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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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