DESIGN OF QUATERNARY LOGICAL CIRCUIT USING VOLTAGE AND CURRENT MODE LOGIC

International Journal of VLSI design & Communication Systems (VLSICS) Vol.8, No.4, August 2017
DOI : 10.5121/vlsic.2017.8402 13
SIMULATION OF FIR FILTER BASED ON
CORDIC ALGORITHM
Shalini Rai and Rajeev Srivastava
Department of Electronics & Communication,
University of Allahabad, Allahabad (UP)
ABSTRACT
Coordinate Rotation Digital Computer (CORDIC) discovered by Jack E Volder. It is a shift-add operation
and iterative algorithm. CORDIC algorithm has wide area for several applications like digital signal
processing, biomedical processing, image processing, radar signal processing, 8087 math coprocessor,
the HP-35 calculator, Discrete Fourier, Discrete Hartley and Chirp-Z transforms, filtering, robotics, real
time navigational system and also in communication systems. In this paper, we discussed about the
CORDIC algorithm and CORDIC algorithm based finite impulse response low pass & high pass filter. We
have generated the M-code for the CORDIC Algorithm and CORDIC Algorithm based FIR filter with the
help of MATLAB 2010a.We also discussed about the frequency response characteristics of FIR filter.
KEYWORDS
CORDIC Algorithm, FIR Filter, MATLAB
1. INTRODUCTION
Filtering process is the process for refining of the signals regarding the applications. We
categorized the filter in main two parts analog filters and digital filters [7, 8]. Digital filters are
more advantageous than analog filters. Digital filters are used in DSP applications [3]. There are
basically four types of filter structure like low pass filter, high pass filter band pass, band reject
filter. The field programmable gate array (FPGA) is bench for the implementation of FIR and IIR
digital filters in the VLSI area. In the last five decades the CORDIC algorithm [1, 2] is the very
popular algorithm to producing the fast VLSI implementations. It is a hardware efficient
algorithm i.e. optimizing the area, speed, power and hardware cost. It is 2-D rotational vector
algorithm. The extended form of algorithm is a unified algorithm [5, 6] for the computations of
rotation in hyperbolic, linear, circular coordinates systems. Basically it is used for the
computations of several trigonometric functions, hyperbolic function and logarithmic functions of
real and complex numbers. As time passes there are several advancements in CORDIC algorithm
for reduction of the number of iterations, like the angle-recording (AR), modified vector rotation,
mixed scaling rotation (MSR) and scaling free CORDIC algorithms have been proposed for
reduction of no of iteration, improving the system performance and speed up the system. In this
paper we have discussed about the CORDIC Algorithm, FIR filter in section II, III respectively.
In section IV we have to deal with designing of FIR filter based on CORDIC algorithm and its
frequency response characteristics with the help of MATLAB 2010a.In section V we have
discussed about the Results and conclusion.

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2. THEORY OF CORDIC ALGORITHM
Coordinate Rotation Digital Computer (CORDIC) was implemented in 1959 by J.E.Volder. It is
used in two different modes one is rotation mode and vectoring mode. Overall the algorithm can
be realized as an iterative sequences of additions or subtractions and shift operations by using the
two modes, which are rotated by a fixed rotation angle (µ- rotations).The CORDIC algorithm is in
the general rotation transform[3,4]-
x’= xcosø – ysinø (1)
y’= xsinø + ycosø (2)
The above equations can readjusted as
x’= [x - ytanø] cosø (3)
y’= [y + xtanø] cosø (4)
These rotation of angles constrained so tan (ø) = ±2-i
. This will reduces tangent multiplication by
simple shift operation. i is the no iteration. The angle ø decomposes into elementary rotations in
sequence manner.
Ø=Ʃαi (5)
So iterative equations of Cordic Algorithm are
xi+1 = [xi –yi tanαi] cosαi (6)
yi+1 = [yi+xi tanαi] cosαi (7)
For the trigonometric identities
cos(αi) = 1/(1+tan2
αi)1/2
(8)
Replace the term 1/ (1+tan2
αi)1/2
= ki
(9)
or ki = 1/(1+2-2i
)1/2
ki denotes as constant multiplication factor .The gain is defined as the inverse of the constant
multiplication factor.
Ai= 1/ki (10)
The system gain An= П[1+2-2i
] ≈1.647
So the above equation no.(6) and (7) becomes
xi+1 = ki.[xi - yi.di.2-i
] (11)
yi+1 = ki.[yi +xi.di.2-i
] (12)
di is the decision function depends the rotational mode.
First is the rotation mode and second is the vectoring mode

15
a) For the rotation mode
di= -1 if zi < 0 (13)
di = +1 if else
after nth
iteration it produces the following results
xn= an[x0cosz0- y0sinz0] (14)
yn= An[x0sinz0 + y0cosz0] (15)
zn =0 (16)
b) For vectoring mode
di = +1 if y<0 (17)
= -1 else
After n iteration it produces the following results
xn = An (x02
+y02
)1/2
(18)
zn= tan-1
(y0/x0)+z0 (19)
yn=0 (20)
3. FINITE IMPLUSE REPONSE FILTER
In many applications of signal processing we want to change the relative amplitudes and
frequency contents of a signal. This process is known as filtering .The ideal filters have a
frequency response that is real and non – negative, i.e. has a zero phase characteristics. A linear
phase characteristics introduces a time shift and this causes no distortion in the shape of the signal
in the pass-band. Since the Fourier transfer of a stable impulse response is continuous function of
ω, cannot get a stable filter.
An ideal frequency selective filter passes complex exponential signal. For a given set of
frequencies and completely rejects the others. In figure 1 shows frequency response for ideal low
pass filter (LPF), ideal high pass filter (HPF), ideal band pass filter (BPF) and ideal backstop filter
(BSF).
(a) Low Pass Filter (b)High Pass Filter (c) Band Pass Filter (d) Band Reject Filter
Figure1: Ideal Filter Frequency Response
Finite Impulse Response: The term digital filter arises because these filters operate on discrete-
time signals. Finite Impulse Response is the filter in which output response is equal to the

16
weighted finite sum of past, present and perhaps future values of the filter input, i.e.
M2
y[n] = Ʃ bkx[n-k] (21)
k=-M1
where both M1 and M2 are finite. An FIR filter is based on a feed-forward filter. Feed forward
means that there is no feedback of past or future to form the present output, just input related
terms. The causal FIR filters has difference equation of the form
M
y[n] = Ʃ bkx[n-k] (22)
k=0
The time domain impulse response of a filter corresponding to a given (desired) frequency
response may be calculated from the inverse Fourier transform of the desired frequency response:
п
Hd(n) = ½*П ∫ Hd(ω) ejnω
dω
-П
The samples hd(n) from the above are time domain values, as indicated by the index n. These are
the time domain samples that would have the frequency response Hd(ω).The conceptual leap is
that we use these numbers as weighting coefficients in a difference equation to form filter itself.
For M=3 the FIR Filter
Figure2: Fir filter for M=3
The impulse response hd(n) thus computed will be infinite in extent. In a practical filter, the order
must be limited. This is obtained by truncating the impulse response. Assuming N is odd, the

17
calculation of hd(n) over the range
-(N-1)/2≤ n≤ (N-1)/2
4. CORDIC BASED HIGH PASS FIR FILTER
In this paper we design the CORDIC algorithm based High pass FIR filter with help of
MATLAB2010a and by its simulink tool. We have chosen the arbitrary frequency response
equation of a High pass FIR filter [10].
Figure 3: Simulation of high pass filter based on CORDIC algorithm
Figure 4: Adder System

18
Figure 5: CORDIC Subsystem
5. RESULTS AND CONCLUSION
PLOT OF MAGNITUDE VALUE VS ANGLE (BY SIMULINK OUTPUT)
MATLAB 2010a M-CODE OUTPUT
a) Filter length N=256 and for CORDIC algorithm no of iteration i=16

19
Plot of h(e) Polar Plot of h(e)
b) Filter length N=126 and for CORDIC algorithm no of iteration i=16

20
c) Filter length N=63 and for CORDIC algorithm no. of iteration i =16

21
d) Filter length N=37 and for CORDIC algorithm no. of iteration i=16
(e) Filter length N=17 and for CORDIC algorithm no. of iteration i=16

22
(f) Filter length N=7 and for CORDIC algorithm no. of iteration i=16
Plot of h(e) polar plot of h(e)

23
We observed from the above results that the FIR filter has linear phase response. In magnitude
response (from simulink) the magnitude of the output is varies from negative value to positive
value and after 120° value the magnitude value is constant.
In magnitude response (from MATLAB) of the filter, the magnitude value of the output is
varies from negative value to the positive value it becomes constant after at angle 75
radian/samples for filter length N=256,after 35 radian/samples for the filter length N=126,after
20 radian/samples for filter length N=63, after 10 radian/samples for filter length N=37,after 5
radian/samples for the filter length N=17 and after 2 radian/samples for the filter length N=7.It
means the cutoff point at the angle(radian/samples) e=2 for the filter length N=7,e=5 for the
filter length N=17, e=10 for the filter length N= 37, e=20 for the filter length N=63 , e=35 for the
filter length N=126 and e=75 for the filter length N=256 respectively, those are very close ideal
characteristics of the high pass filter. There are some distortion is observed in the phase response
characteristics of the filter when we take the filter length N= 17and 7.When we are decreased the
filter length the cutoff point also decreased but some distortion are occurred in phase response
characteristics. Magnitude plot of h versus samples value and the polar plot of the h are denser as
filter length increases.

24
REFERENCES
[1] J.E.Volder, “The CORDIC trigonometric computing technique,” IRE Transactions on Electronic
Computers, vol.8, no.3 pp.330-334, 1959.
[2] J.S.Walther, “A unified algorithm for elementary functions”, in proceedings of AFIPS Spring Joint
Computer Conference, pp.379-385, May 1971.
[3] Y.H.HU, “CORDIC based VLSI architecture for digital signal processing,” IEEE signal processing
Magazine, vol.9, no.3, pp16-35, 1992.
[4] Ray Andraka, Andraka consulting group, “A Survey of CORDIC ALGORITHMS FOR FPGA
based computers” ,in Proceedings of the 6th ACM/SIGDA International Symposium on Field
Programmable Gate Arrays (FPGA‟98),PP.191-200,Feburary 1998.
[5] J.E.Volder, “The birth of CORDIC”, Journal of VLSI Signal Processing, Vol.25, no.2, PP.101- 105,
2000.
[6] J.S.Walther, “The story of Unified CORDIC”, Journal of VLSI signal processing, vol.25, no.2, pp -
107-112, 2000.
[7] Pramod.K.Meher, Javier VALLS, Tso Bing Juang, K.Sridharan, Koushik Maharatna, “50 years of
CORDIC Algorithms, Architectures and Applications”, IEEE Transactions on Circuits and System -1:
Regulars Papers, vol.56, no.9. September 2009.
[8] B.Lakshmi and A.S.Dhar, “CORDIC Architectures: A Survey”, Hindawai Publishing Corporation
VLSI Design Volume 2010, Article ID 794891, 19 Pages.
[9] Richa Upadhyay, Dr.Nisha Sarwade, Shrugal Varde, “Simulink Design of Pipelined CORDIC for
Generation of Sine and Cosine Values”, International Journal of Computing Engineering
ResearchVol3 Issue.3, March 2013.
[10]Nutan Das , Swarnaprabha Jena, Siba Kumar Panda, “ FPGA implementation of Angle Generator for
CORDIC Based High pass Filter Design”, IOSR Journal of Electronics and Communication
Engineering, pp 01-11,2016

DESIGN OF QUATERNARY LOGICAL CIRCUIT USING VOLTAGE AND CURRENT MODE LOGIC

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DESIGN OF QUATERNARY LOGICAL CIRCUIT USING VOLTAGE AND CURRENT MODE LOGIC