Implementation of Rotation and Vectoring-Mode Reconfigurable CORDIC

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ISSN No: 2456
International
Research
Implementation of Rotation and Vectoring
Kothapally Mounika
1
PG Scholar,
Dept of ECE, CMR Institute of Technology, Kandlakoya, Medchal,
ABSTRACT
CORDIC or CO-ordinate Rotation Digital Computer
is a fast, simple, efficient and powerful algorithm
for diverse Digital Signal Processing applications.
Primarily developed for real-time airborne
computations, it uses a unique computing technique
which is especially suitable for solving the
trigonometric relationships involved in plane co
ordinate rotation and conversion from rectangular to
polar form. It comprises a special serial arithmetic
unit having three shift registers, three
adders/subtractors, Look-Up table and special
interconnections. In this project A CORDIC
processor for sine/cosine calculation was designed
using VHDL programming in Xilinx ISE 13.2. The
CORDIC module was tested for its functionality
correctness by test-bench analysis. Subsequently,
FPGA implementation of the CORDIC core followed
by Chip Scope Pro analysis of the
waveforms was performed.
Keywords: Circular Trigonometry, Coordinate
Rotation Digital Computer (CORDIC), Hyperbolic
Trigonometry, Reconfigurable CORDIC
I. INTRODUCTION
For a long time the field of Digital Signal P
has been dominated by Microprocessors. This is
mainly because they provide designers with the
advantages of single cycle multiply
instruction as well as special addressing modes.
Although these processors are cheap and flexible they
are relatively slow when it comes to performing
certain demanding signal processing tasks e.g. Image
Compression, Digital Communication
Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 4 | May-Jun 2018
ISSN No: 2456 - 6470 | www.ijtsrd.com | Volume
International Journal of Trend in Scientific
Research and Development (IJTSRD)
International Open Access Journal
Implementation of Rotation and Vectoring-Mode Reconfigurable
CORDIC
Kothapally Mounika1
, P. Pavan Kumar2
, K. Shobha Rani
PG Scholar, 2
Associate Professor,
Hyderabad, Telangana, India
ordinate Rotation Digital Computer
is a fast, simple, efficient and powerful algorithm used
for diverse Digital Signal Processing applications.
time airborne
computations, it uses a unique computing technique
which is especially suitable for solving the
trigonometric relationships involved in plane co-
otation and conversion from rectangular to
polar form. It comprises a special serial arithmetic
unit having three shift registers, three
Up table and special
interconnections. In this project A CORDIC-based
e calculation was designed
programming in Xilinx ISE 13.2. The
CORDIC module was tested for its functionality and
bench analysis. Subsequently,
CORDIC core followed
output logic
Circular Trigonometry, Coordinate
Rotation Digital Computer (CORDIC), Hyperbolic
Trigonometry, Reconfigurable CORDIC
For a long time the field of Digital Signal Processing
Microprocessors. This is
esigners with the
single cycle multiply-accumulate
cial addressing modes.
these processors are cheap and flexible they
are relatively slow when it comes to performing
certain demanding signal processing tasks e.g. Image
Communication and Video
Processing. Of late, rapid advancements have been
made in the field of VLSI and IC design. As a result
special purpose processors with custom
have come up. Higher speeds can be achieved by
these customized hardware solutions at competitive
costs.
To add to this, various simple and hardware
algorithms exist which map well onto these chips a
can be used to enhance speed and flexibili
performing the desired signal processing tasks. One
such simple and hardware
CORDIC, an acronym for Coordinate Rotation Digital
Computer, proposed by Jack E Volder [7]. CORDIC
uses only Shift-and Add arithmetic with table Look
Up to implement different functions. By making slight
adjustments to the initial conditions and the LUT
values, it can be used to efficiently implement
Trigonometric, Hyperbolic, Exponential functions,
Coordinate transformations
hardware. Since it uses only shift
VLSI implementation of such an
achievable. DCT algorithm has diverse applications
and is widely used for Image compression.
Implementing DCT using CORDIC algorithm reduces
the number of computations during
increases the accuracy of reconstruction of the image,
and reduces the chip area of implementation of a
processor built for this purpose.
This reduces the overall
FPGA provides the hardware environment in which
dedicated processors can be tested for
Jun 2018 Page: 1594
6470 | www.ijtsrd.com | Volume - 2 | Issue – 4
Scientific
(IJTSRD)
International Open Access Journal
Mode Reconfigurable
Shobha Rani2
Processing. Of late, rapid advancements have been
made in the field of VLSI and IC design. As a result
purpose processors with custom-architectures
have come up. Higher speeds can be achieved by
these customized hardware solutions at competitive
To add to this, various simple and hardware-efficient
algorithms exist which map well onto these chips and
can be used to enhance speed and flexibility while
signal processing tasks. One
-efficient algorithm is
CORDIC, an acronym for Coordinate Rotation Digital
Computer, proposed by Jack E Volder [7]. CORDIC
and Add arithmetic with table Look-
Up to implement different functions. By making slight
adjustments to the initial conditions and the LUT
values, it can be used to efficiently implement
Trigonometric, Hyperbolic, Exponential functions,
etc. using the same
uses only shift-add arithmetic,
of such an algorithm is easily
DCT algorithm has diverse applications
and is widely used for Image compression.
ng CORDIC algorithm reduces
e number of computations during processing,
increases the accuracy of reconstruction of the image,
and reduces the chip area of implementation of a
processor built for this purpose.
This reduces the overall power consumption.
FPGA provides the hardware environment in which
dedicated processors can be tested for

International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470
@ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 4 | May-Jun 2018 Page: 1595
their functionality. They perform various high-speed
operations that cannot be realized by a
simple microprocessor. The primary advantage that
FPGA offers is On-site programmability. Thus, it
forms the ideal platform to implement and test the
functionality of a dedicated processor designed using
CORDIC algorithm.
Window filtering techniques are commonly employed
in signal processing paradigm to limit time and
frequency resolution. Various window functions are
developed to suit different requirements for side-lobe
minimization, dynamic range, and so forth.
Commonly, many hardware efficient architectures are
available for realizing FFT, but the same is not true
for windowing–architectures. The conventional
hardware implementation of window functions uses
lookup tables which give rise to various area and time
complexities with increase in word lengths. Moreover,
they do not allow user-defined variations in the
window length. An efficient implementation of
flexible and reconfigurable window functions using
CORDIC algorithm is suggested. Though they allow
user-defined variations in window length, latency is a
major problem. The CORDIC algorithm inherently
suffers from latency issues and using two CORDIC
processors in series, as is done. The overall latency of
the system is hampered.
II. LITERATURE SURVEY
During spectral analysis, the input signals are to be
truncated to fit a finite observation window according
to the length of FFT processor. This direct truncation
using conventional windowing, known as rectangular
window function leads to undesirable effects known
as spectral leakage and picket fence effect in
frequency domain. To minimize these effects during
spectral analysis, researchers have proposed different
kinds of windowing functions such as Hanning,
Hamming and Blackman windowing functions. These
windowing functions are widely adopted because of
their good spectral characteristics like central peak
width, 6-dB point, highest side lobe and rate of side
lobe fall off and equivalent noise bandwidth (ENBW).
Among these, Blackman windowing leads to better
side lobe attenuation. It is needless to present all these
characteristics in detail here, however readers may
refer for the same. Here only Blackman windowing
has been discussed for implementation. Though ROM
based implementation is already existing, which
restricts flexible implementation and also restricts
fitting with the advanced FFT processors in terms of
variable length and speed. Basic idea of this work is
to propose a flexible and fast architecture for
Blackman windowing function to fit with the
advanced FFT processor. Before presenting the
proposed architecture in the next section, Blackman
windowing function has been highlighted here briefly.
A typical block diagram for real time FFT based
spectral analysis system is shown in Fig.1.
Fig.1. Spectral analysis system
The Blackman window, with the above approximation
coefficients, provide attenuation of at least 60dB of
side lobes[1] with only a modest increase in
computation over that required by the Hanning and
Hamming window due to another cosine term as in
equation (2). This windowing function demands the
attention for designing hardware efficient, flexible
window length setting and high throughput VLSI
architecture using CORDIC whose implementation is
quite economic in terms of hardware. Now from
equation (2), we shall have a parallel and pipelined
architecture for aforesaid windowing function, where
the selection of window length (N) is user defined as
per requirement for the application. Since the equation
needs trigonometric computation, so the
implementation using CORDIC algorithm is better
choice in terms of computation and to change the
value of N dynamically. But look up table or ROM
method fails to achieve the same. In case of fixed N
also, though existing implementation is based on look
up table, it consumes more time to access the ROM
and to compute multiplication and addition. Whereas
CORDIC based proposed architecture gives same
result with high throughput and lesser hardware
compared to ROM based computation. Here
multiplication and trigonometric computations are
realized using linear and circular CORDIC algorithm
respectively.

Fig.2. Block representation for spectral analysis
III. PROPOSED SYSTEM
RECONFIGURABLE CORDIC
To design a reconfigurable CORDIC architecture with
minimum reconfiguration overhead, we need to
maximize the sharing of common hardware circuit in
different configurations. Therefore, to explore the
possibility of reconfigurable CORDIC, we examine,
here, the commonalities in three main issues of
CORDIC implementation, namely: 1) the coordinate-
rotation matrix; 2) selection of elementary angles; and
3) direction of micro rotations.
A. Reference Reconfigurable CORDIC
A basic design for reconfigurable CORDIC based on
unified CORDIC algorithm was proposed. The major
concern with the design of conventional
reconfigurable architecture is the incompatibility in
RoC of circular and hyperbolic trajectories. The RoC
of circular CORDIC is [−99°, 99°], while that of
hyperbolic CORDIC is given by |θ|≤1.1182 radians.
This limits the maximum angle of rotation of the
reconfigurable design to 64°. The incompatible RoC
of circular and hyperbolic CORDICs makes it
difficult to implement them in the same circuit to
perform rotation through [−180°, 180°]. Another
major issue with the conventional reconfigurable
CORDIC is scaling. We need to have two different
scaling circuits for circular and hyperbolic CORDIC,
and select the output from one of the scaling circuits
depending on the selection of trajectory of operation.
B. Design Strategy for Proposed Reconfigurable
CORDIC
The circular and hyperbolic CORDICs require two
different scaling circuits, which is quite costly.
Therefore, it is necessary to use a scale-free
implementation in the reconfigurable CORDIC. Here,
we discuss the scaling-free CORDIC and its
limitations, followed by the discussions on our design
strategy for a reconfigurable CORDIC.
1) Scaling-Free CORDIC Algorithm and Its
Limitations: The scaling-free CORDIC [2] employs
second-order Taylor series approximation, where the
rotation matrix is given by
This approximation imposes a restriction on the basic-
shift1i =[(b−2.585/3)]. For 16-bit applications, the
basic-shift is i =4, which reduces the RoC to 7.16°,
which can be extended to 22.5° using multiple
iterations corresponding to the basic-shift i =4. This is
a major drawback, which limits the applicability of
this algorithm. Moreover, the algorithms focus only
on circular rotation-mode, which cannot be directly
extended to hyperbolic CORDIC, since the second
order of approximation of Taylor series expansion of
hyperbolic functions results in a very low RoC (nearly
22.5°). Due to the lack of symmetry in hyperbolic
functions, the RoC cannot be extended to the entire
coordinate space.
2) Reconfigurability of Rotation-Mode CORDIC:
Scaling-free algorithms for circular and hyperbolic
trajectories are proposed. Moreover, in both the
scaling-free algorithms, third order of approximation
of Taylor series is used to derive the CORDIC
rotation-matrices, as
Fig.3. Proposed reconfigurable rotation-mode
CORDIC processor.
Note that the same set of elementary angles is used for
both circular and hyperbolic rotation-modes. This is a
big advantage to derive the reconfigurable CORDIC,
since no differentiation is required to identify the
micro rotations according to the trajectories. For

circular and hyperbolic trajectories, the elementary
angles are redefined as
Whereas i is the number of shifts for the ith
iteration. The RoC for both the trajectories is
compatible and extends to the entire coordinate space.
The design for rotation-mode CORDIC with slight
modification can be extended to support vectoring-
mode as discussed below
3) Re configurability of Vectoring-Mode CORDIC:
To realize a vectoring-mode CORDIC, all the micro
rotations will be performed in the clockwise direction
for both the circular and hyperbolic trajectories. The
rotation matrices are given by
Whereas i is the shift-index ith iteration . The sign-bit
of the y-coordinate over successive iterations
determines the angle of rotation θ. For vectoring-
mode, the maximum angle of rotation that can be
computed lies in the range [0,π/4]. However, this
range can be extended to the entire coordinate space
using octant wave symmetry of sine and cosine
functions for circular trajectory.
Proposed Reconfigurable CORDIC:
The coordinate calculation matrices for circular and
hyperbolic CORDICs differ by the sign of operands,
and to realize that additions are to be replaced by
subtractions and vice-versa. This can be easily
realized by a reconfigurable add/subtract circuit. In
both cases, the basic-shift could be either 2 or 3, but
the number of micro rotations varies with the mode of
operation. Besides, each case will have its own circuit
to enable the extension of RoC. Based on these
observations, we design three reconfigurable
CORDIC architectures:
1) rotation-mode reconfigurable CORDIC;
2) vectoring-mode reconfigurable CORDIC;
3) generalized reconfigurable CORDIC.
A. Rotation-Mode Reconfigurable CORDIC
The proposed design for reconfigurable rotation-mode
CORDIC (shown in Fig. 3) consists of three parts: 1)
preprocessing unit; 2) reconfigurable CORDIC
rotation unit; and 3) post processing unit. The
preprocessing unit ensures that the input rotation
angle to the CORDIC processing structure always lies
in the range [0,π/4], as the maximum rotation angle
that can be handled by micro rotation sequence
generator is π/4. The post processing unit is required
only for circular trajectory to swap/complement the
sine/cosine values depending on the octant of the
rotation angle. The user can control the trajectory of
the reconfigurable CORDIC by changing a 1-bit
signal T. The rotation matrix for reconfigurable
rotation-mode CORDIC is obtained after unifying the
rotation matrices of circular and hyperbolic case given
by (4a) and (4b), respectively, as
Where T= 0 hyperbolic
T=1 circular
Fig.4. Structure of the proposed reconfigurable
recursive CORDIC architectures.
1) Proposed Recursive Architecture:
The recursive architecture (shown in Fig. 4) uses a
single CORDIC micro rotator to perform all the
CORDIC iterations. The circular CORDIC of requires
one iteration less than the hyperbolic CORDIC, but
here we realize the architecture for the same number
of iterations (eight for sbasic=2 and eleven
forsbasic=3) for both circular and hyperbolic
trajectories. The reconfigurable coordinate calculation
unit (RCCU) isshowninFig.3.

Fig.5. RCCU for recursive design
2. Proposed Pipelined Architecture:
Fig. 6 shows the reconfigurable CORDIC rotation unit
for basic-shift 2. The shift-index si Is fixed in every
RCCU, and hence the shifters are hardwired and do
not involve high complexity barrel-shifters. The
implementation of RCCUs varies according to the
basic-shift si. With slight modifications, the pipeline
can be extended for basic-shift 3.
Fig.6. Reconfigurable rotation-mode CORDIC unit
for basic-shift 2
B. Reconfigurable Vectoring-Mode CORDIC
The reconfigurable rotation matrix for vectoring mode
is obtained by unifying (6a) and (6b), as
Where T=0 hyperbolic ----8
T=1 circular
By changing the implementation of the RCCU to
implement, the recursive architecture of Fig. 2 can be
used to realize CORDIC iterations for vectoring-
mode. The rollover counter value is 15 for sbasic=2,
and 17 forsbasic=3. The pipelined architecture of
vectoring-mode reconfigurable CORDIC consists of
eight stages for sbasic = 2, as shown in Fig. 7.
Fig 7 Proposed pipeline reconfigurable vectoring-
mode CORDIC unit for sbasic=2
Similar to reconfigurable rotation-mode CORDIC, for
increasing shift-indices, the implementation of
RCCUs is simplified for reconfigurable vectoring-
mode CORDIC as well. The input coordinates
[Xin’,Yin’] are first preprocessed to obtain
coordinates[xin,yin] and octant mapping signals. The
coordinates [xin,yin]are input to the vectoring-mode
CORDIC pipeline to generate an angle θ ∈[0,π/4].
The rotation angle θ generated by the vectoring-mode
CORDIC pipeline is mapped to the desired octant
using the octant mapping signals generated by the
preprocessing unit. Therefore, the RoC supported by
the proposed vectoring-mode reconfigurable CORDIC
is [−π, π]. C. Proposed Generalized Reconfigurable
CORDIC The generalized reconfigurable CORDIC
can operate either in vectoring-mode or in rotation-
mode for both circular and hyperbolic trajectories.
The user can select the trajectory of operation using a
single bit signal T(T=1 for circular and T=0 for
hyperbolic). Another single bit signal M is used to
control the mode of operation (M=0 for rotation-mode
and M=1 for vectoring-mode). The recursive
architecture of the proposed generalized
reconfigurable CORDIC is implemented by
combining the CORDIC micro rotators for both
rotation-mode and vectoring-mode CORDICs, as
shown in Fig. 8. The throughput of the proposed
recursive generalized reconfigurable CORDIC is the
same as that of the recursive reconfigurable vectoring-
mode CORDIC. The block diagram for pipelined
generalized reconfigurable CORDIC using basic-
shifts basic=2 is shown in Fig. 9. It can be easily
extended to basic-shifts basic=3 as is done for

reconfigurable rotation-mode and vectoring-mode
CORDICs.
Fig.8. Structure of CORDIC micro rotator for the
proposed recursive generalized reconfigurable
CORDIC.
Fig.9. Proposed pipeline generalized
reconfigurable CORDIC unit for sbasic=2.
IV.EXTENSION WORK
Blackman window architecture
Fig. 10 Pipelined CORDIC Architecture (a)
Circular (b) linear
A high throughput VLSI architecture for Blackman
windowing. Since most of the implementation of
windowing functions for real time applications, are
based on either ROM or DSP processor. Here the
proposed architecture is designed using major blocks
like CORDIC(CO-ordinate Rotation DIgital
Computer) and Han-Carlson adder. This architecture
is flexible in terms of window length. So that a single
chip can be used for those applications, where
variable length is required.
The architecture is shown in Fig.10 for VLSI
implementation of Blackman windowing function.
Major blocks of proposed architecture are described
subsequently. Two linear CORDIC blocks are used
for multiplying input samples with constant
coefficients (b0and b2), however the multiplication of
constant coefficient (b1=0.5) with input samples is
done with only hard shifter (1-bit right) and passes
through FIFO_1 for synchronization with other
parallel paths and similarly FIFO_2 is also used for
synchronization. Circular CORDIC has been used to
compute cosine functions given in equation (2) and
multiplication of intermediate values (i.e. values from
lower linear CORDIC and FIFO_1 as shown in
Fig.10). CORDIC blocks used in our proposed
architecture are purely pipelined, where add/sub

circuit is the critical path. Here length of FIFOs is
equal to the number of stages of pipelined CORDIC
minus one, e.g, for 16-bit precision CORDIC, the
number of stages are sixteen and thus FIFO length to
be fifteen.
SIMULATION RESULTS:
Fig: Simulation result of proposed system
RTL SCHEMATIC:
Fig: RTL Schematic of proposed system
The above diagram shows that RTL Schematic for
the proposed method. The number of gates and
other design summary is included in the following
table which can be followed by the technological
diagram of the proposed method.
TECHNOLOGICAL SCHEMATIC:
Figure: Technology schematic of proposed system
Design summary :
The following figure includes all the design summary
of the proposed method.
Figure: Design summary of proposed system
TIMING REPORT:
Figure: Timing Report of proposed system

EXTENSION RESULTS:
Figure: Simulation results of extended system
RTL SCHEMATIC:
Figure. RTL schematic of extended system
The above diagram shows that RTL Schematic for
the Extension method. The number of gates and
other design summary is included in the following
table which can be followed by the technological
diagram of the Extension method.
TECHNOLOGICAL SCHEMATIC:
Figure: Technology schematic of extended system
Design summary :
The following Figure includes all the design summary
of the Extension method.
Fig: Design summary of extended system
TIMING REPORT:
Fig: Timing report of extended system
Comparison Table
The following diagram show that the entire
comparison between proposed method and extension
method.
Design Slices LUT
s
IOB’S Delay
(ns)
Proposed 1650 2876 35 183.4
60ns
Extension 1300 2438 35 161.3
50ns
Fig;: Comparison between proposed system and
extended system.

V.CONCLUSION
CORDIC is a powerful algorithm, and a popular
algorithm of choice when it comes to
various Digital Signal Processing applications.
Implementation of a CORDIC-based processor on
FPGA gives us a powerful mechanism of
implementing complex computations on a platform
that provides a lot of resources and flexibility at a
relatively lesser cost. Further, since the algorithm is
simple and efficient the design and VLSI
implementation of a CORDIC based processor is
easily achievable. In this project a CORDIC module is
designed and simulated using Xilinx ISE using VHDL
as a synthesis tool. The output of the CORDIC core is
analyzed and verified on the test-bench.
REFERENCES
1) Ray Andraka, FPGA '98. Proceedings of the 1998
ACM/SIGDA sixth international
symposium on Field programmable gate arrays,
Feb. 22-24, 1998, Monterey, CA. pp191-200
2) Vikas Sharma,"FPGA Implementation of EEAS
CORDIC based Sine and Cosine
Generator",M.Tech Thesis, Dept. Electron.
Comm. Eng.,Thapar Uni., Patiala, 2009.
3) Satyasen Panda,"Performance Analysis and
Design of a Discrete Cosine Transform
Processor using CORDIC Algorithm",M.Tech
Thesis,Dept. Electron.Comm.Eng.,NIT
Rourkela, Rourkela, Orissa,2010.
4) Kris Raj, Gaurav Doshi, Hiren Shah [online]
Available:
http://teal.gmu.edu/courses/ECE645/projects_S06/
talks/CORDIC.pdf(URL)
5) S.K.Pattanaik and K.K.Mahapatra, "DHT Based
JPEG Image Compression Using a Novel
Energy Quantization Method", IEEE International
Conference on Industrial Technology,pp.2827-
2832,Dec 2006.
6) Volder, J.,"Binary Computation Algorithms for
coordinate rotation and function
generation,"Convair Report IAR-1148
Aeroelectrics Group, June 1956.
7) Volder, J.,"The CORDIC Trigonometric
Computing Technique,"IRE Trans.Electronic
Computing, Vol EC-8, pp330-334 Sept 1959.
8) Walther J.S.,"A unified algorithm for elementary
functions," Spring Joint Computer
Conf.,1971,proc.,pp.379-385.
9) Ahmed,H.M.,Delosme,J.M.,andMorf,M.,"Highly
Concurrent Computing Structure for Matrix
Arithmetic and Signal Processing," IEEE Comput.
ag,,Vol.15,1982,pp.65-82.
10) Depreterre, E.,Dewilde, P.,and Udo, R.,'Pipelined
CORDIC Architecture for Fast VLSI
Filtering and Array Processing," Proc.ICASSP'84,
1984,pp.41.A.6.1-41.A.6.4.
Author’s Profile:
Kothapally Mounika is currently
pursuing MTECH in VLSI system
design in CMR Institute of
Technology, Kandlakoya, Medchal,
Hyderabad. She received her
bachelor’s degree in Electronics and
Communication Engineering in 2015
from Stanley College of engineering
and technology for women affiliated to Osmania
University, Hyderabad. Her current research interests
include very large scale integration (VLSI) low power
design.
P. Pavan Kumar is currently
working as a Associate Professor in
CMR Institute of Technology,
Medchal, Hyderabad. His current
research interests include very large
scale integration (VLSI) low power
design, test automation and fault-
tolerant computing.
K. Shobha Rani is currently working
as a Associate Professor in CMR
Institute of Technology, Medchal,
Hyderabad. Her current research
interests include very large scale
integration (VLSI) low power design,
high performance architectures and
VLSI reliability.

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