The intersample behavior of the input signals influences the estimation, simulation and prediction of continuous-time models. A sampled signal is characterized only by its values at the sampling instants. However, when you apply a continuous-time input to a continuous-time system, the output values at the sampling instants depend on the inputs at the sampling instants and on the inputs between these points.

The `iddata`

and `idfrd`

objects have an `InterSample`

property
which stores how the input behaves between the sampling instants.
You can specify the behavior between the samples to be piecewise constant
(zero-order hold), linearly interpolated between the samples (first-order
hold) or band-limited. A band-limited intersample behavior of the
input signal means:

A filtered input signal (an input of finite bandwidth) was used to excite the system dynamics

The input was measured using a sampling device (A/D converter with antialiasing) that reported it to be band-limited even though the true input entering the system was piecewise constant or linear. In this case, the sampling devices can be assumed to be a part of the system being modeled.

When input signal is band-limited, the estimation is performed as follows:

Time-domain data is converted into frequency domain data using

`fft`

and the sample time of the data is set to zero.Discrete-time frequency domain data (

`iddata`

with`domain = 'frequency'`

or`idfrd`

with sample time*Ts≠0*) is treated as continuous-time data by setting the sample time*Ts*to zero.

The resulting continuous-time frequency domain data is used
for model estimation. For more information, see Pintelon, R. and
J. Schoukens, *System Identification. A Frequency Domain
Approach*, section 10.2, pp-352-356,Wiley-IEEE Press, New
York, 2001.

Similarly, the intersample behavior of the input data affects
the results of simulation and prediction of continuous-time models. `sim`

and `predict`

commands
use the `InterSample`

property to choose the right
algorithm for computing model response.

The following example simulates a system using first-order hold ( `foh`

) intersample behavior for input signal.

sys = idtf([-1 -2],[1 2 1 0.5]); rng('default') u = idinput([100 1 5],'sine',[],[],[5 10 1]); Ts = 2; y = lsim(sys, u, (0:Ts:999)', 'foh');

Create an `iddata`

object for the simulated input-output data.

data = iddata(y,u,Ts);

The default intersample behavior is zero-order hold ( `zoh`

).

data.InterSample

ans = zoh

Estimate a transfer function using this data.

np = 3; % number of poles nz = 1; % number of zeros opt = tfestOptions('InitMethod','all','Display','on'); opt.SearchOption.MaxIter = 100; modelZOH = tfest(data,np,nz,opt)

modelZOH = From input "u1" to output "y1": -217.2 s - 391.6 --------------------------------- s^3 + 354.4 s^2 + 140.2 s + 112.4 Continuous-time identified transfer function. Parameterization: Number of poles: 3 Number of zeros: 1 Number of free coefficients: 5 Use "tfdata", "getpvec", "getcov" for parameters and their uncertainties. Status: Estimated using TFEST on time domain data "data". Fit to estimation data: 81.38% (simulation focus) FPE: 0.1146, MSE: 0.111

The model gives about 80% fit to data. The sample time of the data is large enough that intersample inaccuracy (using `zoh`

rather than `foh`

) leads to significant modeling errors.

Re-estimate the model using `foh`

intersample behavior.

```
data.InterSample = 'foh';
modelFOH = tfest(data, np, nz,opt)
```

modelFOH = From input "u1" to output "y1": -1.197 s - 0.06843 ------------------------------------- s^3 + 0.4824 s^2 + 0.3258 s + 0.01723 Continuous-time identified transfer function. Parameterization: Number of poles: 3 Number of zeros: 1 Number of free coefficients: 5 Use "tfdata", "getpvec", "getcov" for parameters and their uncertainties. Status: Estimated using TFEST on time domain data "data". Fit to estimation data: 60.86% (simulation focus) FPE: 0.001748, MSE: 0.4905

`modelFOH`

is able to retrieve the original system correctly.

Compare the model outputs with data.

compare(data, modelZOH, modelFOH)

`modelZOH`

is compared to data whose intersample behavior is `foh`

. Therefore, its fit decreases to around 70%.

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