FPGA Implementation of FIR Filter using Various Algorithms: A Retrospective

International Journal of Research in Computer Science
eISSN 2249-8265 Volume 4 Issue 2 (2014) pp. 19-24
www.ijorcs.org, A Unit of White Globe Publications
doi: 10.7815/ijorcs.42.2014.081
www.ijorcs.org
FPGA IMPLEMENTATION OF FIR FILTER USING
VARIOUS ALGORITHMS: A RETROSPECTIVE
Jinalkumari K. Dhobi1
, Dr. Y. B. Shukla2
, Dr. K.R.Bhatt3
1
P.G. Student, EC Dept., SVIT, Vasad
Email: jinal.dhobi@gmail.com
2
Assoc. Prof., EC Dept., SVIT, Vasad
Email: ybshukla2003@gmail.com
Assoc. Prof., EC Dept., SVIT, Vasad
Email: krbhattec@gmail.com
Abstract: This manuscript is a thorough study of
FPGA implementation of Finite Impulse response
(FIR) with low cost and high performance. The key
observation of this paper is an elaborate analysis
about hardware implementations of FIR filters using
different algorithm i.e., Distributed Arithmetic (DA),
DA-Offset Binary Coding (DA-OBC), Common Sub-
expression Elimination (CSE) and sum-of-power-of-
two (SOPOT) with less resources and without
affecting the performance of the original FIR Filter.
Keywords: DA, DA-OBC, CSE, SOPOT, FPGA
I. INTRODUCTION
A digital filter is a system that carry out
mathematical operation on a sampled of discrete time
signal to modify or alter the component of the signal in
time or frequency domain. It consist of an analog-to-
digital converter at front end, followed by a
microprocessor and some peripheral component i.e.,
memory to store data & filter coefficients. At the
backend side a digital-to-analog converter is used to
complete the output stage. For a real time applications,
an FPGA or ASIC or Specialized DSP with parallel
architecture is used instead of a general purpose
microprocessor. Digital filters can be implemented in
two ways, by convolution (FIR) and by recursion (IIR)
[1]. A FIR filter has a number of useful properties
compared to an IIR filter i.e. inherently stable, no
feedback require, designed to be linear phase. With
recent trend towards portable computing and wireless
communication systems, power consumption has been
an important design consideration [2] so the field
programmable gate array (FPGA) is an alternative
solution for realization of digital signal processing
task.
This paper is organized as follows: section II
introduced design method of the FIR filter, section III
describes the various algorithms to implement the FIR
filter on FPGA, Section IV discusses the Comparison
of algorithms and finally the conclusion is presented in
section V.
II. DESIGN METHODS
FIR filters are used where exact linear phase
response is required. It is non-recursive filter, consists
of two parts one is Approximation problem and second
one is Realization problem. The steps of
approximation are, first take the Ideal frequency
response after that choose the Class of filter & Quality
of approximation and finally select the Method to find
the filter transfer function. The realization part select
the structure to implement the transfer function. There
are mainly three well-known methods for designing
FIR filter namely the window method, Frequency
sampling technique and Optimal filter design method.
Among these three methods, window method is simple
and efficient way to design an FIR filter [3]. In this
method, first start with the ideal desired frequency
response 𝐻𝐻𝑑𝑑 �𝑒𝑒𝑗𝑗 𝑗𝑗
� of the specified filter then Compute
the inverse DTFT of 𝐻𝐻𝑑𝑑 �𝑒𝑒𝑗𝑗 𝑗𝑗
� i.e., ℎ𝑑𝑑 (𝑛𝑛) which is
infinite in duration after that Choose an appropriate
window function 𝑤𝑤(𝑛𝑛) and calculate the impulse
response ℎ(𝑛𝑛) of specified filter as ℎ(𝑛𝑛) =
ℎ𝑑𝑑 (𝑛𝑛) ∗ 𝑤𝑤(𝑛𝑛) to truncated at some point n=M-1. Once
ℎ(𝑛𝑛) is determined, it’s DTFT 𝐻𝐻�𝑒𝑒𝑗𝑗 𝑗𝑗
� and Z-
transform 𝐻𝐻(𝑧𝑧) can be calculated for any further
analysis. For truncation in third step of ℎ𝑑𝑑 (𝑛𝑛) to M-
terms, direct truncation method using rectangular
window 𝑤𝑤(𝑛𝑛) is multiplied with ℎ𝑑𝑑 (𝑛𝑛) which is
infinite in nature but rectangular window contain sharp
discontinuity which leads to Gibbs Phenomenon effect
[4]. In order to reduce the ripples, instead of sharp
discontinuity function, choose a window function
having taper and decays toward zero gradually [4].
Some of window [5] commonly used are as followed:
1. Bartlett Triangular window:
𝑤𝑤(𝑛𝑛) =
⎩
⎪
⎨
⎪
⎧
2(𝑛𝑛 + 1)
𝑁𝑁 + 1
, 𝑛𝑛 = 0 𝑡𝑡𝑡𝑡 (𝑁𝑁 − 1)/2
2 −
2(𝑛𝑛 + 1)
𝑁𝑁 + 1
, 𝑛𝑛 = (𝑁𝑁 − 1)/2 𝑡𝑡𝑡𝑡 (𝑁𝑁 − 1)
0 𝑂𝑂𝑂𝑂ℎ𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒
… . (1)
2. Generalized cosine windows:

20 Jinalkumari K. Dhobi, Dr. Y. B. Shukla, Dr. K.R.Bhatt
www.ijorcs.org
(Rectangular, Hanning, Hamming and
Blackman)
𝑤𝑤(𝑛𝑛) = 𝑎𝑎 − 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 �
2𝑝𝑝(𝑛𝑛 + 1)
𝑁𝑁 + 1
� + 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 �
4𝑝𝑝(𝑛𝑛 + 1)
𝑁𝑁 + 1
�
Where n = 0 to N-1
…. (2)
3. Kaiser window with parameter β:
𝑤𝑤(𝑛𝑛) =
𝐼𝐼0(𝛽𝛽�1 − �1 −
2(𝑛𝑛)
𝑁𝑁−1
�
2
𝐼𝐼0(𝛽𝛽)
Where n= 0 to N-1
…. (3)
𝑀𝑀 =
δ2 − 7.95
2.286∆ω
…. (4)
β=
⎩
⎨
⎧
0.1102(δ2 − 8.7), δ2 ≥ 50𝑑𝑑𝑑𝑑
0.5842(δ2 − 21)0.4
+ 0.07886(δ2 − 21),
21𝑑𝑑𝑑𝑑 < δ2 < 50𝑑𝑑𝑑𝑑
0, δ2 ≤ 21𝑑𝑑𝑑𝑑
.... (5)
In Rectangular window, due to the direct
truncation of ℎ𝑑𝑑 (𝑛𝑛) leads to the Gibbs phenomenon
effect which apparent itself as a fixed percentage
overshoot and ripple before and after an approximated
discontinuity in the frequency response due to the non-
uniform convergence of the Fourier series at the
discontinuity [5]. According to the simulation result
[5] obtained from FDAtool, the Bartlett window
reduced the overshoot in the designed filter but spreads
the transition region. The generalized cosine window,
Hanning, Hamming and Blackman is complicated but
provide a smooth truncation of ideal impulse response
and a frequency response. The best window method is
Kaiser Window because it has the shape parameter β
which depending on the filter taps M was used to
adjust the main lobe width and side lobe attenuation,
choosing M can produce a variety of transition band
and optimal stopband attenuation [6]. When the
transition band ∆ω(rad) and the stopband attenuation
δ2 = −20𝑙𝑙𝑙𝑙 𝑙𝑙10 𝛼𝛼2(𝑑𝑑𝑑𝑑) were given, the Kaiser FIR
filter taps M and the shape parameter β can be
obtained from equation (4) and (5) [6].
III. FPGA IMPLEMENTATION USING VARIOUS
ALGORITHMS
The application of digital filter are widespread and
include but are not limited to the Communication
System, Audio System, Instrumentation, Image-
Processing and enhancement, Speech synthesis. It is
nowadays convenient to consider computer programs
and digital hardware that can perform digital filtering
as two different implementations of digital filters,
namely, Software digital filters, and Hardware digital
filters. Software digital filters can be implemented in
terms of a high-level language, such as C++ or
MATLAB, on a personal computer or workstation or
by using a low-level language on a general-purpose
digital signal processing chip. Hardware digital filter
can be designed using a number of highly specialized
interconnected VLSI chips like DSP, ASIC and FPGA.
Software digital filter have no counterpart in the
analog world and therefore, for non-real-time
application they are the only choice. ASIC based
Implementation of FIR Filter provides little costlier
and long development time but gives high flexibility.
The realization of FIR filter based on FPGA received
extensive attention because it gives high flexibility,
high performance, low cost and shorter development
time [8]. The core of the FIR filter implementation is
multiplication and accumulation (MAC) operation.
The design method of MAC can be define using
various algorithm and techniques. One is general direct
MAC structure, which are expensive in hardware
because of logic complexity and area usage. The others
are Distributed Arithmetic (DA) [7], DA-Offset Binary
Coding (DA-OBC) [9], Common Subexpression
Elimination (CSE) [10] and Sum-of-power-of-two
(SOPOT) [11] which reduced the required number of
multiplier and adders.
A. Distributed arithmetic and DA-OBC
Distributed Arithmetic is a famous method, which
converts calculation of MAC to a serial of look up
table accesses and summation [8]. It was initially
proposed by Croisier in 1973 [7] and further developed
by Peled and Lui [8].The principle of DA algorithm is
as follows:
An FIR filter of N order is shown as Eq. (6)
𝑦𝑦(𝑛𝑛) = ∑ ℎ𝑘𝑘 𝑥𝑥𝑘𝑘(𝑛𝑛)𝑁𝑁−1
𝑘𝑘=0 …. (6)
Where ℎ𝑘𝑘the set of constant coefficient of the filter
is, 𝑦𝑦(𝑛𝑛) is the output data, 𝑥𝑥𝑘𝑘(𝑛𝑛) is the input data,
which can be expressed as Eq. (7) using m-bit 2’s
complementary binary number.
𝑥𝑥𝑘𝑘 = −𝑥𝑥𝑘𝑘,𝑚𝑚−1 + ∑ 𝑥𝑥𝑘𝑘,𝑚𝑚−1−𝑗𝑗
𝑚𝑚−1
𝑗𝑗=1 2−𝑗𝑗
…. (7)
Where 𝑥𝑥𝑘𝑘,𝑚𝑚−1 is the most significant bit of 𝑥𝑥𝑘𝑘, Eq.
(8) can be derived when 𝑥𝑥𝑘𝑘 of Eq. (7) is substitute into
Eq. (6).
𝑦𝑦(𝑛𝑛) = � ℎ𝑘𝑘[−𝑥𝑥𝑘𝑘,𝑚𝑚−1 + � 𝑥𝑥𝑘𝑘,𝑚𝑚−1−𝑗𝑗
𝑚𝑚−1
𝑗𝑗 =1
2−𝑗𝑗
]
𝑁𝑁−1
𝑘𝑘=0
= � � ℎ𝑘𝑘
𝑚𝑚−1
𝑗𝑗 =1
𝑥𝑥𝑘𝑘,𝑚𝑚−1−𝑗𝑗 2−𝑗𝑗
−
𝑁𝑁−1
𝑘𝑘=0
� ℎ𝑘𝑘 𝑥𝑥𝑘𝑘,𝑚𝑚−1
𝑁𝑁−1
𝑘𝑘=0

FPGA Implementation of Fir Filter using Various Algorithms: A Retrospective 21
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= � � ℎ𝑘𝑘
𝑁𝑁−1
𝑘𝑘=0
𝑥𝑥𝑘𝑘,𝑚𝑚−1−𝑗𝑗 2−𝑗𝑗
−
𝑚𝑚−1
𝑗𝑗=1
� ℎ𝑘𝑘 𝑥𝑥𝑘𝑘,𝑚𝑚−1
𝑁𝑁−1
𝑘𝑘=0
…. (8)
In Eq. (8) the calculation result of
∑ ℎ𝑘𝑘
𝑁𝑁−1
𝑘𝑘=0 𝑥𝑥𝑘𝑘,𝑚𝑚−1−𝑗𝑗 have 2𝑁𝑁
kinds of different results
because of the value of 𝑥𝑥𝑘𝑘,𝑚𝑚−1−𝑗𝑗 is 0 or 1. All the
possible results of ∑ ℎ𝑘𝑘
𝑁𝑁−1
𝑘𝑘=0 𝑥𝑥𝑘𝑘,𝑚𝑚−1−𝑗𝑗 are constructed
in advance and stored into one LUT. The contents of
LUT can be fetch by using 𝑥𝑥𝑘𝑘,𝑚𝑚−1−𝑗𝑗 obtain from shift
register unit as a LUT address signal. So the value of
𝑦𝑦(𝑛𝑛) calculated by shifting and accumulating
operation.
Table 1: LUT contents of DA for 4-tap FIR filter
Address signal (𝑏𝑏3 𝑏𝑏2 𝑏𝑏1 𝑏𝑏0) LUT content
0000 0
0001 ℎ0
0010 ℎ1
0011 ℎ1 + ℎ0
0100 ℎ2
0101 ℎ2 + ℎ0
0110 ℎ2 + ℎ1
0111 ℎ2 + ℎ1 + ℎ0
1000 ℎ3
1001 ℎ3 + ℎ0
1010 ℎ3 + ℎ1
1011 ℎ3 + ℎ1 + ℎ0
1100 ℎ3 + ℎ2
1101 ℎ3 + ℎ2 + ℎ0
1110 ℎ3 + ℎ2 + ℎ1
1111 ℎ3 + ℎ2 + ℎ1 + ℎ0
Hence, DA algorithm reduced the logic resources
by converting complex MAC operation to a simple
look-up table access and summations. The main
disadvantage of DA algorithm is the size of LUT is
large and also capacity of LUT will exponentially
increase with the order of the filter. Bo Hong, et al [9]
proposed a new algorithm which can reduced the
capacity of LUT by half through the use of offset
binary coding. The principle of DA-OBC algorithm is
as follows [12]:
𝑥𝑥𝑘𝑘 can also be expressed as Eq. (9)
𝑥𝑥𝑘𝑘 =
1
2
[𝑥𝑥𝑘𝑘 − (−𝑥𝑥𝑘𝑘)]
=
1
2
�−�𝑥𝑥𝑘𝑘,𝑚𝑚−1 − 𝑥𝑥𝑘𝑘,𝑚𝑚−1��
+ � [�𝑥𝑥𝑘𝑘,𝑚𝑚−1−𝑗𝑗 − 𝑥𝑥𝑘𝑘,𝑚𝑚−1−𝑗𝑗��2−𝑗𝑗
𝑚𝑚−1
𝑗𝑗=1
− 2−(𝑚𝑚−1)
]
…. (9)
They define 𝑑𝑑𝑘𝑘,𝑗𝑗 as follows, the value of 𝑑𝑑𝑘𝑘,𝑗𝑗 is -1 or +1.
𝑑𝑑𝑘𝑘,𝑗𝑗 = �
�𝑥𝑥𝑘𝑘,𝑗𝑗 − 𝑥𝑥𝑘𝑘,𝑗𝑗��, 𝑗𝑗 ≠ 𝑚𝑚 − 1
−�𝑥𝑥𝑘𝑘,𝑚𝑚−1 − 𝑥𝑥𝑘𝑘,𝑚𝑚−1��, 𝑗𝑗 = 𝑚𝑚 − 1
…. (10)
Eq. (9) can be simplified as Eq. (11).
𝑥𝑥𝑘𝑘 =
1
2
[� 𝑑𝑑𝑘𝑘,𝑚𝑚−1−𝑗𝑗 2−𝑗𝑗
− 2−(𝑚𝑚−1)
]
𝑚𝑚−1
𝑗𝑗=0
…. (11)
Put Eq. (11) in to Eq. (6), and get
= � ��
1
2
ℎ𝑘𝑘
𝑁𝑁−1
𝑘𝑘=0
𝑑𝑑𝑘𝑘,𝑚𝑚−1−𝑗𝑗 � 2−𝑗𝑗
−
𝑚𝑚−1
𝑗𝑗 =0
(
1
2
� ℎ𝑗𝑗
𝑁𝑁−1
𝑗𝑗 =0
)2−𝑚𝑚−1
…. (12)
For convenience,
𝐷𝐷𝑗𝑗 = �
1
2
ℎ𝑘𝑘
𝑁𝑁−1
𝑘𝑘=0
𝑑𝑑𝑘𝑘,𝑚𝑚−1−𝑗𝑗 , 𝐷𝐷𝑒𝑒𝑒𝑒 = −
1
2
� ℎ𝑗𝑗
𝑁𝑁−1
𝑗𝑗=0
…. (13)
The value of 𝐷𝐷𝑗𝑗 have 2𝑁𝑁
kinds of different result
which can be pre-computed and stored in a LUT and
the LUT can be reduced by half [9] because the
contents of LUT which are stored in addresses, are 1’s
complement of each other is in same magnitude with
the reverse sign.
Table 2: LUT contents of DA-OBC for 4-tap FIR filter
Address
𝑥𝑥0𝑗𝑗 = 0
𝑥𝑥1𝑗𝑗 𝑥𝑥2𝑗𝑗 𝑥𝑥3𝑗𝑗
LUT
Contents
Address
𝑥𝑥0𝑗𝑗 = 1
𝑥𝑥1𝑗𝑗 𝑥𝑥2𝑗𝑗 𝑥𝑥3𝑗𝑗
LUT
Contents
000
−
ℎ0 + ℎ1 + ℎ2 + ℎ
2
111 ℎ0 + ℎ1 + ℎ2 + ℎ3
2
001
−
ℎ0 + ℎ1 + ℎ2 − ℎ
2
110 ℎ0 + ℎ1 + ℎ2 − ℎ3
2
010
−
ℎ0 + ℎ1 − ℎ2 + ℎ
2
101 ℎ0 + ℎ1 − ℎ2 + ℎ3
2
011
−
ℎ0 + ℎ1 − ℎ2 − ℎ
2
100 ℎ0 + ℎ1 − ℎ2 − ℎ3
2
100
−
ℎ0 − ℎ1 + ℎ2 + ℎ
2
011 ℎ0 − ℎ1 + ℎ2 + ℎ3
2
101
−
ℎ0 − ℎ1 + ℎ2 − ℎ
2
010 ℎ0 − ℎ1 + ℎ2 − ℎ3
2
110
−
ℎ0 − ℎ1 − ℎ2 + ℎ
2
001 ℎ0 − ℎ1 − ℎ2 + ℎ3
2
111
−
ℎ0 − ℎ1 − ℎ2 − ℎ
2
000 ℎ0 − ℎ1 − ℎ2 − ℎ3
2

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B. Common Subexpression Elimination (CSE)
M. Thenmozhi, et al [10] proposed an algorithm
which reduce the complexity is Common
subexpression elimination (CSE). In this algorithm, the
coefficient is based on canonical signed digit (CSD),
which minimize the number of adder/subtractor used
in each coefficient multiplier. The aim of CSE
algorithm is to identify multiple event of identical bit
patterns present in coefficients to remove the
redundant multiplications which results in considerable
reduction of adders as well as the complexity of FIR
filter compared to the conventional implementation.
This algorithm using binary representation of
coefficients for the implementation of higher order FIR
filter with less number of adder than CSD based CSE
algorithm. The Binary CSE (BCSE) algorithm
technique focuses on reducing redundant computations
in coefficient multipliers by reusing the most common
binary bit patterns (BCSs) presents in coefficients [13].
In n-bit binary number, the number of BCSs is2𝑛𝑛
−
(𝑛𝑛 + 1).
For example: A 3 bit binary representation can
form four BCSs.
[0 1 1] = 𝑥𝑥2 = 2−1
𝑥𝑥 + 2−2
𝑥𝑥
[1 0 1] = 𝑥𝑥3 = 𝑥𝑥 + 2−2
𝑥𝑥
[1 1 0] = 𝑥𝑥4 = 𝑥𝑥 + 2−1
𝑥𝑥
[1 1 1] = 𝑥𝑥5 = 𝑥𝑥 + 2−1
𝑥𝑥 + 2−2
𝑥𝑥
Where x is the input signal. The other BCSs such
as [0 0 1], [0 1 0] and [1 1 0] do not required any
adder for implementation because they have only one
nonzero bit. From above four BCSs they conclude that
𝑥𝑥2 can be obtained by right shift operation without
using any extra adders and also 𝑥𝑥5 can be obtained
from 𝑥𝑥4 using an adder.
C. Sum-of-power-of-two (SOPOT)
Catalin Damian, et al [11] proposed a high speed
and low area architecture for the implementation of
FIR filter without any multiplication block.
Figure 1: Direct FIR filter
Figure 2. describe the Architecture unit of FIR
filter with SOPOT type coefficient. In this algorithm
the coefficient values are integer’s power-of-two or
sum-of- power-of-two (SOPOT) with two or three
terms and the multipliers can be replaced by shifters.
In [15] and [16], FIR architectures presented based on
theory of SOPOT of two terms. Because of error
occurred in two terms the approximation of filter’s
coefficient is extended the algorithm to SOPOT with
three terms.
Figure 2. Architecture unit of FIR filter with SOPOT type
coefficient
𝐻𝐻[𝑗𝑗] = ∑ 𝑎𝑎𝑖𝑖,𝑗𝑗 . 2𝑏𝑏𝑖𝑖,𝑗𝑗2
𝑖𝑖=0 …. (13)
Where 𝑎𝑎𝑖𝑖,𝑗𝑗 = {−1,1} and𝑏𝑏𝑖𝑖,𝑗𝑗 = {−𝑡𝑡, … ,0, … , 𝑢𝑢}; t
and u determine the word length dynamic range of
each filter coefficient. Larger the numbers t and u will
gives the closer approximation to its original number.
Computational algorithm: The Computational
algorithm is shown in figure 3. The algorithm consist
of two repetitive structure; one by i is to calculate
a[i,j] and b[i,j] for the f[j] co-efficient.
Figure 3: SOPOT 𝑎𝑎𝑖𝑖,𝑘𝑘 and 𝑏𝑏𝑖𝑖,𝑘𝑘 terms calculation algorithm

FPGA Implementation of Fir Filter using Various Algorithms: A Retrospective 23
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Table 3: Example of Two and Three SOPOT
Integer Two SOPOT Three SOPOT
13 23
+ 22
=12 (8%) 23
+ 22
+ 20
=13
(0%)
25 24
+ 23
=24 (4%) 24
+ 23
+ 20
=25
(0%)
67 26
+ 21
=66 (2%) 26
+ 21
+ 20
=67
(0%)
IV. COMPARISION OF ALGORITHMS BASED ON
SURVEY
DA algorithm reduces the complexity of FIR filter
by converting MAC operation to serial of look up
table. CSE algorithm reduces the adder by just finding
the multiple event and SOPOT is converting
coefficient into power of two format and eliminate the
complete Multiplication block. Among the three
algorithm DA is easy to implement at the cost of
storage resources.
V. CONCLUSION
The realization structure of FIR filter consists of a
MAC operation which is made up of multipliers and
adders. DA algorithm, it completely eliminate the
multiplication block by converting complicated MAC
operation to look-up table access and summation at the
cost of increasing storage resource where DA-OBC
based architecture will decreases the LUT size by half
and make operation speed faster. In CSE algorithm, It
eliminate the multipliers and adders by identifying the
multiple event that have an identical bit pattern and
SOPOT algorithm diminished the complexity of
working task by completely abolishing the
multiplication block by only adders and shifter. So
According to our survey, DA structure is easy to
implement on FPGA because of pre-calculated results
are already stored in LUTs and in FPGA it is easy to
design but the main drawback of this algorithm is that,
it uses an extra resources. Where the SOPOT is used
for low power and low area application as it uses
shift/add multiplexer based multiplier. The CSE
algorithm is used to find and eliminate most common
event among filter co-efficient which results in power
and area saving by reducing multiplier with a small
number of adders while implemented in FIR filters.
VI. REFERENCES
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www.ijorcs.org
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How to cite
Jinalkumari K. Dhobi, Dr. Y. B. Shukla, Dr. K.R.Bhatt, “FPGA Implementation of Fir Filter using Various
Algorithms: A Retrospective”. International Journal of Research in Computer Science, 4 (2): pp. 19-24, March
2014. doi: 10.7815/ijorcs.42.2014.081

FPGA Implementation of FIR Filter using Various Algorithms: A Retrospective

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