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# Matched Z-transform method

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### Matched Z-transform method

The s-plane poles and zeros of a 5th-order Chebyshev type II lowpass filter to be approximated as a discrete-time filter
The z-plane poles and zeros of the discrete-time Chebyshev filter, as mapped into the z-plane using the matched Z-transform method with T = 0.1 second. The labeled frequency points and band-edge dotted lines have also been mapped through the function exp(sT), to show how frequencies along the axis in the s-plane map onto the unit circle in the z-plane.

The matched Z-transform method, also called the pole–zero mapping[1][2] or pole–zero matching method,[3] is a technique for converting a continuous-time filter design to a discrete-time filter (digital filter) design.

The method works by mapping all poles and zeros of the s-plane design to z-plane locations z = exp(sT), for a sample interval T.[4]

Alternative methods include the bilinear transform and impulse invariance methods.

Responses of the filter (dashed), and its discrete-time approximation (solid), for nominal cutoff frequency of 1 Hz, sample rate 10 Hz. The discrete-time filter does not reproduce the Chebyshev equiripple property in the stopband due to the interference from cyclic copies of the response.

## References

1. ^ Won Young Yang (2009). Signals and Systems with MATLAB. Springer. p. 292.
2. ^ Bong Wie (1998). Space vehicle dynamics and control. AIAA. p. 151.
3. ^ Arthur G. O. Mutambara (1999). Design and analysis of control systems. CRC Press. p. 652.
4. ^ S. V. Narasimhan and S. Veena (2005). Signal processing: principles and implementation. Alpha Science Int'l Ltd. p. 260.
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