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 sample time.

Input Arguments

`tau`	Time delay to discretize.
`Ts`	Sample 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
 
Sample 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
 
Sample time: 1

Tips

If tau is an integer multiple of Ts, then sys represents the pure discrete delay z^–N, 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:

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

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

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

where D = τ/T_s 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.

Documentation