## Documentation |

This example shows how to use the commands for continuous/discrete, discrete/continuous, and discrete/discrete conversions.

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Continuous/Discrete Conversion Discrete/Continuous Conversion |

Control System Toolbox™ offers extensive support for discretization and resampling of linear systems including:

`c2d`discretizes continuous-time models`d2c`compute continuous-time extensions of discrete-time models`d2d`resamples discrete-time models.

Several algorithms are available to perform these operations, including:

Zero-order hold

First-order hold

Impulse invariant

Tustin

Matched poles/zeros.

**Continuous/Discrete Conversion**

For example, consider the second-order system with delay:

To compute its zero-order hold (ZOH) discretization with sampling rate of 10 Hz, type

G = tf([1 -2],[1 3 20],'inputdelay',1); Ts = 0.1; % sampling interval Gd = c2d(G,Ts)

Gd = 0.07462 z - 0.09162 z^(-10) * ---------------------- z^2 - 1.571 z + 0.7408 Sample time: 0.1 seconds Discrete-time transfer function.

Compare the continuous and discrete step responses:

step(G,'b',Gd,'r') legend('Continuous','Discretized')

**Discrete/Continuous Conversion**

Conversely, you can use `d2c` to compute a continuous-time "interpolant" for a given discrete-time system. Starting with the discretization `Gd` computed above, convert it back to continuous and compare with the original model `G`:

Gc = d2c(Gd); step(G,'b',Gd,'r',Gc,'g--') legend('Original','Discretized','D2C Interpolant')

The two continuous-time responses match perfectly. You may not always obtain a perfect match especially when your sampling interval `Ts` is too large and aliasing occurs during discretization:

Ts = 1; % 10 times larger than previously Hd = c2d(G,Ts); Hc = d2c(Hd); step(G,'b',Hd,'r',Hc,'g--',10) legend('Original','Discretized','D2C Interpolant')

**Resampling of Discrete-Time Systems**

Resampling consists of changing the sampling interval of a discrete-time system. This operation is performed with `d2d`. For example, consider the 10 Hz discretization `Gd` of our original continuous-time model `G`. You can resample it at 40 Hz using:

Gr = d2d(Gd,0.025)

Gr = 0.02343 z - 0.02463 z^(-40) * ---------------------- z^2 - 1.916 z + 0.9277 Sample time: 0.025 seconds Discrete-time transfer function.

Compare this with a direct discretization at 40 Hz:

step(G,'b',Gr,'r',c2d(G,0.025),'g--',4) legend('Continuous','Resampled from 0.1 to 0.025','Discretized with Ts=0.025')

Notice that both approaches lead to the same answer.

**Which Algorithm and Sampling Rate to Choose?**

See the example entitled "Discretizing a Notch Filter" for more details on how the choice of algorithm and sampling rate affect the discretization accuracy.

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