VHDL code implementation

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CHAPTER 1: INTRODUCTION 1.1 FAST FOURIER TRANSFORM A Fast Fourier Transform (FFT) is an efficient algorithm to compute the Discrete Fourier Fourier Trans Transfor form m (DFT) (DFT) and its its invers inverse. e. There There are many many distin distinct ct FFT algori algorithm thmss involving a wide range of mathematics, from simple complex-numer arithmetic to group theory and numer theory. The fast Fourier Transform is a highly efficient procedure for  computing the DFT of a finite series and re!uires less numer of computations than that of direct evaluation of DFT. "t reduces the computations y ta#ing advantage of the fact that the calculation of the coefficients of the DFT can e carried out iteratively. Due to this, FFT computation techni!ue is used in digital spectral analysis, filter simulation, autocorrelation and pattern recognition. The FFT is ased on decomposition and rea#ing the transform into smaller  transforms and comining them to get the total transform. FFT reduces the computation time re!uired to compute a discrete Fourier transform and improves the performance y a factor of $%% or more over ov er direct evaluation of the DFT. A DFT DFT deco decomp mpos oses es a se!u se!uen ence ce of valu values es into into comp compon onen ents ts of diff differ eren entt fre!uencies. This operation is useful in many fields ut computing it directly from the definition is often too slow to e practical. An FFT is a way to compute the same result more !uic#ly& computing a DFT of N  of N  points   points in the ovious way, using the definition,  ta#es '(   ) arithmetical operations, while an FFT can compute the same result in only '( N   log N ) operations.  N  log N  The difference in speed can e sustantial, especially for long data sets where N  where N  may e in the thousands or millions*in practice, the computation time can e reduced y several several orders orders of magnitude magnitude in such cases, and the improvement improvement is roughly roughly proportiona proportionall to N to N +log ( N   N ). ). This huge improvement made many DFT-ased DFT-ased algorithms practical. FFTs are of great importance importance to a wide variety of applicatio applications, ns, from digital digital signal processing and solving solving partial partial different differential ial e!uations to algorithms algorithms for !uic# multiplic multiplication ation of large large integers. The most well #nown FFT algorithms algorithms depend upon the factoriat factoriation ion of N  of N , ut there are FFT with ' ( N   log log  N ) complexi complexity ty for all  N , even for prime prime  N . any FFT algorithms only depend on the fact that is an N an N th primitive root of unity, and thus can e appli applied ed to analo analogou gouss tran transf sfor orms ms over over any fini finite te fiel field, d, such such as num numer er-t -the heor oret etic ic transforms. The Fast Fourier Transform algorithm exploit the two asic properties of the twiddl twiddlee factor factor - the symmet symmetry ry proper property ty and period periodici icity ty proper property ty which which reduce reducess the numer of complex multiplications re!uired to perform DFT. FFT algori algorithm thmss are ased ased on the fundam fundament ental al princi principle ple of decomp decomposi osing ng the computation of discrete Fourier Transform of a se!uence of length  into successively smaller discrete Fourier transforms. There are asically two classes of FFT algorithms. A) Decimation "n Time (D"T) algorithm /) Decimation "n Fre!uency (D"F) algorithm. "n decimation-in-time, the se!uence for which we need the DFT is successively divided into smaller se!uences and the DFTs of these suse!uences are comined in a certain pattern to otain the re!uired DFT of the entire se!uence. "n the decimation-in-

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fre!uency approach, the fre!uency samples of the DFT are decomposed into smaller and smaller suse!uences in a similar manner. The numer of complex multiplication and addition operations re!uired y the simple forms oth the Discrete Fourier Transform (DFT) and "nverse Discrete Fourier  Transform ("DFT) is of order N  order  N  as there are N are  N data points to calculate, each of which re!uires N re!uires N complex arithmetic operations. The discrete Fourier transform (DFT) is defined y the formula& −  j Π nK   N −$ N  0  X ( K ) = ∑  x(n) • e n =% 1here 2 is an integer ranging from % to N  to N  3  3 $. The algorithmic complexity of DFT will '( N ) ) and hence is not a very efficient method. "f we can4t do any etter than this then the DFT will not e very useful for the ma5ority of practical D67 application. 8owever, there are a numer of different 4Fast Fourier Transform4 (FFT) algorithms that enale the calculation the Fourier transform of  a signal much faster than a DFT. As the name suggests, FFTs are algorithms for !uic#  calculation of discrete Fourier transform of a data vector. The FFT is a DFT algorithm which reduces the numer of computations needed for N for N points  points from '( N  N 2) 2 ) to '( N log  N ) wher wheree log log is the the ase ase- - loga logari rith thm. m. "f the the func functi tion on to e tran transf sfor orme med d is not not harmonically related to the sampling fre!uency, fre!uenc y, the response of an FFT loo#s li#e a 9sinc function (sin x (sin x)) + x. + x. The :adix- D"T algorithm rearranges the DFT of the function x function xn into two parts& a sum over the even-numered indices n ; m m and a sum over the odd-numered indices n ; m m < $&

'ne 'ne can can fact factor or a comm common on mult multip ipli lier er out out of the the seco second nd sum sum in the the e!uation. "t is the two sums are the DFT of the even-indexed part x part xm and the DFT of  odd-indexed part x part xm < $ of the function x function xn. Denote the DFT of the  E ven-indexed ven-indexed inputs  xm y E   y E k  and the DFT of the Odd-indexed inputs x inputs xm < $ y Ok  and we otain&

8owever, these smaller DFTs have a length of N  of N +, +, so we need compute only N  only N + + outputs& than#s to the periodicity properties of the DFT, the outputs for + = # =  from a DFT of length N  length N + + are identical identical to the outputs outputs for %= # = +. That is, E  is, E k   3 ? ?iexp> 3 ?ik  oeys the relation& exp> 3 ?i ?i(k