 Open Access
 Total Downloads : 19
 Authors : Kiran K, Dr. Radhakrishna Rao K A
 Paper ID : IJERTCONV2IS13015
 Volume & Issue : NCRTS – 2014 (Volume 2 – Issue 13)
 Published (First Online): 30072018
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
FPGA implementation of InterpolationBased Chase BCH Decoding
Kiran k1
1 Final year M.Tech student, E&C Engg. Dept., PES College of Engineering, Mandya kiran.kkiru@gmail.com
Dr. Radhakrishna Rao K A PhD 2 2 Professor, E&C Engg. Dept.,
PES College of Engineering, Mandya karkrao@yahoo.com
Abstract BCH codes are is widely used in digital telecommunication systems, with satellite communications and data recording systems such as CD and DVD. Compared with harddecision decoding of BCH codes, the soft decision Chase algorithm can achieve significant coding gain by carrying out decoding trials on 2 test vectors. Earlier the one pass Chase schemes find the error locators based on the Berlekamps algorithm and need hardwaredemanding selection methods to decide which locator corresponds to the correct code word.
In this paper, an interpolationbased onepass Chase decoder is
proposed for BCH codes. By using of the binary property of BCH codes, a lowcomplexity method is developed to select the interpolation output leading to successful decoding. The code word recovery step is also significantly simplified. From architectural analysis, the proposed decoder with = 4 for a (4200, 4096) BCH code has higher efficiency in terms of throughputoverarea ratio than the prior onepass Chase decoder based on the Berlekamps algorithm
Index Terms BCH codes, Chase decoding, interpolation.

INTRODUCTION
BCH CODES can be found in many applications, including flash memory, optical transport network, and digital video broadcasting. For a BCH code with minimum distance dmin, traditional harddecision decoding (HDD) algorithms, such as the Berlekamps algorithm, can correct t = dmin/2 errors. Through flipping the n least reliable bits and trying 2 test vectors, the softdecision Chase algorithm can correct up
t + errors. It also has better errorcorrecting performance than the generalized minimum distance (GMD) decoder and the soft decision decoder in [1], which assumes that all but one error are located in the 2t least reliable bits.
To reduce the complexity of the Chase BCH decoding, one pass schemes have been proposed to derive the error locators for all test vectors in one run based on the Berlekamps algorithm [2], [3]. Further simplifications and implementation architectures of this scheme were developed in [4]. It was observed that selecting the error locator corresponding to the correct code word requires expensive parallel Chien search and accounts for a majority part of the overall decoder.
This brief proposes a novel interpolationbased Chase BCH decoder. By making use of the binary property of BCH codes, substantial modifications and simplifications are developed in this brief. In particular, instead of employing expensive parallel Chien search, the selection of the interpolation output
leading to successful decoding is achieved by simple evaluation value computation without any errorcorrecting performance or code rate loss.
In addition, the recovery of each code word bit is done through testing the evaluation values of two lowdegree polynomials. From architectural analysis, the proposed decoder with = 4 for an example (4200, 4096) BCH code has 2.3 times higher efficiency in terms of throughputover area ratio than the Chase decoder based on the Berlekamps algorithm [4], while achieving the same errorcorrecting performance.
The structure of this brief is as follows. Section II introduces the Chase decoding. The interpolationbased Chase BCH decoding and its implementation architectures are proposed in Sections III and IV, respectively. Section V provides the hardware complexity analysis, and Simulation results and conclusions are drawn in Section VI and VII respectively.

CHASE DECODING
A terrorcorrecting (n, k) BCH code over finite field GF (2p) (p Z+) is considered in this brief. In the Chase algorithm, 2 test vectors are formed by flipping each of the least reliable bits in the received word, and decoding is done for each vector. As a result, up to t + errors can be corrected. It can be observed, that the Chase decoding can significantly outperform both the GMD and softdecision decoding in [1]. Carrying out the decoding for each test vector separately in the Chase algorithm leads to high hardware complexity.
Instead, the error locator polynomials of all test vectors can be
derived using the onepass schemes in [2] and [3] for BCH decoding. These schemes find the error locator for the first test vector using the Berlekamps algorithm. Then, the error locator for each successive vector is computed by updating the polynomials derived from the Berlekamps algorithm. To decide which error locator corresponds to the correct code word, the Chien search is employed previously to select the one whose root number equals its degree.
Binary BCH codes can be considered as subfield subcodes of RS codes. Interpolationbased onepass Chase RS decoders have been developed. For an (n, ) RS code, the symbols of a codeword c can be considered as the evaluation values of a degree 1 message polynomial, f(x), over n distinct nonzero finite field elements 0, 1. . . n1.
Assume that the hard decision of the received vector is r. A polynomial Q(x, y) that passes each point, (i, ri), with (1,
1) minimum weighted degree can be found through interpolation. If the number of errors does not exceed
(n )/2, then Q(x, y) would have a factor yf(x), and r is decoded.
The interpolationbased Chase decoder can be implemented according to Fig. 1. First, a code word is derived by applying systematic encoding on the last symbols in r. Then, = r + = (c + ) + e = + e is another code word, , corrupted by the same error vector e, and the decoding is done on instead. This follows the reencoding technique introduced in [5].
The interpolation problem can be solved by the Koetters algorithm [6]. For highrate codes, it starts with a pair of polynomials {1, y} and iteratively forces them to pass one more point each time with minimum increase in the weighted degree. At the end, the polynomial with lower weighted degree is the interpolation output. Denote the set of the last k code positions, also called the systematic positions, by S.
Since = 0 for i S, the interpolation over the points in these positions can be presolved as v(x) =i S (x + i), and this factor is taken out of the interpolation process using a coordinate transformation. As a result, the real interpolation only needs to be done on the remaining n k points. The test vectors in the Chase algorithm can be arranged in a Gray code manner, i.e., adjacent vectors only have a pair of different points (i , ri) and (i, ri).
The unified backwardforward interpolation [7] eliminates (i, ri) from a given interpolation result and adds (i, ri) in one iteration. Accordingly, the interpolation results for all test vectors are computed in one run. Moreover, when a code position in S needs to be flipped, the coordinate transformation and interpolation can be modified using the
An (n, k) tbit errorcorrecting BCH code over GF(2p) is a subfield subcode of an (n, ) tsymbol errorcorrecting RS code over GF(2p). In another word, all the (n, k) BCH code words form a subset of the (n, ) RS code words. n = 2t, and n k is equal to or slightly less than pt.
The interpolation based decoding is developed based on the interpretation that the code word symbols are evaluation values of the message polynomial. BCH codes cannot be enoded this way since the evaluation values of a binary message polynomial over finite field elements are usually not binary. Hence, BCH code words are considered as RS code words in order to apply the interpolation based decoding.
Applying RS systematic reencoding to the last code positions, n points remain to be interpolated for each test vector. The same backwardforward interpolation scheme [7], [8] can be adopted to derive the interpolation results of all vectors in one run. Nevertheless, by making use of the property that r is binary in BCH decoding, substantial simplifications can be made on the polynomial selection and code word recovery steps.
Therefore, there are 2pk code words for the (n, ) RS code over GF (2p). However, only a small proportion, 2k of them, are also code words of the (n, k) binary BCH code. Hence, the chance of returning a binary BCH code word is extremely small if the test vector is undecodable, particularly for long codes. Inspired by this, the interpolation output polynomials can be selected based on whether they will lead to binary code words.
Nevertheless, testing if each symbol in the decoded word is binary requires the code word recovery step to be completed first. Instead, we propose to check only a few symbols that are easy to compute from the interpolation output. Which symbols to test depends on the specifics of the computations used in the decoding.
Adopting the reencoding technique, the decoding is actually carried out on = r + . Then, the returned code word is added up with to compute the code word c. i = ri is binary for i S. In addition, using the code word recovery scheme in [8], i for i S is zero unless the corresponding code position is a root of q1(x). q1(x) has at most t roots and t <<
for highrate codes. Therefore, ci for i S is mostly binary and cannot be used to tell whether the entire c is binary.
methods in [8].
On the other hand,
and i for i are mostly nonbinary
Compared to the schemes based on the Berlekamps algorithm, the interpolationbased scheme leads to more efficient Chase decoders for RS codes. However, the application of this scheme for BCH codes has not been studied. In the next section, an efficient onepass Chase BCH decoder based on interpolation is proposed.

INTERPOLATIONBASED CHASE BCH DECODER
for undecodable cases, and the chance that i + i (i ) is binary is extremely small. Here, denotes the first n k code positions. Accordingly, we propose to select the interpolation output whose corresponding + is binary in the first two symbols. is available, and the two symbols of can be easily computed from the interpolation output.
In our simulations, the test vectors are ordered according to reducing reliability as much as possible, and the first interpolation output passing the test is selected.
It is possible that q1() = 0. In this case, q0() is also zero, and the LHopital rule needs to be applied. It says that, if
d(x) = a(x)/b(x) and a(i) = b(i) = 0 for a certain i, then d(i) = a(x)/b(x)i. The derivative of a polynomial over finite fields of characteristic two is the collection of the odd degree terms.

VLSI ARCHITECTURES
Systematic RS reencoding can be done by linear feedback shift registers (LFSRs), and an efficient parallel architecture is available in [9]. The modified unified backwardforward interpolator in [8] is among the most efficient and is adopted in the proposed decoder.
The proposed polynomial selection tests whether ( 1) + 0 and ( ) + 1 are binary. ( 1) and ( ) can be derived using (1). As aforementioned, v(1) and v() are precomputed and are nonzero and can be computed on the fly using an adder and a multiplierregister loop in at most 2 clock cycles.
The architecture in Fig. 2 tells whether f() + 1 is binary, and ( 1) + 0 can be tested by a similar architecture, in which the constant multipliers in the four feedback loops are replaced by hardwiring since they implement multiplications by one. The unified interpolator in [8] outputs the coefficients of q0(x) and q1(x) serially. q0 () and q1() are computed by the first and third feedback loops in Fig.2 and are available after max(deg(q0(x)), deg(q1(x))) clock cycles.
With the help of the signal a that flips every clock cycle, q 0() and q 1() are computed by the second and fourth loops in Fig. 2, respectively. q' 1() is nonzero since (x + ) can only be a simple factor of q1(x), and the common scaler does not affect the value of f(). Denote the outputs of the two multiplexors in Fig. 2 by qu and ql.
The testing can be done equivalently as whether qu v() + ql sf ()1 equals zero or ql sf () as shown in Fig. 2 to avoid expensive finite field inverters.
The architecture in Fig. 4 implements BCH code word recovery according to (3). The evaluation values of q1(x) and q0(x) over the k code positions in S are computed through the Chien search. Similarly, the polynomial coefficients are divided into even and odd parts so that the evaluation values Of 0(x) are also available. Then, whether i is 1 or 0 is decided based on the evaluation values and if i SF. Such a decision unit can be implemented by very simple hardware. To achieve high speed, the Chien search can be implemented by the parallel architecture in [10].

HARDWARE COMPLEXITY ANALYSIS
Taking a t = 8 (4200, 4096) BCH code over GF(213) as an example, the hardware complexity of the proposed interpolation based Chase BCH decoder with = 4 is analyzed and listed in Table I.
Fig. 4. Table 1.
This code is considered as a subfield sub code of a
(4200, 4184) RS code over GF(213) in the decoding. To increase the throughput, pipelining is applied along the dashed lines in Fig. 1. Each pipelining stage should take similar number of clock cycles to improve the hardware utilization efficiency. This is achieved through adjusting the parallel processing factors used in the functional blocks. The
critical path of each block is designed not to exceed one finite field multiplier, one adder, and one multiplexor.
The unified backwardforward interpolator in [8] for the systematically reencoded decoder processes polynomial coefficients serially.
The number of clock cycles spent for an interpolation iteration is dx + , where dx is the maximum x degree of the polynomials during that iteration and is the pipelining latency of the interpolator.
The worst case interpolation latency happens when all the 2 test vectors need to be interpolated. In this case, dx increases from 1 to (n )/2 gradually during the n forward interpolation iterations for the first test vector and remains at (n )/2 during the backwardforward interpolation iterations for later vectors.
To further increase the interpolation speed, multiple copies of the interpolator can be employed. When two copies are used, each copy needs to take care of eight test vectors. Accordingly, the worst case interpolation latency is ((1 + 3) + (8 + 3)) Ã— 8/2 Ã— 2 + (8 + 5) Ã— 7 + 1 + 2 + 2 + 3 + 3 + 3 + 3 =
228 clock cycles.
To match the speed of the interpolation, the parallel architecture in [9] can be adopted to implement the systematic RS reencoder. In addition, the interpolation points in SF and need to be modified according to the coordinate transformation. The modifications on the points in SF can be done simultaneously as the systematic reencoding using one multiplier and one adder [8]. The reencoding and coordinate transformation are completed in [4184/20] + 16 + 4 = 230 clock cycles.
Each backwardforward interpolation iteration takes as short as 13 clock cycles. The polynomial selection should be done in the same amount of time to avoid slowing down the decoder. max{deg(q1(x)), deg(q0(x))} = (n )/2 = 8 in the backward forward iterations, and the data paths in Fig. 2 are divided into four pipelining tages. Hence, the computation of c1 + 1 can be finished in 12 clock cycles, and another copy of the architecture is used to compute c0 + 0 in parallel.
Considering the latency of the polynomial selection, the proposed code word recovery should be finished in about 22812= 216 clock cycles. Accordingly 4096/216= 19 parallel processing is adopted in the Chien search engines in Fig. 3, and 19 copies of the other units are employed.
A GF (213) multiplier can be implemented by 192 XOR gates and 169 AND gates, and a squarer has 20 XOR gates. On average, a constant multiplier over GF (213) needs 40 XOR gates. An inversion in GF (213) can be completed in 13 clock cycles using a squarermultiplierregister loop.
Each AND or OR gate has 3/4 the area of an XOR, and each multiplexor or storing a bit in memory requires the same area
as an XOR. Also, each register takes about three times the area of an XOR. Using these assumptions, the total equivalent gate counts of the BCH decoders are computed and listed in TableI.
Compared to the Chase BCH decoder based on the Berlekamps algorithm [4], the proposed interpolationbased Chase decoder can achieve around 2.3 times higher efficiency in terms of throughputoverarearatio, while having the same errorcorrecting performance.

SIMULATION RESULTS
The proposed circuit is modeled using verilog language and simulated on Xilinx ISE DESIGN suite 14.3 and implemented on Xilinx Spartan3e FPGA device. Fig.5. shows the simulated waveforms.
Fig. 5. Simulation Result of Decoder

CONCLUSION

This brief developed an efficient interpolationbased one pass Chase BCH decoder. By making use of the binary property of BCH codes, novel polynomial selection and code word recovery schemes were proposed. In particular, the proposed polynomial selection led to significant complexity reduction without sacrificing the errorcorrecting performance. Future work will be directed to further speeding up the interpolation based BCH decoder without incurring large area overhead.
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Web source: the error correcting page

Web source: NPTEL

Error control coding by Shu Lin, Daniel J Costello

Fundamentals of Error correcting codes by Cary Huffman and vera pless

Art of Error correcting codes by Robert, Zarangoza
MANJUNATH K
Department of Electronics and communication Engg.
PESCE, Mandya.
MAHESH GOWDA N M
Assistant Professor,
Department of Electronics and communication Engg.
PESCE, Mandya.
makes real time processing a reality. Further, the butterfly
Abstract Digital Signal Processing Systems have wide
complex multiplications to (N/2) log N2 and additions to N
range of application in the image processing, voice processing , radars and biodata signal compression. Basically these applications are developed using fast fourier and inverse fourier transform concepts. The computational complexity associated with these FFT and IFFT algorithm is very high and also speed is one
of the important factor we should consider while providing the VLSI implementation. This paper present a new algorithm for 2D FFT and IFFT that overcome the irregularities that present in the butterfly structure VLSI implementation. The design is coded and implemented on Spartan 3 FPGA and it is tested for 2D color images.
KeywordsDiscrete Fourier Transform,Fast Fourier Transform ,twiddle factor.
INTRODUCTION
The field of Digital Signal Processing (DSP) has grown enormously in the past decade and is playing a significant role in driving the technology. The DSP applications reach out to our everyday life such as medicine, surveillance, authentication and many more areas. In all these applications, Discrete Fourier has been widely used for efficient implementations. The Fast Fourier Transform is an efficient algorithm to compute the Discrete Fourier Transform (DFT) fast and its inverse. The evaluation of both produces
same results but FFT is much faster. The DFT involves N2 complex multiplications and N(N1) complex additions, where
log2N. Thus there is a large reduction in the calculation which
structure makes hardware realizations difficult since regularity of expressions are broken.
This Paper presents a novel algorithm that can overcome the irregularity encountered in the butterfly structural implementations in VLSI realization. In contrast to the butterfly structure, the proposed algorithm is highly regular and hence suitable for VLSI implementations. The proposed design incorporates a high degree of pipelining and massive parallelism and hence offers high throughputs. Section I presents the basics of Discrete Fourier Transforms.
The regular FFT/IFFT algorithm outlined earlier is proposed in Section II. The quantitative and qualitative analysis of the algorithm is summarized and tabulated in Section III successively.

1D Discrete Fourier Transform Pair
One dimensional signal is described using function that depends on one independent variable with reference to time. Audio signals, radar signals and biomedical signals are examples of one dimensional signal. A transform changes one domain value such as time in to frequency components and makes the signal processing far more easily than the time domain components.
The Discrete Fourier Transform for a 1D signal is given
N is 8 for 8×8 pixels block of an image. As the value of Nby:
increases, the number complex calculations also increase resuIting in high processing time.
A Fast Fourier Algorithm (FFT) was discovered in
1965 by Cooley and Tukey, which reduced the number of calculations drastically and paved the way for real time processing of discrete signals which revolutionized the field of Digital Signal Processing [1]. Most of FFT algorithrns decompose the overall Npoint DFT into successively smaller and smaller expressions popularly known as butterfly structure. The FFT algorithrns achieve its computational efficiency through divide and conquer strategy. The fundamental prnciple of FFT algorithm is that of dividing the computation of DFT sequence of length N into successively smaller DFTs by exploiting the Symmetry property and Periodicity property of DFT. The FFT reduces the number
The Inverse Fourier Transform is given by

2D Discrete Fourier Transform Pair
Two dimensional signals is described using function
that depends on two variables. Images are examples of two dimensional signals. The Discrete Fourier Transform for a 2D signal is given by the expression:
The Inverse Fourier Transform is given by
Fast Fourier transform algorithrns resorts to a variety of tricks for reducing the time required to compute a DFT [2]. Several methods for computing FFT and IFFT are discussed in Ref. [1]. Decimation In Time (DIT) divides a sequence x (n) in to odd and even sequences in successive stages to realize the whole DFT sequence. Likewise, Decimation In Frequency (DIF) divides the number of frequency components X(k) in to odd and even components to get the DFT in frequency domain [3]. FFT helps to transform the signal from time domain to frequency domain, where filtering and correlation can be performed with fewer operations [4].
A Radix sorting is a fast and stable algorithm that sorts keys, say, numerical digits. The processing of keys begins at the Least significant Digit and proceeds to the Most Significant Digit. In DFT applications, the radixn algorithm divides a DFT of size N in to n interleaved DFTs of size N/n with each recursive stages. Radix 2 first divides the DFT of size N in to two interleaved DFTs of size N/2 and then computes the DFT of even indexed inputs X2m and odd indexed input X2m+l. These resuIts are combined to produce the DFT of the sequence [5].
The implementation of Radix 2 algorithm is centered on conjugating the twiddle factors of the corresponding forward FFT. It has single path delay feedback pipelined processor [6]. The Radix 2 algorithrns are simplest FFT algorithms. The Radix 2 DIT FFT can be applied recursively to the two length N/2 DFTs to save computation and it has to be successively applied to reach DFTs of length2 [5]. Floating point arithmetic has been virtually impossible to implement on FPGA based systems due to the inherent complexity of floating point arithmetic. With the introduction of high level languages such as VHDL, rapid prototyping of floating point formats has become possible making such complex structures more feasible to implement [7]. For this a
VHDL library has to be formed which includes addition, subtraction, multiplication and divisionmodules. This situation inevitably utilizes large amount of FPGA resources.
Most of the research to date for the implementation of FFT algorithrns has been performed using general purpose processors, Digital Signal Processors (DSPs) and dedicated FFT Processors [3]. However, as Field Programmable GateArrays (FPGAs) have grown in capacity, improved in performance, and decreased in cost, they have become a viable solution for performing computationally intensive tasks, with the ability to tackle applications for custom chips and programmable DSP devices [8].

Proposed Regular, FFT Algorithm for Image Processing
Twiddle Factor is a key component in FFT/IFFT computation. It is any of the trigonometric constant coefficients that are multiplied by the data in the course of the FFT algorithrn. They are the coefficients used to combine results from a previous stage to form inputs to the next stage. Twiddle factor term was apparently coined by Gentleman and Sande in 1963. It is the root of unity complex multiplication constants in the butterfly operations and also it describes the rotating vector which rotates in increments according to the number of samples. The twiddle factor is expressed as
By Euler's formula, the exponential term can also be written as
The exponential term (j2rrnk/N) of the DFT equation may be written using the Euler's formula:
The RHS of the Euler's equation is expressed as (m*n) matrix forms for various values of (u, x) and (v, y)
as Sine and cosine terms. The matrices and their transposes may be stored as lookup tables on ROMs and accessed block by block for hardware implementation [9].
As we have observed in the FFT equation, it consists of complex additions, complex multiplications and twiddle factors irregularities to compute the FFT. Also in hardware realization, computation speed is of utmost importance and hence the algorithm should be amenable for parallelism and pipelining to speed up the computation.
The irregularities of the twiddle factors in FFT can
also be overcome by the Cosine and Sine transforms of the signal. The FFT for 2D signal is obtained by adding the Sine and Cosine transforms of the input signal. The transforms in this algorithm is obtained in matrix form by varying the values of u and x from 0 to 7. The transposed
Sine and Cosine matrices are obtained by varying the values of v and y from 0 to 7.The required Cosine matrix and Sine matrix for FFT and IFFT are obtained as shown
in Eq. (8) and Eq. (9)
The proposed, regular, 2D FFT can be obtained by subtracting the Cosine and Sine transform as per Eq. 10
Similarly, the IFFT for 2D is computed by subtracting the Cosine and Sine transform of the FFT of the signal and maybe expressed as:

Proposed Regular FFT/IFFT Algorithm
The proposed FFT/IFFT algorithm is also folIows:

Read the input file from the host system.

Evaluate the (8×8) Cosine and Sine matrix using Equations 8 and 9.

The image is accessed as (8×8) block successively, and the Cosine and Sine transforms are obtained.

Compute the FFT of the image using Eq. 10.

Verify the obtained FFT values with buiIt in FFT functional values of MATLAB.

Compute the IFFT of the processed image in step 4 using the Eq. 11.

Verify visually the reconstructed image with the original image.

Calculate the Power Signal to Noise Ratio (PSNR) of the processed signals for validation. Note: A PSNR value of 35 dB and above implies the reconstructed image is indistinguishable from the original image.



Experimental Results and Comparative Study
Due to the importance and use of FFT in many signal and Image processing applications, a novel FFT and
IFFT algorithm has been presented in this paper. This
algorithm exploits the Regularity, Parallelism and Pipelined architecture for the implementation.
The FFT and IFFT are coded in MATLAB in order to establish the correct working of the proposed algorithrn. It also serves as a reference for checking the hardware results after the algorithm is implemented on FPGA.
The algorithm is designed using MATLAB (R2008a)and applied on to various images. MATLAB is considered as industry standard for this field of research [9].
The reconstructed images are of good quality with PSNR value
Table 1. Reconstructed Quality of SampIe Images using the Proposed FFT/IFFT Algorithrns
b
better than 35 dB. Table 3 shows the output quality of the proposed Regular FFT/IFFT algorithm. Figures 1 and 2 show the sample reconstructed images.

CONCLUSION
A new algorithm has been presented in this paper to compute FFT and IFFT of the image signals. The proposed algorithm has been coded in Matlab and verilog. The design is synthesized and implemented on spartan 3 FPGA., and the reconstructed images are good with a PSNR value of 35 dB or above.
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Alan V. Oppenheim, Ronald W. Schafer, John R. Buck,\ Discrete Time Signal Processing, Prentice Hall, Second Edition, pp. 646652, 1999.

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