# Complex-Order Models: A System Identification Point of View
## 2 Operators of Complex Order
### Complex-Order Real-Valued Differintegral
#### 2.2.1 Type 1 CO-RV Differential Operator  
One possibility of defining a complex-order differential operator is to take the mean value of differential operators of conjugate complex orders.  
**Definition 1**.: _Given \(\alpha>0\) and \(\beta>0\), the complex-order real-valued differential operator of the first type (CO-RV-1) is defined as:_  
\[\begin{split}\tfrac{1}{0}\mathfrak{D}_{t}^{\alpha,\beta}:=\frac{1 }{2}\Big{(}\partial\mathcal{D}_{t}^{\alpha+j\beta}+{}_{0}\mathcal{D}_{t}^{ \alpha-j\beta}\Big{)}\,\end{split} \tag{6}\]  
_with the associated Laplace domain operator:_  
\[\begin{split}\mathfrak{D}^{\alpha,\beta}(s):=\frac{1}{2}(s^{ \alpha+j\beta}+s^{\alpha-j\beta})\.\end{split} \tag{7}\]  
Figure 2: Real (**left**) and imaginary (**right**) parts of the impulse response (kernel) of the complex-order integral.  
Such an operator has been previously used in [12]. By the direct application of (3), we see that:  
\[{}^{1}\mathfrak{D}^{\alpha,\beta}(j\omega) =\frac{1}{2}\left(e^{\alpha\ln\omega-\beta\frac{\pi}{2}}e^{j(\beta \ln\omega+\alpha\frac{\pi}{2})}+e^{\alpha\ln\omega+\beta\frac{\pi}{2}}e^{j(- \beta\ln\omega+\alpha\frac{\pi}{2})}\right)\] \[=\frac{1}{2}e^{\alpha\ln\omega}e^{j\alpha\frac{\pi}{2}}\left(e^{- \beta\frac{\pi}{2}}e^{j(\beta\ln\omega)}+e^{+\beta\frac{\pi}{2}}e^{-j\beta\ln \omega}\right)\]  
In order to lighten the notation slightly, let us introduce:  
\[\kappa(p,q)\coloneqq\frac{e^{-p}e^{iq}+e^{p}e^{-jq}}{2}=\cos q\cosh p-j\sin q \sinh p\, \tag{8}\]  
for which it is easy to conclude that:  
\[|\kappa(p,q)| =\sqrt{\cos^{2}q+\sinh^{2}p}\, \tag{9}\] \[\arg^{*}\kappa(p,q) =-\atan^{*}(\tan q\tanh p)\, \tag{10}\]  
where \(\arg^{*}\) and \(\atan^{*}\) denote the unwrapped argument and arcus tangent, respectively. The introduced complex-order real-valued differential operator can now be expressed as:  
\[{}^{1}\mathfrak{D}^{\alpha,\beta}(j\omega)=(j\omega)^{\alpha}\kappa\left( \frac{\beta\pi}{2},\beta\ln\omega\right)\, \tag{11}\]  
with:  
\[|{}^{1}\mathfrak{D}^{\alpha,\beta}(j\omega)| =\omega^{\alpha}\sqrt{\cos^{2}(\beta\ln\omega)+\sinh^{2}(\frac{ \beta\pi}{2})}\, \tag{12}\] \[\arg^{*}{}^{1}\mathfrak{D}^{\alpha,\beta}(j\omega) =\frac{\alpha\pi}{2}-\atan^{*}(\tan(\beta\ln\omega)\tanh(\frac{ \beta\pi}{2})). \tag{13}\]  
We may finally conclude that the operator \({}^{1}\mathfrak{D}\) of orders \(\alpha,\beta>0\) has an amplitude characteristic with an approximately constant logarithmic slope equal to \(20\alpha\) dB per decade, and a phase characteristic with an approximately constant logarithmic slope of \(-\beta\) radians per decade. Note that the phase slope is negative. As will be demonstrated in the sequel, the sign of the amplitude slope is significant: we expect all CO-RV-1 differential operators to have a positive slope, and all CO-RV-1 integral operators to have a negative slope of the amplitude characteristics. The same does not hold for the slope of the phase. We will demonstrate that it is possible to have both differential and integral operators with either positive or negative phase slopes.