## Transforming Between Discrete-Time and Continuous-Time Representations

### Why Transform Between Continuous and Discrete Time?

Transforming between continuous-time and discrete-time representations is useful, for example, if you have estimated a discrete-time linear model and require a continuous-time model instead for your application.

You can use `c2d`

and `d2c`

to transform any linear
identified model between continuous-time and discrete-time representations.
`d2d`

is useful is you want to
change the sample time of a discrete-time model. All of these operations change
the sample time, which is called *resampling* the
model.

These commands do not transform the estimated model uncertainty. If you want to
translate the estimated parameter covariance during the conversion, use
`translatecov`

.

**Note**

`c2d`

and `d2d`

correctly
approximate the transformation of the noise model only when the sample time
`T`

is small compared to the bandwidth of the
noise.

### Using the c2d, d2c, and d2d Commands

The following table summarizes the commands for transforming between continuous-time and discrete-time model representations.

Command | Description | Usage Example |
---|---|---|

`c2d` |
Converts continuous-time models to discrete-time models. You cannot use |
To transform a continuous-time model
mod_d = c2d(mod_c,T) where |

`d2c` |
Converts parametric discrete-time models to continuous-time models. You cannot use |
To transform a discrete-time model
mod_c = d2c(mod_d) |

`d2d` |
Resample a linear discrete-time model and produce an equivalent discrete-time model with a new sample time. You can use the resampled model to simulate or predict output with a specified time interval. |
To resample a discrete-time model
mod_d2 = d2d(mod_d1,Ts) |

The following commands compare estimated model `m`

and its
continuous-time counterpart `mc`

on a Bode plot:

% Estimate discrete-time ARMAX model % from the data m = armax(data,[2 3 1 2]); % Convert to continuous-time form mc = d2c(m); % Plot bode plot for both models bode(m,mc)

### Specifying Intersample Behavior

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. Thus, the
`InterSample`

data property describes how the algorithms
should handle the input between samples. For example, you can specify the
behavior between the samples to be piece-wise constant (zero-order hold,
`zoh`

) or linearly interpolated between the samples (first
order hold, `foh`

). The transformation formulas for
`c2d`

and `d2c`

are affected by the
intersample behavior of the input.

By default, `c2d`

and `d2c`

use the
intersample behavior you assigned to the estimation data. To override this
setting during transformation, add an extra argument in the syntax. For
example:

% Set first-order hold intersample behavior mod_d = c2d(mod_c,T,'foh')

### Effects on the Noise Model

`c2d`

, `d2c`

, and
`d2d`

change the sample time of both the dynamic model
and the noise model. Resampling a model affects the variance of its noise
model.

A parametric noise model is a time-series model with the following mathematical description:

$$\begin{array}{l}y(t)=H(q)e(t)\\ E{e}^{2}=\lambda \end{array}$$

The noise spectrum is computed by the following discrete-time equation:

$${\Phi}_{v}(\omega )=\lambda T{\left|H\left({e}^{i\omega T}\right)\right|}^{2}$$

where $$\lambda $$ is the variance of the white noise *e(t)*,
and $$\lambda T$$ represents the spectral density of *e(t)*.
Resampling the noise model preserves the spectral density $$\lambda $$*T *. The spectral density $$\lambda $$*T * is invariant up to the Nyquist
frequency. For more information about spectrum normalization, see Spectrum Normalization.

`d2d`

resampling of the noise model affects simulations with
noise using `sim`

. If you resample a model to a faster sampling
rate, simulating this model results in higher noise level. This higher noise
level results from the underlying continuous-time model being subject to
continuous-time white noise disturbances, which have infinite, instantaneous
variance. In this case, the *underlying continuous-time
model* is the unique representation for discrete-time models. To
maintain the same level of noise after interpolating the noise signal, scale the
noise spectrum by $$\sqrt{{\scriptscriptstyle \raisebox{1ex}{${T}_{New}$}\!\left/ \!\raisebox{-1ex}{${T}_{Old}$}\right.}}$$, where *T _{new}* is the
new sample time and

*T*is the original sample time. before applying

_{old}`sim`

.