thiran

Generate fractional delay filter based on Thiran approximation

Syntax

sys= thiran(tau,Ts)

Description

sys= thiran(tau,Ts)discretizes the continuous-time delaytauusing a Thiran filter to approximate the fractional part of the delay.Tsspecifies the sample time.

Input Arguments

`tau`	Time delay to discretize.
`Ts`	Sample time.

Output Arguments

sys

Discrete-timetfobject.

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 Sample time: 1

The time delay is 2.4 s, and the sample time is 1 s. Therefore,sys1是一个订单3的离散传递函数。

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

sys2 = thiran(10, 1) Transfer function: 1 ---- z^10 Sample time: 1

Tips

Iftauis an integer multiple ofTs, thensysrepresents the pure discrete delayz^–N, withN=tau/Ts. Otherwise,sysis a discrete-time, all-pass, infinite impulse response (IIR) filter of orderceil(tau/Ts).
thiranapproximates and discretizes a pure time delay. To approximate a pure continuous-time time delay without discretizing, usepade. To discretize continuous-time models having time delays, usec2d.

Algorithms

The Thiran fractional delay filter has the following form:

$H (z) = \frac{a_{N} z^{N} + a_{N - 1} z^{N - 1} + \dots + a_{1}}{a_{0} z^{N} + a_{1} z^{N - 1} + \dots + a_{N}} .$

The coefficientsa₀, ...,a_Nare given by:

$\begin{array}{l} a_{k} = {(- 1)}^{k} (\begin{matrix} N \\ k \end{matrix}) \prod_{i = 0}^{N} \frac{D - N + i}{D - N + k + i}, \forall k : 1, 2, \dots, N \\ a_{0} = 1 \end{array}$