DIF-FFT

Presented by :Aleem Alsanbani

Saleem Almaqashi

Fast Fourier Transform FFT

- A fast Fourier transform (FFT) is an efficient algorithm to compute the discrete Fourier transform (DFT) and inverse of DFT.

- There are many FFT algorithms which involves a wide range of mathematics,. A Discrete Fourier transform decomposes a sequence of values into components of different frequencies.

- This operation is very useful in many fields but computing it directly from the definition is often too slow to be practical .

Cont ..

• FFT are special algorithms for speedier implementation of DFT.

• FFT requires a smaller number of arithmetic operations such as multiplications and additions than DFT.

• FFT also requires lesser computational time than DFT .

Fast Fourier Transform Algorithms

• Direct computation of the DFT is less efficient because it does not exploit the properties of symmetry and periodicity of the phase factor WN = e-j2π/N  .

• These properties are:

 

- Symmetry property.

- Periodicity property.

• As we already know that all computationally efficient algorithms for DFT are collectively known as FFT Algorithms and these algorithms exploit the above two properties of phase factor, WN.

FFT Algorithms Classification Based On Decimation

• Another classification of FFT algorithms based on Decimation of s(n) :r S(K). Decimation means decomposition into decimal parts. On the basis of decimation process, FFT algorithms are of two types: 

• 1. Decimation-in-Time FFT algorithms.

• 2. Decimation-in-Frequency FFT algorithms.                              

Cont..

• Decimation-in-Time Algorithms: sequence s(n) will be broken up into odd numbered and even numbered subsequences.

• This algorithm was first proposed by Cooley and Tukey in 1965.

• Decimation-in-Frequency Algorithms. the sequence s(n) will be. Broken up into two equal halves.

• This algorithm was first proposed by Gentlemen and Sande in 1966. 

• Computation reduction factor of FFT algorithms .• Number of computations required for direct DFT / Number of

computations required for FFT algorithm 

•             = N2 / N / 2 log2 (N)

decimation-in-frequency FFT algorithm

• In decimation-in-frequency FFT algorithm, the output DFT sequence S(K) is broken into smaller and smaller subsequences. For the derivation of this algorithm, the number of points or samples in a given sequence should be N = 2r where r > 0. For this purpose, we can first-divide the input sequence into the first-half and the second-half of the points.

• Flow graph of complete decimation-in-frequency (DIF) decomposition of an N-point DFT computation (N = 8).

Steps for Computation of Decimation in Frequency FFT Algorithm 

• Given below are the important steps for the computation of DIF FFT algorithms.

• 1. Data shuffling is not required but whole sequence is divided in two parts: first half and second half. From these we calculate g(n) and h(n) as follows: 

•                    g(n) = s(n) +s(n+N/2 ) 

• and             h(n) = s(n)-s(n+N/2 )

• where             n  = 0, 1, ..., N/2 -1

     Finally data shuffling is performed. It is also performed by Bit reversal.

 

Number Of Complex Multiplications Required In DIF- FFT Algorithm

• Number of complex multiplications required in decimation-m-. FFT algorithm are the same as that required in decimation-in-time FFT algorithm.

• Number of complex multiplication required in these DFT algorithms are N/2 log2iV, where N= 2r, r>0 and N is the total number of points (or samples) in a discrete-time sequence. Thus the total computations (number of multiplication and addition operations) are the same in both FFT algorithms.

• Now we will compare the computational complexity for the direct computation of the DFT and FFT algorithm. This comparison is given in Table

Number Of Complex Multiplications Required In DIF- FFT Algorithm

No. of points"or samples)

in a sequences)n), N

  

Complexmultiplication

sin direct

computation of

DFT=NN =A

Complexmultiplication

sin FFT

algorithmsN/2 log2 N = B

SpeedimprovementFactor -A/B

  

22- 4 16 4 4.0=

23- 8 64 12 5.3=

24 - 16 256 32 8.0=

First stage of the decimation-in-frequency FFT algorithm..

Alternate DIT FFT structures

• DIT structure with input natural, output bit-reversed (OSB 9.14):

Alternate DIT FFT structures

• DIT structure with input bit-reversed, output natural

Radix-2 Decimation-In-Frequency Solved Example Part1

• Example Find the DFT of the following discrete-time sequence .  

•                    s(n) = {1, -1, -1, -1, 1, 1, 1, -1} using Radix-2 decimation-in-frequency FFT algorithm.

• Solution. The Twiddle factor or phase rotation factor WN= involved in the FFT calculation are found out as follows for N= 8.                

Example Part1

Example Part1

Radix-2 decimation-in-frequency Solved Example Part2

• Fig.Flow graph of Radix-2 decimation-in-frequency (DIF) FFT algorithm   N = 8. In Radix-2 decimation-in-frequency (DIF) FFT algorithm, original sequence s(n) is decomposed into two subsequences as first half and second half of a sequence. There is no need of reordering (shuffling) the original sequence as in Radix-2 decimation-in-time (DIT) FFT algorithm. But resultant discrete frequency sequence is shuffled (reordered) into natural order because these are obtained in unnatural order. Flow graph of Radix-2 decimation-in-frequency (DIF) FFT algorithm for N= 8 is shown in Fig. Determination of DFT using Radix-2 DIF FFT algorithm requires three stages because the number of points in a given sequence is 8, i.e., = 2r — N — 8, where r is number of stages required = 3.

Solv.Stage I :                        A0 = s(0) + s(4) = 1 + 1 = 2                             A1 = s(l)   + s(5) = -1 + 1 =0              A2 = s(2) + s(6) = -1 + 1 = 0                           A3 = s(3) + s(7) = -1 - 1 = -2                          A4 =  [s(0)+(-1) s(4)] W8

0 = [1 + (-1) (1)]  x 1 =0 A5 =  [s(1) +  (-1) s(5)]W8

1 = [-1 + (-1)(1)]((1-j) /√2= - √2(l - j)                A6 =  [s(2) +  (-1) s(6)]W8

2 = [-1 + (-1) x 1] (- j) =2j                                         

  A7 =  [s(3) +  (-1) s(7)]W83 = [-1 + (-1)(-1)]{(-(1-j) /√2} = 0

Solv….

S(K) = {S(0), S(l), S(2), S(3), S(4), S(5), S(6), S(6), S(7)}

Or  S(X) = {0-√2+(2 + √2  )j, 2 -2j √2+(-2 + √2)j,4,

                   √2+ (2 - √2  )j,2 + 2j,- √2  -(2 + √2)7}

21

Conclusions

• Radix 22, 24… Structure uses less adders and multipliers but still has good efficiency processing DIF DFT

• Common Factor Algorithm and Butterfly Structure enable this architecture to reuse its modules numerous times

22

References

• [1],Shousheng He and Torkelson, M. “A new approach to pipeline FFT processor,” Proceedings of IPPS '96, 15-19, pp766 –770. April 1996

• [2] Alan V.Oppenheim, Ronald W. Schafer, “ Discrete-time signal processing “ 2nd edition

• [3] Zhangde Wang “INDEX MAPPING FOR ONE TO MULTI DIMENSIONS “Electronics Letters Publication Volume: 25, pp: 781-782 Jun 1989

• [4] He, S. & Torkelson, M., A systolic array implementation of common factor algorithm to compute DFT, Proc. Int. Symp. on Parallel Architectures, Algorithms and Networks, Kanazawa, Japan, pp. 374-381, 1994.

• [5]IJung-YeolOH and Myoung-Seob LIM , ‘Fast Fourier Transform Algorithm for Low-Power and Area-Efficient Implementation’EICETRANS.COMMUN.,VOL.E89–B, APRI

• [6]BURRUS, c. s.: 'Index mappings for multidimensional formulation of the DFT and convolution',IEEE Trans., 1977, ASSP-25, (6), pp. 239-242

_________

_____

__