gives the discrete-time approximation, with sampling period τ, of the continuous-time systems models lsys.


specifies the transform variable z.

Details and Options


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Basic Examples  (1)

A discrete-time approximation of a continuous-time system:

Scope  (6)

Convert a continuous-time transfer-function model to the discrete-time domain:

Convert a continuous-time state-space model to discrete time:

Convert a multiple-input, multiple-output system to discrete time:

Convert a symbolic system:

Convert a time-delay TransferFunctionModel:

Convert a singular descriptor StateSpaceModel:

Options  (5)

Method  (5)

By default, the approximation is based on the bilinear transformation:

Specify the desired approximation method:

Compare the various approximation methods:

An approximation that preserves the transmission at the specified frequency:

The bilinear and backward Euler methods may add states to descriptor state-space models:

Applications  (1)

Various approximations to a fourth-order Butterworth lowpass filter:

Bode plots:

Properties & Relations  (5)

A stable transfer-function model:

The "ForwardRectangularRule" method may not give a stable approximation:

The stability of the "BilinearTransform" approximation depends on the critical frequency:

Critical frequencies less than the Nyquist frequency give stable approximations:

When the critical frequency is more than the Nyquist frequency, the approximation is unstable:

All other approximations give a stable system if the continuous-time system is stable:

ToContinuousTimeModel is essentially the inverse of ToDiscreteTimeModel:

Time delays in the resulting system are given relative to the sampling period:

ToDiscreteTimeModel may add states to systems with neutral time delays:

Wolfram Research (2010), ToDiscreteTimeModel, Wolfram Language function,


Wolfram Research (2010), ToDiscreteTimeModel, Wolfram Language function,


Wolfram Language. 2010. "ToDiscreteTimeModel." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (2010). ToDiscreteTimeModel. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_todiscretetimemodel, author="Wolfram Research", title="{ToDiscreteTimeModel}", year="2010", howpublished="\url{}", note=[Accessed: 17-July-2024 ]}


@online{reference.wolfram_2024_todiscretetimemodel, organization={Wolfram Research}, title={ToDiscreteTimeModel}, year={2010}, url={}, note=[Accessed: 17-July-2024 ]}