a modified split-radix fft with fewer arithmetic operations

5/26/2018 A Modified Split-radix Fft With Fewer Arithmetic Operations

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A MODIFIED SPLIT-RADIX

WITH FEWER ARITHMETOPERATIONS

M. KARISHMA & M. CHARISHMA

N B K R I S T, VIDYANAGAR


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INTRODUCTIONFourier Series

The first four partial sums of the

Fourier series for a square wave

A Fourier series decomposes periodic functions or periodic signals into the sum of a (possibly infinite) set of simple

namely sine and cosine terms (or complex exponentials). The Discrete-time Fourier transform is a periodic function, often

Fourier series. It is named after Joseph Fourier.

The Z-transform reduces to a Fourier series for the important case |z|=1. Fourier series is also central to the original proof of

sampling theorem.

The study of Fourier series is a branch of Fourier analysis.


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Definition : Fourier Series

where:

When the coefficients (known as Fourier coefficients) are computed as follows:

approximates on and the approximation improves as N

The infinite sum, is called the Fourier seriesrepresentation of


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Introduction

Fourier Series is Complicated Fourier Series is applicable for Infinite Series

Fourier Series is applicable for Periodic Series only.

So, Fourier Transform is developed.


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Fourier Transform - Definition

Fourier Transformation is a mathematical transformation emtransform signals between time (or spatial) domain and domain.

In the case of a periodic function over time the Fourier transfosimplified to the calculation of a discrete set of complex called Fourier series coefficients. They represent the frequency s

the original time-domain signal. Also, when a time-domaiis sampled to facilitate storage or computer-processing, it is stillrecreate a version of the original Fourier transform according to summation formula, also known as discrete-time Fourier transfor


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Fourier Transform - Limitation

Fourier Series or Fourier Transforms are applicable Continuous signals.

Most of the cases, we deal with Discrete or Digital signals (Computer Processing)

Therefore, Discrete Fourier Transformation (DFT)

developed.


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Discrete Fourier Transformation (D

Forward DFT

Inverse DFT

The computation of DFT coefficients require N2 complex multiplications and N(N-1) complex

(Each complex multiplication require 4 real multiplications and two real additions, each comple

require two real additions. )

In other words, computation of DFT is cumbersome process.

This complexity can be reduced by using Fast Fourier Transformation.


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FAST FOURIER TRANSFORMATION

The Fast Fourier Transformation is defined as

[1] 10;

1

0

NkWnxkX

N

n

nk

N

x[n] = x[0], x[1], , x[N-1]

By dividing the sequence x[n] into even and

odd sequences as x[2n] = x[0], x[2], , x[N-2]

x[2n+1] = x[1], x[3], , x[N-1]

The number of complex multiplications and additions can be reduced to

(N/2)log2(N) and (N)log2(N) ,provided N is a power of 2 (i.e., 2v). This is

known as Radix -2 FFT.


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Radix-2 FFT - Butterfly Diagram


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Radix3 FFT

Based on the N value, Radix can be chosen appropriately.

For example: If N is a multiple of 3 or power of 3, then Raused. That means, N can be divided into three terms, as sh


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Radix4 FFT


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Split Radix To evaluate larger N value, Split Radix or Mixed Radix can be used.

For example if N=32, the split-radix FFT (SRFFT) algorithms exploit this idea by using bradix-4 decomposition in the same FFT algorithm.

First, we recall that in the radix-2 decimation-in-frequency FFT algorithm, the even-number

theN-point DFT are given as

A radix-2 suffices for this computat

If we use a radix-4 decimation-in-frequency FFT algorithm for the odd-numbered samples of

DFT, we obtain the followingN/4-point DFTs:


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Butterfly for SRFFT algorithm

S li R di FFT B fl Di


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Split Radix FFT Butterfly DiagramLength 32 split-radix FFT algorithms from paper by Duhamel (1986)


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a modified split-radix fft with fewer arithmetic operations

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