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HolderModel[]

represents the single-input, single-output model of a zero-order hold.

HolderModel[specs]

represents a holder with specifications specs.

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HolderModel

HolderModel[]

represents the single-input, single-output model of a zero-order hold.

HolderModel[specs]

represents a holder with specifications specs.

Details

  • HolderModel is known as zero-order hold (ZOH), first-order hold (FOH) and inverse sampling.
  • HolderModel is typically used to convert a discrete-time signal coming from a digital microcontroller or a digital signal processor to a continuous-time signal used when actuating in the physical world or with a model of the physical world.
  • A holder is the model of an operation that takes a discrete-time signal defined for integers and generates a continuous-time signal f(t)=g(TemplateBox[{{t, /, T}}, Floor]), where is the sampling period. This holds the value constant between samples.
  • The specification spec is an association and can have the following keys:
  • "InputVariables"{u[t]}input variables
    "Order"0hold order, a non-negative integer
    "OutputVariables"{y[t]}output variables
    "SamplingPeriod"1sampling period
    "SignalCount"1number of inputs and outputs
    "TemporalVariable"ttemporal variable
  • The "Order" property specifies the order of the extrapolation to use. An order extrapolator uses the previous points to derive an interpolating polynomial of degree . This polynomial is then used to extrapolate the value over the next sampling period.
  • HolderModel[…]["prop"] can be used to obtain various properties of the model.
  • The value of "prop" can be any of the keys in spec and the following:
  • "Properties"list of property names
    "PropertyAssociation"property names and values as an association
    "PropertyDataset"property names and values as a dataset
    {p1,p2,…}values of properties pi

Examples

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Basic Examples  (2)

A zero-order hold (ZOH) model:

Wolfram Language code: HolderModel[]

Its properties:

Wolfram Language code: %["PropertyDataset"]

Simulate the response of a ZOH to a sinusoidal input:

Wolfram Language code: r = InputOutputResponse[HolderModel[], Table[Sin[t], {t, 0, 3π}]]

Plot the response and the input sequence:

Wolfram Language code: Show[Plot[r, {, 0, 3π}], ListPlot[Table[{t, Sin[t]}, {t, 0, 3π}], IconizedObject[«opts»]]]

Scope  (9)

A zero-order hold model:

Wolfram Language code: HolderModel[]

A first-order hold:

Wolfram Language code: HolderModel[<|"Order" -> 1|>]

A hold with custom properties:

Wolfram Language code: HolderModel[<|"InputVariables" -> u[t], "OutputVariables" -> y[t], "TemporalVariable" -> t|>]
Wolfram Language code: %["Dataset"]

List all available properties:

Wolfram Language code: HolderModel[Association["DiscreteVariables" -> {Subscript[, 1][t]}, "InputVariables" -> {u[t]}, "OutputVariables" -> {y[t]}, "InitialStateValues" -> {}, "TemporalVariable" -> t, "WhenEvent" -> Mod[t, 1] == 0, "WhenEventAction" -> {Subscript[ ... tion", "StateEquations", "OutputExpressions", "SignalCount", "Type", "PropertyFunction", "Order", "SamplingPeriod", "SummaryItems"}], {"InputVariables", "Order", "OutputVariables", "SamplingPeriod", "SignalCount", "TemporalVariable"}]["Properties"]

Obtain a specific property:

Wolfram Language code: HolderModel[Association["DiscreteVariables" -> {Subscript[, 1][t]}, "InputVariables" -> {u[t]}, "OutputVariables" -> {y[t]}, "InitialStateValues" -> {}, "TemporalVariable" -> t, "WhenEvent" -> Mod[t, 1] == 0, "WhenEventAction" -> {Subscript[ ... tion", "StateEquations", "OutputExpressions", "SignalCount", "Type", "PropertyFunction", "Order", "SamplingPeriod", "SummaryItems"}], {"InputVariables", "Order", "OutputVariables", "SamplingPeriod", "SignalCount", "TemporalVariable"}]["SamplingPeriod"]

Obtain several properties:

Wolfram Language code: HolderModel[Association["DiscreteVariables" -> {Subscript[, 1][t]}, "InputVariables" -> {u[t]}, "OutputVariables" -> {y[t]}, "InitialStateValues" -> {}, "TemporalVariable" -> t, "WhenEvent" -> Mod[t, 1] == 0, "WhenEventAction" -> {Subscript[ ... tion", "StateEquations", "OutputExpressions", "SignalCount", "Type", "PropertyFunction", "Order", "SamplingPeriod", "SummaryItems"}], {"InputVariables", "Order", "OutputVariables", "SamplingPeriod", "SignalCount", "TemporalVariable"}][{"Order", "SignalCount"}]

Obtain the properties as an association:

Wolfram Language code: HolderModel[Association["DiscreteVariables" -> {Subscript[, 1][t]}, "InputVariables" -> {u[t]}, "OutputVariables" -> {y[t]}, "InitialStateValues" -> {}, "TemporalVariable" -> t, "WhenEvent" -> Mod[t, 1] == 0, "WhenEventAction" -> {Subscript[ ... tion", "StateEquations", "OutputExpressions", "SignalCount", "Type", "PropertyFunction", "Order", "SamplingPeriod", "SummaryItems"}], {"InputVariables", "Order", "OutputVariables", "SamplingPeriod", "SignalCount", "TemporalVariable"}]["PropertyAssociation"]

As a dataset:

Wolfram Language code: HolderModel[Association["DiscreteVariables" -> {Subscript[, 1][t]}, "InputVariables" -> {u[t]}, "OutputVariables" -> {y[t]}, "InitialStateValues" -> {}, "TemporalVariable" -> t, "WhenEvent" -> Mod[t, 1] == 0, "WhenEventAction" -> {Subscript[ ... tion", "StateEquations", "OutputExpressions", "SignalCount", "Type", "PropertyFunction", "Order", "SamplingPeriod", "SummaryItems"}], {"InputVariables", "Order", "OutputVariables", "SamplingPeriod", "SignalCount", "TemporalVariable"}]["PropertyDataset"]

Simulate the response of a first-order hold to a sinusoidal sequence:

Wolfram Language code: Short[r1 = InputOutputResponse[HolderModel[<|"Order" -> 1|>], Table[Sin[t], {t, 0, 2.5π}]]]

Plot the response and the sequence:

Wolfram Language code: Show[Plot[r1, {, 0, 2.5π}, PlotRange -> All], ListPlot[Table[{t, Sin[t]}, {t, 0, 2.5π}], IconizedObject[«opts»]]]

Decrease the sampling period to obtain a better approximation of the sinusoid:

Wolfram Language code: Short[r2 = InputOutputResponse[HolderModel[<|"Order" -> 1, "SamplingPeriod" -> 0.5|>], Table[Sin[t], {t, 0, 2.5π, 0.5}]]]

Plot the response and the input sequence of the better approximation:

Wolfram Language code: Show[Plot[r2, {, 0, 2.5π}, PlotRange -> All], ListPlot[Table[{t, Sin[t]}, {t, 0, 2.5π, 0.5}], IconizedObject[«opts»]]]

The lower sampling period results in a better approximation:

Wolfram Language code: NIntegrate[Abs /@ {Sin[] - r1[[1]], Sin[] - r2[[1]]}, {, 0, 2.5π}]

Applications  (1)

A holder model is a subsystem of a sampled data system:

Wolfram Language code: csys = SystemsConnectionsModel[{TransferFunctionModel[{{{1.}}, (1 + s)^3}, s], TransferFunctionModel[{{{4.539074548064891 - 9.145098981931369* + 4.606282997931085*^2}}, {{0.23358983848622455 - 0.5071796769724491* + 0.2735898384862246*^2}}}, , ... ummaryItems"}], {"InputVariables", "OutputVariables", "SamplingPeriod", "SignalCount", "TemporalVariable"}]}, {{4, 1} -> {3, 1}, {3, 1} -> {2, 1}, {2, 1} -> {5, 1}, {5, 1} -> {1, 1}, {1, 1} -> {6, 1}, {6, 1} -> {3, 2}}, {{4, 1}}, {{1, 1}}];

Simulate the system:

Wolfram Language code: r = InputOutputResponse[csys, 1, {t, 0, 10}]

Plot the response:

Wolfram Language code: Plot[r, {t, 0, 10}]

Properties & Relations  (3)

A sampler model is essentially a time-based action:

Wolfram Language code: u = Tanh[t]; sp = 0.5;

The response based on WhenEvent:

Wolfram Language code: r1 = NDSolveValue[{WhenEvent@@{Mod[t, sp] == 0, y[t] -> u}, y[0] == 0, Derivative[1][$][t] == 0, $[0] == 0}, y[t], {t, 0, 5}, DiscreteVariables -> {y[t]}]

The response of sampler model:

Wolfram Language code: r2 = InputOutputResponse[SamplerModel[<|"SamplingPeriod" -> sp|>], u, {t, 0, 5}]

They are the same:

Wolfram Language code: Plot[{r1, r2}, {t, 0, 5}, PlotRange -> All]

A first-order hold is based on a linear interpolation of the current and previous input values:

Wolfram Language code: u = Tanh[t]; sp = 0.5;

The variable is the current input value, is the previous input value and is the total time elapsed:

Wolfram Language code: action = {Subscript[x, 1][t], Subscript[x, 2][t], 𝒯[t]} -> {u, Subscript[x, 1][t], 𝒯[t] + sp};

The discrete variables , and are updated at each sampling instant:

Wolfram Language code: we = WhenEvent@@{Mod[t, sp] == 0, action}

Solve for the slope and intercept of the linear interpolation:

Wolfram Language code: sol = Solve[{Subscript[x, 1][t] == m 𝒯[t] + c, Subscript[x, 2][t] == m(𝒯[t] - sp) + c}, {m, c}]

The output expression:

Wolfram Language code: out = m t + c /. sol//Simplify

Solve for the output signal:

Wolfram Language code: r1 = NDSolveValue[{we, Derivative[1][$][t] == 0, $[0] == 0, Subscript[x, 1][0] == 0, Subscript[x, 2][0] == 0, 𝒯[0] == 0}, out, {t, 0, 5}, DiscreteVariables -> {Subscript[x, 1][t], Subscript[x, 2][t], 𝒯[t]}]

The solution using the holder model:

Wolfram Language code: r2 = InputOutputResponse[HolderModel[<|"SamplingPeriod" -> sp, "Order" -> 1|>], u, {t, 0, 5}]

They are the same:

Wolfram Language code: Plot[{r1, r2}, {t, 0, 5}, PlotRange -> All]

A holder model inverses the operation of a sampler model:

Wolfram Language code: f = 3; u = Sech[f t]; sp = 1 / ( 50f);

A sampler model in series with a holder model:

Wolfram Language code: sys = SystemsConnectionsModel[{SamplerModel[<|"SamplingPeriod" -> sp|>], HolderModel[<|"SamplingPeriod" -> sp|>]}, {{1, 1} -> {2, 1}}, {{1, 1}}, {{2, 1}}]

The output signal is the same as the input signal:

Wolfram Language code: InputOutputResponse[sys, u, {t, 0, π}]; Plot[{u, %[[1]]}, {t, 0, π}, PlotRange -> All]
Wolfram Research (2024), HolderModel, Wolfram Language function, https://reference.wolfram.com/language/ref/HolderModel.html.

Text

Wolfram Research (2024), HolderModel, Wolfram Language function, https://reference.wolfram.com/language/ref/HolderModel.html.

CMS

Wolfram Language. 2024. "HolderModel." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/HolderModel.html.

APA

Wolfram Language. (2024). HolderModel. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HolderModel.html

BibTeX

@misc{reference.wolfram_2026_holdermodel, author="Wolfram Research", title="{HolderModel}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/HolderModel.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_holdermodel, organization={Wolfram Research}, title={HolderModel}, year={2024}, url={https://reference.wolfram.com/language/ref/HolderModel.html}, note=[Accessed: 12-June-2026]}