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# thiran

Generate fractional delay filter based on Thiran approximation

## Syntax

sys = thiran(tau, Ts)

## Description

sys = thiran(tau, Ts) discretizes the continuous-time delay tau using a Thiran filter to approximate the fractional part of the delay. Ts specifies the sampling time.

## Input Arguments

 tau Time delay to discretize. Ts Sampling time.

## Output Arguments

 sys Discrete-time tf object.

## Examples

Approximate and discretize a time delay that is a noninteger multiple of the target sample time.

```sys1 = thiran(2.4, 1)

Transfer function:
0.004159 z^3 - 0.04813 z^2 + 0.5294 z + 1
-----------------------------------------
z^3 + 0.5294 z^2 - 0.04813 z + 0.004159

Sampling time: 1```

The time delay is 2.4 s, and the sample time is 1 s. Therefore, sys1 is a discrete-time transfer function of order 3.

Discretize a time delay that is an integer multiple of the target sample time.

```sys2 = thiran(10, 1)

Transfer function:
1
----
z^10

Sampling time: 1```

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### Tips

• If tau is an integer multiple of Ts, then sys represents the pure discrete delay zN, with N = tau/Ts. Otherwise, sys is a discrete-time, all-pass, infinite impulse response (IIR) filter of order ceil(tau/Ts).

• thiran approximates and discretizes a pure time delay. To approximate a pure continuous-time time delay without discretizing, use pade. To discretize continuous-time models having time delays, use c2d.

### Algorithms

The Thiran fractional delay filter has the following form:

$H\left(z\right)=\frac{{a}_{N}{z}^{N}+{a}_{N-1}{z}^{N-1}+\cdots +{a}_{1}}{{a}_{0}{z}^{N}+{a}_{1}{z}^{N-1}+\cdots +{a}_{N}}.$

The coefficients a0, ..., aN are given by:

$\begin{array}{l}{a}_{k}={\left(-1\right)}^{k}\left(\begin{array}{c}N\\ k\end{array}\right)\prod _{i=0}^{N}\frac{D-N+i}{D-N+k+i},\text{ }\forall k:1,2,\dots ,N\\ {a}_{0}=1\end{array}$

where D = τ/Ts and N = ceil(D) is the filter order. See [1].

## References

[1] T. Laakso, V. Valimaki, "Splitting the Unit Delay", IEEE Signal Processing Magazine, Vol. 13, No. 1, p.30-60, 1996.