e. Low Pass to High Pass Filter etc.

Frequency Transformations• We need to apply a suitable frequency transformation, if we wish to

designbandpass, bandstop and high-pass filters, using the low-pass approximatingfunction analysis

Filter Transformations• We can use the concept of filter transformations to determine the

new filter designs from a lowpass design. As a result, we can construct a 3rd-order Butterworth high-pass filter or a 5th-order Chebychev bandpass filter!

Filter Transformations

Normalized Lowpass Filter• When designing a filter, it is common practice to first design a

normalized low-pass filter, and then use a spectral transform to transform that low-pass filter into a different type of filter (high-pass, band-pass, band-stop).

• The reason for this is because the necessary values for designing lowpass filters are extensively described and tabulated. From this, filter design can be reduced to the task of looking up the appropriate values in a table, and then transforming the filter to meet the specific needs

Lowpass to Lowpass Transformation

• Having a normalized transfer function, with cutoff frequency of 1 Hz, one can modify it in order to move the cutoff frequency to a specified value

Transfer Function

Analog Element Values

Lowpass to Highpass

Lowpass to Highpass

Normalised HPF response

Lowpass to Highpass

Equivalent LPF response

we define values for the transformed frequency Ω as

Lowpass to Highpass

but with frequency transformation, we substitute for ω/ωp with

In the Chebychev case, we apply the substitution

Conversion of Low-pass and High-pass Filter transfer functions from continuous time to discrete time difference equations.

• The following converts two filter transfer function that are represented in the Laplace Space

• (Continuous time) into their discrete time equivalents in the Z-space using the Bilinear Transform

• (AKA Tustin’s Method), then converts them to difference equations expressing the current output as a combination of previous inputs and outputs.

Definition of Values

Definition of Values

Low Pass Filter

Conversion

Low Pass Filter

Conversion

High Pass Filter

Conversion

High Pass Filter

Conversion

High Pass Filter

Conversion