# Radix 4 Code

**Topics:**Fast Fourier transform, Discrete Fourier transform, Digital signal processing

**Pages:**10 (2991 words)

**Published:**April 25, 2013

Researchers Debalina Ghosh

Microelectronics and VLSI Design Techno India, Salt Lake Kolkata, India Email: debolina1512@gmail.com

FPGA Based Implementation of FFT Processor Using Different Architectures Abstract:

The Fast Fourier Transform (FFT) is an efficient algorithm for computing the Discrete Fourier Transform (DFT) and requires less number of computations than that of direct evaluation of DFT. It has several applications in signal processing. Because of the complexity of the processing algorithm of FFT, recently various FFT algorithms have been proposed to meet real-time processing requirements and to reduce hardware complexity over the last decades. This is in two directions. One related to the algorithmic point of view and the other based on ASIC architecture. The last one was pushed by VLSI technology evolution. In this work, we present three different architectures of FFT processor to perform 1024 point FFT analysis. The designs have been simulated and its FPGA based implementation has been verified successfully using Xilinx ISE 11.1 tool using VHDL. There are also comparative studies among those architectures. The objective of this work was to get an area & time efficient architecture that could be used as a coprocessor with built in all resources necessary for an embedded DSP application.

Depanwita Debnath

Electronics and Communication Engineering Techno India, Salt Lake Kolkata, India Email: depanwita_sarkar@yahoo.co.in

Dr. Amlan Chakrabarti

A.K.Choudhury, School of Information Technology University of Calcutta Kolkata, India Email: acakcs@caluniv.ac.in

Keywords: Fast Fourier Transform, FFT butterfly radix 2 & 4, CORDIC, SineCosine lookup table, Xilinx Core.

Introduction:

Audio and communications signal processing are well developed lines massively used now a days in many application lines and products. Since digital communications are quite active fields, the arithmetic complexity of the Discrete Fourier Transform (DFT) algorithm becomes a significant factor with impact in global computational costs. Cooley and Tukey [1] developed the wellknown radix-2 Fast Fourier Transform (FFT) algorithm to reduce the computational load of the DFT. The Discrete Fourier Transform (DFT) X(k) of N points is given by

N-1 X(k) =Σ x(n)WNnk n =0 { 0≤k< N-1}, WNnk = exp(-j2πnk/N) (1)

Where the X(k) and x(n) are frequency-domain sequences and time-domain sequence. Instead of the direct implementation of the equation (1), the FFT algorithm factorizes a large point DFT recursively into many small point DFT in order to reduce the overall operations. There are two well-known types of decompositions called Decimation in Time (DIT) and Decimation In Frequency (DIF) FFT. The only difference between these two algorithms is that, DIT starts

1|© I JAI TI 2 0 1 2

with bit reverse order input and generates normal order output. Nevertheless DIF starts with normal order input and generates bit reverse order output. Throughout this paper DIF algorithm is used. The conventional method of Fast Fourier Transform FFT calculation involves N2complex multiplications and N(N-1) complex additions. The radix-2 Cooley-Tukey algorithm performs the same computation involving (N/2)log2N complex multiplications and (N)log2N complex additions. But it is more efficient computationally to employ a radix-4 FFT algorithm other than radix -2 logarithms. The radix-4 decimation-in frequency DFT is given by

(2) Note that the input to each N/4-pointDFT is a linear combination of four signal samples scaled by a twiddle factor. This procedure is repeated v times, where v = log 4N. The complete butterfly operation for Radix-4 DIF is shown in figure 1 (a) and in a more compact form in figure 1(b).

Figure1.The basic butterfly for radix-4 DIF FFT algorithm

In this paper, we present radix-4 FFT processor using different architectures that allows any size points to transform, fixed point arithmetic, pipeline...

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