Generate fractional delay filter based on Thiran approximation
sys = thiran(
the continuous-time delay
sys = thiran(
tau using a Thiran filter
to approximate the fractional part of the delay.
the sample time.
Time delay to discretize.
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,
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
tau is an integer multiple of
sys represents the pure discrete delay z–N,
with N =
a discrete-time, all-pass, infinite impulse response (IIR) filter
thiran approximates and discretizes
a pure time delay. To approximate a pure continuous-time time delay
without discretizing, use
To discretize continuous-time models having time delays, use
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 .
 T. Laakso, V. Valimaki, “Splitting the Unit Delay”, IEEE Signal Processing Magazine, Vol. 13, No. 1, p.30-60, 1996.