Fast Fourier Transform • • • The fast Fourier transform using radix-2 and radix-4 Decimation or decomposition in frequency and in time Programming examples The fast Fourier transform (FFT) is an efficient algorithm that is used for converting a time-domain signal into an equivalent frequency-domain signal, based on the discrete Fourier transform (DFT). Several real-time programming examples on FFT are included. INTRODUCTION The discrete Fourier transform converts a time-domain sequence into an equivalent frequency-domain sequence. The inverse discrete Fourier transform performs the reverse operation and converts a frequency-domain sequence into an equivalent time-domain sequence. The fast Fourier transform (FFT) is a very efficient. | DSP Applications Using C and the TMS320C6x DSK. Rulph Chassaing Copyright 2002 John Wiley Sons Inc. ISBNs 0-471-20754-3 Hardback 0-471-22112-0 Electronic 6 Fast Fourier Transform The fast Fourier transform using radix-2 and radix-4 Decimation or decomposition in frequency and in time Programming examples The fast Fourier transform FFT is an efficient algorithm that is used for converting a time-domain signal into an equivalent frequency-domain signal based on the discrete Fourier transform DFT . Several real-time programming examples on FFT are included. INTRODUCTION The discrete Fourier transform converts a time-domain sequence into an equivalent frequency-domain sequence. The inverse discrete Fourier transform performs the reverse operation and converts a frequency-domain sequence into an equivalent time-domain sequence. The fast Fourier transform FFT is a very efficient algorithm technique based on the discrete Fourier transform but with fewer computations required. The FFT is one of the most commonly used operations in digital signal processing to provide a frequency spectrum analysis 1-6 . Two different procedures are introduced to compute an FFT the decimation-in-frequency and the decimation-in-time. Several variants of the FFT have been used such as the Winograd transform 7 8 the discrete cosine transform DCT 9 and the discrete Hartley transform 10-12 . Programs based on the DCT FHT and the FFT are available in Ref. 9. 182 Development of the FFT Algorithm with Radix-2 183 DEVELOPMENT OF THE FFT ALGORITHM WITH RADIX-2 The FFT reduces considerably the computational requirements of the discrete Fourier transform DFT . The DFT of a discrete-time signal x nT is X k N x n Wnk k 0 1 . N -1 n 0 where the sampling period T is implied in x n and N is the frame length. The constants W are referred to as twiddle constants or factors which represent the phase or W e -j 2pN and is a function of the length N. Equation can be written for k 0 1 . N -1 as
