Sampling Frequency Of Bilinear Calculator

Sampling Frequency Of Bilinear Formula:

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

Sampling Frequency (fe):

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1. What is the Sampling Frequency Of Bilinear Formula?

The Sampling Frequency Of Bilinear formula calculates the sampling frequency required for digital signal processing when using the bilinear transform method. This transformation maps the analog s-plane to the digital z-plane, preserving stability and frequency response characteristics.

2. How Does the Calculator Work?

The calculator uses the Sampling Frequency Of Bilinear formula:

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

Where:

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

Explanation: The formula accounts for the frequency warping effect that occurs during the bilinear transform, ensuring accurate digital representation of analog filters.

3. Importance of Sampling Frequency Calculation

Details: Proper sampling frequency calculation is crucial for digital filter design, preventing aliasing, maintaining frequency response characteristics, and ensuring system stability in digital signal processing applications.

4. Using the Calculator

Tips: Enter distortion frequency and bilinear frequency in Hz. Both values must be positive numbers greater than zero for valid calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is the bilinear transform?
A: The bilinear transform is a method for converting continuous-time systems (analog filters) to discrete-time systems (digital filters) while preserving stability and frequency response characteristics.

Q2: Why is frequency warping important?
A: Frequency warping occurs during the bilinear transform, where the analog frequency axis is non-linearly compressed. This formula accounts for that warping to determine the appropriate sampling frequency.

Q3: What are typical applications of this calculation?
A: This calculation is essential in digital filter design, audio processing, telecommunications, and any application requiring conversion from analog to digital domain using bilinear transform.

Q4: What happens if the denominator becomes zero?
A: If the denominator approaches zero, the sampling frequency becomes very large or undefined, indicating an impractical or impossible sampling scenario.

Q5: How does this relate to the Nyquist frequency?
A: The calculated sampling frequency must be at least twice the highest frequency component of interest (Nyquist criterion), and this formula ensures proper frequency mapping during bilinear transformation.