A Low-Power CMOS Programmable CNN Cell and its Application to Stability of CNN with Opposite-Sign Templates

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(IJCSIS) International Journal of Computer Science and Information Security,
Vol. 9, No. 5, May 2011

A Low-Power CMOS Programmable CNN Cell and
its Application to Stability of CNN with Opposite-
Sign Templates
S. El-Din, A. K. Abol Seoud, and A. El-Fahar M. El-Sayed Ragab
Electrical Engineering Department School of Electronics, Comm. and Computer Eng.
University of Alexandria E-JUST.
Alexandria, Egypt. Alexandria, Egypt.
E-mail: eng_salah_alx@yahoo.com E-mail: m.ragab@ejust.edu.eg

Abstract--In this paper, a novel VLSI architecture adaptation of power four quadrant multipliers using MOSFET's operating in
the Cellular Neural Network (CNN) paradigm is described. It is the weak inversion regime, where the small currents
based on a combination of MOS transistors operating in weak contribute to the low- power consumption [13]. The multiplier
inversion regime. This combination has enabled a CMOS also has a variable transconductance characteristic for the
implementation of a simplified version of the original CNN
programmability of the CNN structure. The proposed cell has
model with the main characteristics of low-power consumption.
Digitally selectable template coefficients are employed and a been applied to study the stability [14, 15], and oscillation of a
local logic and memory are added into each cell providing a CNN paradigm [16, 17]. The performance of the proposed
simple dual computing structure (analog and digital). A four- circuit has been evaluated using PSPICE simulations.
quadrant analog multiplier is used as a voltage controlled
current source which is feeding from the weighting factors of the II. The General Framework
template elements. The main feature of the multiplier is the high A cellular neural network [1] is a special type of neural
value of the weight voltage range which varies between the networks, where the analog processing elements on one layer
ground voltage and the supply voltage. A simulation example for are arranged in a two-dimensional grid having cell
stability of a class of nonreciprocal cellular neural network with
interconnections with nearest neighbors only. Consider the
opposite-sign template is presented.
analog processing cell circuit, henceforth called a cell, as
Keywords: Cellular Neural Network, Low-power CNN, Opposite- shown in Fig.1(a), with only one nonlinear element whose
Sign Template. characteristics is shown in Fig.1(b). This cell is located in the
( i , j ) position of a two-dimensional regular array of M  N
N i, j  of a typical cell Ci, j 
I. INTRODUCTION
cells. The r-neighborhood
Cellular Neural Networks (CNNs), introduced by Chua and r
Yang in 1988 [1], have been extensively studied in the past is defined as:
two decade [2, 3, 4]. All such studies have been focused on
four special topics: 1) the CNN functions; 2) hardware
N i, j    Ck , l , max  k  i , l  j   r (integer )
r
(1)
An r =1 neighborhood of a cell within a cell array consists of
implementation; 3) software systems; and 4) various
all those cells shown shaded in Fig.1(c).
engineering and scientific applications [5]. CNNs have been
successfully applied to signal processing systems, especially
in static image treatment [3], and to solve nonlinear algebraic
equations [6]. It has also been shown that the process of
moving images requires the introduction of time delays in the
signals transmitted through the network [7,8,9].Through VLSI
technology and using switching circuit techniques such delays
can be introduce in the interaction between neurons [8]. To
realize the CNN on a silicon chip, the CNN cell is required to
have low power consumption. Various analog VLSI (a)
implementations of CNN building locks have been previously
implemented and tested [10, 11]. Such implementations have
served to build CNNs under different constraints concerning
the size of the network, the kind of cell input and state
(analog/digital), the power consumption, and the
programmability features of the network allowing more
compact VLSI implementations [12].
The aim of this paper is to design and implement a new
low-power CMOS CNN cell. The circuit employs low-
(b)

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(IJCSIS) International Journal of Computer Science and Information Security,
Vol. 9, No. 5, May 2011

is assured even if the symmetry condition is not met. In [15] a
through stability analysis of cellular neural networks with
opposite-sign templates has been presented. In this analysis,
the dependence of complete stability on the template values,
and the parameter regions for complete stability and
instability have been determined. This class is defined by the
template values which satisfy the following structures and
sign conditions [14]:

0 0 0
A  s p s
(c)
(9)
Figure 1. The cell circuit model and its neighborhood in a cell array. (a) The
cell circuit model (b) The characteristics of the single nonlinear element of
 
the cell (a voltage-controlled current source). (c) An r =1 neighborhood in a 0
 0 0

part of a cell array.
1
where p and s0
The dynamical system equations describing a cellular neural
network consist of the following equations and constraints:
R x
The complete stability of the system defined by (2) has been
(1) State Equation:
proven to be strongly conjectural if [15]:
dV 1 ( p  1)
 (t )  
xij
C V xij
A(i, j; k , l )V ykl (t )
(2) i) B0 ii )  s  ( p  1) (10)
dt R x C ( k ,l )
Nr (i , j )
2
  B(i, j; k , l )V ukl (t )  I Also, the network will oscillate periodically if [16]:
C ( k ,l ) N r (i , j ) i) B 0 ii ) s  p  1 (11)
where
1  i  M; 1 j  N IV. Low-Power CMOS Programmable CNN Cell
(2) Output Equation: The block diagram of a continuous time CNN cell is shown
V yij
(t )  0.5 V xij
(t )  1  V xij (t )  1  f (V xij ).  (3) in Fig.2.

(3) Input Equation:
V uij
 Eij . (4)
(4) Constraint Equations:

V xij
(0)  1 (5)

V uij
 1. (6)
(5) Parameter Assumptions:
A(i, j; k , l )  A(k , l; i, j ) Symmetry condition (7) Figure2. Block diagram of CNN cell.

C  0, Rx  0. (8) Vxij is the state of cell Cij, with an initial condition Vxij(0),
RxC conforms the integration time constant of the system. The
III. Stability of Cellular Neural Networks cell output is Vyij (t) = f (Vxij (t)), where f can be any
A necessary condition for the proper operation of a cellular convenient non-linear function. The block A can be
neural network is that it be completely stable within the implemented using a set of four quadrant multipliers whose
dynamic range of prescribed inputs. A circuit is said to be inputs are the outputs of the cells within the assumed
completely stable if every trajectory tends to an equilibrium neighborhood and the template A values. Similarly, block B
state. The complete stability of a subclass of cellular neural can be implemented using a set of four quadrant multipliers
networks is defined by symmetric templates [1]. The whose inputs are the inputs of the cells within the assumed
symmetry condition means that the feedback values between neighborhood and the template B values. The outputs of
any two cells are reciprocal in the sense that corresponding blocks A and B are (in the current form) Ixy and Ixu,
values are the same; i.e., A(i, j; k , l )  A(k , l; i, j ) . The respectively. Those currents are summed with the bias current
assumption ( 7 ) implies the perfect symmetry of the I of the cell and then integrated in the RxC circuit, to result in
feedback-template values between any two cells within a the cell state voltage Vxij. The output voltage of the cell Vyij is
neighborhood. From theorem 4 in [1], if the parameters satisfy obtained through the limiting transfer function f(Vxij).
the symmetry condition, the circuit will be completely stable. Alternatively, the nonlinear transfer function f(Vxij) can be
But many unsymmetrical templates have been found for some incorporated in the multiplier circuits themselves, resulting in
important applications [3]. In [14] it has been shown that for a a small area CNN cell. This can be realized using low-power
class of practically important templates (positive / negative CMOS four quadrant multipliers operating in weak inversion
and opposite-sign templates), the complete stability property regime.

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(IJCSIS) International Journal of Computer Science and Information Security,
Vol. 9, No. 5, May 2011


 I b tanh(k (
V 1  V 2 )) if (12)
 2 V 3
is high and V 4 is low
A. Programmable Low-power CMOS four quadrant I o  I1  I 2  
multiplier  tanh(k (V 1  V 2 )) if
 Ib
 2 V 3
is low and V 4 is high
Fig. 3 shows the proposed programmable low-power CMOS
four quadrant multiplier circuit and its sub-circuit where k  1 , with n is a slope factor ( in practice it
representation. nU T

Vdd Vdd
lies between 1 and 2 and is close to 1 for high values of gate
voltage), and UT is the thermal voltage whose value is 26mv at
I1 Io
room temperature. Current switching logic controlled by V3
I2 and V4 enables the output to change sign. The transfer
 Vdd Vdd characteristic of the multiplier circuit is shown in figure 4. It
is noted that the output transfer characteristic is linearly
V3 V3 proportional to one of the multiplier inputs, I b, and varies
Va V4 nonlinearly with the other input, (V1-V2).


V1 V2

B
Ib
Vdd
B0 B1 B2 B3 B4
Vgg
I0 I0 I1 I2 I3 I4

(a)

V3 V4

V1 Io
Figure 4. Transfer characteristic of the proposed four quadrant multiplier.
V2
Ib
B. Complete CNN CMOS Implementation
Fig. 5 shows a complete implementation of a CNN cell using
the proposed multiplier circuit.

V3,u1 V4,u1
Vcom
(b) Vu1 Io,u1

Vcom
Figure 3. (a) Four quadrant multiplier schematic (b) Multiplier sub-circuit Ib,u1

representation
Cells’ inputs V3,u2 V4,u2
This circuit represents a trade-off between digital and analog u(Nr)
Vu2
Vcom
Io,u2

techniques. It is composed of registers which store the weight Vcom
Ib,u2
Vdd

values, a linear DAC and a tranconductance multiplier. The I

DAC has five bits plus sign weight storage which sets the tail Vun Vx

bias current Ib. The least significant bit bias current has been Vyn

set to 40 pA. The DAC has shown good monotonicity in the Mr C

weak inversion regime. Each bit (B0-B4) of the DAC is Vcom
V3,y2 V4,y2

Vy2 Io,y2
controlled by a pass transistor which can be turned on or off Cells’ outputs
y(Nr) Vcom
Ib,y2
depending on the value stored in the corresponding CMOS
latch.I0-I4 are the current sources which contribute to the bias
current Ib in a successive power of two fashion. The DAC is Vcom
V3,y1 V4,y1
Io,y1
connected to a transconductance amplifier to form a four Vy1
Vcom
quadrant multiplier. Assuming weak inversion operation for Ib,y1

all MOS devices in the multiplier circuit, it can be shown that
the output current Io is expressed as:
Figure5. Complete CNN cell.

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The sets of multipliers in the lower and upper parts of Fig.5 where Rx is the resistance of the diode-connected transistor,
represent the second and third terms in the left hand side of
equation (2), respectively. Each multiplier in the lower set f (V x)  tanh ( kV x ) ,
accepts one of the cells' outputs within the given 2
neighborhood, as one input, and the corresponding template and a1 and a2 represent the template-A values of the network.
value A ( ) as the other input. The A- template values are
determined by the programmable tail current sources I b,y and
their signs are controlled by the multiplier control inputs V 3's Vcom
and V4's. On the other hand, each multiplier in the upper set
accepts one of the cell's inputs within the given neighborhood Vcom
as one input, and the corresponding template value B( ) as the
other input. Also, those B- template values are determined by
the programmable tail current source, Ib,u and their signs are
Vx1
controlled by the corresponding multiplier control inputs V3's
and V4's. The output currents of the two multiplier sets are Mr
C
Vcom
summed together and applied to the RxC current integrator.
The resistor Rx is implemented using the diode-connected
transistor Mr. Vcom

V. SIMULATION EXAMPLE
To test the validity of the proposed CNN cell, a cellular neural
network with two cells using an opposite-sign template is
Vcom
considered [15]. The network is shown in Fig.6.
Vcom

Vx2

Mr
Vcom C

Vcom

Figure 6. CNN with opposite-sign template. Figure7. Low-power CMOS implementation of the CNN of Fig. 6

The cells" inputs and their bias terms are set to zero, in order AS mentioned previously, the network is considered to be
to remove the possibility of forced stability. The state
( a  1)
equations of the system can be described by: conjecturally stable if 1
  ( a1  1) and will
a
x1  x1  p f ( x1)  s f ( x2)

2
2
(13) oscillate periodically if a  ( a1  1)  0 . Fig .8 shows the
x x
 2 2
 s f ( x1)  p f ( x2). 2
transient behavior of the network with a1= 2.0 and a2 = 0.99.
Fig. 7 shows how the CNN of Fig.6 can be implemented using A complete stability is observed in such a case where state
the proposed CNN cell. Note that in such an architecture the voltages Vx1 and Vx2 are converged to constant values. Fig. 9
cell's state voltages Vx1 and Vx2 are directly fedback to the Shows the transient behavior of the network with a1=2.0 and
two cells and the nonlinear functions f (x1) and f(x2) are a2 =1.2. Periodic oscillations of state voltages Vx1 and Vx2 are
already embedded in the multipliers' transfer characteristics. observed in this case.
As previously stated, this would guarantee compact CNN
design architectures. The state equations resulting from such
an implementation are then expressed as:

C dV x1  
1
V x1  a1 I b0 f (V x1)  a2 I b0 f (V x 2) (14)
dt Rx
and, C dV x 2   1
V  a I f (V )  a I f (V ) (15)
dt Rx x 2 1 b0 x 2 2 b0 x 2

Figure8. Transient behavior of the network with a1=2.0, a2=0.99.

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