Master’s Thesis
Hardware Implementation of the
Exponential Function Using Taylor
Series and Linear Interpolation
Ateeq Ur Rahman Shaik
Department of Electrical and Information Technology,
Faculty of Engineering, LTH, Lund University, May 2014.
Master’s Thesis Report
HARDWARE IMPLEMENTATION
OF THE EXPONENTIAL
FUNCTION USING TAYLOR
SERIES AND LINEAR
INTERPOLATION
By
Ateeq Ur Rahman Shaik
Department of Electrical and Information Technology
Faculty of Engineering, LTH, Lund University
SE-221 00 Lund, Sweden
1
2
Popular Scientific Essay of
”Hardware Implementation of Exponential Function
using Taylor Series and Linear Interpolation”
This Master thesis is about ASIC implementation of Exponential
Function using two different approaches. Exponential functions are
incremental functions which are useful in areas such as
Communication circuits, computer graphics, signal and image
processing applications. More precisely the unary functions such as
sine, cosine, exponential, logarithmic functions are applicable in
signal generational circuits such as oscillators, variable gain circuits.
Sometimes software implementation does not meet the requirements
in some applications, in case of speed is of utmost importance it has
to be implemented in hardware. The two methods used for
implementation of the function such as Taylor series method and
Linear Interpolation.
In Taylor series approach two different solutions as suggested.
Each one has its own advantages of its own. Linear Interpolation
method is memory intensive design, in which pre-calculated values
are stored in an Look-up Table for future calculations of unknown
values.
A total of five designs are synthesized using synopsis Design
Vision STM065nm process technology using two different methods.
Both the architectures are mainly focused on low power, minimizing
area and better performance. A comparison is made between these
architectures. The target libraries used are high and low threshold
voltage. Synopsis PrimeTime is used for the estimation of dynamic
power consumption.
3
4
Abstract
This thesis work is targeted towards ASIC implementation of the
exponential function. Generally, unary functions like trigonometric,
logarithmic and exponential functions are useful in areas such as
computer graphics, signal processing and image processing for high
speed applications. Sometimes software implementation does not
meet the requirements in some applications, in case of speed is of
utmost importance it has to be implemented in hardware. In this work
the exponential function is implemented using two different
architectures that are using Taylor series and Linear Interpolation.
Both architectures are mainly focused on low power, minimizing area
and better performance. A comparison is done between the two
architectures. The designs are synthesized in synopsis Design Vision
STM065nm process technology with target libraries with high and
low threshold voltage. Synopsis PrimeTime is used for the estimation
of dynamic power consumption.
5
Acknowledgments
First of all it is my immense pleasure to have the opportunity
working under Prof. Peter Nilsson, sincere words are not sufficient to
describe his caring nature, patience and his expertise in the field of
digital design. I would like to express my heartfelt gratitude towards
my supervisors Erik Hertz, Rakesh Gangarajaiah their timely
availability for solving my petty issues and faults during the course of
work. Again I would like to say I cannot forget the rest of my life
their favor to finishing the final part of Master’s degree. It was highly
impossible to finish up this work without their guidance and support
during this work.
Secondly I would like to thank Prof. Pietro Andreani, Prof.
Henrik Sjöland, Prof. Viktor Öwall, and Associate Prof. Joachim N.
Rodrigues for completing my MS successfully in the EIT department.
I would like to thank my professors for teaching me all the mandatory
as well as elective courses, which are part of my degree.
Special thanks to Pia Bruhn for solving all the issues related to the
academic management.
I would like to thank all my friends who provided me constant
support, caring, without whom the stay in Sweden would not be
possible. For the fear of missing I am not naming anybody in the list.
Last but not least thanking almighty for his grace upon and giving
me a chance to improve to be better human being. In the end I would
like to thank my parents who made me possible to come into this
world and pleasant stay till now, as well as all my family members for
their support in all times.
Thank you all who supported me directly or indirectly in the
process of this journey.
6
Contents
Abstract.........................................................................................................5
Acknowledgments.........................................................................................6
1.
2.
3.
Introduction.........................................................................................11
1.1.
ProjectSpecification....................................................................11
1.2.
ThesisOrganization.....................................................................12
Theory..................................................................................................13
2.1.
TaylorSeries................................................................................13
2.2.
Interpolation................................................................................15
2.2.1.
LinearInterpolation.............................................................15
2.2.2.
PolynomialInterpolation.....................................................16
2.2.3.
SplineInterpolation.............................................................17
2.2.4.
PiecewiseLinearinterpolation............................................18
PerformanceMetrics...........................................................................19
3.1.
MaximumAbsoluteError:...........................................................19
3.2.
MeanError:.................................................................................19
3.3.
Median:.......................................................................................20
3.4.
Standarddeviation:.....................................................................20
3.5.
RootͲmeanͲsquare(RMS)Error:..................................................20
3.6.
ErrorProbabilityDistribution......................................................21
3.7.
Decibel(dB).................................................................................21
3.7.1.
Powermeasurement...........................................................22
3.7.2.
Amplitudemeasurement....................................................22
3.7.3.
ErrorindBscale...................................................................22
3.8.
PowerAnalysis:...........................................................................23
3.8.1.
StaticPowerConsumption:.................................................23
7
3.8.2.
ShortͲCircuitPowerConsumption:......................................23
3.8.3.
DynamicPowerConsumption:............................................24
CHAPTER4...................................................................................................25
4.
HardwareImplementation..................................................................25
4.1.
TaylorSeriesArchitectures..........................................................27
4.1.1.
TaylorSeriesArchitecture1................................................27
4.1.2.
TaylorSeriesArchitecture2................................................29
4.1.3.
TaylorSeriesArchitecture2withpipelinedstage...............32
4.2.
LinearInterpolationArchitectures..............................................34
4.2.1.
ExampleofasimpleLUT:....................................................34
4.2.2.
LinearinterpolationArchitecturewith1LUT:.....................36
4.2.3. LinearinterpolationArchitecturewith2LUTswith256
Intervals:38
4.2.4. LinearinterpolationArchitecturewith2LUTswith128
intervals:40
5.
6.
EstimationofError..............................................................................43
5.1.
TaylorseriesArchitecture1:.......................................................43
5.2.
TaylorseriesArchitecture2:.......................................................45
5.3.
LinearInterpolationarchitecturewith1LUT:.............................48
5.4.
LinearInterpolationwith2LUTsand256intervals:...................51
5.5.
LinearInterpolationwith2LUTsand128intervals:...................54
Results.................................................................................................59
6.1.
Area.............................................................................................59
6.2.
Power..........................................................................................63
6.3.
Timing..........................................................................................64
6.4. PowerestimationofTaylorseriesarchitecture2using
PrimeTime:..............................................................................................65
8
6.5. PowerEstimationforLinearInterpolationwith2LUTsusing
PrimeTime:..............................................................................................67
6.6. ComparisonofTaylorseriesApproachandLinearInterpolation
Approach.................................................................................................68
7.
Conclusions..........................................................................................71
8.
Futurework.........................................................................................73
REFERENCES................................................................................................75
ListofFigures...............................................................................................77
ListofTables................................................................................................79
ListofAcronyms..........................................................................................81
9
10
CHAPTER 1
1. Introduction
Unary functions i.e. trigonometric, logarithmic and exponential
functions are very useful in digital design applications. Their
applications can be found in digital signal processing, communication
systems, robotics, computer graphics etc. Generally for high-speed
applications a simple software solution does not necessarily meet the
speed requirements. So in order to achieve the desired speed, a
hardware solution is investigated. In this regard optimization for
speed and area are the major design challenges in a digital application
specific integrated circuits (ASIC) implementation.
Researchers are currently trying to find out efficient hardware
implementations of these functions to be used in signal and image
processing applications. In this thesis, hardware implementations for
exponential function using linear interpolation method and Taylor
series have been investigated.
1.1.
Project Specification
This thesis project is an investigation study and there are no fixed
design specifications. The proposed design methods, which are
explained in forthcoming chapters, are compared using parameters
area, speed, and power consumption. In the design process minimum
area and low-power consumption has primarily been taken in account.
Table I shows some of the fixed design specifications.
11
TABLE I.
SPECIFICATIONS OF THE HARDWARE
IMPLEMENTATION OF EXPONENTIAL FUNCTION
Parameter
Value
Supply Voltage
1.2V
Accuracy
>15 bits
Cell Library
LPHVT, LPLVT
Temperature
25°C
Process
STM065nm CMOS process
1.2. Thesis Organization
This thesis report is comprised of ten chapters. Chapter 2
explains the theoretical aspects of Taylor’s series and interpolation.
Chapter 3 discusses some error metrics required to qualify the
designs. In chapter 4 the proposed architectures are implemented on
hardware level. In chapter 5 error metrics of implemented designs are
demonstrated. Chapter 6 discusses and compares the results of
proposed architectures behavioral models and hardware level
implementations. The thesis is concluded in chapter 7 and future
recommendations are provided in chapter 8.
12
CHAPTER 2
2. Theory
The proposed architectures in chapter 3 use Taylor series and
linear interpolation for implementing the exponential function. Using
Taylor series, complex functions are translated into series of low level
functions (terms) which can be mapped on hardware. Whereas in
interpolation a function is rebuilt using predefined data points which
are stored in a memory. This chapter will explain the basics of Taylor
series and the interpolation method to form a theoretical background.
2.1. Taylor Series
Taylor series is the representation of a function into an infinite
summation of terms of function that are obtained by differentiating
that function at any given value [1]. To represent a real function f(x)
at any arbitrary point x = a, is given by equation (2.1).
݂ሺݔሻ ൌ ݂ሺܽሻ ݂ ᇱ ሺ ݔെ ܽሻ ᇲᇲ ሺሻ
ଶǨ
ǥ ሺ ݔെ ܽሻଶ ሺሻ ሺሻ
Ǩ
ሺయሻ ሺሻ
ଷǨ
ሺ ݔെ ܽሻଷ
ሺ ݔെ ܽሻ
(2.1)
The generalized notation of Taylor’s series is shown in (2.2).
ஶ
݂ሺݔሻ ൌ ݂ሺܽሻ ቆ
ሺ݂ሺሻ ሺܽሻ כሺ ݔെ ܽሻ
ቇ
݊Ǩ
ୀ
(2.2)
Taylor series expansion of most common functions such as
exponential, logarithmic and trigonometric functions are expressed as
follows:
The exponential function ݁ ௫ is expressed in Taylor series as
equation (2.3).
13
ଵ
ଵ
ଶ
݁ ௫ ൌ ݁ ቂͳ ሺ ݔെ ܽሻ ሺ ݔെ ܽሻଶ ሺ ݔെ ܽሻଷ ڮቃ
(2.3)
The Taylor series expansion of a logarithmic function is
expressed as shown in (2.4):
ሺݔሻ ൌ ሺܽሻ ሺ௫ିሻ
െ ሺ௫ିሻమ
ଶమ
ሺ௫ିሻయ
ଷయ
െ ڮ
(2.4)
Similarly trigonometric functions sin(x) and cos(x) can be
expressed in Taylor series expansion as shown in equations (2.5) and
(2.6).
ͳ
ሺݔሻ ൌ
ሺܽሻ െ ሺܽሻሺ ݔെ ܽሻ െ
ሺܽሻሺ ݔെ ܽሻଶ ʹ
ͳ
ሺܽሻሺ ݔെ ܽሻଷ ڮ
(2.5)
ͳ
ሺݔሻ ൌ ሺܽሻ
ሺܽሻሺ ݔെ ܽሻ െ ሺܽሻሺ ݔെ ܽሻଶ
ʹ
ͳ
െ
ሺܽሻሺ ݔെ ܽሻଷ ڮ
(2.6)
14
2.2. Interpolation
Interpolation is a process of estimating unknown values from a set
of known values provided the unknown values are within the range of
known values [2]. There are a variety of interpolation techniques
among them the most common are linear interpolation, polynomial
interpolation, spline interpolation and piecewise linear interpolation.
These interpolation techniques are explained in detail in the following
subsections [2].
2.2.1. Linear Interpolation
In this Linear Interpolation method the unknown value is
estimated based upon known data points provided the unknown value
is in between the two known points [2]. For example, consider two
known data points A(xa, ya) and B(xb, yb). A third unknown data
point C(x, y) is to be calculated as shown in Fig. 1. C(x, y) can be
calculated for a given value of x and y using equation (2.7) [2].
ݕൌ ݕ ሺ ݔെ ݔ ሻ כ
ሺ௬್ ି௬ೌ ሻ
ሺ௫್ ି௫ೌ ሻ
(2.7)
PointB
yb
PointC
y
ya
PointA
xa
Fig. 1.
x
xb
The illustration of Linear Interpolation Method.
15
Suppose A(xa, ya) and B(xb, yb) represents A(1, 5) and B(2, 8). For
x = 1.5 the value of y at point C can be calculated using equation (2.7)
as shown as (2.8).
ݕൌ ͷ ሺͳǤͷ െ ͳሻ כ
ሺ଼ିହሻ
ሺଶିଵሻ
(2.8)
ݕൌ Ǥͷ
From the equation (2.8), the result of y-coordinate at x = 1.5 is
6.5.
This method is known for its simplicity of all interpolation
techniques and noted for its accuracy and speed of retrieving results.
This method is most commonly used in computer graphics. In this
method, the intermediate intervals are chosen such that the number is
a power of 2 such as 2, 4, 8, 16, 32, 64, 128, 256, in a linear
interpolation method.
The main advantages of this method are the computation speed of
the operation and that it is easy to use. The gradient of difference
plays an important role in evaluating the interpolating value.
2.2.2. Polynomial Interpolation
In Polynomial Interpolation method, a set of polynomial functions
are used to solve the unknown interpolant value.
Consider a given set of (n+1) data points. Based upon these values
there exists an interpolation polynomial function as shown in (2.9).
ሺݔሻ ൌ ܽ ݔ ܽିଵ ݔିଵ ܽିଶ ݔିଶ ܽିଷ ݔିଷ
ǥ ܽଶ ݔଶ ܽଵ ݔ ܽ Ǥ
(2.9)
Upon construction of (n+1) equations based on the given set of
data points, in generalized form as shown in (2.10).
ሺݔ ሻ ൌ ݕ i { א0,1,2,……,n}
(2.10)
16
Upon including all the polynomial equations into (2.10), a system
of linear equations is formed. After arranging the set of (n+1) it
deduces into three matrices as shown in (2.10).
ܽ
ݕ
ݕଵ
ܽ
ିଵ
ିଵ
ݔ ݔ ͳ ڮ
ۇ
ۊ
ۇ
ݕଶ ۊ
ڰ
ڭቍ * ܽۈିଶ
ቌݔଵ
=
ǥ ۋ
ۈǥۋ
ݔଶ ݔଶ ିଵ ͳ ڮ
ǥ
ǥ
ܽ ۉ ی
ݕۉ ی
(2.10)
In the above matrices the left matrix is known as Vandermonde
Matrix [2]. Solving the equations, there exists a unique set of
solutions for the derived matrices. The solution to the polynomial
equation is achieved by solving the Vandermonde matrix.
There are limitations related to the convergence of the polynomial
equation. The uniform convergence cannot be guaranteed when the
function is indefinitely differentiable. This condition leads to
oscillatory form of solution and also complexity can be highly
increased in-terms of functions where the order (n) of polynomial is
large.
2.2.3. Spline Interpolation
In cases when the roots of polynomial interpolation are oscillatory
in nature, then there is a solution for this equation which is known as
the spline interpolation method [3]. In this interpolation method the
interpolation is subdivided into various intervals or sub functions i.e.,
different polynomials define the nature of the function between each
interval. The nature of the function is based upon the function
between each interval.
There exists various forms of spline interpolation methods such as
linear method, quadratic method, cubic method; in most cases spline
interpolation refers to cubic method. Upon solving the coefficients of
the functions leads to solution of the spline interpolation method. In
this method it structures a smooth curve which is continuous in first
and second derivatives of the function. This method also reduces
complexity of computations in comparison with the polynomial
interpolation method.
17
2.2.4. Piecewise Linear interpolation
Piecewise linear interpolation is a broad method of applying a
simple linear interpolation to all the continuous intervals in a series of
given data points [2].
Piecewise linear interpolation is generally achieved by executing
the process of linear interpolation between each interpolation interval.
It is one of the quickest and simplest interpolation techniques. In this
method the first step is to locate the interpolant value inside the
interval, then apply linear interpolation procedure. The appearance of
this method resembles a set of straight lines.
Consider a given set of data points (1,20), (2,7), (3,4), (4,9), (5,7),
(6,3) after interpolation the desired piecewise linear interpolations is
shown in Fig. 2.
Fig. 2.
Illustration of piecewise linear interpolation with given set of data
points.
18
CHAPTER 3
3. Performance Metrics
Performance of hardware architecture can be characterized by its
error behavior and its power analysis. In this regard error
characterization of any design generally uses five basic metrics.
These metrics are maximum absolute error, mean error, median,
standard deviation, root mean square error and error probability
distribution. In power analysis, static power consumption, short
circuited power consumption and dynamic power consumption are
considered.
Following sections will define these metrics very briefly.
3.1. Maximum Absolute Error:
The maximum Absolute Error is a metric which defines worst
possible error in any design [4]. It is defined as the difference
between the approximated value and actual value, as shown in (3.1).
οݔ ൌ ȁݔෝప െ ݔ ȁ
(3.1)
Where, ݔෝప is approximation value,
ݔ is actual value,
3.2. Mean Error:
Consider given n terms in a sequence of error approximating
function. The Mean Error is defined as the average of the individual
error of each specific quantity, as shown in (3.2) [4].
ͳ
ݔҧ ൌ כሺݔෝప െ ݔ ሻ
݊
ୀଵ
(3.2)
19
ݔҧ is mean value
3.3. Median:
Median is defined as the mid value of any sequence of sample
errors. In case there is even number of error samples, median will be
the average value of the two middle values in the sequence [4].
3.4. Standard deviation:
The Standard Deviation is defined as the average of the square of
the individual error term and mean error, as shown in (3.3). In simple
words standard deviation represents how much error variations exist
from the average value. Low standard deviation means that difference
between any given value and the mean value is close to zero [4].
ଶ
ଵ
ߪ ଶ ൌ כσୀଵൣݔෝప Ȃݔҧ ൧
(3.3)
3.5. Root-mean-square (RMS) Error:
The Root Mean Square value is defined as the square root of the
average of the sum of the squared terms of individual errors, as
shown (3.4).
ଵ
ݔ௦ ൌ ට כσୀଵሺݔෝప െ ݔ ሻଶ
(3.4)
RMS error is very useful when the error is varying in positive and
negative values.
20
3.6. Error Probability Distribution
Error Probability distribution is a graph interpreting the error
diagram to visualize the deviation of absolute error. This diagram
simplifies the interpretation of standard deviation and RMS values
[4].
The following Fig.3 shows an example of Error probability
distribution diagram.
Fig. 3.
Error Probability distribution diagram
3.7. Decibel (dB)
In statistical measurement quantities dB scale is mostly used in
order to provide greater accuracy, high resolution and easier
understanding of the metrics.
21
The dB-scale is generally used for power and amplitude
measurements.
3.7.1. Power measurement
To analyze the power or intensity measurement of any device, in
dB units [4], it is expressed as shown in (3.5)
ܲௗ ൌ ͳͲ ݈݃ כଵ ቆ
݉݁ܽݎ݁ݓ݀݁ݎݑݏሺܲሻ
ቇ
ݎ݁ݓ݁ܿ݊݁ݎ݂݁݁ݎሺܲ ሻ
(3.5)
3.7.2. Amplitude measurement
To analyze the amplitude measurement of characteristic such as
voltage or current of any device, it is represented as shown in (3.6)
[4].
ܣௗ ൌ ʹͲ ݈݃ כଵ ቆ
݉݁ܽ݁݀ݑݐ݈݅݉ܽ݀݁ݎݑݏሺܣሻ
ቇ
݁݀ݑݐ݈݅݉ܽ݁ܿ݊݁ݎ݂݁݁ݎሺܣ ሻ
(3.6)
3.7.3. Error in dB scale
Generally the Signal to Noise Ratio (SNR) is expressed in dB
scale rather than linear scale to accommodate large error deviations
ranging from tenth (10e-1) to billionth (10e-9) in order. The dB scale
gives the freedom to express large variations in short notation.
However, the error of dB scale can be further expressed in terms of
number of bits as shown in (3.7).
ܰ ݏݐ݅ܤ݂ݎܾ݁݉ݑൌ ݎݎݎܧ
ݎݎݎܧ
ൎ ʹͲ כଵ ሺʹሻ
ǤͲʹ
(3.7)
22
3.8. Power Analysis:
Power analysis is an important design parameter in which the total
power consumption of a system is estimated. Enormous effort is
invested in optimizing power utilization of ASIC devices. In this
project designs are optimized for Low-power to ultra-low power
design [6] [7].
In this regard the total power consumption in any circuit is
expressed as shown in (3.8):
P total
=
P leakage +
P short-circuit + P dynamic
(3.8)
Where
P leakage
=
VDD*I leak
P short-circuit =
I sc T sc VDD*Į
P dynamic
Į*CtotalVDD2fclk
=
Where, P leakage is leakage power consumption,
P short-circuit is short circuit power consumption,
P dynamic is short dynamic power consumption,
VDD is supply voltage,
I leak is leakage current,
I sc is short-circuit current,
T sc is short-circuit duration of the circuit,
f clk frequency of clock
3.8.1. Static Power Consumption:
This is the power consumption of a circuitry during powered-ON
state. It is defined as shown in (3.9).
P static
=
VDD * (I substrate + I oxide)
(3.9)
3.8.2. Short-Circuit Power Consumption:
The short-circuit power consumption can be expressed in (3.10):
23
ܲ௦௧ି௨௧ ൌ ߙ ܫ כ௦ ܶ௦ ܸ
(3.10)
Where Iscis the short circuit current from VDD to ground, Tsc is
the short circuit duration and ߙis the switching factor.
3.8.3. Dynamic Power Consumption:
Power consumption of logic circuitry during the transitional
phases of a digital design is known as dynamic power consumption. It
is expressed in (3.11):
ܲௗ௬ ൌ ߙ ܥ כ௧௧ ܸ ଶ ݂
(3.11)
Where Ctotal is the total circuit capacitance, fclk is the switching
frequency and VDD is the supply voltage.
24
CHAPTER 4
4. Hardware Implementation
ASIC implementation of a digital design involves a design flow as
shown in Fig. 4. It starts with design specifications which are often
defined for a project. Then a reference model is created in MATLAB
according to design specifications. In the next step a behavioral
model is designed in Modelsim and its results are compared to the
results of the MATLAB model. The behavioral model is fine-tuned
duing functional verification to meet the design specifications. Once
the functional requirements of the design are met the design is
synthesized on gate level net-list in Synopsis design vision. After this,
timing simulations are performed on the synthesized design during
post-synthesis verification. In case the design doesn’t meet the timing
requirements it is fine tuned in the behavioral model again and the
process is followed until the timing requirements are met. When the
design has acceptable performance it is ready for sign off.
25
DesignSpecifications
MakingReferenceModel
(MATLABModel)
Fault
BehavioralModelling
(VHDLmodelin
Modelsim)
PostͲSynthesis
Verification
(Modelsim)
Fault
FunctionalVerification
(VHDLTestbench)
Correct
PowerEstimation
(SynopsisPrimeTime)
Correct
SignOff
Synthesis(DesignVision)
Fig. 4.
Flow Chart of ASIC Design.
In this chapter two different techniques have been used to
implement the exponential function on hardware level. The first
technique uses Taylor series expansion method whereas the second
technique uses the linear interpolation method. The theoretical
significance of these techniques has been discussed in Chapter 2. In
this chapter, different hardware implementations using these
techniques will be suggested to find a better compromise between
area and speed.
26
4.1. Taylor Series Architectures
In this section, two different architectures, architecture 1 and 2 are
suggested using Taylor series expansion method. Architecture 2 is
derived based on the limitations of architecture 1 and has reduced
area and power consumption.
4.1.1. Taylor Series Architecture 1
To implement architecture 1 of exponential function, a Taylor
series expansion is applied to exponential function ex, substituting f(x)
= ݁ ௫ in (2.1) reveals;
ଵ
ଵ
݁ ௫ ൌ ݁ ቂͳ ሺ ݔെ ܽሻ ሺ ݔെ ܽሻଶ ሺ ݔെ ܽሻଷ
ଶ
ଵ
ଶସ
ଵ
ସ
ሺ ݔെ ܽሻ ଵଶ
ହ
ሺ ݔെ ܽሻ ଵ
ଶ
ሺ ݔെ ܽሻ ଵ
ହସ
ሺ ݔെ ܽሻ ቃ
(4.1)
In (4.1) when factoring (x-a) out of all terms and rearranging their
order we get, as shown in (4.2).
ଵ
ଵ
ଶ
݁ ௫ ൌ ݁ ൦ͳ ሺ ݔെ ܽሻ ൮ͳ ሺ ݔെ ܽሻ ቌ ሺ ݔെ ܽሻ ൭
ଵ
ଵ
ଶସ
ଵଶ
ሺ ݔെ ܽሻ ቆ ሺ ݔെ ܽሻ ൬
ଵ
ଶ
ሺ ݔെ ܽሻ൰ቇ൱ቍ൲൪
(4.2)
The hardware of architecture1, having ‘x’ as input and ‘exp(x)’ as
output, is generated based on equation (4.2) and is shown in Fig. 5. It
has 7 multipliers and 7 adders.
27
Fig. 5.
Hardware implementation of Exponential function using Taylor
series architecture 1.
In this architecture, the input x has been specified to vary between
0 and 1. After MATLAB modelling of the design the estimated
wordlength is 19 bits at each stage, to achieve desired precision. This
architecture uses 2’s compliment fixed-point number representation
for positive and negative values. The wordlength of each fixed
coefficient of multipliers and adders is 18 bits. The precision
achieved for architecture 1 is 15.72 bits.
28
4.1.2. Taylor Series Architecture 2
To derive the hardware architecture 2 using Taylor series method
consider (4.2).
ଵ
ଵ
ଶ
݁ ௫ ൌ ݁ ൦ͳ ሺ ݔെ ܽሻ ൮ͳ ሺ ݔെ ܽሻ ቌ ሺ ݔെ ܽሻ ൭
ଵ
ଵ
ଶସ
ଵଶ
ሺ ݔെ ܽሻ ቆ ሺ ݔെ ܽሻ ൬
ଵ
ଶ
ሺ ݔെ ܽሻ൰ቇ൱ቍ൲൪
(4.2)
Substituting the inner most part of (4.2) with݇ଵ ,
ଵ
ଵ
ଶ
݁ ௫ ൌ ݁ ͳ ሺ ݔെ ܽሻ൭ͳ ሺ ݔെ ܽሻቆ ሺ ݔെ ܽሻ ൬
ଵ
ሺ ݔെ ܽሻቀ ሺ ݔെ ܽሻ݇ଵ ቁ൰ቇ൱൩
ଶସ
(4.3)
Where ݇ଵ ൌ ଵ
ଵଶ
ଵ
ଶ
ሺ ݔെ ܽሻ ൌ ଵ
ଵଶ
െ
ଶ
௫
ଶ
Equation (4.3) can be further reduced by substituting the innermost factor with k2 as shown in (4.4),
ଵ
ଵ
ଶ
݁ ௫ ൌ ݁ ͳ ሺ ݔെ ܽሻ ൭ͳ ሺ ݔെ ܽሻ ቆ ሺ ݔെ ܽሻ ൬
ሺ ݔെ ܽሻ ቀ
ଵ
ଶସ
݇ଶ ቁ൰ቇ൱൩
(4.4)
Where ݇ଶ ൌ ሺ ݔെ ܽሻ ቀ
ൌ െ
ଵ
ଵଶ
െ
ଶ
௫
ቁ
ଶ
ܽ
ܽଶ
ݔ
ʹܽݔ
ݔଶ
െ
ͳʹͲ ʹͲ ͳʹͲ
ʹͲ ʹͲ
29
Equation (4.4) can be re-arranged to a simpler form as shown
(4.5).
௫
݁ ൌ ݁ ሺܿ ܿଵ ݔ ܿଶ ݔଶ ܿଷ ݔଷ ܿସ ݔସ ܿହ ݔହ ሻ
(4.5)
Where the constant values (Ci) are expressed in table II as follows:
TABLE II.
COEFFICIENTS USED IN TAYLOR ARCHITECTURE 2
Constant
C0
C1
Coefficient
a 2 a3 a 4 a5
a6
2
6 24 120 720
0.606532118055556
2a 3a 2 4a3 31a 4 6a5
2
6
24
720 720
0.606597222222222
1 a 1
Numerical Value
C2
1 3a 6a 2 64a3 14a 4
2 6
24
720
720
0.302604166666667
C3
1 4a 66a2 16a3
6 24 720 720
0.103472222222222
C4
1 34a 9a 2
24 720 720
0.021180555555556
C5
7
2a
720 720
0.008333333333333
In architecture 2, all the coefficients of (ܥ ) can be hardwired in
implementation. Equation (4.5) can be further reduced as expressed in
(4.6),
݁ ௫ ൌ ݁ ቆܿ ݔ൬ܿଵ ݔቀܿଶ ݔ൫ܿଷ ݔሺܿସ ܿݔହ ሻ൯ቁ൰ቇ
(4.6)
30
Now the hardware of architecture 2, having ‘x’ as input and
‘exp(x)’ as output, is generated based on equation (4.6) and is shown
in Fig. 6. It has 6 multipliers and 5 adders.
Fig. 6.
Hardware implementation of Exponential function using Taylor
series Architecture 2.
The input x has been specified in ranges from 0 to 1. All the
coefficients are positive in Taylor series architecture 2. In this
architecture unsigned fixed point number representation is used. Each
stage has a wordlength of 20 bits, 1 bit for integer part and 19 bits
fractional part. The coefficients are specified in Table II and the word
length of each coefficient is 20 bits. The wordlength at the output
stage is 18 bits. The precision achieved is 16.33 bits.
Fig. 7 shows the partial synthesized hardware of Taylor series
architecture 2 in stm65nm process. This shows the gate level
connections of all the components in architecture 2.
31
Fig. 7.
Synthesized design (partial) of Taylor series architecture 2.
4.1.3. Taylor Series Architecture 2 with pipelined stage
Taylor series architecture 2 has a limitation of working up to a
few MHz of frequency range. In order to calculate the dynamic power
consumption of Taylor series architecture 2 up to 1 GHz it is
pipelined as shown in Fig. 8.
32
Fig. 8.
Taylor Series Architecture 2 with pipelined stage
Fig. 8 shows that a pipeline stage i.e. registers, is inserted in the
middle of Taylor series architecture 2 which divides the critical path
into half. Taylor series architecture 2 with pipeline stage has
increased operating frequency which enables the dynamic power
analysis to be performed at frequency of around 1GHz.
33
4.2. Linear Interpolation Architectures
Besides Taylor series architectures, interpolation based
architectures are implemented to compare various performance
metrics. In this regard three different architectures are suggested.
Implementing Linear Interpolation based architecture needs Look-up
Table (LUT) to store the pre-calculated values. To understand the
basic concept of using an LUT, for linear interpolation architecture, a
simple example is demonstrated below.
4.2.1. Example of a simple LUT:
In this section a simple 8x8 LUT is demonstrated. Firstly an 8x8
LUT is designed in VHDL. The design is synthesized in Synopsis
Design Vision to generate the hardware architecture.
The following VHDL code shows an LUT design:
type t_lut_data is array (0 to 7) of
std_logic_vector(7 downto 0);
constant data1:t_lut_data:= (
"01000000",
"01000000",
"01000001",
"01000001",
"01000010",
"01000010",
"01000011",
"01000011"
);
List1: Listing of VHDL code for a single 8x8 LUT
This design is also synthesized at gate level in Design Vision
which is shown in Fig. 9.
34
Fig. 9.
An 8x8 LUT after synthesis in Design Vision.
The gate-level net-list of the LUT is generated using LPHVT
technology library of stm065nm process. The following table III
shows the resources (cells used) in the hardware of an 8x8 LUT.
TABLE III.
CELLS USED IN HARDWARE OF AN 8X8 LUT.
Cell
Attributes
Reference
Library
U8
HS65_LH_IVX9
CORE65LPHVT 1.560
U10
HS65_LH_IVX9
CORE65LPHVT 1.560
t_reg[1]
HS65_LH_DFPRQX4 CORE65LPHVT 10.400
t_reg[2]
HS65_LH_DFPRQX4 CORE65LPHVT 10.400
x_out_reg[0]
HS65_LH_DFPRQX4 CORE65LPHVT 10.400
x_out_reg[1]
HS65_LH_DFPRQX4 CORE65LPHVT 10.400
x_out_reg[6]
HS65_LH_DFPRQX4 CORE65LPHVT 10.400
Total 7cells
Area
55.120
35
4.2.2. Linear interpolation Architecture with 1 LUT:
In this architecture, the exponential function is implemented using
linear Interpolation approach as described in section 4.3. The
following mathematical equation (4.7) is a starting point of hardware
architecture 1.
ݕൌ ݕ ሺ݊ሻ ൫ݔሺ݊ሻ െ ݔ ሺ݊ሻ൯ כ
ሺ௬್ ሺାଵሻି௬್ ሺሻሻ
ሺ௫್ ሺାଵሻି௫್ ሺሻሻ
(4.7)
Generally, in-terms of hardware implementation, division is a
highly complex function because it demands more resources than any
other logical or arithmetic function. In order to avoid direct division,
there is more investigation in various techniques, to find a simple
logical solution. As there are fixed number of intervals in the
architecture which means that the denominator in (4.7) is always a
fixed fractional number.
The equation (4.7) is reformed as (4.8) after fixing the
denominator as v which is the inverse of difference of intervals.
ݕሺ݊ሻ ൌ ݕ ሺ݊ሻ כ ݒ൫ݔሺ݊ሻ െ ݔ ሺ݊ሻ൯ כ൫ݕ ሺ݊ ͳሻ െ ݕ ሺ݊ሻ൯
(4.8)
Substituting כ ݒ൫ݔሺ݊ሻ െ ݔ ሺ݊ሻ൯ ൌ ݂ሺ݊ሻ into (4.8).
ݕሺ݊ሻ ൌ ݕ ሺ݊ሻ ݂ሺ݊ሻ כ൫ݕ ሺ݊ ͳሻ െ ݕ ሺ݊ሻ൯
(4.9)
The hardware of interpolation architecture 1 using 1 LUT, with
x as input and y(n) as output, is generated based on equation (4.9).
The architecture is shown in Fig. 10.
36
Fig. 10. Hardware implementation of exponential function using Linear
Interpolation with 1 LUT.
In this architecture, a single LUT is used to store the 256
intervals of the exponential function. Each value has a precision of 18
bits. The values of intermediate intervals are generated using
MATLAB. Both the base value and gradient difference are deduced
from an LUT.
37
The designs are meant to verify 1024 intervals between 0 and 1.
From the input ‘x’ is of word length 20 bits, 8 Most Significant
Bit (MSB) bits are used for fetching the base value and next base
value. The next base value is fetched out to evaluate the gradient
difference which is multiplied with the 12 Least Significant Bit (LSB)
bits of input ‘x’. The base value and gradient difference are evaluated
using a single LUT as shown in Fig. 5. The hardware consists of an
LUT, two adders and one multiplier. The precision at the output is
17.73 bits.
4.2.3. Linear interpolation Architecture with 2 LUTs with
256 Intervals:
After discussing the interpolation architecture with one LUT, a
new architecture was proposed by thesis supervisor wherein the aim
is to reduce the latency of the architecture. Upon investigation of the
solution it is decided to use two LUTs in order to store base value and
gradient difference separately.
To implement Linear Interpolation architecture with 2 LUTs
consider (4.9).
ݕሺ݊ሻ ൌ ݕ ሺ݊ሻ ݂ሺ݊ሻ כ൫ݕ ሺ݊ ͳሻ െ ݕ ሺ݊ሻ൯
¾ In the first LUT base value is stored. Each value has a
precision of 18 bits.
¾ In the second LUT gradient difference is stored. Each
value has a precision of 12 bits.
The hardware architecture generated using this scheme is shown
in the Fig. 11. It has two LUTs, one multiplier and one adder.
38
Fig. 11. Hardware implementation of exponential function using
Linear Interpolation method with 2 LUTs and 256 intervals.
The input x has a word length of 20 bits. The first 8 MSB bits are
used to address the base value and gradient coefficient from LUT1
and LUT2 respectively. The remaining 12 bits of LSB of x are used
to multiply with the gradient coefficient to form the gradient
difference which has a word length of 19 bits. The base value is
added with the gradient difference to form the final result. The word
length of the output is 19 bits. In this architecture, latency is reduced
to large extent, however memory size is increased. The precision
achieved is 17.27 bits, which is less than the precision of architecture
1 of 17.73 bits.
39
4.2.4. Linear interpolation Architecture with 2 LUTs with
128 intervals:
The precision achieved in the previous two architectures is more
than 17 bits however the requirement of this work is 15 bits.
.
Fig. 12. Hardware implementation of exponential function
Linear Interpolation with 2 LUTs using 128 intervals.
using
In order to reduce the size of the memory used, the number of
intervals in this architecture is reduced to 128, compared to 256 in
architecture 2. Both the base value and gradient difference are
40
calculated using two separate LUTs with 128 intervals in each LUT.
The hardware of this method is shown in Fig. 12.
In this architecture two LUTs are used, each with the size of 128
intervals.
¾ In LUT1 base value is stored. Each value has word length
of 18 bits.
¾ In LUT2 gradient difference coefficients are stored. Each
value has word length of 12 bits.
The input x has a word length of 20 bits. First 7 bits MSB are used
to address the base value and gradient difference coefficient in LUT1
and LUT2 respectively. The gradient difference coefficient is
multiplied with the 13 bits LSB of input x. The wordlength of
gradient difference is 20 bits. The base value and gradient difference
are added to form the final result. The output wordlength is 19 bits
consisting 2 bits of integer part and 17 bits of mantissa part. The
hardware resources used in this architecture is 2 LUTs, one multiplier
and one adder. The precision achieved using architecture 3 is 15.72
bits which is better than architecture 1 and architecture 2. Hence
architecture 3 is a better compromise between size of memory and
precision.
41
42
CHAPTER 5
5. Estimation of Error
In this chapter error estimation metrics, which measure the
accuracy of a system, are summarized for the architectures designed
in chapter 4. The error results discussed here are after post synthesis
simulations of the architectures using Design Vision.
5.1. Taylor series Architecture 1:
The upper half of Fig. 12 shows the error difference in linear
scale. Whereas, the equivalent number of bits of the Taylor series
architecture 1 with respect to real function are shown in the lower half
of Fig. 13.
Fig. 13. Error difference of Taylor series architecture 1 in linear scale and
total number of bits.
43
The details of error metrics are shown in Table IV.
TABLE IV.
ERROR METRICS OF TAYLOR ARCHITECTURE 1.
Metric
Quantity
Linear scale
in Equivalent
bits
Max Error
0.00000165592
19.20
Min Error
-0.00001857604
15.71
Mean Error
-0.00000697915
17.13
Median Error
-0.00000682426
Standard
Deviation
0.00000314758
RMS Error
0.00000765546
System has mostly negative errors and the interesting thing is that
the worst case error is equivalent to 15.71 which satisfies the system
requirements.
Fig. 14 shows the histogram of probability error distribution of
Taylor series architecture 1. It can also be noticed in the figure that
the errors terms are negative in this architecture.
44
Fig. 14. Histogram of probability error distribution of Taylor Architecture
1.
5.2. Taylor series Architecture 2:
The upper half of Fig. 15 shows the error difference in linear
scale. Whereas, the equivalent number of bits of the Taylor series
architecture 2 with respect to real function are shown in the lower half
of Fig. 15.
45
Fig. 15. Error difference of Taylor series architecture 2 in linear scale and
total number of bits.
46
TABLE V.
ERROR METRICS OF TAYLOR ARCHITECTURE 2
Metric
Quantity
Linear scale
in Equivalent
bits
Max Error
0.00000530197
17.53
Min Error
-0.00001213279
16.33
Mean Error
-0.00000169389
19.17
Median Error
-0.00000247776
Standard Deviation
0.00000287994
RMS Error
0.00000333994
Table V shows that the equivalent bits for the worst case error is
16.33 which is better than Taylor’s series architecture 1. Also there is
very small difference between standard deviation and RMS error
which shows the distribution to be even.
Fig. 16 illustrates the probability error distribution of Taylor
series architecture 2. The distribution is on both sides of zero, with
more peaks in the negative side than on the positive side.
47
Fig. 16. Histogram of probability error distribution of Taylor Architecture 2
for exponential function implementation.
5.3. Linear Interpolation architecture with 1 LUT:
The upper half of Fig. 17 shows the error difference in linear
scale. Whereas, the equivalent number of bits of the Linear
Interpolation architecture 1 with respect to real function are shown in
the lower half of Fig. 17.
48
Fig. 17. Error difference of linear interpolation architecture (using 1 LUT)
in linear scale and total number of bits.
49
TABLE VI.
ERROR METRICS OF LINEAR INTERPOLATION METHOD USING
1 LUT
Metric
Quantity
Linear scale
in Equivalent
bits
Max Error
0.00000432
17.82
Min Error
-0.00000459
17.73
Mean Error
-0.00000085
20.16
Median Error
-0.00000091
Standard Deviation
0.00000164
RMS Error
0.00000184
Table VI shows that the equivalent bits for the worst case error is
17.73 which is better than Taylor’s series architectures. It can also be
noticed that due to very small difference between standard deviation
and RMS error, the distribution is more or less symmetric.
But from Fig. 18 of probability error distribution the peaks are
more in the negative side compared to positive.
50
Fig. 18. Histogram showing error distribution of interpolation
method for exponential function implementation using 1 LUT.
5.4. Linear Interpolation with 2 LUTs and 256
intervals:
The upper half of Fig. 19 shows the error difference in linear
scale. Whereas, the equivalent number of bits of the Linear
Interpolation with 2 LUTs and 256 intervals with respect to real
function are shown in the lower half of Fig. 19.
51
Fig. 19. Error difference of linear interpolation architecture (with 2 LUTs
and 256 intervals) in linear scale and total number of bits.
52
TABLE VII. ERROR METRICS OF LINEAR INTERPOLATION METHOD USING
2 LUTS (256 INTERVALS)
Metric
Quantity
Linear scale
in Equivalent
number of bits
Max Error
0.000004323
17.82
Min Error
-0.000006351
17.26
Mean Error
-0.000001466
19.38
Median Error
-0.000001443
Standard Deviation
0.000001775
RMS Error
0.000002302
Table VII shows that the equivalent bits for the worst case error is
17.26 which is better than Taylor’s series architectures. The mean
error is located in the negative side.
The following histogram in Fig. 20 illustrates the probability error
distribution of Linear Interpolation method with two LUTs (using 256
intervals). The error distributions mean value is negative and hence
the error peaks are also negative.
53
Fig. 20. Histogram of probability error distribution of interpolation method
with 2 LUTs (using 256 intervals).
5.5. Linear Interpolation with 2 LUTs and 128
intervals:
The upper half of Fig. 21 shows the error difference in linear
scale. Whereas, the equivalent number of bits of the Linear
Interpolation with 2 LUTs and 128 intervals with respect to real
function are shown in the lower half of Fig. 21.
54
Fig. 21. Error difference of linear interpolation architecture (with 2 LUTs
and 128 intervals) in linear scale and total number of bits.
55
TABLE VIII. ERROR METRICS OF LINEAR INTERPOLATION METHOD USING
2 LUTS USING 128 INTERVALS
Metric
Quantity
Linear scale
in Equivalent
number of
bits
Max Error
0.000018522
17.52
Min Error
-0.000005334
15.72
Mean Error
0.000004556
17.74
Median Error
0.000004195
Standard Deviation
0.000004827
RMS Error
0.000006636
Table VIII shows that the equivalent bits for the worst case error
is 15.72 which satisfies the system specifications. The mean error is
located in the positive side.
Fig. 22 shows the probability error distribution which has more
peaks on the positive side compared to the negative side of the
distribution.
56
Fig. 22. Histogram showing probability error distribution of linear
interpolation architecture with 2 LUTs and 128 intervals.
57
58
CHAPTER 6
6. Results
In this chapter area, timing and power results of architectures
based on Taylor series and Linear Interpolation methods are
presented and discussed. In this regard Design vision tool is used for
area, timing and power results. Apart from Design Vision, Primetime
is used for more accurate power results. These results are described in
the following sections.
6.1. Area
Comparison of area between Taylor series architectures and
Interpolations based architectures can be summarized as follows;
x
Taylor series hardware architecture 1 consists of 7
multipliers and 7 adders.
x
Taylor series hardware architecture 2 consists of 6
multipliers and 5 adders.
x
Interpolation architecture with 1 LUT and 256 intervals
has 1 LUT, 1 multiplier and 2 adders.
x
Interpolation architecture with 2 LUTs and 256 intervals
has 2 LUTs, 1 multiplier and 1 adder.
x
Interpolation architecture with 2 LUTs and 128 intervals
has 2 LUTs, 1 multiplier and 1 adder.
In the Tables IX and X, the
architecture is presented using the
High Threshold Voltage (LPHVT)
Voltage (LPLVT) respectively, at
operating frequency of 10MHz.
area occupied by each one of
technology libraries Low Power
and Low Power Low Threshold
a supply voltage of 1.2V at an
59
TABLE IX.
AREA REPORT USING LIBRARY LPHVT USING 1.2V
Design
Minimum Area Cells
Occupied (µm2)
Taylor Arch 1
22386
4949
Taylor Arch 2
20692
4531
1 LUT
6010
1566
2 LUTs
4568
1297
2 LUTs (128
intervals)
4192
1157
Areainμm2
AreausingLPHVT
25000
20000
15000
10000
5000
0
Taylor
Arch1
Taylor
Arch2
1LUT
Designs
Fig. 23. Area comparison using LPHVT library
60
2LUTs
2LUTs
(128
intervals)
TABLE X.
AREA REPORT USING LIBRARY LPLVT USING 1.2V
Design
Minimum Area Cells
Occupied (µm2)
Taylor Arch 1
22169
4917
Taylor Arch 2
20652
4543
1 LUT
6095
1588
2 LUTs
4622
1335
2 LUTs (128
intervals)
4222
1161
Areainμm2
AreausingLPLVT
25000
20000
15000
10000
5000
0
Taylor
Arch1
Taylor
Arch2
1LUT
Designs
Fig. 24. Area comparison using LPLVT library
61
2LUTs
2LUTs
(128
intervals)
The above Fig. 23 and Fig. 24 represent the area comparison
graphs of all the designs using technology libraries LPHVT and
LPLVT respectively.
In the table XI and XII, it can be noticed that the size occupied by
the architecture using 1 LUT is large compared to the architecture
using 2 LUTs. It is due to fact that during synthesis stage flatten
process is duplicating one more LUT with 18 bits word length. It can
be concluded that Taylor series architecture has approximately 4
times more area than that of linear interpolation architecture.
For further information about hardware resources of Taylor series
architecture 2 and Linear Interpolation, Table XI and Table XII are
presented. The operating frequency is 10MHz and supply voltage of
1.2V using technology library LPHVT.
TABLE XI.
RESOURCES USED FOR TAYLOR ARCHITECTURE 2 USING
TECHNOLOGY LIBRARY LPHVT AT 1.2V.
Number of ports:
46
Number of nets:
4554
Number of cells:
4531
Number of combinational cells:
4508
Number of sequential cells:
Number of buf/inv:
23
500
62
TABLE XII. RESOURCES USED FOR LINEAR INTERPOLATION
ARCHITECTURE WITH 2 LUTS AND 256 INTERVALS USING TECHNOLOGY
LIBRARY LPHVT AT 1.2V.
Number of ports:
42
Number of nets:
1328
Number of cells:
1297
Number of combinational cells:
1277
Number of sequential cells:
Number of buf/inv:
20
208
6.2. Power
Low power consumption is an important design criterion in a
hardware design. The power results presented in this section are
approximated using Design Vision.
The power consumption of a design consist of total dynamic
power, cell internal power, and cell leakage power. Table XIII shows
the approximate power consumption of all designs using technology
library LPHVT.
TABLE XIII. POWER REPORT USING LIBRARY LPHVT AT 10MHZ
Design with
Total power
(uW)
Cell internal
power (uW)
Cell leakage
power (nW)
Taylor 1
117.6216
59.7387
75.4156
Taylor 2
115.9396
57.6809
66.4293
1LUT
20.5189
11.0545
22.6205
2LUTs
16.3632
9.0386
16.8428
2LUTs
(128 intervals)
15.7136
8.7496
15.4545
Table XIV shows the power consumption results using
technology library LPLVT.
63
TABLE XIV. POWER REPORT USING LIBRARY LPLVT AT 10MHZ
Design with
Total power
(uW)
Cell internal
power (uW)
Cell leakage
power
(uW)
Taylor 1
144.0615
75.9570
17.1698
Taylor 2
144.5094
75.0068
15.8027
1LUT
24.6240
13.3810
4.7604
2LUTs
18.4378
10.1787
3.3317
2LUTs (128
intervals)
17.7640
9.8814
2.8861
It can be deduced from Table XIII and XIV that Taylor series
architecture has at-least six times more total dynamic power
consumption than the linear interpolation architecture.
6.3. Timing
Timing is one of the key parameters in analyzing the performance
of a design. To estimate the maximum speed of a circuit, the critical
path is crucial aspect of a digital hardware. Critical path decides
maximum speed of a design. Tables XV shown the critical path of all
the designs using LPHVT technology library. Also Table XVI shows
the critical path of all the designs using LPLVT technology library.
TABLE XV. TIMING REPORT USING LIBRARY LPHVT
Design
Critical path (ns)
Taylor Arch1
49.77
Taylor Arch 2
40.32
Interpolation with 1 LUT
9. 77
Interpolation with 2 LUT
7.23
Reduced LUT (128 intervals)
7.23
64
TABLE XVI. TIMING REPORT USING LIBRARY LPLVT
Design
Critical path (ns)
Taylor Arch1
34.83
Taylor Arch 2
24.84
Interpolation with 1 LUT
7.17
Interpolation with 2 LUT
5.17
Reduced LUT (128 intervals)
5.17
From the above tables it can be concluded that, interpolation
method is approximately 5 times faster compared to Taylor series
method. However, interpolation method is a memory based design.
As the number of intervals are increasing, the area and complexity
increases.
6.4. Power estimation of Taylor series architecture
2 using PrimeTime:
In this section, power estimations using Synopsis PrimeTime is
utilized. A logarithmic scale of frequency vs power is estimated in
PrimeTime for Taylor series architecture 2:
65
Fig. 25. A log-log graph of frequency versus power calculation for
Taylor series architecture 2.
Fig. 25 shows logarithmic plot of frequency vs. power. The power
calculations are estimated for cell’s internal power and leakage
power. At low frequencies such as near DC level the cell leakage
power is of significant value compared to its internal power. At
higher frequencies the leakage power is constant but the cell internal
power increases exponentially. Using same set of constraints the
power consumption at low frequencies is only that of static power
consumption. However with the increase in frequency the dynamic
power consumption comes into effect. Dynamic power consumption
exponentially increases at higher frequency ranges.
66
6.5. Power Estimation for Linear Interpolation
with 2 LUTs using PrimeTime:
Fig. 26 shows the power results of linear interpolation architecture
2 (with 2 LUTs and 256 intervals) in PrimeTime.
Fig. 26. A log-log graph of frequency versus power calculation for a linear
interpolation architecture with 2 LUTs and 256 intervals.
In Fig. 26 a logarithmic plot of power consumption using linear
interpolation architecture with 2 LUTs and 256 intervals is plotted. In
this plot the power consumption for frequencies below 1 KHz is
static. At higher frequencies the dynamic power consumption
increases at a rate of 10 times per 10 fold increase in frequency.
67
6.6. Comparison of Taylor series Approach and
Linear Interpolation Approach
To compare power consumption between interpolation and Taylor
series method, a combined graph is plotted as shown Fig. 27.
Fig. 27. Comparison of power estimation between
methodology and linear interpolation approach.
Taylor
series
From the Fig. 27 it can be noticed that the power consumption of
Taylor series architecture method is approximately ten times that of
linear interpolation method, at any given frequency. Table XVII
shows the dynamic power consumption results of Taylor series
architecture 2 and linear interpolation method.
68
TABLE XVII. TABLE OF COMPARISON OF POWER CONSUMPTION OF TWO
ARCHITECTURES USING PRIMETIME.
Frequency
(MHz)
Total power consumption
Total power
consumption in Taylor Interpolation architecture
with 2 LUTs (W)
architecture 2
(W)
0.000001
7.6885e-08
1.7047e-08
0.00001
7.7749e-08
1.7116e-08
0.0001
8.6383e-08
1.7809e-08
0.0002
9.5980e-08
1.8579e-08
0.0004
1.1516e-07
2.0118e-08
0.0005
1.2475e-07
2.0888e-08
0.001
1.7272e-07
2.4736e-08
0.002
2.6869e-07
3.2430e-08
0.005
5.563e-07
5.5520e-08
0.01
1.0361e-06
9.3999e-08
0.1
9.6698e-06
7.8664e-07
1
9.6007e-05
7.7130e-06
10
9.5938e-04
7.6977e-05
100
0.0096
7.6961e-04
1000
0.0959
0.0076
69
70
CHAPTER 7
7. Conclusions
Hardware implementation of exponential function had been
successfully implemented using two different methods. In the first
approach two architectures were implemented using Taylor series
expansions;
x
One with general expansion of Taylor series application.
x
Second is a novel architecture with reduced number of
multipliers and adders.
The second one is a Linear Interpolation method which has three
different approaches;
x
Architecture with one LUT and 256 intervals.
x
Architecture with two LUTs and 256 intervals.
x
The third architecture uses two LUTs and 128 intervals.
The emphasis is laid on minimum area, low-power design and
with desired performance. The main aim of the thesis is to achieve
precision of 15 bits however the designed methods are able to achieve
more than 17 bits of accuracy in various approaches and its error
behavior is studied. All the designs are successfully synthesized in
stm65nm for minimum area, low-power and possible maximum
speed. The dynamic power is recorded using Synopsis PrimeTime.
Comparison of novel methodology of Taylor series is compared with
the Linear Interpolation techniques, conclusions are drawn in terms of
area, speed and power.
It can be concluded from this work that the Taylor series method
is a less memory intensive design compared to the Linear
Interpolation method. However the Taylor series architecture
consumes an area five times that of Interpolation approach. The
dynamic power consumption is ten times more in the Taylor series
architecture than the linear interpolation architecture. The Linear
Interpolation method is at-least five times faster than that of the
Taylor series implementation.
71
72
CHAPTER 8
8. Future work
By using the novel methodology of Taylor series approximation,
there is scope of implementing various other unary functions such as
logarithmic function or trigonometric function. The LUT approach
can also be substituted by other highly advanced approach like
improved parabolic synthesis technique to increase the efficiency of
the algorithm.
73
74
REFERENCES
[1] http://mathworld.wolfram.com/TaylorSeries.html
[2] T. W. Roberts, “Non-oscillatory interpolation for the Semi-Lagrangian
scheme”, Dissertation.
[3] G. Muntingh, “Topics In Polynomial Interpolation
Dissertation presented for the degree of Ph D.
Theory,”
[4] Erik Hertz, “Article on Error Evaluation”, EIT, Lund, March 18, 2013.
[5] J Lai, “Hardware Implementation of the Logarithm Function”,
Masters thesis, Lund University, Sep 2013.
[6] D. M. Lurascu, A. V. Bofill, “Hardware Approximation of the Square
Root Function”, Masters Thesis, Lunds Universitet, Jan 2014.
[7] E. Hertz, “Parabolic Synthesis,” Thesis for the degree of Licentiate
in Engineering.
[8] www.math.smith.edu/Local/cicintro/ch10.pdf
[9] http://www.haverford.edu/physics/MathAppendices/Taylor_Series.pdf
[10] http://math.stackexchange.com/questions/218421/what-are-thepractical-applications-of-the-taylor-series.
75
76
List of Figures
Fig. 1. The illustration of Linear Interpolation Method. ................ 15
Fig. 2. Illustration of piecewise linear interpolation with given set
of data points. .................................................................................... 18
Fig. 3. Error Probability distribution diagram ............................... 21
Fig. 4. Flow Chart of ASIC Design. .............................................. 26
Fig. 5. Hardware implementation of Exponential function using
Taylor series architecture 1. .............................................................. 28
Fig. 6. Hardware implementation of Exponential function using
Taylor series Architecture 2. ............................................................. 31
Fig. 7. Synthesized design (partial) of Taylor series architecture 2.
32
Fig. 8. Taylor Series Architecture 2 with pipelined stage.............. 33
Fig. 9. An 8x8 LUT after synthesis in Design Vision. .................. 35
Fig. 10. Hardware implementation of exponential function using
Linear Interpolation with 1 LUT. ...................................................... 37
Fig. 11. Hardware implementation of exponential function using
Linear Interpolation method with 2 LUTs and 256 intervals. ........... 39
Fig. 12. Hardware implementation of exponential function using
Linear Interpolation with 2 LUTs using 128 intervals. ..................... 40
Fig. 13. Error difference of Taylor series architecture 1 in linear
scale and total number of bits. ........................................................... 43
Fig. 14. Histogram of probability error distribution of Taylor
Architecture 1. ................................................................................... 45
FIG. 15. Error difference of Taylor series architecture 2 in linear
scale and total number of bits. ........................................................... 46
Fig. 16. Histogram of probability error distribution of Taylor
Architecture 2 for exponential function implementation. ................. 48
Fig. 17. Error difference of linear interpolation architecture (using
1 LUT) in linear scale and total number of bits. ............................... 49
Fig. 18. Histogram showing error distribution of interpolation
method for exponential function implementation using 1 LUT. ....... 51
Fig. 19. Error difference of linear interpolation architecture (with 2
LUTs and 256 intervals) in linear scale and total number of bits. .... 52
77
Fig. 20. Histogram of probability error distribution of interpolation
method with 2 LUTs (using 256 intervals). ...................................... 54
Fig. 21. Error difference of linear interpolation architecture (with 2
LUTs and 128 intervals) in linear scale and total number of bits. .... 55
Fig. 22. Histogram showing probability error distribution of linear
interpolation architecture with 2 LUTs and 128 intervals................. 57
Fig. 23. Area comparison using LPHVT library .......................... 60
Fig. 24. Area comparison using LPLVT library .......................... 61
Fig. 25. A log-log graph of frequency versus power calculation for
Taylor series architecture 2. .............................................................. 66
Fig. 26. A log-log graph of frequency versus power calculation for
a linear interpolation architecture with 2 LUTs and 256 intervals. ... 67
Fig. 27. Comparison of power estimation between Taylor series
methodology and linear interpolation approach. ............................... 68
78
List of Tables
TABLE I. specifications of the hardware
implementation of exponential function ........................................... 12
TABLE II. Coefficients used in Taylor architecture 2 .................. 30
TABLE III. Cells used in hardware OF an 8x8 LUT. ................. 35
TABLE IV. Error metrics of Taylor architecture 1. .................... 44
TABLE V. Error metrics of Taylor architecture 2 ......................... 47
TABLE VI. Error metrics of Linear interpolation method using 1
LUt
50
TABLE VII. Error metrics of Linear interpolation method using 2
LUts (256 intervals) .......................................................................... 53
TABLE VIII. Error metrics of Linear interpolation method using 2
LUts using 128 intervals ................................................................... 56
TABLE IX. Area Report using library lphvt using 1.2v ............. 60
TABLE X. Area report using library lplvt using 1.2v ................... 61
TABLE XI. resources used for taylor architecture 2 using
technology library lphvt at 1.2v. ....................................................... 62
TABLE XII. resources used for linear interpolation architecture
with 2 luts and 256 intervals using technology library lphvt at 1.2v.
63
TABLE XIII. power report using library lphvt at 10mhz .............. 63
TABLE XIV.
power report using library lplvt at 10mhz............ 64
TABLE XV. timing report using library lphvt .............................. 64
TABLE XVI.
timing report using library lplvt ........................... 65
TABLE XVII. Table of comparison of power consumption of two
architectures using PrimeTime. ......................................................... 69
79
80
List of Acronyms
ASIC
Application Specific Integrated Circuit
CMOS
Complementary Metal Oxide Semiconductor
ENOB
Effective Number of Bits
FPGA
Field Programmable Gate Array
GIDL
Gate-Induced Drain Leakage
IC
Integrated circuit
LUT
Look-up table
LPHVT
Low Power High Threshold Voltage
LPLVT
Low Power Low Threshold Voltage
NMOS
N-type Metal Oxide Semiconductor
PMOS
P-type Metal Oxide Semiconductor
RBDL
Reverse-Biased Diode Leakage
RTL
Register Transfer Level
SNR
Signal to Noise Ratio
SNDR
Signal to Noise and Distortion Ratio
VCD
Value Change Dump
VHDL
VHSIC Hardware Descriptive Language
VHSIC
Very High-Speed Integrated Circuit
81
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