The complex order fractional derivatives and systems are non hermitian

The complex order fractional derivatives and systems are non hermitian Manuel D. Ortigueira 1,*+ 1 Centre of Technology and Systems-UNINOVA and Department of Electrical Engineering, NOVA School of Science and Technology of NOVA University of Lisbon, Portugal. * Correspondence: mdo@fct.unl.pt † Current address: Campus of NOVA School of Science and Technology, Quinta da Torre, 2829–516 Caparica, Portugal.

4 we analyse the complex order derivatives and its behaviour with the use of the Hilbert transform. 37 The hermitian component of the complex order derivative as proposed by [15] is studied in section 5. 38 Finally, some conclusions are presented.
where x(t) is the input, y(t) is the output, and h(t) is the impulse response of the system. Assume that exists and the convergence of the integral in (2) is uniform on ]−∞, +∞[. The function H(ω) is called frequency response of the system, since, if the input of the system is x(t) = e iω 0 t , ω 0 , t ∈ R with i = √ −1, then the output is given by provided that H(iω 0 ) is finite, [25], which happens always if the system in BIBO stable. The concept of frequency response can be extended to systems not BIBO stable, but with h(t) being AI only on finite length intervals. In such cases, we say that the system is wide sense stable and H(iω 0 ) can have singularities on isolated points. The relation (2) is the called analysis equation of the FT. The corresponding synthesis equation reads Among the large number of properties of the FT, there is one very interesting: the Hermitean symmetry that we can state as where (.) * means "conjugate". Relation (4) implies that H(ω) = A(ω)e iφ(ω) , where A(ω) and φ(ω) are called amplitude and phase spectra. These are even and odd functions, respectively: These functions will be the features characterizing the hermitian fractional derivatives. We will say 46 that a derivative is hermitian if its frequency response verifies the hermitian property. Each derivative will be treated as an elemental system. For each set we will compute the corresponding 56 frequency response.

Real order fractional derivatives
The Riemann-Liouville derivative is given by where m − 1 < α ≤ m and m ∈ Z + .

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Definition 2. We define the Caputo derivative by where m − 1 < α ≤ m and m ∈ Z + .
Let a = 0 for simplicity, ω ∈ R (always in the following), and y(t) = e iωt , t ≥ 0 and consider the expression in (7). We have The last integral is the FT of a real function. Therefore, it is hermitian. This can be repeated to the derivatives. We can go further by writing The second parcel, leads to the function η(ω) = 1 Γ(α) ∞ t ξ α−1 e iωξ dξ that is also hermitian and decreases asymptotically to zero. Then, if α ∈ R, we obtain where the factor (iω) α − η(ω) is hermitian. So, the real order RL derivative is hermitian and, as 64 the second parcel tends to zero, que can say that approximately the frequency response of the RL 65 derivative can be represented by (iω) α at least for the analysis we will perform. Similarly, we can 66 prove that the Caputo derivative is also hermitian, if the order is real.

Liouville and Grünwald-Letnikov derivatives
68 Assume that f (t) is an AI function defined on R.
Applying the FT to derivatives (11) to (13) we obtain, [24,28], showing that, if the order is real, the derivative is hermitian. With these results we can conclude that 75 the two-sided derivatives, [29], namely, Riesz' and Feller's, of real order are also hermitian. As seen above the real order derivatives are hermitian. On the other hand, most results in section 3 remain valid in the complex order case. In particular, the result stated in (15) where we will set α = a + ib. The frequency response to the complex order derivative reads: The first term corresponds to a real order derivative and then is hermitian. Therefore, second term that corresponds to an imaginary order derivative [4] has to be studied separately. Let Φ(ω) = (iω) ib be the corresponding frequency response. As it is not difficult to note, iω = |ω|e i π 2 sgn(ω) , where sgn(.) is the signum function. Then Thus, the amplitude (gain) spectrum of the complex order fractional derivative is given by and a phase φ(ω) = arg (Φ(ω)) = aπ 2 sgn(ω) + b ln |ω| = −φ(−ω).
As the gain is not symmetric and the phase is not anti-symmetric we conclude that the imaginary order . Thus the FT of the derivative is Using the properties of the Dirac impulse we are allowed to write that, after FT inversion, leads to y 1 (t) = |ω 0 | a e −b π 2 e i(ω 0 t+ aπ 2 +b log |ω 0 |) However, taking x 2 (t) = e −iω 0 t , ω 0 > 0, t ∈ R, we obtain y 2 (t) = |ω 0 | a e b π 2 e −i(ω 0 t+ aπ A comparison of y 1 (t) and y 2 (t) shows that they have different amplitudes and the phases are 82 not anti-symmetric. As consequence, the derivatives of cos(ω 0 t) or of sin(ω 0 t) are not real since they result from adding or subtracting y 1 (t) and y 2 (t). On the other hand, we could obtain other derivatives 84 by taking the real (imaginary) parts of y 1 (t) and y 2 (t). They are different.  86 Makris seemed to be the first to note the abonormality of the complex order derivatives. In [8], he supposedly found a way of avoiding the problem made clear in Example 1: instead of working with a real signal, x(t), he proposed the use of a complex one

Using the Hilbert transform
wherex(t) was constructed in such a way that f (t) could be expressed in terms of sisoids as x 1 (t) in Example 1 [9]. It is not difficult to verify that he was computing the so-called analytical signal, traditionally used in Telecommunications for implementing the "single side band modulation" that was the base of the telephone trafic before the digital age [31]. In this case,x(t) is the Hilbert transform of x(t) [30]. Apparently, this solves the problem, since we only involve positive or negative, not both, types of frequencies. However, we continue to have two different possibilities as the above example shows. In fact, we have cos(ω 0 t) = Re(x 1 (t)) = Re(x 2 (t)), but we can define two different derivatives of for cos(ω 0 t): D a+ib cos(±ω 0 t) = |ω 0 | a e −b π 2 cos(ω 0 t + aπ 2 + b log |ω 0 |) positive frequency (+ω 0 ) |ω 0 | a e +b π 2 cos(ω 0 t + aπ 2 − b log |ω 0 |) negative frequency (−ω 0 ).

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The single sideband signals f ± (t) agree with and generalize Makris' proposal. However, we have two 92 different solutions for each problem as we can see in the following.

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Example 2. Let a system defined by the differential equation D a+ib y(t) + y(t) = x(t) and x(t) = 1 1+t 2 . What is the output?
The frequency response of the system is H(ω) = 1 (iω) a+ib + 1 that is not hermitian having different behavior for positive or negative frequencies as shown in Figure 1  We have then f ± (t) = 1 ± it that will be the input to the differential equation. 3. The FT of the "analytical signals" are given by: Therefore, the output frequency responses will be given by that give rise to two different outputs g + (t) = Attending to (18) and (19) the ouputs will be different as well as their real parts.
Remark 1. This situation can be found also in the two-sided derivatives. For the Riesz case, that has frequency response given by H(ω) = |ω| α , [29], we are led to H(ω) = |ω| a |ω| ib = |ω| a e ib ln(|ω|) We observe that the phase is even when it should be odd. Therefore, the complex order Riesz derivative is not 101 hermitian too. 102 We come to the conclusion that the complex order derivatives, on being non hermitian, do not 103 lead to unique solutions to linear systems, therefore with limited usefulness. The non-hermitian character of the complex order derivative can be avoided by a cascade of a complex order derivative and its conjugatē which is equivalent to a real order derivative. Therefore the use of complex order derivative in this scheme is without particular interest.
These relations allow us to conclude that 106 1.
The operators with frequency responses given by (28) only one is hermitian: Its Bode amplitude spectrum is bounded and oscilating, but behaving almost as an allpass filter 108 (see Figure 2); 109 3. The corresponding Bode phase spectrum decreases linearly (see Figure 2); To have an idea of the action ofD ib , we used the periodic function x(t) so that y(t) =D ib x(t), so that where ω 0 = 4π and The function x(t) is represented in the upper strip in Figure 3. The other strips represent y(t) for 120 b = 0.5, 1, 2. As it can be seen, the action of theD operator is essentially a distortion in the signal that 121 is aconsequence of different delays suffered by the harmonics. 122 6. Conclusions 123 We just showed that the complex order fractional derivatives are not heritian, therefore not 124 suitable for modeling real life phenomena. Their hermitian parts can be used, but they behave like 125 allpass filters. Their amplitudes do not decrease when the frequency increases, meaning that they do 126 not filter out any band of the signal. Therefore they are also not useful in modeling real phenomena 127 that are essentially bandlimited.