# thiran

Generate fractional delay filter based on Thiran approximation

## Syntax

## Description

discretizes the continuous-time delay `sys`

= thiran(`tau`

,`Ts`

)`tau`

using a Thiran filter to
approximate the fractional part of the delay. `Ts`

specifies the sample
time of the filter, which is returned as a discrete-time transfer function model. The Thiran
fractional-delay filter has the following form:

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

The coefficients *a*_{0}, ..., *a _{N}* are given by:

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

where *D* = *τ*/*T _{s}* and

*N*= ceil(

*D*) is the filter order. See [1].

## Examples

## Input Arguments

## Output Arguments

## Tips

## References

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

## Version History

**Introduced in R2010a**