# Real time digital signal processing P7

## Fast Fourier Transform and Its Applications Frequency analysis of digital signals and systems was discussed in Chapter 4. To perform frequency analysis on a discrete-time signal, we converted the time-domain sequence into the frequency-domain representation using the z-transform, the discrete-time Fourier transform (DTFT), or the discrete Fourier transform (DFT). The widespread application of the DFT to spectral analysis, fast convolution, and data transmission is due to the development of the fast Fourier transform (FFT) algorithm for its computation. The FFT algorithm allows a much more rapid computation of the DFT, was developed in the mid-1960s by Cooley and Tukey. It is. | Real-Time Digital Signal Processing. Sen M Kuo Bob H Lee Copyright 2001 John Wiley Sons Ltd ISBNs 0-470-84137-0 Hardback 0-470-84534-1 Electronic 7 Fast Fourier Transform and Its Applications Frequency analysis of digital signals and systems was discussed in Chapter 4. To perform frequency analysis on a discrete-time signal we converted the time-domain sequence into the frequency-domain representation using the z-transform the discrete-time Fourier transform DTFT or the discrete Fourier transform DFT . The widespread application of the DFT to spectral analysis fast convolution and data transmission is due to the development of the fast Fourier transform FFT algorithm for its computation. The FFT algorithm allows a much more rapid computation of the DFT was developed in the mid-1960s by Cooley and Tukey. It is critical to understand the advantages and the limitations of the DFT and how to use it properly. We will discuss the important properties of the DFT in Section . The development of FFT algorithms will be covered in Section . In Section we will introduce the applications of FFTs. Implementation considerations such as computational issues and finite-wordlength effects will be discussed in Section . Finally implementation of the FFT algorithm using the TMS320C55x for experimental purposes will be given in Section . Discrete Fourier Transform As discussed in Chapter 4 we perform frequency analysis of a discrete-time signal x n using the DTFT defined in . However X w is a continuous function of frequency w and the computation requires an infinite-length sequence x n . Thus the DTFT cannot be implemented on digital hardware. We define the DFT in Section for N samples of x n at N discrete frequencies. The input to the -point DFT is a digital signal containing N samples and the output is a discrete-frequency sequence containing N samples. Therefore the DFT is a numerically computable transform and is suitable for DSP implementations. .

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