System Definition
Presets
Numerator b (b₀, b₁, …)
Comma-separated, up to 5 values
Denominator a (a₀, a₁, …)
Recommended: a₀ = 1 (normalized)
Stability Check
Transfer Function Preview
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DC Gain [dB]
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Peak Gain [dB]
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Cutoff [π rad]
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No. of Poles
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No. of Zeros
Pole-Zero Plot (z-plane)
Impulse Response (32 samples)
Frequency Response (0 to π rad)
Key Equations
Z-transform definition (bilateral):
$$X(z) = \sum_{n=-\infty}^{\infty} x[n]\, z^{-n}$$Transfer function: $H(z) = \dfrac{B(z)}{A(z)} = \dfrac{b_0 + b_1 z^{-1} + \cdots}{a_0 + a_1 z^{-1} + \cdots}$
Relation to DTFT: substitute $z = e^{j\omega}$ to obtain $H(e^{j\omega})$
Bilinear transform: $s = \dfrac{2}{T}\cdot\dfrac{z-1}{z+1}$ (analog-to-digital conversion)
Applications: IIR filter design (Butterworth, Chebyshev), digital filtering of vibration data, digital PID controllers in the z-domain, window function design for pre-FFT processing.