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Fractional-order multi-phase oscillators design and analysissuitable for higher-order PSK applications
Mohammed E. Fouda1 • Ahmed Soltan2 • Ahmed G. Radwan1,3 •
Ahmed M. Soliman4
Received: 1 May 2015 / Revised: 15 January 2016 / Accepted: 10 March 2016
� Springer Science+Business Media New York 2016
Abstract Recently, multi-phase oscillator design wit-
nesses a lot of progress in communication especially phase
shift keying based systems. Yet, there is a lack in design
multi-phase oscillator with different fractional phase shifts.
Thus, in this paper, a new technique to design and analyze
a multi-phase oscillator is proposed. The proposed proce-
dure is built based on the fractional-order elements or
constant phase elements in order to generate equal or dif-
ferent phase shifts. The general characteristics equation for
any oscillator is studied to derive expressions for the
oscillation conditions and oscillation frequency. Also, sta-
bility analysis is introduced to guarantee the oscillation.
Then, different examples of oscillators for equal and dif-
ferent phase shifts are introduced with their simulations.
Keywords Oscillator � Multi-phase � Fractional � PSK �Stability � Oscillator analysis
1 Introduction
Phase shift keying PSK is one of famous digital modulation
techniques where the data are conveyed by changing the
phase of the reference signal. In PSK, a finite number of
phases are assigned to encode unique pattern binary digits
where each pattern is assigned by a symbol and each
symbol is assigned by a certain phase. The most common
examples are binary phase shift keying (BPSK) which uses
two phases, and quadrature phase-shift keying (QPSK)
which uses four phases. However, any number of phases
can be used [10].
PSK modulation technique is used in different wireless
standards such as IEEE 802.11b,g wireless LAN standard
[20, 36], where a variety of different PSKs are used
depending on the data-rate required, RFID standards such
as ISO/IEC14443 (adopted for biometric passports), credit
cards such as American Express [25], and IEEE 802.15.4
standard (such as ZigBee). BPSK and QPSK are the most
commonly used in communication applications due to their
implementation simplicity but 8-PSK (p=4 PSK) is not
used due to the implementation complexity of the
oscillator.
Indeed, many techniques were proposed to implement a
multi-phase oscillator [11]. Yet, most of these circuits
suffer from a poor phase noise and can be used only for
generating equally spaced phase difference points [14, 16].
So, fractional order calculus could be used to implement a
non-equally spaced phase difference points [8]. However,
most of is the previously proposed fractional order oscil-
lators were using only two fractional order elements which
& Mohammed E. Fouda
m_elneanaei@ieee.org
Ahmed Soltan
a.s.a.abd-el-aal@newcastle.ac.uk
Ahmed G. Radwan
agradwan@ieee.org
Ahmed M. Soliman
asoliman@ieee.org
1 Engineering Mathematics and Physics Department, Faculty
of Engineering, Cairo University, Giza 12613, Egypt
2 School of Electrical, Electronic and Computer Engineering,
Newcastle University, Newcastle upon Tyne,
United Kingdom
3 Nano-electronic Integrated Systems Center, Nile University,
Giza, Egypt
4 Electronics and Communication Engineering Department,
Cairo University, Giza 12613, Egypt
123
Analog Integr Circ Sig Process
DOI 10.1007/s10470-016-0716-2
limits the number of output nodes to two [23]. So, the goal
of this work is to generalize the design of the fractional
order oscillator to any number of output nodes by using any
number of fractional order elements. Hence, an oscillator
with any number of nodes and equal or unequal phase
difference can be implemented using the design procedure
introduced in this paper.
Fractional calculus is a field of mathematics which is
considered an extension of the traditional definitions of the
integer-order calculus [3] which has the potential to
accomplish what integer-order calculus cannot. Fractional
models provide an excellent instrument for the description
of memory and hereditary properties of various materials
and processes. This is the main advantage of fractional
derivatives in comparison with classical integer-order
models, in which such effects are in fact neglected. The
advantages of fractional calculus become apparent in
modeling mechanical and electrical properties of real
materials, as well as in the description of rheological
properties of rocks, and in many other fields. Many phys-
ical phenomena have intrinsic fractional order description
and so fractional order calculus is necessary in order to
explain them. In addition, many researchers have many
trials to implement electrical elements with fractional order
behavior. The element is equivalent to the traditional
electrical elements resistors, capacitors and inductors but in
the fractional order domain.
The basic definition for the fractional derivative of order
a is well known as Riemann-Liouville definition which is
given by [3]:
Daf ðtÞ ¼
1
C m�að Þdm
dtm
Z t
0
f sð Þt�sð Þaþ1�m
ds m� 1\a\m;
dm
dtmf tð Þ a¼ m:
8>><>>:
ð1Þ
where m is the first integer number greater than fractional-
order a, and for the passive electrical elements a is
enclosed between �1 and 1. Applying the Laplace trans-
form to (1) assuming zero initial conditions yields:
Lf0Dat f ðtÞg ¼ saFðsÞ ð2Þ
So, the definition of (2) leads to the existence of the linear
fractional-order elements. these elements are either
capacitive for a\0, or inductive for a[ 0, or resistive for
a ¼ 0. To model the fractional order element, a finite ele-
ment approximation of the special case Z ¼ 1=CffiffiffiS
pwas
reported in [13]. The technique was later developed by the
authors of [15, 18, 27] for any order.
Different approximation techniques of the fractional-
order derivative in terms of a complicated system of integer
orders were proposed long time ago such as Carlson,
Oustaloup and Matsuda approximations. Carlson approxi-
mation is derived from a regular Newton process [4].
Oustaloup approximation provides a continuous approxi-
mation based on a recursive distribution of zeros and poles
at well chosen intervals [21]. While, Matsuda approxima-
tion provides continuous approximation by calculating gain
at logarithmically spaced frequencies [35]. In addition,
There are many other methods have been proposed [34].
Finite element approximations offer a valuable tool by
which the effect of a fractance device can be simulated
using a standard circuit simulator, or studied experimen-
tally. So, many researchers have investigated the emulating
the fractional-order element depending on the aforemen-
tioned approximation techniques. A finite element
approximation of the special case Z ¼ 1=Cffiffis
pwas reported
by Saito and Sugi [27]. This finite element approximation
is based on the possibility of emulating a fractional-order
capacitor via semi-infinite RC trees as shown in Fig. 1(a).
This technique was later developed [15] to include any
fractional order less than unity. The circuit diagram of the
RC equivalent circuit for the fractional order element of
any order is presented as shown in Fig. 1(b). All the pre-
vious equivalent circuits do not offer a simple practical
two-terminal device. Therefore, many trials had been per-
formed by the researchers to implement a two port frac-
tional order elements such as:
– creating deterministic fractal structures realized by a
metal oxide semiconductor (MOS) technology [9],
– using the frequency dependent dielectric properties of
some materials like LiN2H5SO4 or using chemical
reaction probe [1],
Fig. 1 Equivalent RC tree
circuit of the fractional order
element of order 0.5 [27], and bequivalent RC tree circuit of the
fractional order element of any
order [15]
Analog Integr Circ Sig Process
123
– using microporous PMMA-film coating on the elec-
trode surface [2], and
– using graphene-percolated polymer composites [6].
Recently, Many books and researches during the last
three decades have aimed to increase the accessibility of
fractional calculus for remodeling most of the existing
applications and analyzing new models in basic natural
sciences [26]. Many applications based on fractional-order
systems have been recently discussed such as in the fields
of bioengineering [7, 12, 17], chaotic systems [24], elec-
tromagnetic and Smith-chart [24]. In addition, many fun-
damentals in the conventional circuit theories and stability
techniques have been generalized into the fractional-order
domain [22, 32, 33]. Moreover, fractional order electrical
circuits such as filters [29–31, 33].
This paper is organized as follows; Sect. 2 discusses the
proposed design procedure. Then, some case studies are
presented in Sect. 4 and the numerical simulations are
discussed in Sect. 5. After that, circuit simulations are
presented in Sect. 6. Finally, the conclusion is given.
2 The proposed design procedure
A linear fractional order differential equations, having n
fractional-order elements, is represented by the following
matrix:
Da1x1
Da2x2
..
.
Dan�1xn�1
Danxn
0BBBBBBB@
1CCCCCCCA
¼
0 1 � � � 0 0
0 0 � � � 0 0
..
. ... . .
. ... ..
.
0 0 � � � 0 1
�ao � a1 � � � � an�2 � an�1
0BBBBBB@
1CCCCCCA
x1
x2
..
.
xn�1
xn
0BBBBBBB@
1CCCCCCCA
ð3Þ
By taking the Laplace transform for both sides and sub-
tracting the right half side from the left. The determinant of
this matrix gives the characteristic equation of this system,
which is given by
DðsÞ ¼ sPn
i¼1ai þ an�1s
Pn�1
i¼1ai þ . . .þ a1s
a1 þ a0 ð4Þ
For oscillation, the real and imaginary parts of the char-
acteristic of (4) should equal zero at s ¼ jx, then, the
conditions for oscillation are given as follows:
Xni¼0
aixPi
j¼1ajcos
Xij¼1
ajp2
!¼ 0 ð5aÞ
Xni¼1
aixPi
j¼2ajsin
Xij¼1
ajp2
!¼ 0 ð5bÞ
where ao should be greater than zero for stable periodic
response. Since, in case of ao [ 0, a real pole in W-plane
exists in unstable region.
On the other hand, in order to realize a generic system
for this characteristic equation, a simple state space rep-
resentation of (3) is illustrated in Fig. 2(b) to guarantee
observability. Therefore, the system is represented by the
following set of equations:
y ¼ x1 ¼1
sa1x2; x2¼
1
sa2x3; � � � xn�1¼
1
san�1xn;
xn¼�1
san
Xni¼1
ai�1xi
ð6Þ
where, the required output nodes are at the integrators
output and the phase difference is relative to the node of X1
as shown in Fig. 2(b). Yet, the summation of the overall
phase shift of the circuit should be 360�. This is divided
into two parts; the maximum phase shift and the comple-
mentary phase, obtained due to the feedback coefficients
vector ða ¼ ðan�1 � � � aoÞTÞ which may be either negative or
positive. So, for positive coefficient the order of the system
should be greater than or equal 2.
In order to design an oscillator with certain fraction
phase shifts, the following steps should be followed.
1. Specify the phase shifts of x2; x3; . . .;xN relative to x1such that the required phase shifts are h1; h2; . . .;hN�1
respectively so the fractional orders are a1¼h1 2p and
generally ai¼ 2p hi�hi�1ð Þ.
2. Substitute into (5) by the values of a and oscillation
frequency x after scaling, and solving 2 equations in N
unknown to get the coefficients vector ðaÞ which gives
less than N � 2 degree of freedom for the system
3. Simulate the behavior model of the system using
SIMULINK using the fractional-order integrator [34].
4. Implement the oscillator with the suitable circuits.
3 Case studies
3.1 Equal phase shifts
In order to obtain equal phase shifts, an equal order frac-
tional-order elements are needed. Hence, from (5), the
condition for oscillation and the oscillation frequency are
determined as follows:
Analog Integr Circ Sig Process
123
Xnk¼0
akxakcos ak
p2
� �¼ 0; ð7aÞ
Xnk¼1
akxa k�1ð Þsin ak
p2
� �¼ 0 ð7bÞ
For the traditional case ða ¼ 1Þ, the maximum output
nodes are four and hence the phase shift between each
output is p2and a1 ¼ a2 ¼ a3 ¼ a4 ¼ 1. By solving (7), the
condition for oscillation and oscillation frequency are the
same as obtained using Routh criterion for the traditional
case which is as follows [19]:
xosc ¼ffiffiffiffiffia1
a3
rð8aÞ
a1a2a3 � a21 � aoa23 ¼ 0 ð8bÞ
On the other hand, in order to design oscillators with
certain phase difference, a can take any value. For exam-
ple, consider an oscillator with three phase shifts is
required. So, for a1 ¼ a2 ¼ a3 ¼ a4 ¼ a and a1 ¼ a2 ¼a3 ¼ a, then
x4acos 2pað Þþax3acos 3ap2
� �þ ax2acos pað Þ
þ axacos ap2
� �þao ¼ 0
ð9aÞ
x3asin 2pað Þ þ ax2asin 3ap2
� �þ axasin apð Þ
þ asin ap2
� �¼ 0
ð9bÞ
From (9), there is a single solution for xo at a� 1 as
shown in Fig. 3. For a[ 1, there are two solutions, one of
these solutions is rejected due to its unstable response
because its poles lie in the right half plane of the physical s-
plane [22]. As clear from Fig. 4, the obtained frequency for
equal phase shifts has single value for a\1. But, for a[ 1,
there are two solutions; one of them is stable and the other
one is not which is discussed in details in [8].
Hence, the pole movement for the proposed oscillator at
equal phase difference is illustrated in Fig. 5. Although, the
conjugate pair of �jxosc is always exist in the oscillator
poles, the oscillator is not always stable. Accordingly, the
oscillator poles start at the stable region of the s-plan and
begins to move towards the unstable region as the value of
a increase as depicted in Fig. 5. Consequently, the poles
(a) (b)
Fig. 2 a Multi-phase oscillator diagram, and b state space representation of the fractional order system
0.4 0.6 0.8 1 1.2 1.4
0
0.2
0.4
0.6
0.8
1
1.2
th 0.6 0.8 1 1.2 1.40
10
20
30
40
50
60
a
0.6 0.8 1 1.2
0
2
4
6
8
10
a(a) (b)
Fig. 3 Change of a x and b a
with respect to the fractional
order a for the case of three
output phase differences
Analog Integr Circ Sig Process
123
appears in the stable region and then moves toward the
unstable region of the oscillator as the value of a increase
of the phase difference increase.
On the other hand, the oscillator could be designed for
non-equal coefficients which increases the design degree of
freedom. Hence, another example is considered for
designing an oscillator with different parameters. For
example, assume an oscillator with three phases with phase
shift between each output is 99� starting from the reference
signal. The conditions for oscillations are not satisfied for
a\0:5, so let’s take a ¼ 0:5 and study the oscillation
conditions and oscillation frequency depending on
a1; a2; a3 and a0. By solving (7) for this case; the oscillation
frequency and conditions for oscillations are given by (10),
respectively
xo ¼1
a23
�a22 � a1a3 � a2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia22 � 2a1a3
q �; a22 [ 2a1a3
ð10aÞ
ao ¼1
2a43
�a42 þ 2a21a
23 � 8a1a
22a3 � 2a32a
23 þ 4a1a2a
33
þ�4a1a2a3 þ 2a22a
23 � 2a1a
33 � 4a32
� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia22 � 2a1a3
q �
ð10bÞ
In order to simplify the previous problem, lets take a1 ¼a3 and a2 ¼ ka1, (10) is reduced to (11). Figure 6(a) shows
the oscillation frequency increases with increasing k within
the its range. While Fig. 6(a) shows the values for ao to
obtain an oscillation as previously discussed ao [ 0 so the
negative values for ao have been removed.
xo ¼ k2 � 1� kffiffiffiffiffiffiffiffiffiffiffiffiffik2 � 2
p; k2 [ 2 ð11aÞ
ao ¼1
2
�k4 � 8k2 þ 2þ 4ðk � 1Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffik2 � 2
pþ a1�
4k � 2k3 þ 2ðk2 � 1Þffiffiffiffiffiffiffiffiffiffiffiffiffik2 � 2
p � ð11bÞ
Fig. 4 The oscillation frequency versus a for different values of a
Fig. 5 Effect of the phase difference on the oscillator poles at x ¼ 100 rad/sec and ða2; a3; a4Þ ¼ ð2:5e3; 50; 2:5e� 5Þ
Analog Integr Circ Sig Process
123
Consequently, the proposed design procedure can be
used to design fractional order oscillators with equal phase
difference by using equal and non-equal coefficients.
3.2 Different Phase Shifts
In this subsection, oscillators with different phase shifts at
the same frequency are presented. So, the fractional orders
are not equal to satisfy the requirement of unequal phase
difference. It is important to note here that, the summation
of all the phases must equal 360�. Assume for simplicity, it
is required to design an oscillator with the phase shifts
22:5�; 67:5�; 135�; 225�, hence the fractional orders are as
follows a1 ¼ 0:25; a2 ¼ 0:5, a3 ¼ 0:75 and a4 ¼ 1. Then,
the characteristic equation is written as follow:
s2:5 þ a3s1:5 þ a2s
0:75 þ a1s0:25 þ ao ¼ 0 ð12Þ
Consequently, the condition of oscillation and the oscilla-
tion frequency are determined from the following
equations:
x2:5cos5p4
� �þ a3x
1:5cos3p4
� �þ a2x
0:75cos3p8
� �
þ a1x0:25cos
p8
� �þ ao ¼ 0
ð13aÞ
x2:25sin5p4
� �þ a3x
1:25sin3p4
� �þ a2x
0:5sin3p8
� �
þ a1sinp8
� �¼ 0
ð13bÞ
By solving (13) for oscillation frequency equal 1 rad/sec
and at a3 ¼ a2 ¼ a1 ¼ a as shown Fig. 7 then the value of
the parameters a and ao are given as follows 0.3512 and
0.4966 respectively. So, a fractional order oscillator with
different phase shifts can be designed using the proposed
procedure and this increases the design degree of freedom
and flexibility.
4 Numerical simulation
Verifying the oscillation is vital in order to make sure that
previous conditions are correct. The first step is through the
numerical simulation. The numerical simulation can be
performed based on two ways. The first way is simulation
the system numerically using one of the popular fractional-
order numerical techniques such as predictor-corrector
approach [5].
The other way is simulating the system using already
built blocks such as non-integer control toolbox for
MATLAB [34]. This toolbox contains a fractional-order
differentiator so in order to build an fractional-order inte-
grator, we use an integer integrator followed by the built
fractional-order differentiator with complementary part of
the fractional-order value. For instance, if we need to build
a fractional-order integrator with a ¼ 0:75 so we put first
order integrator followed by fractional-order differentiator
with order a ¼ 0:25. Moreover, in the differentiator, it is
required to choose the fractional approximation technique
and the the fractional-order bandwidth where the fractional
phase has minmum variations around the required phase
shift order ðap2Þ.
In this work, we used the second way where the system
is built using SIMULINK Since is easier to build and is
more generic. The used approximation technique is
−10 −5 0 5 100
50
100
150
200
(a) (b)
Fig. 6 a The obtained
oscillation frequency versus the
ratio a2=a1, and b the required
ao for sustained oscillation
a3=a2=a1
1 2 3 4 5
2
4
6
rad/
sec
Fig. 7 The oscillation frequency versus a1 ¼ a2 ¼ a3
Analog Integr Circ Sig Process
123
Simpson with bandwidth [0.001-1000] Hz. Also, an initial
condition should be added to initiate the oscillation, so an
initial value has added to the first integrator and equals one.
The numerical simulation for equal phase shift of 72 �where a = 0.8 is shown in Fig. 8(a). By using the previous
analysis at x ¼ 1 rad=sec, the value of ao ¼ 1. Also,
Fig. 8(b) and (c) show the transient simulation for a ¼ 0:5
where the obtained oscillation frequency is 0.26795 rad/
sec and 3.732 rad/sec respectively. Furthermore, the
transient simulation for the case of a ¼ 1:1 is illustrated in
Fig. 9(a) for the stable solution of x. This figure is plotted
for a ¼ 2 and ao ¼ 0:346 where the obtained oscillation
frequency is 1.061 rad/sec. In addition, the simulation for
the unequal phase shifts is depicted in Fig. 9(b) with the
same values which is aforementioned in the previous
section. From the simulations, it is obvious that the
oscillations of each case have the same frequency with
different phase shifts.
5 Circuit realization
Now, it is important to prove the reliability of the previous
analysis using the circuit simulations. The circuit can be
obtained by replacing each integrator of Fig. 2 by an CCII
based fractional order integrator with a time constant of s
0 5 10 15 20
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Time sec0 20 40 60 80 100
-1
-0.5
0
0.5
1
1.5
Time Sec
X 1 X 2 X 3 X 4X 1 X 2 X 3 X 4
0 2 4 6 8 10-4-3
-2-10
123
4
5
Time sec
X 1 X 2 X 3 X 4
(a) (b)
(c)
Fig. 8 Numerical simulation
for a a ¼ 0:8 and phase shift of
72 at ao ¼ 1, b a ¼ 0:5 and
a3 ¼ a2 ¼ �a1 ¼ 1, and c a ¼0:5 and a1 ¼ a2 ¼ �a3 ¼ 1
0 5 10 15 20
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.81
X 1 X 2 X 3 X 4
Time sec(a) (b)
Fig. 9 a Numerical simulation
for a ¼ 1:1 and a ¼ 2, bnumerical simulation at
different phase shifts at
a1 ¼ 0:25; a2 ¼ 0:5, a3 ¼ 0:75and a4 ¼ 1
Analog Integr Circ Sig Process
123
where s equals RC. In addition, the feedback factors an are
replaced with a resistors and their value is calculated using
the previous analysis.
For instance, in case of designing four phases oscillator,
four CCII based integrators are needed as depicted in
Fig. 10. The characteristics equation of this oscillator is
given by
sP4
i¼1ai þ a3
R4C4
sP3
i¼1ai þ a2
R4R3C4C3
sa1þa2
þ a1Q4i¼2 RiCi
sa1 þ aoQ4i¼1 RiCi
¼ 0ð14Þ
As an examples for different cases, let’s design oscillator
with oscillation frequency equals 100 krad/sec.
– Integer example: In case of integer order, the phase
shifts are 0; p=2; p and 3p=2. The circuit is imple-
mented as depicted in Fig. 11(a). The transient
simulation in the traditional case of a1 ¼ a2 ¼ a3 ¼a4 ¼ 1 is performed for R1 ¼ R2 ¼ R3 ¼ R4 ¼ R, C1 ¼C2 ¼ C3 ¼ C4 ¼ C and ao ¼ a1 ¼ 1; a2 ¼ 2. Thus, the
characteristics equation is simplified to
s4 þ a3
RCs3 þ a2
R2C2s2 þ a1
R3C3sþ ao
R4C4¼ 0 ð15Þ
In order to design oscillator with this frequency , then
the time constant is s ¼ RC ¼ 10�5. Figure 11(b) show
Fig. 10 CCII based circuit schematic of four phases oscillator
0 50us 100us 150us 200us-8.0V
-4.0V
0V
4.0V
8.0V X1 X2 X3 X4
(a)
(b)
Fig. 11 Transient simulation of four phases shifted by p=2: a oscillator circuit, and b SPICE circuit simulation of the oscillator in the traditional
case
Analog Integr Circ Sig Process
123
the transient simulation of the circuit where the for
equal phase shifts are obtained.
– Equal fractional phases example: Let’s design an equal
phase shift of 72� with oscillation frequency
xo ¼ 100 krad=sec, then a1 ¼ a2 ¼ a3 ¼ a4 ¼ 0:8.
But, here we have many degree of freedom. So let
R1 ¼ R2 ¼ R3 ¼ R4 ¼ R and C1 ¼ C2 ¼ C3 ¼ C4 ¼C. Thus, the characteristics equation is simplified to
s3:2 þ a3
RCs2:4 þ a2
R2C2s1:6 þ a1
R3C3s0:8 þ ao
R4C4¼ 0
ð16Þ
Hence, let’s assume that the time constant
s ¼ RC ¼ 10�4, then a2 ¼ 1; a2 ¼ 1; a1 ¼ �1; and
ao ¼ 1:618034.
– Different fractional phases example: Let’s design four
phase shifts oscillator with phase shifts 45�; 108�; 180�
and 270�. Then, the required fractional-order capacitors
are a4 ¼ 0:5; a3 ¼ 0:7; a2 ¼ 0:8 and a1 ¼ 1 respec-
tively. And, let’s assume equal time constant so R1 ¼R2 ¼ R3 ¼ R4 ¼ R and C1 ¼ C2 ¼ C3 ¼ C4 ¼ C.
Then the characteristics equation is
s3 þ a3
RCs2:5 þ a2
R2C2s1:8 þ a1
R3C3s1 þ ao
R4C4¼ 0 ð17Þ
This proposed oscillator doesn’t the generality of the
analysis or the procedure. This procedure can be applied
for any sinusoidal oscillatory structure since this analysis is
based on the characteristic equation not derived for specific
oscillator. So, let’s take a simple oscillator with the sim-
plest characteristic equation as an example. The charac-
teristic equation is given by
saþb þ a2sa þ a1s
b þ ao ¼ 0 ð18Þ
For instance, the oscillator,shown in Fig. 12(a), is based on
using three op-amps with two integrator loops where a2 ¼1
C1R4� 1
C1R3; a1 ¼ 0 and a0 ¼ 1
R1R2C1C2[28]. This circuit has
three outputs output of each opamp with two phase shifts,
The phase shift between V1 and V2, and the inverted V1. By
following the proposed procedure to design an oscillator
has fo ¼ 2:5kHz with 63� phase shift between V1 and V2.
Here, many design parameters can be selected as we have
many degrees of freedom. In case of choosing equal-order
fractional elements ða; bÞ ¼ ð0:8; 0:8Þ, the corresponding
Fig. 12 a Simple oscillator
circuit diagram, and SPICE
transient simulation of the
oscillator outputs at bða; bÞ ¼ ð0:8; 0:8Þ, c ða;bÞ ¼ð0:5; 0:8Þ and dða; bÞ ¼ ð1:2; 1:5Þ
Analog Integr Circ Sig Process
123
circuit parameters are R1 ¼ 193 kX;R2 ¼ R3 ¼ 10 kX;R4 ¼ 11:6 kX and C1 ¼ C2 ¼ 10 nF. Moreover,
Fig. 12(b) shows the transient outputs of the oscillator.
While Fig. 12(c) and (d) show the case of different frac-
tional-order elements. Figure 12(c) and (d) are plotted for
ðR1;R2;R3;R5;C1;C2Þ equal ð403:25; 10; 10; 11:6 kX;10; 117 nFÞ and ðR1;R2;R3;R5;C1;C2Þ equal ð31:5X; 10;10 kX; 264X; 2; 10 nFÞ. So, using the same oscillator
structure, it is possible to design required phases depending
on choosing its parameters.
This proposed circuit realization has many advantages
over the others realizations. this realization is simple to
realize for instance, in [16], a multiphase oscillator has
been realized using all pass filters with floating fractional
elements with equal phase shifts. This design has two
drawbacks; added extra hardware and containing floating
elements. It is more simple to realize accurate grounded
fractional element rater than floating fractional element.
6 Conclusion
The design and analysis of multi-phase oscillator for high
order PSK applications have been introduced. Moreover, a
procedure to design any distribution of phase shifts either
equal or different shifts is proposed. several examples and
stability analysis are introduced to obtain the conditions for
oscillation and the oscillation frequency. In designing this
oscillator, many parameters are available to be chosen
without a lot of constrains where the designer can select the
suitable values for his design. The proposed realization is
more reliable due to its simplicity and the grounded float-
ing fractional elements.
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Mohammed E. Fouda received
the B.Sc. degree (honors), in
Electronics and Electrical
Communications Engineering
and the M.Sc. in Engineering
Mathematics from Faculty of
Engineering, Cairo University,
Cairo, Egypt in 2011 and 2014
respectively. He is an assistant
lecturer, Faculty of Engineering,
Cairo University, Egypt. His
research interests include Mem-
element-based Circuits and
Analog Circuits. He has
authored and coauthored more
than 30 journal and conference papers. Fouda won the physical sci-
ences award from Misr El-Khair Institution for international pub-
lishing in 2013. Also, He won the best paper award in ICM 2013 in
Lebanon.
Ahmed Soltan received the
B.Sc. and M.Sc. degrees from
the University of Cairo, Cairo,
Egypt, in 2004 and 2008,
respectively, and got his Ph.D.
degree in electronics and com-
munication at Cairo University,
Cairo, Egypt in 2014. He is
currently working as a Teacher
Assistant in Department of
Electronics and Communica-
tions Engineering, Fayoum
University, Fayoum, Egypt,
since his graduation. His current
research interests include the
investigation of fractional circuits and systems, specifically in frac-
tional order analog filters for signal processing. Also he is interested
in the analog circuits with particular emphasis on current-mode
approach, RF power amplifiers, and VCO.
Ahmed G. Radwan (M’96–
SM’12) received the B.Sc.
degree in Electronics, and the
M.Sc. and Ph.D. degrees in Eng.
Mathematics from Cairo
University, Egypt, in 1997,
2002, and 2006, respectively.
He is an Associate Professor,
Faculty of Engineering, Cairo
University, and also with the
Nanoelectronics Integrated Sys-
tems Center, Nile University,
Egypt. From 2008 to 2009, he
was a Visiting Professor in the
ECE Dept., McMaster Univer-
sity, Canada. From 2009 to 2012, he was with King Abdullah
University of Science and Technology (KAUST), Saudi Arabia. His
research interests include chaotic, fractional order, and memristor-
based systems. He is the author of more than 125 international papers,
six US patents, three books, two chapters, and h-index=17. Dr.
Radwan was awarded the Egyptian Government first-class medal for
achievements in the field of Mathematical Sciences in 2012, the Cairo
University achievements award for research in the Engineering Sci-
ences in 2013, and the Physical Sciences award in the 2013 Inter-
national Publishing Competition by Misr El-Khair Institution. He won
the best paper awards in many international conferences as well as the
best thesis award from the Faculty of Engineering, Cairo University.
He was selected to be among the first scientific council of Egyptian
Young Academy of Sciences (EYAS), and also the first scientific
council of the Egyptian Center for the Advancement of Science,
Technology and Innovation (ECASTI).
Analog Integr Circ Sig Process
123
Ahmed M. Soliman (LSM’09)
was born in Cairo Egypt, on
November 22, 1943. He
received the B.Sc. degree with
honors from Cairo University,
Cairo, Egypt, in 1964, the M.S.
and Ph.D. degrees from the
University of Pittsburgh, Pitts-
burgh, PA, USA, in 1967 and
1970, respectively, all in elec-
trical engineering. He is cur-
rently Professor Electronics and
Communications Engineering
Department, Cairo University,
Cairo, Egypt. From September
1997 to September 2003, he served as Professor and Chairman
Electronics and Communications Engineering Department, Cairo
University, Egypt. From 1985 to 1987, he served as Professor and
Chairman of the Electrical Engineering Department, United Arab
Emirates University, and from 1987 to 1991 he was the Associate
Dean of Engineering at the same University. He has held visiting
academic appointments at San Francisco State University, Florida
Atlantic University, and the American University in Cairo. He was a
visiting scholar at Bochum University, Germany (Summer 1985) and
with the Technical University of Wien, Austria (Summer 1987). He is
Associate Editor of the Journal of Circuits, Systems and Signal Pro-
cessing from January 2004 to present. He is Associate Editor of the
Journal of Advanced Research Cairo University. Dr. Soliman was
decorated with the First Class Science Medal, from President El-Sadat
of Egypt, for his services to the field of Engineering and Engineering
Education, in 1977. In 2008, he received the State Engineering Sci-
ence Excellency Prize Award from the Academy of Scientific
Research Egypt. In 2010, he received the State Engineering Science
Appreciation Prize Award from the Academy of Scientific Research
Egypt. He is amember of the Editorial Board of the IET Proceedings
Circuits Devices and Systems. He is a Member of the Editorial Board
of Electrical and Computer Engineering (Hindawi). He is a Member
of the Editorial Board of Analog Integrated Circuits and Signal
Processing. He is also a Member of the Editorial Board of Scientific
Research and Essays. He served as Associate Editor of the IEEE
TRANSACTIONS ON CIRCUITS AND SYSTEMS–PART I from
December 2001 to December 2003. In 2013, he was decorated with
the First Class Science Medal, from the President of Egypt, for his
services to the country.
Analog Integr Circ Sig Process
123
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