# How to Use a Frequency Transformation in Filter Design

January 20, 2021

### Key Takeaways

• Signal processing makes use of sampling, impulse response functions, and transfer functions to produce desired filtering behavior.

• New filter transfer functions can be designed from low pass filter transfer functions using a frequency transformation.

• Once you use a frequency transformation in filter design, you can apply standard simulation tools to examine electrical behavior for your filter stages and analog circuit networks.

A frequency transformation in filter design lets you convert between transfer functions for standard filter topologies.

Analog and digital signal processing in networking cables or connections make copious use of filters to attenuate signals in various bandwidths. Your next analog, digital, or mixed-signal linear system can employ many different filters, and filter responses can be related using a frequency transformation. The central idea in a frequency transformation in filter design is to convert impulse response for one filter topology into the analogous filter response for a different topology which, in turn, improves signal strength.

Once the impulse responses for different filter topologies are known, measurements can be incorporated into larger circuit networks involving multiple filters, amplifiers, or other signal processing stages. The goal of using a frequency transformation in filter design is to examine how a filter’s individual response plays a role in the larger circuit response in a larger electrical network. To get started, you’ll need the formulas for a frequency transformation in filter design.

## Formulas for Frequency Transformation in Filter Design

A frequency transformation is used in LTI system design and analysis to compute the transfer function of a high pass, bandpass, or bandstop filter directly from the transfer function for a low pass filter. As long as the low pass filter transfer function is known, the conjugate filter’s transfer function can be determined. The conventional method is to take a low pass filter transfer function of any order, determine its poles and zeros, determine its bandwidth, and then transform these quantities into the values you would see in the other types of fundamental filters.

The conventional approach is to start with a low pass filter that has a known transfer function. The equations used for frequency transformation in filter design are shown below.

In this set of equations, the pole of the initial low pass filter is:

and the frequencies are denoted using F. For the high pass filter, any zeros are denoted using ⍵. Using these formulas, you can calculate the important points you would see in a Bode plot for the new filter (poles, zeros, and resonances).

Formulas needed to convert a low pass filter transfer function into transfer functions for other filters.

In digital filters that use discrete sampling, the standard filter topologies have definitive transfer functions, and conversion between different types of filters is very useful for designing a signal processing algorithm. For example, you may need to determine whether a low-pass filter or band-pass filter is more desirable for processing an arbitrary waveform in the time domain. These are normally part of a larger algorithm that is modeled as a cascaded network.

## Modeling Digital Filters in Cascaded Networks

If you’re modeling a cascaded network with a filter stage, you can use a frequency transformation with a transfer function to generate new transfer functions as part of a cascaded network. Using a frequency transformation in a cascaded network with a filter lets you experiment with different filter topologies and transfer functions within the network (normally a 2-port network). This type of task is important in system-level design to ensure the combined transfer function of a circuit network produces the desired electrical behavior.

When designing a digital filter, the emulated analog response can be determined from the filter’s impulse response using the convolution theorem, but with discrete quantized samples of the analog signal. The response can then be returned to the transfer function for your digital filter. Once the impulse response is converted to a transfer function in the frequency domain, the transfer function can be modified to a different type of filter using a frequency transformation.

When experimenting with different types of filters, the following frequency transformation process can be used to see how different filters affect signals in a cascaded network.

1. Design an initial filter impulse response and calculate the filter’s transfer function.

2. Apply a frequency transformation to convert the filter’s transfer function to a different transfer function for another filter topology.

3. Calculate the impulse response from the transformed filter transfer function.

4. Simulate the network with the new impulse response.

You can then use standard simulation techniques to determine the impulse response of the cascaded network. ABCD network parameters are the easiest mathematical tool for simulating cascaded networks as the filter’s network parameters can be determined from the transfer function.

The filter occupies stage 2 in this cascaded network, and various filter transfer functions can be examined using a frequency transformation for filter design. Note that we are using ABCD parameters, which are numbered from output to input in ascending order. The ABCD parameters for the filter stage can be determined directly from its transfer function.

If you’re not familiar with network parameter analysis or if you don’t have access to mathematical tools for these types of systems, you can use a SPICE simulator with custom component model files to create simulations for networks with digital filters. You can create a component in your schematic with SPICE models containing the filter’s transformed impulse response functions, and you can apply these various impulse responses to simulate multiple filters in your filter networks during circuit design. Not all signal processing emulation software makes this so easy, and you’ll need the right simulation and analysis tools to help you quickly actualize these simulations in filter design.

After you’ve used a frequency transformation in filter design to create impulse response functions, use the front-end design features from Cadence to build your circuit models with transformed filters, and run your simulations. You can use the modeling and simulation features in PSpice Simulator to simulate the electrical behavior of circuit networks with digital or analog circuits. Once you’re ready to create a PCB from your system, simply capture your schematics in a blank layout and use Cadence’s board design utilities to finish your design.

If you’re looking to learn more about how Cadence has the solution for you, talk to us and our team of experts.

###### Previous Article
How to Use a Transmission Line Reflection Coefficient Correctly

There is more than one way to examine how the transmission line reflection coefficient affects your signals.

###### Next Article
Achieving PWM with a Microcontroller in Your PCB Design

Understand PWM, how to send signals from a microcontroller, and the EMI considerations when planning the du...