bilinear transformation

Bilinear Transformations: Bridging the Gap between Analog and Digital Filters

The world of signal processing relies heavily on filters, which selectively modify the frequencies present in a signal. While analog filters operate on continuous-time signals, digital filters work with discrete-time signals sampled at specific intervals. A crucial tool connecting these two domains is the bilinear transformation, a powerful mathematical tool for transforming analog filters into their digital equivalents.

Understanding the Bilinear Transformation

At its core, the bilinear transformation is a conformal mapping of the complex plane, represented by the function:

f(z) = (az + b) / (cz + d)

where a, b, c, and d are real numbers satisfying the condition ad - bc ≠ 0. This transformation is also known as a linear fractional transformation or Möbius transformation.

The significance of this mapping lies in its ability to preserve angles and shapes, crucial properties in signal processing. It transforms points and lines in the complex plane, allowing for the manipulation of frequency characteristics.

From Analog to Digital: The Key to Filter Design

A special case of the bilinear transformation plays a vital role in digital filter design. It maps the imaginary axis (jω) in the complex s-plane, representing analog frequencies, to the unit circle (|z| = 1) in the complex z-plane, representing digital frequencies. This mapping is defined by:

*s = (2/T) * (1 - z⁻¹) / (1 + z⁻¹) *

where T is the sampling interval.

This transformation acts as a bridge between the analog and digital domains, allowing for the design of digital filters from equivalent analog filters. The process involves four key steps:

1. Define characteristic digital frequencies (ωi): These frequencies represent the desired filter characteristics in the digital domain.
2. Prewarp digital frequencies to analog frequencies (ωi): This crucial step ensures accurate frequency mapping using the formula ωi = (2/T) * tan(ωi * T / 2).
3. Design an analog filter with the prewarped frequencies (ωi): This step uses established analog filter design techniques to create the desired filter behavior.
4. Replace 's' in the analog filter with the bilinear transformation: This final step transforms the analog filter function into its digital equivalent, ready for implementation.

Advantages and Limitations

The bilinear transformation offers several advantages in digital filter design:

• Straightforward conversion: It allows for direct transformation of analog filter designs to their digital counterparts.
• Frequency preservation: It preserves the relative frequency characteristics of the original analog filter, ensuring accurate filter behavior in the digital domain.
• Flexibility: It can be applied to various filter types, including low-pass, high-pass, band-pass, and band-stop filters.

However, the bilinear transformation also has limitations:

• Frequency warping: It introduces a non-linear mapping of frequencies, potentially leading to slight frequency distortions.
• Limited accuracy: It can introduce inaccuracies, especially at higher frequencies, due to the frequency warping effect.

Despite these limitations, the bilinear transformation remains a powerful tool for digital filter design, enabling the development of efficient and effective digital filters from existing analog filter designs. It plays a vital role in bridging the gap between analog and digital signal processing, paving the way for the widespread use of digital filters in diverse applications.