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.

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123