A Closed-Form IIR Approximation of Fractional Operator $s^{\nu }$ Around a Prescribed Low Frequency

This letter proposes a closed-form design method for an infinite impulse response (IIR) digital approximation of the fractional-order operator
$s^{\nu }$
,
$\nu \in (-1,1)$
, with improved performance around a predefined low frequency
$\beta$
,
$0 < \beta < 1$
. The proposed method utilizes a modified indirect discretization strategy, where it generates a rational
$s$
-domain expression using the continued fraction expansion (CFE) of a scaled operator
$(s/\beta)^{\nu }$
to minimize the truncation error around
$\beta$
. The
$s$
-domain expression is then discretized using a well-matched
$s$
-to-
$z$
transformation and scaled back by
$\beta ^{\nu }$
to fit
$s^{\nu }$
. The new approximations are compared with recently-proposed models by simulation in terms of frequency response as well as QRS complex detection. The proposed method shows a great potential to attain low-order accurate models that can be immediately used for hardware realizations in a wide range of applications.