Bilinear Transformation Frequency Calculator

Bilinear Transformation Formula:

\[ f_b = \frac{2 \pi f_c}{\tan\left(\frac{\pi f_c}{f_e}\right)} \]

Bilinear Frequency (fb):

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1. What is Bilinear Transformation?

The Bilinear Transformation is a method used in digital signal processing to convert analog filter designs into digital filter designs. It maps the continuous-time domain to the discrete-time domain while preserving stability and frequency response characteristics.

2. How Does the Calculator Work?

The calculator uses the Bilinear Transformation formula:

\[ f_b = \frac{2 \pi f_c}{\tan\left(\frac{\pi f_c}{f_e}\right)} \]

Where:

\( f_b \) — Bilinear Frequency (Hz)
\( f_c \) — Distortion Frequency (Hz)
\( f_e \) — Sampling Frequency (Hz)
\( \pi \) — Archimedes' constant (approximately 3.14159)
\( \tan \) — Trigonometric tangent function

Explanation: This formula performs a numerical integration of the analog transfer function into the digital domain, mapping the frequency response while avoiding aliasing effects.

3. Importance of Bilinear Transformation

Details: The bilinear transformation is crucial in digital filter design as it provides a stable mapping from the s-plane to the z-plane, preserves system stability, and avoids the frequency warping effects that occur with other transformation methods.

4. Using the Calculator

Tips: Enter the distortion frequency and sampling frequency in Hertz. Both values must be positive numbers. The calculator will compute the corresponding bilinear frequency.

5. Frequently Asked Questions (FAQ)

Q1: What is frequency warping in bilinear transformation?
A: Frequency warping is a nonlinear relationship between analog and digital frequencies caused by the tangent function in the transformation. Higher frequencies get compressed in the digital domain.

Q2: When should bilinear transformation be used?
A: It's particularly useful when designing digital filters from analog prototypes, especially for IIR (Infinite Impulse Response) filters where stability preservation is critical.

Q3: What are the advantages over other transformation methods?
A: The bilinear transformation preserves stability, maps the entire jω-axis to the unit circle, and avoids aliasing problems that occur with impulse invariance method.

Q4: Are there limitations to bilinear transformation?
A: The main limitation is frequency warping, which distorts the frequency axis. This can be pre-warped to compensate for specific frequency points.

Q5: How does sampling frequency affect the result?
A: Higher sampling frequencies reduce the effects of frequency warping and provide better approximation of the analog filter characteristics in the digital domain.