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Generate fractional delay filter based on Thiran approximation


sys = thiran(tau, Ts)


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


Time delay to discretize.


Sampling time.

Output Arguments


Discrete-time tf object.


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:
Sampling time: 1

More About

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  • 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.


The Thiran fractional delay filter has the following form:

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

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


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

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