Abstract
This chapter discusses the Fourier representations for finite-length time sequences, referred to as the discrete Fourier transform (DFT), which is not a continuous-frequency function but a discrete-frequency sequence obtained by sampling the discrete-time Fourier transform of the sequence with an equally spaced frequency interval. It shows some relationships between DFT and the Fourier series representation of periodic sequences, some properties of DFT, efficient algorithms for computing DFT, and applications of DFT to the convolution computations.. The chapter also shows the properties of the DFT, including the circular shift and the circular convolution properties. The chapter introduces an efficient algorithm for computing DFTs, which is called the fast Fourier transform (FFT).. It also proves that the computing time required for linear convolutions in the frequency domain is less than that which is necessary for the direct computation of the linear convolution in the time domain. Finally, the chapter discusses practical methods for computing linear convolutions where an indefinitely long sequence is convolved with a finite-length sequence.
Published Version
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