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IOSR Journal of Electronics and Communication Engineering (IOSR-JECE)
e-ISSN: 2278-2834,p- ISSN: 2278-8735.
PP 01-11
FPGA implementation of Angle Generator for CORDIC Based
High pass FIR Filter Design
Nutan Das1, Swarnaprabha Jena2, Siba Kumar Panda2
M.Tech Scholar,VLSI Design, Centurion University of Technology & Management, BBSR, Odisha
Assistant Professor, Department of ECE, Centurion University of Technology & Management, BBSR, Odisha
Abstract: The design of high speed VLSI architecture introduced a large number of algorithmes for real-time
Digital Signal Processing(DSP) and in the VLSI Technology, the realizations of this DSP algorithmes have been
directed in many of the computations in Signal Processing and wireless communication appilications such as
trigonometic and complex functions. This paper proposed a hardware efficient CORDIC based parallel
architechture for the calculation of cosine and sine functions and High Pass FIR (Finite Impulse Response
)Filter design using Very High Speed Integrated Circuit Hardware Description Language (VHDL). The
generation of cosine sine functions of CORDIC algorithm have synthesized, simulated and tested it on Xilinx
FPGA SPARTAN 3E kit.The code have synthesized using Xilinx ISim 14.4 simulator software. This paper also
described the angle to 15bit binary conversion, analysis of all output values and the power utilization summary
.Finally, magnitude plot of HighPass FIR Filter is plotted and analysed.
In advanced VLSI technology have stimulated a great interest in developing special purpose processor
architecture arrays to facilitate real-time signal processing. The reductions in the hardware cost, less area, high
speed and less power consumption are motivated the development of various architectures to design filters in
Digital Signal Processing. Generally, DSP algorithms use the calculation of elementary functions which require
high computational power.
Now, Field-Programmable Gate Array (FPGA) is introduced hardware technology for DSP systems
offer the capability to develop the most suitable architecture for the computational, memory and power
requirements of the DSP applications. For FPGA implementation the DSP system produced highly parallel,
pipelined and hybrid architecture. These Filters are used to demonstrate the mapping, introduced retiming and
the low power optimizations are demonstrated by using a FFT-based applications and development.
In the CORDIC(Coordinate Rotation Digital Computer), J.E. Volder [1], Rotation is a 2-D(
dimensional) vector and a target angle is divided into desired rotation angle into the weighted sum of a set of
predefined elementary rotation angles in which through each rotation can be realized with shift and add/sub
operations. Also, the CORDIC algorithm is again extended [2] to proposed a unified algorithm for computation
of rotation in and hyperbolic coordinate systems, circular, linear, embedding coordinate system. These functions
are implemented for many applications like digital signal processing [3].
CORDIC used mainly simple shift and add operations[4], that performed several computational tasks
are calculation of trigonometric functions(cosine,sine), hyperbolic functions(htan,hcos) and logarithmic
functions, real and complex functions, and simple arithmetic functions are multiplications, division, square-root
functions Now CORDIC Algorithms are used in the area of signal and image processing and communication
systems, robotics and 3-D(Dimensional) graphics, Fast Fourier Transforms(FFT), Filters, Computer Graphics[5]
and Robotics [6]. CORDIC offers high computational applications: computer graphics, in which a combination
of scaling and rotations are required for real time world. For fast computation programming CORDIC is again
used in the field of Robotics.
Background And Literature Review
Amritakar Mandal etal. [7] described the design of pipelined architecture for the computation of Sine
and Cosine values and it is based on application of a specific CORDIC processor. The design of CORDIC is
mainly based on the circular rotation mode gives a high system throughput by reducing latency in each
individual pipelined stage.
Lakshmi.B etal. [8] explained a low latency field programmable gate array implementation of an
unfolded architecture for the implementation of rotational CORDIC algorithm. Here the computational device
was highly suitable for the implementation of customized hardware in portable devices and in which a large
parallelism and low clock rate were utilized for low power consumption.
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FPGA implementation of Angle Generator for CORDIC Based High pass FIR Filter Design
Siba Ku panda etal. [9, 10] explained an innovative idea of designing multiplier circuit for various
VLSI signal processing applications.
Ray Andraka [11] has summarized that by using CORDIC Algorithm some functions like sine,
cosine, tangent, circular, linear, hyperbolic functions, Cartesian to polar transformations and inverse functions
was implemented. For FPGA Implementation next time this paper also described as bit serial, iterative and
online CORDIC architecture design.
Velmurugan [12] explained a proposed low power analog and mixed mode implementation of particle
filter for target tracking. Based on CORDIC Algorithm[13] the particle filter algorithm was used to exploits
addition, multiplication, Gaussian and Arctan calculations.
X. Hu. etal. [14] has described a proposed unified CORDIC algorithm based on the parallel and
pipelined CORDIC processor. J. Sudhaa etal.[15] has proposed the potential application of the proposed 2D
Gaussian function includes image enhancement, smoothing, edge detection, filtering etc and to gain real time
performance, the architectures developed exploits high degrees of pipelining and parallel processing .
N. Takagi, etal [16] described a proposed Double rotation and correcting rotation methods to
implement constant scale factor CORDIC in which result was 50% increasing in number of iterations. For that
this increase in latency is reduced by proposed branching algorithm.
J. Duprat etal. [17] described additional CORDIC module to perform rotations in both directions, if
the direction cannot be determined using intermediate results. The main demerits of branching method is the
necessity of performing two conventional CORDIC iterations in parallel, which consumes more silicon area
than the conventional methods. Also, this method gives a faster implementation than [16].
M. Chakraborty, etal.[18] explained a class of pipelined CORDIC architectures for the LMS-based
transversal adaptive filter. Introducing an alternate formulation of the LMS algorithm, obtained by expressing
the mean square error as a convex function of a set of angle variables and monotonically related to the filter tap
weights. Also proposed architectures employed a micro-level pipelining and are adjustable to strike tradeoffs
between throughput efficiency vis-a-vis hardware complexity.
Shanmuga Kumar M. etal. [19] summarized a proposed the high precision and hardware efficient
CORDIC structure for handheld scientific calculator. Tsao Y C, etal.[20] presented the realization of area
efficient architectures using Distributed Arithmetic (DA) for implementation of Finite Impulse Response (FIR)
filter.Also described the performance of the bit-serial and bit parallel DA along with pipelining architecture and
different quantized versions are analyzed for FIR filter Design.
Shrikant Patel ,etal [21] presented the implementation of Modified Distributed Arithmetic, and
introduce it into the FIR filters design, and then presents a 31-order FIR low-pass filter using Modified
Distributed Arithmetic and save considerable MAC blocks to decrease the circuit scale.
M.Chakrapani ,etal [22] explained the mode of operation of CORDIC algorithm and Control CORDIC
Angle correction and Quadrant correction. Latency can be further reduced by another CORDIC algorithm
techniques. By angle selection scheme the Overshoot problem in original CORDIC has been overcome and
complexity of hardware also increases for the increasing of number of bits.
The contribution of this paper is based on a new and modified architecture in which it needs less area
for implementation and less delay output. The need of hardware efficient algorithms in the advanced
technologies For the fast signal processing requirements .So, the current trend is back toward hardware efficient
algorithms, many of those applications require elementary calculations like sin and cosine function. So, in this
paper i am presenting a angle generator using CORDIC algorithm and its performance reports. The parallel
architecture takes the angle as input and gives both sine and cosine for the given input in predetermined number
of micro rotations. These micro rotations are decided by the accuracy demanded by the application. Here our
objective is design to plot the magnitude curve using the frequency response equation of a FIR Filter using
CORDIC algorithm in VHDL.
Section I gives brief introduction, section II defines the Background and Literature Review. The Basic
CORDIC algorithm and architecture are briefly studied in section III, The FPGA Implementation, Simulation
and Synthesis results are discussed in section IV and finally conclusion is explained in section V.
Basic Cordic Algorithm And Architechture
Coordinate Rotation Digital Computer (CORDIC) ,firstly implemented in1959 by J.E.Volder [1],is an
ancient principles of two-dimensional geometry. CORDIC is a typical iterative algorithm, is used to accelerate
many digital functions and applications in signal processing. This algorithm may be realized as an iterative form
of add, sub and shift operations. The two basic modes in which the different functions are computing and these
are a the rotation mode and the vectoring mode. The algorithm can be realized as an iterative sequence of
additions or subtractions and shift operations by using the two modes, which are rotated by a fixed rotation
angle (micro rotations) .
In Rotation mode is shown in Fig-2 , we can rotate (1,0) by an angle φ degrees to get (x, y)
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FPGA implementation of Angle Generator for CORDIC Based High pass FIR Filter Design
x  x. cos( )
y  y. sin(  )
x  x. cos( )  y. sin(  )
y   y. cos( )  x. sin(  )
Rearranged as the equations are
x  cos( ).[ x  y. tan( )]
y   cos( ).[ yi  x. tan( )]
Rewrite in terms of ai: (0  i  n)
xi 1  cos(i ).[ xi  yi .d i . tan(i )]
yi 1  cos(i ).[ yi  xi .d i . tan(i )]
Figure- 1 (a) and (b) Vector mode by the angle φ
Figure- 2 (a) and (b) Rotation mode by the angle ϴ
We can restrict the rotation angle
tan( i ) to tan(i )  2 i , then the multiply operation by tangent part is
reduced to shift operations. The new iterative rotation can be defined as
xi 1  Ki .[ xi  yi .di .2i ]
yi 1  Ki .[ yi  xi .di .2i ]
Where, i=0,1,2,3,4……n.
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FPGA implementation of Angle Generator for CORDIC Based High pass FIR Filter Design
Removing the scaling factor Ki from the iterative part yields a simple shift-add algorithm for rotation
CORDIC. After several iterations, the product of Ki factors will converge to a constant coefficient 0.607252.The
exact gain depends on the number of iterations:
 2i
An  n 1  2
 1.646762 
Then the angle accumulator is added to the Equation [7],below
xi 1  xi  yi .d i .2 i
yi 1  yi  xi .d i .2 i
zi 1
1  i
 zi _ d i . tan
 1,2z i  0
di  
 1, otherwise
The angle accumulator is initialized with the input rotation angle and the rotation direction at each iteration i is
decided by the magnitude of the residual angle in the angle accumulator. If the residual angle is positive, then
 1, z i  0
di  
 1, otherwise
A CORDIC has three inputs x0,y0,z0 and depending on the inputs to the Rotation CORDIC and various results
can be produced at the outputs are Xn, Yn, Zn shown in Fig:2
After several iterations it will produce the results according to a CORDIC Block is shown in Fig-3.
x n  An x0  y 0 tan z 0 
zn  0
y n  An  y 0  x0 tan z 0 
Figure-3 A CORDIC Block
For convergence of φ to 0 choose di=sign Zi
If we can start with X0=1/k and y=0, at the end of the process,we find Xn=Cos Z0 and Yn= SinZ0 and the
domain of convergence is lies between -99.7<Z0<99.7.
A. Using CORDIC in Vectoring mode is shown in Fig-2, we can write the expression as below:
xi 1  xi  yi .d i .2 i
yi 1  yi  xi .d i .2 i
zi 1  zi _ d i . tan 1 2 i
And we can solve as
x n  An x 2  y 2
yn  0
z n  z 0  tan 1 (
For convergence of Yn to 0, choose
d i   sgn( xi yi )
z n  tan 1 y0
If we start with x0=1 and z0=0, we find
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FPGA implementation of Angle Generator for CORDIC Based High pass FIR Filter Design
 1, yi  0
di  
 1, otherwise
3.1 Required Angle
At first pre -computation of tan  i  (2 ) or  i  tan (2 ) value then we obtained binary value of
each required angle. Here we use 10 iterations, so each iteration we use the αi value from the TABLE-1 as
shown below.
Table-1 Required Angle For Each Iteration
 i  tan 1 (2 1 )
tan  i
3.2 Working Cordic Architechture
The working CORDIC parallel Architecture figure is given below. This architecture consists of Add
block, add/sub block, comparator and shift blocks as shown in Fig-4.We have assumed that, if the input angular
value<=45 degree, then inputs (x, y) are (1,0) or else (0,1).Then compare this angle with a reference angle (30
degree) and right bit-shifted operation is occurred depending upon the iteration step(i), x(i) and y(i) and then
added or subtracted depending upon the value of comparator circuit(z0) with x(i) and y(i) respectively to
generate x(i+1) and y(i+1).Again we repeat the 10 iterations to getting the desire output values.
Figure-4 Working Parallel CORDIC Architecture
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FPGA implementation of Angle Generator for CORDIC Based High pass FIR Filter Design
3.3 Cordic Based High Pass FIR Filter
The Block diagram of CORDIC based high pass FIR filter is shown in FIG-5.Here the main blocks are Bit-shift,
Add/Sub and CORDIC and the given inputs to the HP FIR filter design is in a 15-bit binary format .The angular
inputs w,2w,3w are provided through the CORDIC blocks and generate the Cosine angle and finally using the
Add/Sub blocks produce the desire output . Here we have chosen an arbitrary frequency response equation of an
High Pass FIR filter is given in equation (11).
H (e jw )  0.75  0.45 cos w  0.31cos 2w  0.15 cos 3w
CORDIC based High Pass FIR Filter
Figure-5 Block diagram of High pass FIR Filter
FPGA Implementation And Simulation Result
In this paper, the FPGA implementation is done for generating the sine and cosine values in binary
format of given angle (30 degree) and taking the input (1,0)of X and Y value in 2bit form using CORDIC
algorithm as shown in Fig-6 and RTL schematic is also shown in Fig-7.Both 15 bit CORDIC architecture and
CORDIC based FIR Filter module have implemented by using Xilinx ISE Design 14.4 as shown in Fig-8 and
Fig-9, described in VHDL language and verified and synthesized the functionality using Xilinx ISim simulator
as shown in Fig-10 and Fig-11. The top-level RTL schematic figures are shown below.
Figure-6 FPGA tested 2bit parallel CORDIC Algorithm
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FPGA implementation of Angle Generator for CORDIC Based High pass FIR Filter Design
Figure-7 RTL Schematic of Parallel CORDIC Top level 2bit Architecture
Figure-8 Top- level RTL CORDIC Architecture
Figure-9 Top- level RTL CORDIC FIR Filter
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FPGA implementation of Angle Generator for CORDIC Based High pass FIR Filter Design
Figure-10 RTL Schematic of CORDIC Architecture
Figure-11 RTL Schematic of FIR Filter
4.1 Output Simulation Result
The output simulation result for parallel CORDIC Architecture (CORDIC_NEW) is shown in Fig-12 and also
output simulation result for FIR Filter(30 degree) and (25 degree) are shown in Fig-13 and Fig-14.
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FPGA implementation of Angle Generator for CORDIC Based High pass FIR Filter Design
Figure-12 Simulation Result For Parallel CORDIC Architecture (CORDIC_NEW)
Figure-13 Simulation Result For(30 Degree) FIR Filter
Figure14 Simulation Result For(25 Degree) FIR Filter
4.2 Magnitude plot of High Pass FIR Filter
The Magnitude plot of high pass filter is shown in Fig-15, described the accuracy between the
educational value and output calculated values of generated angles (10 degree to85 degree) from CORDIC
based HP FIR Filter Design circuit using VHDL language. The plot clearly reflects the accuracy of the output
which is generated from the CORDIC based FIR Filter frequency response equation (11).
Figure-15 Magnitude Plot Of High Pass FIR Filter
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FPGA implementation of Angle Generator for CORDIC Based High pass FIR Filter Design
4.3 Power Summry
The Power Summery of Parallel CORDIC Algorithm is given in Fig-16.The total Power utilization is 0.031W.
Optimization Op
Dynamic (W)
Total (W)
Figure-16 Power Summery of Parallel CORDIC Algorithm
We have successfully simulated and imlpemented the CORDIC algorithm in FPGA for calculating the
Cosine and Sine of an angle in 15 bit binary format on Xilinx Spartan3E (XC3S100Ecp132-5) device using ISE
Design Suite 14.4,ISim Simulator of VHDL language.Also we have plotted the magnitude curve of a High Pass
FIR Filter and this error less curve was generated for output accuracy between equational calculation output
values and generated output of CORDIC_FIL values .In future CORDIC algorithm can be further extended for
clock- pipellined architecture in VERILOG language and to calculate more complex and higher order problems
and digital filter design in DSP and VLSI domain with more acuracy ,high speed and low cost.The number of
Input Output Blocks utilization in both cases(CORDIC_NEW ,CORDIC_FIR) are 90% and 68% respectfully
and the total power utilization is 0.030.
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