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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 ๐ป๐ป๐‘‘๐‘‘ ๏ฟฝ๐‘’๐‘’๐‘—๐‘— ๐‘—๐‘—
๏ฟฝ of the specified filter then Compute
the inverse DTFT of ๐ป๐ป๐‘‘๐‘‘ ๏ฟฝ๐‘’๐‘’๐‘—๐‘— ๐‘—๐‘—
๏ฟฝ i.e., โ„Ž๐‘‘๐‘‘ (๐‘›๐‘›) which is
infinite in duration after that Choose an appropriate
window function ๐‘ค๐‘ค(๐‘›๐‘›) and calculate the impulse
response โ„Ž(๐‘›๐‘›) of specified filter as โ„Ž(๐‘›๐‘›) =
โ„Ž๐‘‘๐‘‘ (๐‘›๐‘›) โˆ— ๐‘ค๐‘ค(๐‘›๐‘›) to truncated at some point n=M-1. Once
โ„Ž(๐‘›๐‘›) is determined, itโ€™s DTFT ๐ป๐ป๏ฟฝ๐‘’๐‘’๐‘—๐‘— ๐‘—๐‘—
๏ฟฝ and Z-
transform ๐ป๐ป(๐‘ง๐‘ง) can be calculated for any further
analysis. For truncation in third step of โ„Ž๐‘‘๐‘‘ (๐‘›๐‘›) to M-
terms, direct truncation method using rectangular
window ๐‘ค๐‘ค(๐‘›๐‘›) is multiplied with โ„Ž๐‘‘๐‘‘ (๐‘›๐‘›) 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:
๐‘ค๐‘ค(๐‘›๐‘›) =
โŽฉ
โŽช
โŽจ
โŽช
โŽง
2(๐‘›๐‘› + 1)
๐‘๐‘ + 1
, ๐‘›๐‘› = 0 ๐‘ก๐‘ก๐‘ก๐‘ก (๐‘๐‘ โˆ’ 1)/2
2 โˆ’
2(๐‘›๐‘› + 1)
๐‘๐‘ + 1
, ๐‘›๐‘› = (๐‘๐‘ โˆ’ 1)/2 ๐‘ก๐‘ก๐‘ก๐‘ก (๐‘๐‘ โˆ’ 1)
0 ๐‘‚๐‘‚๐‘‚๐‘‚โ„Ž๐‘’๐‘’๐‘’๐‘’๐‘’๐‘’๐‘’๐‘’๐‘’๐‘’๐‘’๐‘’
โ€ฆ . (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)
๐‘ค๐‘ค(๐‘›๐‘›) = ๐‘Ž๐‘Ž โˆ’ ๐‘๐‘๐‘๐‘๐‘๐‘๐‘๐‘ ๏ฟฝ
2๐‘๐‘(๐‘›๐‘› + 1)
๐‘๐‘ + 1
๏ฟฝ + ๐‘๐‘๐‘๐‘๐‘๐‘๐‘๐‘ ๏ฟฝ
4๐‘๐‘(๐‘›๐‘› + 1)
๐‘๐‘ + 1
๏ฟฝ
Where n = 0 to N-1
โ€ฆ. (2)
3. Kaiser window with parameter ฮฒ:
๐‘ค๐‘ค(๐‘›๐‘›) =
๐ผ๐ผ0(๐›ฝ๐›ฝ๏ฟฝ1 โˆ’ ๏ฟฝ1 โˆ’
2(๐‘›๐‘›)
๐‘๐‘โˆ’1
๏ฟฝ
2
๐ผ๐ผ0(๐›ฝ๐›ฝ)
Where n= 0 to N-1
โ€ฆ. (3)
๐‘€๐‘€ =
ฮด2 โˆ’ 7.95
2.286โˆ†ฯ‰
โ€ฆ. (4)
ฮฒ=
โŽฉ
โŽจ
โŽง
0.1102(ฮด2 โˆ’ 8.7), ฮด2 โ‰ฅ 50๐‘‘๐‘‘๐‘‘๐‘‘
0.5842(ฮด2 โˆ’ 21)0.4
+ 0.07886(ฮด2 โˆ’ 21),
21๐‘‘๐‘‘๐‘‘๐‘‘ < ฮด2 < 50๐‘‘๐‘‘๐‘‘๐‘‘
0, ฮด2 โ‰ค 21๐‘‘๐‘‘๐‘‘๐‘‘
.... (5)
In Rectangular window, due to the direct
truncation of โ„Ž๐‘‘๐‘‘ (๐‘›๐‘›) 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 ฮฒ
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 โˆ†ฯ‰(rad) and the stopband attenuation
ฮด2 = โˆ’20๐‘™๐‘™๐‘™๐‘™ ๐‘™๐‘™10 ๐›ผ๐›ผ2(๐‘‘๐‘‘๐‘‘๐‘‘) were given, the Kaiser FIR
filter taps M and the shape parameter ฮฒ 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)
๐‘ฆ๐‘ฆ(๐‘›๐‘›) = โˆ‘ โ„Ž๐‘˜๐‘˜ ๐‘ฅ๐‘ฅ๐‘˜๐‘˜(๐‘›๐‘›)๐‘๐‘โˆ’1
๐‘˜๐‘˜=0 โ€ฆ. (6)
Where โ„Ž๐‘˜๐‘˜the set of constant coefficient of the filter
is, ๐‘ฆ๐‘ฆ(๐‘›๐‘›) is the output data, ๐‘ฅ๐‘ฅ๐‘˜๐‘˜(๐‘›๐‘›) is the input data,
which can be expressed as Eq. (7) using m-bit 2โ€™s
complementary binary number.
๐‘ฅ๐‘ฅ๐‘˜๐‘˜ = โˆ’๐‘ฅ๐‘ฅ๐‘˜๐‘˜,๐‘š๐‘šโˆ’1 + โˆ‘ ๐‘ฅ๐‘ฅ๐‘˜๐‘˜,๐‘š๐‘šโˆ’1โˆ’๐‘—๐‘—
๐‘š๐‘šโˆ’1
๐‘—๐‘—=1 2โˆ’๐‘—๐‘—
โ€ฆ. (7)
Where ๐‘ฅ๐‘ฅ๐‘˜๐‘˜,๐‘š๐‘šโˆ’1 is the most significant bit of ๐‘ฅ๐‘ฅ๐‘˜๐‘˜, Eq.
(8) can be derived when ๐‘ฅ๐‘ฅ๐‘˜๐‘˜ of Eq. (7) is substitute into
Eq. (6).
๐‘ฆ๐‘ฆ(๐‘›๐‘›) = ๏ฟฝ โ„Ž๐‘˜๐‘˜[โˆ’๐‘ฅ๐‘ฅ๐‘˜๐‘˜,๐‘š๐‘šโˆ’1 + ๏ฟฝ ๐‘ฅ๐‘ฅ๐‘˜๐‘˜,๐‘š๐‘šโˆ’1โˆ’๐‘—๐‘—
๐‘š๐‘šโˆ’1
๐‘—๐‘— =1
2โˆ’๐‘—๐‘—
]
๐‘๐‘โˆ’1
๐‘˜๐‘˜=0
= ๏ฟฝ ๏ฟฝ โ„Ž๐‘˜๐‘˜
๐‘š๐‘šโˆ’1
๐‘—๐‘— =1
๐‘ฅ๐‘ฅ๐‘˜๐‘˜,๐‘š๐‘šโˆ’1โˆ’๐‘—๐‘— 2โˆ’๐‘—๐‘—
โˆ’
๐‘๐‘โˆ’1
๐‘˜๐‘˜=0
๏ฟฝ โ„Ž๐‘˜๐‘˜ ๐‘ฅ๐‘ฅ๐‘˜๐‘˜,๐‘š๐‘šโˆ’1
๐‘๐‘โˆ’1
๐‘˜๐‘˜=0
FPGA Implementation of Fir Filter using Various Algorithms: A Retrospective 21
www.ijorcs.org
= ๏ฟฝ ๏ฟฝ โ„Ž๐‘˜๐‘˜
๐‘๐‘โˆ’1
๐‘˜๐‘˜=0
๐‘ฅ๐‘ฅ๐‘˜๐‘˜,๐‘š๐‘šโˆ’1โˆ’๐‘—๐‘— 2โˆ’๐‘—๐‘—
โˆ’
๐‘š๐‘šโˆ’1
๐‘—๐‘—=1
๏ฟฝ โ„Ž๐‘˜๐‘˜ ๐‘ฅ๐‘ฅ๐‘˜๐‘˜,๐‘š๐‘šโˆ’1
๐‘๐‘โˆ’1
๐‘˜๐‘˜=0
โ€ฆ. (8)
In Eq. (8) the calculation result of
โˆ‘ โ„Ž๐‘˜๐‘˜
๐‘๐‘โˆ’1
๐‘˜๐‘˜=0 ๐‘ฅ๐‘ฅ๐‘˜๐‘˜,๐‘š๐‘šโˆ’1โˆ’๐‘—๐‘— have 2๐‘๐‘
kinds of different results
because of the value of ๐‘ฅ๐‘ฅ๐‘˜๐‘˜,๐‘š๐‘šโˆ’1โˆ’๐‘—๐‘— is 0 or 1. All the
possible results of โˆ‘ โ„Ž๐‘˜๐‘˜
๐‘๐‘โˆ’1
๐‘˜๐‘˜=0 ๐‘ฅ๐‘ฅ๐‘˜๐‘˜,๐‘š๐‘šโˆ’1โˆ’๐‘—๐‘— are constructed
in advance and stored into one LUT. The contents of
LUT can be fetch by using ๐‘ฅ๐‘ฅ๐‘˜๐‘˜,๐‘š๐‘šโˆ’1โˆ’๐‘—๐‘— obtain from shift
register unit as a LUT address signal. So the value of
๐‘ฆ๐‘ฆ(๐‘›๐‘›) calculated by shifting and accumulating
operation.
Table 1: LUT contents of DA for 4-tap FIR filter
Address signal (๐‘๐‘3 ๐‘๐‘2 ๐‘๐‘1 ๐‘๐‘0) LUT content
0000 0
0001 โ„Ž0
0010 โ„Ž1
0011 โ„Ž1 + โ„Ž0
0100 โ„Ž2
0101 โ„Ž2 + โ„Ž0
0110 โ„Ž2 + โ„Ž1
0111 โ„Ž2 + โ„Ž1 + โ„Ž0
1000 โ„Ž3
1001 โ„Ž3 + โ„Ž0
1010 โ„Ž3 + โ„Ž1
1011 โ„Ž3 + โ„Ž1 + โ„Ž0
1100 โ„Ž3 + โ„Ž2
1101 โ„Ž3 + โ„Ž2 + โ„Ž0
1110 โ„Ž3 + โ„Ž2 + โ„Ž1
1111 โ„Ž3 + โ„Ž2 + โ„Ž1 + โ„Ž0
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]:
๐‘ฅ๐‘ฅ๐‘˜๐‘˜ can also be expressed as Eq. (9)
๐‘ฅ๐‘ฅ๐‘˜๐‘˜ =
1
2
[๐‘ฅ๐‘ฅ๐‘˜๐‘˜ โˆ’ (โˆ’๐‘ฅ๐‘ฅ๐‘˜๐‘˜)]
=
1
2
๏ฟฝโˆ’๏ฟฝ๐‘ฅ๐‘ฅ๐‘˜๐‘˜,๐‘š๐‘šโˆ’1 โˆ’ ๐‘ฅ๐‘ฅ๐‘˜๐‘˜,๐‘š๐‘šโˆ’1๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
+ ๏ฟฝ [๏ฟฝ๐‘ฅ๐‘ฅ๐‘˜๐‘˜,๐‘š๐‘šโˆ’1โˆ’๐‘—๐‘— โˆ’ ๐‘ฅ๐‘ฅ๐‘˜๐‘˜,๐‘š๐‘šโˆ’1โˆ’๐‘—๐‘—๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ2โˆ’๐‘—๐‘—
๐‘š๐‘šโˆ’1
๐‘—๐‘—=1
โˆ’ 2โˆ’(๐‘š๐‘šโˆ’1)
]
โ€ฆ. (9)
They define ๐‘‘๐‘‘๐‘˜๐‘˜,๐‘—๐‘— as follows, the value of ๐‘‘๐‘‘๐‘˜๐‘˜,๐‘—๐‘— is -1 or +1.
๐‘‘๐‘‘๐‘˜๐‘˜,๐‘—๐‘— = ๏ฟฝ
๏ฟฝ๐‘ฅ๐‘ฅ๐‘˜๐‘˜,๐‘—๐‘— โˆ’ ๐‘ฅ๐‘ฅ๐‘˜๐‘˜,๐‘—๐‘—๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ, ๐‘—๐‘— โ‰  ๐‘š๐‘š โˆ’ 1
โˆ’๏ฟฝ๐‘ฅ๐‘ฅ๐‘˜๐‘˜,๐‘š๐‘šโˆ’1 โˆ’ ๐‘ฅ๐‘ฅ๐‘˜๐‘˜,๐‘š๐‘šโˆ’1๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ, ๐‘—๐‘— = ๐‘š๐‘š โˆ’ 1
โ€ฆ. (10)
Eq. (9) can be simplified as Eq. (11).
๐‘ฅ๐‘ฅ๐‘˜๐‘˜ =
1
2
[๏ฟฝ ๐‘‘๐‘‘๐‘˜๐‘˜,๐‘š๐‘šโˆ’1โˆ’๐‘—๐‘— 2โˆ’๐‘—๐‘—
โˆ’ 2โˆ’(๐‘š๐‘šโˆ’1)
]
๐‘š๐‘šโˆ’1
๐‘—๐‘—=0
โ€ฆ. (11)
Put Eq. (11) in to Eq. (6), and get
= ๏ฟฝ ๏ฟฝ๏ฟฝ
1
2
โ„Ž๐‘˜๐‘˜
๐‘๐‘โˆ’1
๐‘˜๐‘˜=0
๐‘‘๐‘‘๐‘˜๐‘˜,๐‘š๐‘šโˆ’1โˆ’๐‘—๐‘— ๏ฟฝ 2โˆ’๐‘—๐‘—
โˆ’
๐‘š๐‘šโˆ’1
๐‘—๐‘— =0
(
1
2
๏ฟฝ โ„Ž๐‘—๐‘—
๐‘๐‘โˆ’1
๐‘—๐‘— =0
)2โˆ’๐‘š๐‘šโˆ’1
โ€ฆ. (12)
For convenience,
๐ท๐ท๐‘—๐‘— = ๏ฟฝ
1
2
โ„Ž๐‘˜๐‘˜
๐‘๐‘โˆ’1
๐‘˜๐‘˜=0
๐‘‘๐‘‘๐‘˜๐‘˜,๐‘š๐‘šโˆ’1โˆ’๐‘—๐‘— , ๐ท๐ท๐‘’๐‘’๐‘’๐‘’ = โˆ’
1
2
๏ฟฝ โ„Ž๐‘—๐‘—
๐‘๐‘โˆ’1
๐‘—๐‘—=0
โ€ฆ. (13)
The value of ๐ท๐ท๐‘—๐‘— have 2๐‘๐‘
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 1โ€™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
๐‘ฅ๐‘ฅ0๐‘—๐‘— = 0
๐‘ฅ๐‘ฅ1๐‘—๐‘— ๐‘ฅ๐‘ฅ2๐‘—๐‘— ๐‘ฅ๐‘ฅ3๐‘—๐‘—
LUT
Contents
Address
๐‘ฅ๐‘ฅ0๐‘—๐‘— = 1
๐‘ฅ๐‘ฅ1๐‘—๐‘— ๐‘ฅ๐‘ฅ2๐‘—๐‘— ๐‘ฅ๐‘ฅ3๐‘—๐‘—
LUT
Contents
000
โˆ’
โ„Ž0 + โ„Ž1 + โ„Ž2 + โ„Ž
2
111 โ„Ž0 + โ„Ž1 + โ„Ž2 + โ„Ž3
2
001
โˆ’
โ„Ž0 + โ„Ž1 + โ„Ž2 โˆ’ โ„Ž
2
110 โ„Ž0 + โ„Ž1 + โ„Ž2 โˆ’ โ„Ž3
2
010
โˆ’
โ„Ž0 + โ„Ž1 โˆ’ โ„Ž2 + โ„Ž
2
101 โ„Ž0 + โ„Ž1 โˆ’ โ„Ž2 + โ„Ž3
2
011
โˆ’
โ„Ž0 + โ„Ž1 โˆ’ โ„Ž2 โˆ’ โ„Ž
2
100 โ„Ž0 + โ„Ž1 โˆ’ โ„Ž2 โˆ’ โ„Ž3
2
100
โˆ’
โ„Ž0 โˆ’ โ„Ž1 + โ„Ž2 + โ„Ž
2
011 โ„Ž0 โˆ’ โ„Ž1 + โ„Ž2 + โ„Ž3
2
101
โˆ’
โ„Ž0 โˆ’ โ„Ž1 + โ„Ž2 โˆ’ โ„Ž
2
010 โ„Ž0 โˆ’ โ„Ž1 + โ„Ž2 โˆ’ โ„Ž3
2
110
โˆ’
โ„Ž0 โˆ’ โ„Ž1 โˆ’ โ„Ž2 + โ„Ž
2
001 โ„Ž0 โˆ’ โ„Ž1 โˆ’ โ„Ž2 + โ„Ž3
2
111
โˆ’
โ„Ž0 โˆ’ โ„Ž1 โˆ’ โ„Ž2 โˆ’ โ„Ž
2
000 โ„Ž0 โˆ’ โ„Ž1 โˆ’ โ„Ž2 โˆ’ โ„Ž3
2
22 Jinalkumari K. Dhobi, Dr. Y. B. Shukla, Dr. K.R.Bhatt
www.ijorcs.org
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 is2๐‘›๐‘›
โˆ’
(๐‘›๐‘› + 1).
For example: A 3 bit binary representation can
form four BCSs.
[0 1 1] = ๐‘ฅ๐‘ฅ2 = 2โˆ’1
๐‘ฅ๐‘ฅ + 2โˆ’2
๐‘ฅ๐‘ฅ
[1 0 1] = ๐‘ฅ๐‘ฅ3 = ๐‘ฅ๐‘ฅ + 2โˆ’2
๐‘ฅ๐‘ฅ
[1 1 0] = ๐‘ฅ๐‘ฅ4 = ๐‘ฅ๐‘ฅ + 2โˆ’1
๐‘ฅ๐‘ฅ
[1 1 1] = ๐‘ฅ๐‘ฅ5 = ๐‘ฅ๐‘ฅ + 2โˆ’1
๐‘ฅ๐‘ฅ + 2โˆ’2
๐‘ฅ๐‘ฅ
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
๐‘ฅ๐‘ฅ2 can be obtained by right shift operation without
using any extra adders and also ๐‘ฅ๐‘ฅ5 can be obtained
from ๐‘ฅ๐‘ฅ4 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 integerโ€™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 filterโ€™s
coefficient is extended the algorithm to SOPOT with
three terms.
Figure 2. Architecture unit of FIR filter with SOPOT type
coefficient
๐ป๐ป[๐‘—๐‘—] = โˆ‘ ๐‘Ž๐‘Ž๐‘–๐‘–,๐‘—๐‘— . 2๐‘๐‘๐‘–๐‘–,๐‘—๐‘—2
๐‘–๐‘–=0 โ€ฆ. (13)
Where ๐‘Ž๐‘Ž๐‘–๐‘–,๐‘—๐‘— = {โˆ’1,1} and๐‘๐‘๐‘–๐‘–,๐‘—๐‘— = {โˆ’๐‘ก๐‘ก, โ€ฆ ,0, โ€ฆ , ๐‘ข๐‘ข}; 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 ๐‘Ž๐‘Ž๐‘–๐‘–,๐‘˜๐‘˜ and ๐‘๐‘๐‘–๐‘–,๐‘˜๐‘˜ terms calculation algorithm
FPGA Implementation of Fir Filter using Various Algorithms: A Retrospective 23
www.ijorcs.org
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
[1] Steven W. Smith, โ€œThe Scientist and Engineerโ€™s Guide
to Digital Signal Processingโ€. California Technical
Publishing San Diego, California, Second Edition 1999.
[2] Wang, Wei, Swamy, M.N.S., Ahmad, M.O., "Low
power FIR filter FPGA implementation based on
distributed arithmetic and residue number
system" Circuits and Systems, 2001. MWSCAS 2001.
Proceedings of the 44th IEEE 2001 Midwest
Symposium on, vol.1, no., pp.102, 105 vol.1, 2001. doi:
10.1109/MWSCAS.2001.986125
[3] Saurabh Singh Rajput, Dr.S.S.Bhadauria,
โ€œImplementation of Fir Filter Using Efficient Window
Function and Its Application in Filtering a Speech
Signalโ€ International Journal of Electronics and
Mechanical Controls, Volume 1 Issue 1 November
2012.
[4] Sen M. Kuo, Bob. H. Lee and Wenshun Tian, โ€œReal-
Time Digital Signal Processing Implementations and
Applicationsโ€ John Wiley & Sons, Ltd, England,
Second Edition 2006.
[5] Sonika Gupta, Aman Panghal, โ€œPerformance Analysis
of FIR Filter Design by Using Rectangular, Hanning
and Hamming Windows Method,โ€ International
Journal of Advanced Research in Computer Science
and Software Engineering, Volume 2, Issue 6, June
2012.
[6] Gao Jinding, Hou Yubao, Su Long, "Design and FPGA
Implementation of Linear FIR Low-pass Filter Based
on Kaiser Window Function", International Conference
on Intelligent Computation Technology and Automation
(ICICTA), vol.2, no., pp.496-498, 28-29 March 2011.
doi: 10.1109/ICICTA.2011.408
[7] Croisier, D.J.Esteban, M.E.Levilion, V.Rizo, "Digital
Filter for PCM Encoded Signals" U.S.Patent, No.3, 777,
130, 1973.
[8] A. Peled and B. Liu, โ€œA new hardware realization of
digital filtersโ€ IEEE Transactions on A.S.S.P., vol.
ASSP-22, pp. 456โ€“462, December 1974.
[9] Bo Hong, Haibin Yin, Xiumin Wang, Ying Xiao,
"Implementation of FIR filter on FPGA using DA-OBC
algorithm", 2nd International Conference on
Information Science and Engineering (ICISE), vol., no.,
pp.3761,3764, 4-6 Dec. 2010.
[10] M. Thenmozhi, N. Kirthika, "Analysis of Efficient
Architectures for FIR filters using Common
Subexpression Elimination Algorithm,โ€ International
Journal of Scientific & Technology Research, Volume
1, Issue 4, May 2012.
[11] Damian, C. Lunca, E., "A low area FIR filter for FPGA
implementation", 34th International Conference on
Telecommunications and Signal Processing (TSP),
pp.521-524, 18-20 Aug. 2011. doi: 10.1109/TSP.
2011.6043675
[12] Heejong Yoo, David V.anderson, "Hardware-efficient
distributed arithmetic architecture for high-order digital
filters" in proc. IEEE International conference on
Acoustics, speech, and signal processing
(ICASSPโ€™05),vol.5, pp.125-128 2005. doi:
10.1109/ICASSP.2005.1416256
[13] Mahesh, R. Vinod, A.P., "A New Common
Subexpression Elimination Algorithm for Realizing
Low-Complexity Higher Order Digital Filters", IEEE
Transactions on Computer-Aided Design of Integrated
Circuits and Systems , vol.27, no.2, pp.217,229, Feb.
2008. doi: 10.1109/TCAD.2007.907064
[14] John G. Proakis., Dimitris G. Manokalis, โ€œDigital
Signal Processing Principle algorithms and
24 Jinalkumari K. Dhobi, Dr. Y. B. Shukla, Dr. K.R.Bhatt
www.ijorcs.org
applicationsโ€ PHI publication, New Jersey, Third
Edition 2004.
[15] Catalin Damian, Cristian Zet, Cristian Fosalau, Mihai
Cretu, โ€œReal, Reactive and Apparent Power Computing
Using FPGA and PWM Intermediary Conversionโ€ XX
IMEKO TC4, Florence, Italy, Sept. 2008.
[16] K. S. Yeung, S. C. Chan, โ€œMultiplier-Less Digital
Filters Using Programmable Sum-Of-Power-Of-Two
(Sopot) Coefficientsโ€, IEEE International Conference
on Field Programmable Technology, pp.78 โ€“ 84, Dec.
2002. doi: 10.1109/FPT.2002.1188667
How to cite
Jinalkumari K. Dhobi, Dr. Y. B. Shukla, Dr. K.R.Bhatt, โ€œFPGA Implementation of Fir Filter using Various
Algorithms: A Retrospectiveโ€. International Journal of Research in Computer Science, 4 (2): pp. 19-24, March
2014. doi: 10.7815/ijorcs.42.2014.081

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FPGA Implementation of FIR Filter using Various Algorithms: A Retrospective

  • 1. 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 ๐ป๐ป๐‘‘๐‘‘ ๏ฟฝ๐‘’๐‘’๐‘—๐‘— ๐‘—๐‘— ๏ฟฝ of the specified filter then Compute the inverse DTFT of ๐ป๐ป๐‘‘๐‘‘ ๏ฟฝ๐‘’๐‘’๐‘—๐‘— ๐‘—๐‘— ๏ฟฝ i.e., โ„Ž๐‘‘๐‘‘ (๐‘›๐‘›) which is infinite in duration after that Choose an appropriate window function ๐‘ค๐‘ค(๐‘›๐‘›) and calculate the impulse response โ„Ž(๐‘›๐‘›) of specified filter as โ„Ž(๐‘›๐‘›) = โ„Ž๐‘‘๐‘‘ (๐‘›๐‘›) โˆ— ๐‘ค๐‘ค(๐‘›๐‘›) to truncated at some point n=M-1. Once โ„Ž(๐‘›๐‘›) is determined, itโ€™s DTFT ๐ป๐ป๏ฟฝ๐‘’๐‘’๐‘—๐‘— ๐‘—๐‘— ๏ฟฝ and Z- transform ๐ป๐ป(๐‘ง๐‘ง) can be calculated for any further analysis. For truncation in third step of โ„Ž๐‘‘๐‘‘ (๐‘›๐‘›) to M- terms, direct truncation method using rectangular window ๐‘ค๐‘ค(๐‘›๐‘›) is multiplied with โ„Ž๐‘‘๐‘‘ (๐‘›๐‘›) 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: ๐‘ค๐‘ค(๐‘›๐‘›) = โŽฉ โŽช โŽจ โŽช โŽง 2(๐‘›๐‘› + 1) ๐‘๐‘ + 1 , ๐‘›๐‘› = 0 ๐‘ก๐‘ก๐‘ก๐‘ก (๐‘๐‘ โˆ’ 1)/2 2 โˆ’ 2(๐‘›๐‘› + 1) ๐‘๐‘ + 1 , ๐‘›๐‘› = (๐‘๐‘ โˆ’ 1)/2 ๐‘ก๐‘ก๐‘ก๐‘ก (๐‘๐‘ โˆ’ 1) 0 ๐‘‚๐‘‚๐‘‚๐‘‚โ„Ž๐‘’๐‘’๐‘’๐‘’๐‘’๐‘’๐‘’๐‘’๐‘’๐‘’๐‘’๐‘’ โ€ฆ . (1) 2. Generalized cosine windows:
  • 2. 20 Jinalkumari K. Dhobi, Dr. Y. B. Shukla, Dr. K.R.Bhatt www.ijorcs.org (Rectangular, Hanning, Hamming and Blackman) ๐‘ค๐‘ค(๐‘›๐‘›) = ๐‘Ž๐‘Ž โˆ’ ๐‘๐‘๐‘๐‘๐‘๐‘๐‘๐‘ ๏ฟฝ 2๐‘๐‘(๐‘›๐‘› + 1) ๐‘๐‘ + 1 ๏ฟฝ + ๐‘๐‘๐‘๐‘๐‘๐‘๐‘๐‘ ๏ฟฝ 4๐‘๐‘(๐‘›๐‘› + 1) ๐‘๐‘ + 1 ๏ฟฝ Where n = 0 to N-1 โ€ฆ. (2) 3. Kaiser window with parameter ฮฒ: ๐‘ค๐‘ค(๐‘›๐‘›) = ๐ผ๐ผ0(๐›ฝ๐›ฝ๏ฟฝ1 โˆ’ ๏ฟฝ1 โˆ’ 2(๐‘›๐‘›) ๐‘๐‘โˆ’1 ๏ฟฝ 2 ๐ผ๐ผ0(๐›ฝ๐›ฝ) Where n= 0 to N-1 โ€ฆ. (3) ๐‘€๐‘€ = ฮด2 โˆ’ 7.95 2.286โˆ†ฯ‰ โ€ฆ. (4) ฮฒ= โŽฉ โŽจ โŽง 0.1102(ฮด2 โˆ’ 8.7), ฮด2 โ‰ฅ 50๐‘‘๐‘‘๐‘‘๐‘‘ 0.5842(ฮด2 โˆ’ 21)0.4 + 0.07886(ฮด2 โˆ’ 21), 21๐‘‘๐‘‘๐‘‘๐‘‘ < ฮด2 < 50๐‘‘๐‘‘๐‘‘๐‘‘ 0, ฮด2 โ‰ค 21๐‘‘๐‘‘๐‘‘๐‘‘ .... (5) In Rectangular window, due to the direct truncation of โ„Ž๐‘‘๐‘‘ (๐‘›๐‘›) 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 ฮฒ 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 โˆ†ฯ‰(rad) and the stopband attenuation ฮด2 = โˆ’20๐‘™๐‘™๐‘™๐‘™ ๐‘™๐‘™10 ๐›ผ๐›ผ2(๐‘‘๐‘‘๐‘‘๐‘‘) were given, the Kaiser FIR filter taps M and the shape parameter ฮฒ 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) ๐‘ฆ๐‘ฆ(๐‘›๐‘›) = โˆ‘ โ„Ž๐‘˜๐‘˜ ๐‘ฅ๐‘ฅ๐‘˜๐‘˜(๐‘›๐‘›)๐‘๐‘โˆ’1 ๐‘˜๐‘˜=0 โ€ฆ. (6) Where โ„Ž๐‘˜๐‘˜the set of constant coefficient of the filter is, ๐‘ฆ๐‘ฆ(๐‘›๐‘›) is the output data, ๐‘ฅ๐‘ฅ๐‘˜๐‘˜(๐‘›๐‘›) is the input data, which can be expressed as Eq. (7) using m-bit 2โ€™s complementary binary number. ๐‘ฅ๐‘ฅ๐‘˜๐‘˜ = โˆ’๐‘ฅ๐‘ฅ๐‘˜๐‘˜,๐‘š๐‘šโˆ’1 + โˆ‘ ๐‘ฅ๐‘ฅ๐‘˜๐‘˜,๐‘š๐‘šโˆ’1โˆ’๐‘—๐‘— ๐‘š๐‘šโˆ’1 ๐‘—๐‘—=1 2โˆ’๐‘—๐‘— โ€ฆ. (7) Where ๐‘ฅ๐‘ฅ๐‘˜๐‘˜,๐‘š๐‘šโˆ’1 is the most significant bit of ๐‘ฅ๐‘ฅ๐‘˜๐‘˜, Eq. (8) can be derived when ๐‘ฅ๐‘ฅ๐‘˜๐‘˜ of Eq. (7) is substitute into Eq. (6). ๐‘ฆ๐‘ฆ(๐‘›๐‘›) = ๏ฟฝ โ„Ž๐‘˜๐‘˜[โˆ’๐‘ฅ๐‘ฅ๐‘˜๐‘˜,๐‘š๐‘šโˆ’1 + ๏ฟฝ ๐‘ฅ๐‘ฅ๐‘˜๐‘˜,๐‘š๐‘šโˆ’1โˆ’๐‘—๐‘— ๐‘š๐‘šโˆ’1 ๐‘—๐‘— =1 2โˆ’๐‘—๐‘— ] ๐‘๐‘โˆ’1 ๐‘˜๐‘˜=0 = ๏ฟฝ ๏ฟฝ โ„Ž๐‘˜๐‘˜ ๐‘š๐‘šโˆ’1 ๐‘—๐‘— =1 ๐‘ฅ๐‘ฅ๐‘˜๐‘˜,๐‘š๐‘šโˆ’1โˆ’๐‘—๐‘— 2โˆ’๐‘—๐‘— โˆ’ ๐‘๐‘โˆ’1 ๐‘˜๐‘˜=0 ๏ฟฝ โ„Ž๐‘˜๐‘˜ ๐‘ฅ๐‘ฅ๐‘˜๐‘˜,๐‘š๐‘šโˆ’1 ๐‘๐‘โˆ’1 ๐‘˜๐‘˜=0
  • 3. FPGA Implementation of Fir Filter using Various Algorithms: A Retrospective 21 www.ijorcs.org = ๏ฟฝ ๏ฟฝ โ„Ž๐‘˜๐‘˜ ๐‘๐‘โˆ’1 ๐‘˜๐‘˜=0 ๐‘ฅ๐‘ฅ๐‘˜๐‘˜,๐‘š๐‘šโˆ’1โˆ’๐‘—๐‘— 2โˆ’๐‘—๐‘— โˆ’ ๐‘š๐‘šโˆ’1 ๐‘—๐‘—=1 ๏ฟฝ โ„Ž๐‘˜๐‘˜ ๐‘ฅ๐‘ฅ๐‘˜๐‘˜,๐‘š๐‘šโˆ’1 ๐‘๐‘โˆ’1 ๐‘˜๐‘˜=0 โ€ฆ. (8) In Eq. (8) the calculation result of โˆ‘ โ„Ž๐‘˜๐‘˜ ๐‘๐‘โˆ’1 ๐‘˜๐‘˜=0 ๐‘ฅ๐‘ฅ๐‘˜๐‘˜,๐‘š๐‘šโˆ’1โˆ’๐‘—๐‘— have 2๐‘๐‘ kinds of different results because of the value of ๐‘ฅ๐‘ฅ๐‘˜๐‘˜,๐‘š๐‘šโˆ’1โˆ’๐‘—๐‘— is 0 or 1. All the possible results of โˆ‘ โ„Ž๐‘˜๐‘˜ ๐‘๐‘โˆ’1 ๐‘˜๐‘˜=0 ๐‘ฅ๐‘ฅ๐‘˜๐‘˜,๐‘š๐‘šโˆ’1โˆ’๐‘—๐‘— are constructed in advance and stored into one LUT. The contents of LUT can be fetch by using ๐‘ฅ๐‘ฅ๐‘˜๐‘˜,๐‘š๐‘šโˆ’1โˆ’๐‘—๐‘— obtain from shift register unit as a LUT address signal. So the value of ๐‘ฆ๐‘ฆ(๐‘›๐‘›) calculated by shifting and accumulating operation. Table 1: LUT contents of DA for 4-tap FIR filter Address signal (๐‘๐‘3 ๐‘๐‘2 ๐‘๐‘1 ๐‘๐‘0) LUT content 0000 0 0001 โ„Ž0 0010 โ„Ž1 0011 โ„Ž1 + โ„Ž0 0100 โ„Ž2 0101 โ„Ž2 + โ„Ž0 0110 โ„Ž2 + โ„Ž1 0111 โ„Ž2 + โ„Ž1 + โ„Ž0 1000 โ„Ž3 1001 โ„Ž3 + โ„Ž0 1010 โ„Ž3 + โ„Ž1 1011 โ„Ž3 + โ„Ž1 + โ„Ž0 1100 โ„Ž3 + โ„Ž2 1101 โ„Ž3 + โ„Ž2 + โ„Ž0 1110 โ„Ž3 + โ„Ž2 + โ„Ž1 1111 โ„Ž3 + โ„Ž2 + โ„Ž1 + โ„Ž0 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]: ๐‘ฅ๐‘ฅ๐‘˜๐‘˜ can also be expressed as Eq. (9) ๐‘ฅ๐‘ฅ๐‘˜๐‘˜ = 1 2 [๐‘ฅ๐‘ฅ๐‘˜๐‘˜ โˆ’ (โˆ’๐‘ฅ๐‘ฅ๐‘˜๐‘˜)] = 1 2 ๏ฟฝโˆ’๏ฟฝ๐‘ฅ๐‘ฅ๐‘˜๐‘˜,๐‘š๐‘šโˆ’1 โˆ’ ๐‘ฅ๐‘ฅ๐‘˜๐‘˜,๐‘š๐‘šโˆ’1๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ + ๏ฟฝ [๏ฟฝ๐‘ฅ๐‘ฅ๐‘˜๐‘˜,๐‘š๐‘šโˆ’1โˆ’๐‘—๐‘— โˆ’ ๐‘ฅ๐‘ฅ๐‘˜๐‘˜,๐‘š๐‘šโˆ’1โˆ’๐‘—๐‘—๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ2โˆ’๐‘—๐‘— ๐‘š๐‘šโˆ’1 ๐‘—๐‘—=1 โˆ’ 2โˆ’(๐‘š๐‘šโˆ’1) ] โ€ฆ. (9) They define ๐‘‘๐‘‘๐‘˜๐‘˜,๐‘—๐‘— as follows, the value of ๐‘‘๐‘‘๐‘˜๐‘˜,๐‘—๐‘— is -1 or +1. ๐‘‘๐‘‘๐‘˜๐‘˜,๐‘—๐‘— = ๏ฟฝ ๏ฟฝ๐‘ฅ๐‘ฅ๐‘˜๐‘˜,๐‘—๐‘— โˆ’ ๐‘ฅ๐‘ฅ๐‘˜๐‘˜,๐‘—๐‘—๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ, ๐‘—๐‘— โ‰  ๐‘š๐‘š โˆ’ 1 โˆ’๏ฟฝ๐‘ฅ๐‘ฅ๐‘˜๐‘˜,๐‘š๐‘šโˆ’1 โˆ’ ๐‘ฅ๐‘ฅ๐‘˜๐‘˜,๐‘š๐‘šโˆ’1๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ, ๐‘—๐‘— = ๐‘š๐‘š โˆ’ 1 โ€ฆ. (10) Eq. (9) can be simplified as Eq. (11). ๐‘ฅ๐‘ฅ๐‘˜๐‘˜ = 1 2 [๏ฟฝ ๐‘‘๐‘‘๐‘˜๐‘˜,๐‘š๐‘šโˆ’1โˆ’๐‘—๐‘— 2โˆ’๐‘—๐‘— โˆ’ 2โˆ’(๐‘š๐‘šโˆ’1) ] ๐‘š๐‘šโˆ’1 ๐‘—๐‘—=0 โ€ฆ. (11) Put Eq. (11) in to Eq. (6), and get = ๏ฟฝ ๏ฟฝ๏ฟฝ 1 2 โ„Ž๐‘˜๐‘˜ ๐‘๐‘โˆ’1 ๐‘˜๐‘˜=0 ๐‘‘๐‘‘๐‘˜๐‘˜,๐‘š๐‘šโˆ’1โˆ’๐‘—๐‘— ๏ฟฝ 2โˆ’๐‘—๐‘— โˆ’ ๐‘š๐‘šโˆ’1 ๐‘—๐‘— =0 ( 1 2 ๏ฟฝ โ„Ž๐‘—๐‘— ๐‘๐‘โˆ’1 ๐‘—๐‘— =0 )2โˆ’๐‘š๐‘šโˆ’1 โ€ฆ. (12) For convenience, ๐ท๐ท๐‘—๐‘— = ๏ฟฝ 1 2 โ„Ž๐‘˜๐‘˜ ๐‘๐‘โˆ’1 ๐‘˜๐‘˜=0 ๐‘‘๐‘‘๐‘˜๐‘˜,๐‘š๐‘šโˆ’1โˆ’๐‘—๐‘— , ๐ท๐ท๐‘’๐‘’๐‘’๐‘’ = โˆ’ 1 2 ๏ฟฝ โ„Ž๐‘—๐‘— ๐‘๐‘โˆ’1 ๐‘—๐‘—=0 โ€ฆ. (13) The value of ๐ท๐ท๐‘—๐‘— have 2๐‘๐‘ 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 1โ€™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 ๐‘ฅ๐‘ฅ0๐‘—๐‘— = 0 ๐‘ฅ๐‘ฅ1๐‘—๐‘— ๐‘ฅ๐‘ฅ2๐‘—๐‘— ๐‘ฅ๐‘ฅ3๐‘—๐‘— LUT Contents Address ๐‘ฅ๐‘ฅ0๐‘—๐‘— = 1 ๐‘ฅ๐‘ฅ1๐‘—๐‘— ๐‘ฅ๐‘ฅ2๐‘—๐‘— ๐‘ฅ๐‘ฅ3๐‘—๐‘— LUT Contents 000 โˆ’ โ„Ž0 + โ„Ž1 + โ„Ž2 + โ„Ž 2 111 โ„Ž0 + โ„Ž1 + โ„Ž2 + โ„Ž3 2 001 โˆ’ โ„Ž0 + โ„Ž1 + โ„Ž2 โˆ’ โ„Ž 2 110 โ„Ž0 + โ„Ž1 + โ„Ž2 โˆ’ โ„Ž3 2 010 โˆ’ โ„Ž0 + โ„Ž1 โˆ’ โ„Ž2 + โ„Ž 2 101 โ„Ž0 + โ„Ž1 โˆ’ โ„Ž2 + โ„Ž3 2 011 โˆ’ โ„Ž0 + โ„Ž1 โˆ’ โ„Ž2 โˆ’ โ„Ž 2 100 โ„Ž0 + โ„Ž1 โˆ’ โ„Ž2 โˆ’ โ„Ž3 2 100 โˆ’ โ„Ž0 โˆ’ โ„Ž1 + โ„Ž2 + โ„Ž 2 011 โ„Ž0 โˆ’ โ„Ž1 + โ„Ž2 + โ„Ž3 2 101 โˆ’ โ„Ž0 โˆ’ โ„Ž1 + โ„Ž2 โˆ’ โ„Ž 2 010 โ„Ž0 โˆ’ โ„Ž1 + โ„Ž2 โˆ’ โ„Ž3 2 110 โˆ’ โ„Ž0 โˆ’ โ„Ž1 โˆ’ โ„Ž2 + โ„Ž 2 001 โ„Ž0 โˆ’ โ„Ž1 โˆ’ โ„Ž2 + โ„Ž3 2 111 โˆ’ โ„Ž0 โˆ’ โ„Ž1 โˆ’ โ„Ž2 โˆ’ โ„Ž 2 000 โ„Ž0 โˆ’ โ„Ž1 โˆ’ โ„Ž2 โˆ’ โ„Ž3 2
  • 4. 22 Jinalkumari K. Dhobi, Dr. Y. B. Shukla, Dr. K.R.Bhatt www.ijorcs.org 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 is2๐‘›๐‘› โˆ’ (๐‘›๐‘› + 1). For example: A 3 bit binary representation can form four BCSs. [0 1 1] = ๐‘ฅ๐‘ฅ2 = 2โˆ’1 ๐‘ฅ๐‘ฅ + 2โˆ’2 ๐‘ฅ๐‘ฅ [1 0 1] = ๐‘ฅ๐‘ฅ3 = ๐‘ฅ๐‘ฅ + 2โˆ’2 ๐‘ฅ๐‘ฅ [1 1 0] = ๐‘ฅ๐‘ฅ4 = ๐‘ฅ๐‘ฅ + 2โˆ’1 ๐‘ฅ๐‘ฅ [1 1 1] = ๐‘ฅ๐‘ฅ5 = ๐‘ฅ๐‘ฅ + 2โˆ’1 ๐‘ฅ๐‘ฅ + 2โˆ’2 ๐‘ฅ๐‘ฅ 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 ๐‘ฅ๐‘ฅ2 can be obtained by right shift operation without using any extra adders and also ๐‘ฅ๐‘ฅ5 can be obtained from ๐‘ฅ๐‘ฅ4 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 integerโ€™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 filterโ€™s coefficient is extended the algorithm to SOPOT with three terms. Figure 2. Architecture unit of FIR filter with SOPOT type coefficient ๐ป๐ป[๐‘—๐‘—] = โˆ‘ ๐‘Ž๐‘Ž๐‘–๐‘–,๐‘—๐‘— . 2๐‘๐‘๐‘–๐‘–,๐‘—๐‘—2 ๐‘–๐‘–=0 โ€ฆ. (13) Where ๐‘Ž๐‘Ž๐‘–๐‘–,๐‘—๐‘— = {โˆ’1,1} and๐‘๐‘๐‘–๐‘–,๐‘—๐‘— = {โˆ’๐‘ก๐‘ก, โ€ฆ ,0, โ€ฆ , ๐‘ข๐‘ข}; 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 ๐‘Ž๐‘Ž๐‘–๐‘–,๐‘˜๐‘˜ and ๐‘๐‘๐‘–๐‘–,๐‘˜๐‘˜ terms calculation algorithm
  • 5. FPGA Implementation of Fir Filter using Various Algorithms: A Retrospective 23 www.ijorcs.org 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 [1] Steven W. Smith, โ€œThe Scientist and Engineerโ€™s Guide to Digital Signal Processingโ€. California Technical Publishing San Diego, California, Second Edition 1999. [2] Wang, Wei, Swamy, M.N.S., Ahmad, M.O., "Low power FIR filter FPGA implementation based on distributed arithmetic and residue number system" Circuits and Systems, 2001. MWSCAS 2001. Proceedings of the 44th IEEE 2001 Midwest Symposium on, vol.1, no., pp.102, 105 vol.1, 2001. doi: 10.1109/MWSCAS.2001.986125 [3] Saurabh Singh Rajput, Dr.S.S.Bhadauria, โ€œImplementation of Fir Filter Using Efficient Window Function and Its Application in Filtering a Speech Signalโ€ International Journal of Electronics and Mechanical Controls, Volume 1 Issue 1 November 2012. [4] Sen M. Kuo, Bob. H. Lee and Wenshun Tian, โ€œReal- Time Digital Signal Processing Implementations and Applicationsโ€ John Wiley & Sons, Ltd, England, Second Edition 2006. [5] Sonika Gupta, Aman Panghal, โ€œPerformance Analysis of FIR Filter Design by Using Rectangular, Hanning and Hamming Windows Method,โ€ International Journal of Advanced Research in Computer Science and Software Engineering, Volume 2, Issue 6, June 2012. [6] Gao Jinding, Hou Yubao, Su Long, "Design and FPGA Implementation of Linear FIR Low-pass Filter Based on Kaiser Window Function", International Conference on Intelligent Computation Technology and Automation (ICICTA), vol.2, no., pp.496-498, 28-29 March 2011. doi: 10.1109/ICICTA.2011.408 [7] Croisier, D.J.Esteban, M.E.Levilion, V.Rizo, "Digital Filter for PCM Encoded Signals" U.S.Patent, No.3, 777, 130, 1973. [8] A. Peled and B. Liu, โ€œA new hardware realization of digital filtersโ€ IEEE Transactions on A.S.S.P., vol. ASSP-22, pp. 456โ€“462, December 1974. 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