VHDL code implementation

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CHAPTER 1: INTRODUCTION CHAPTER 1: INTRODUCTION 1.1 FAST FOURIER TRANSFORM 1.1 FAST FOURIER TRANSFORM

A Fast Fourier Transform (FFT) is an efficient algorithm to compute the Discrete A Fast Fourier Transform (FFT) is an efficient algorithm to compute the Discrete Four

Fourier ier TrTransansforform m (D(DFT) FT) and and itits s invinverserse. e. TheThere re are are manmany y disdistintinct ct FFT FFT algalgoriorithmthmss involving a wide range of mathematics, from simple complex-numer arithmetic to group 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 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 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 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 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, this, FFT computation techni!ue is used in digital spectral analysis, filter simulation, autocorrelation and pattern recognition.

autocorrelation and pattern recognition.

The FFT is ased on decomposition and rea#ing the transform into smaller 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 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 time re!uired to compute a discrete Fourier transform and improves the performance y a factor of $%% or more ov

factor of $%% or more over direct evaluation of the DFT.er direct evaluation of the DFT. A

A DFDFT T dedecocompmpososes es a a sese!u!uenence ce of of vavalulues es ininto to cocompmpononenents ts of of didiffffererenentt fre!uencies. This operation is useful in many fields ut computing it directly from the 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 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

more !uic#ly& computing a DFT of N N points in the ovious way, using the definition, points in the ovious way, using the definition, ta#es '( 

ta#es '( _{) arithmetical operations, while an FFT can compute the same result in only}_{) arithmetical operations, while an FFT can compute the same result in only} '(

'( N N log log N N ) operations.) operations.

The difference in speed can e sustantial, especially for long data sets where The difference in speed can e sustantial, especially for long data sets where N N may e in the thousands or millions*in practice, the computation time can e reduced y may e in the thousands or millions*in practice, the computation time can e reduced y severa

several l orderorders of s of magnimagnitude in tude in such cases, and such cases, and the improvemthe improvement is ent is roughlroughly y proporproportionationall to

to N N +log (+log ( N N ). This huge ). This huge improvement made many improvement made many DFT-DFT-ased algorithms practical. FFTsased algorithms practical. FFTs are of

are of great importgreat importance to ance to a a wide variety of wide variety of appliapplicatiocations, from ns, from digitdigital signal processingal signal processing and

and solvisolving ng partipartial al diffdifferenterential e!uations to ial e!uations to algorialgorithms for thms for !uic# !uic# multmultipliciplication of ation of larglargee integers.

integers.

The most well #nown FFT algori

The most well #nown FFT algorithms depend upon the thms depend upon the factofactoriatriation ofion of N N , ut, ut there are FFT with ' (

there are FFT with ' ( N N l logog N N ) compl) complexiexity for allty for all N N , , even for even for priprimeme N N . any FFT. any FFT algorithms only

algorithms only depend on depend on the fact the fact that that is anis an N N th primitive root of unity, and thus can eth primitive root of unity, and thus can e app

applilied ed to to anaanalologougous s trtranansfsfororms ms ovover er any any fifininite te fifieleld, d, susuch ch as as nunummerer-t-theheororeteticic transforms.

transforms.

The Fast Fourier Transform algorithm exploit the two asic properties of the The Fast Fourier Transform algorithm exploit the two asic properties of the twi

twiddlddle e facfactor tor - - the the symsymmetmetry ry proproperperty ty and and perperiodiodiciicity ty proproperperty ty whiwhich ch redreduceuces s thethe numer of complex

numer of complex multiplications re!uired to perform DFT.multiplications re!uired to perform DFT. FFT

FFT algalgoriorithmthms s are are asased ed on on the the funfundamdamentental al priprincinciple ple of of decdecompomposiosing ng thethe computation of discrete Fourier Transform of a se!uence of length  into successively computation of discrete Fourier Transform of a se!uence of length  into successively smaller discrete Fourier transforms. There are asically

smaller discrete Fourier transforms. There are asically two classes of FFT algorithms.two classes of FFT algorithms. A) Decimation "n Time (D"T) algorithm

A) Decimation "n Time (D"T) algorithm

/) Decimation "n Fre!uency (D"F) algorithm. /) Decimation "n Fre!uency (D"F) algorithm.

"n decimation-in-time, the se!uence for which we need the DFT is successively "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 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 certain pattern to otain the re!uired DFT of the entire se!uence. "n the

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decimation-in-fre!uency approach, the decimation-in-fre!uency samples of the DFT are decomposed into smaller and fre!uency approach, the fre!uency samples of the DFT are decomposed into smaller and smaller suse!uences in a

smaller suse!uences in a similar manner.similar manner.

The numer of complex multiplication and addition operations re!uired y the 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 simple forms oth the Discrete Fourier Transform (DFT) and "nverse Discrete Fourier Transform ("DFT) is of order

Transform ("DFT) is of order N N _{as there are}_{as there are N}_{N data points to calculate, each of which}_{data points to calculate, each of which} re!uires

re!uires N N complex arithmetic operations.complex arithmetic operations.

The discrete Fourier transform (DFT) is defined y the formula& The discrete Fourier transform (DFT) is defined y the formula&

00   $ $ % % )) (( )) (( N N nK nK j j N N n n e e n n x x K K X X Π Π − − ∑ ∑−− = = • • = = 1here 2 is an integer ranging

1here 2 is an integer ranging from % tofrom % to N N 3 $. 3 $. The algorithmic complexity of DFT will '(

The algorithmic complexity of DFT will '( N N ) and hence is not a very efficient) 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 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 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 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# 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 calculation of discrete Fourier transform of a data vector. The FFT is a DFT algorithm which reduces the numer of computations needed for

which reduces the numer of computations needed for N N points from points from '('( N 2 N 2) to '() to '( N N loglog N

N ) ) whwherere e lolog g is is ththe e aasese- - lologagaririththm. m. "f "f ththe e fufuncnctition on to to e e trtranansfsforormemed d is is nonott harmonically related to the sampling fre!uenc

harmonically related to the sampling fre!uency, the response of an FFT loo#s li#e a 9sincy, the response of an FFT loo#s li#e a 9sinc function (sin

function (sin x x) +) + x. x.

The :adix- D"T algorithm rearranges the DFT of the function

The :adix- D"T algorithm rearranges the DFT of the function x_{xnn into two parts&} into two parts& a sum over the even-numered indices

a sum over the even-numered indices nn ;  ; mm and a sum over the odd-numered indices and a sum over the odd-numered indices n

n ;  ; mm < $& < $&

'n

'ne e cacan n fafactctor or a a cocommmmon on mumultltipiplilier er ouout t of of ththe e sesecocond nd susum m in in ththee e!uation. "t is the two sums are the DFT of the even-indexed part

e!uation. "t is the two sums are the DFT of the even-indexed part x_xm_{m and the DFT of} and the DFT of odd-indexed part

odd-indexed part x_xm_{m < $}_{< $ of the function} of the function x_{xnn. Denote the DFT of the}. Denote the DFT of the E E _{ven-indexed inputs}_{ven-indexed inputs}

x

xmm y y E E k k and the DFT of the and the DFT of the OOdd-indexed inputsdd-indexed inputs x xmm < $ < $ y y OOk k and we otain& and we otain&

8owever, these smaller DFTs have a length of

8owever, these smaller DFTs have a length of N N +, so we need compute only+, so we need compute only N N ++ outputs& than#s to the periodicity properties of the DFT, the outputs for + = # =  from outputs& than#s to the periodicity properties of the DFT, the outputs for + = # =  from a DFT of length

a DFT of length N N + are identic+ are identical to the al to the outputoutputs for %= s for %= # = # = +. That is,+. That is, E _E k_k <_< N_N +_{+ }_ ; ; E _E k_k and

and O_Ok_{k <}_< N_N +_{+ }_{ ;}; O_Ok_{k . The phase factor exp> 3 ?}. The phase factor exp> 3 ?ik ik ++ N N @ @ callecalled a twidd a twiddle fadle factor wctor whichhich oeys the relation& exp> 3 ?

oeys the relation& exp> 3 ?ii((k k < < N N + ) + + ) + N N @ ;@ ; ee 3 ?3 ?iiexp> 3 ?exp> 3 ?ik ik + + N N @ ; 3 exp> 3 ?@ ; 3 exp> 3 ?ik ik + + N N @,@, fli

flippipping the ng the sigsign n of theof the O_Ok_{k <}_{< N}_{N +}_{+ }_{ terms. Thus, the whole DFT can e calculated as} terms. Thus, the whole DFT can e calculated as follows&

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fre!uency approach, the fre!uency samples of the DFT are decomposed into smaller and fre!uency approach, the fre!uency samples of the DFT are decomposed into smaller and smaller suse!uences in a