Generate fractional delay filter based on Thiran approximation
discretizes the continuous-time delay
sys = thiran(
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:
The coefficients a0, ..., aN are given by:
where D = τ/Ts and N = ceil(D) is the filter order. See .
Approximate and Discretize Time Delay
Approximate and discretize a time delay of 2.4 s with a sample time of 1 s.
sys = thiran(2.4,1)
sys = 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 seconds Discrete-time transfer function.
The time delay is not an integer multiple of the sample time. Therefore, the approximation
sys is a discrete-time transfer function of order 3.
Time Delay of Integer Multiple of Sample Time
Discretize a time delay of 2.5 s with a sample time of 0.5 s.
sys = thiran(2.5,0.5)
sys = 1 --- z^5 Sample time: 0.5 seconds Discrete-time transfer function.
Here, the time delay is an exactly five times the target sample time. Therefore,
sys is a pure discrete delay of order 5.
tau — Time delay
Time delay to discretize, specified as a positive scalar value.
thiran assumes that
tau is in
Ts — Sample time
Sample time of discretized approximation of time delay, specified as a positive
thiran assumes that
Ts is in
sys — Discretized approximation of time delay
Discretized approximation of time delay, returned as a discrete-time transfer
tf) model with sample time
tauis an integer multiple of
sysrepresents the pure discrete delay z–N, with N =
sysis a discrete-time, all-pass, infinite impulse response (IIR) filter of order
The filter is a discrete-time, all-pass, infinite impulse response (IIR) filter of
ceil(tau/Ts). The unit
 T. Laakso, V. Valimaki, “Splitting the Unit Delay”, IEEE Signal Processing Magazine, Vol. 13, No. 1, p.30-60, 1996.
Introduced in R2010a