IEEE Circuits and Systems Magazine - Q2 2022 - 51

N
N
/ / k
k=0 m 0
=
M
aa DR t
bb DR t
m km
km
aa
aa
-
+
M
/ / k
k=0 m 0
=
that defines a new LS relating the autocorrelation functions
of input and output signals. Noting that
W ~~ ~~=- =
ab
ab
-
+
() () ()
jj je() sgn( )
j
ab ab ab r
-
+
2
the frequency response of such system is
M
M
Gj~ 2
() = N
/ /
/ /
=
=
bb
k= 0 m 0
N
aa
k= 0 m 0
In the following we study briefly the two-sided derivatives.
These were formally introduced in [63, 107, 108]
and were unified in a formulation that included the onesided
(forward/backward) Gr├╝nwald-Letnikov derivatives
[109, 110].
Definition III.1. Let (),,ft t R! be a real function and
ci , R! two real parameters. We define a two-sided GL
type FD of f(t) by
i
c
() = lim+
h " 0
n ccmm-+1
+
=+
3
Dfth
/-c
3
CC
C
()( )(-+ -
-
ci ci
c
11
n
2
2
nn
ft nh
)
,
++1
(94)
where c is the derivative order, and i is an asymmetry
parameter. The situation corresponding to
c NN , N!=deserves
particular attention [23], [110].
For absolutely or square integrable functions, the FT
of (94) is given by
FF6() ()@,
6Dftft
i
c
where
W ~~ sgn ()
~
i
c
() || j
=
rivative (TSFD),
D ,i
c
e
cir
2
.
(96)
Definition III.2. The general two-sided fractional decan
be expressed by its FT [109], [110]
F6Dfte F()sgn
i
c
()@ =~~$
j
c
ri~
2
(),
()@ = W ~
i
c
(95)
k m
k m
~
~
+
a
e
+
a
e
~
,
(92)
m km
km
aa
aa
-
+
xx(),
(91)
yy () =
with stationary increments. These are suitable to understand
phenomena exhibiting long range or /f1 a
dences. Let H (, )01!
[111] and let b R0
is defined by
.
Bt B ()HH 0
'#+3
Bb,
H
() () sgn(~)
-
km jk m ar
2
.
() () sgn(~)
-
km jk m ar
2
(93)
()-= 1
C`H+
0 8() ()-- -
#t
tB
tB
() () ,
xx x
xx
HH
H
-- d ()
1
2
1
- d
2
1
2
1
B
(98)
where () = and B(t) is the standard Brownian
motion. Note that B(t) is not differentiable, but we
can assign it a generalized derivative, the white noise,
(),,
wt t R! so that Bt wt tdd= () () . We can show that the
fBm can be defined in a way similar to the classic Brownian
motion:
vt Bt BDw0dHH H=- = #t
() ()
where
rt Dwtw t()() ,
C() 3
a ()== 1
a
f
()
-a #t
-
is the Liouville forward derivative of order -+H /12
(55). As H (, ),01!
then (/ ,/ ).1212!a -
a 0= (i.e. that H /).12=
rta
xx x
-
-- d
a
1
(100)
Expression (99)
is similar to the current definition of Brownian motion,
provided that
This formula
suggests the use of other FD definitions alternative to
the Liouville definition, as the GL formulation. If the
white noise is gaussian, then ()
rta
ian noise (FGN). The signal () has an infinite power,
but its mean value is constant (and null). Moreover, the
autocorrelation function,
Rt Er tr @ der
xx x=+aa
(, )( )( ),
6
pends only on t, not on .x Therefore, the fractional noise
()
rta
its autocorrelation function is
()
Rt = v2
a
t
-
22 cos
C aar
() ()
||
--a
21
.
(101)
Relation (101) shows that we only have a (wide sense)
stationary (hyperbolic) noise if
(97) +-C () () .22 (102)
21 02 0aa ar
and
where c and i are the derivative order and asymmetry
parameter, respectively.
3) The Fractional Brownian Motion
As an application of the presented formalism, we consider
the fractional Brownian motion (fBm). This process
was studied first by Mandelbrot and Van Ness [13] that
suggested it as a model for non-stationary signals, but
SECOND QUARTER 2022
cos
The other cases do not lead to a valid autocorrelation
function of a stationary stochastic process, since
it does not have a maximum at the origin. Then, for
12 /
- 110a
and (,a!!Z+
22 1),
nn n
+
, we obtain
valid autocorrelation functions. We conclude that, if
|| /,
a 112 we obtain a stationary process in the anti-derivative
case, a10 , and a nonstationary process in the
derivative case, a2 0 [112], [113].
IEEE CIRCUITS AND SYSTEMS MAGAZINE
51
is a wide sense stationary stochastic process and
is a fractional Gauss()
a
() ,
f xx
(99)
2
1
j
depenbe
the so-called Hurst parameter
! The fBm, BtH (), with parameter H

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