## The fast Fourier transform |

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2nft ALGOL aliasing amplitude application array Audio and Electroacoustics band-limited Compute the discrete computer program consider continuous convolution continuous Fourier transform convolution and correlation convolution integral convolution results convolve Cooley Cooley-Tukey correlation integral defined determine the Fourier develop discrete convolution discrete Fourier transform discrete transform dual node Electroacoustics June Equation example factor fast Fourier transform FFT algorithm Figure Fourier integral Fourier series Fourier trans Fourier transform pair frequency convolution theorem frequency domain frequency function frequency sampling function h(t function illustrated graphical h(kT IEEE Trans illustrated in Fig imaginary impulse functions inverse Fourier transform limit of integration linear Note odd function Parseval's Theorem periodic function problem procedure rectangular function relationship sample interval sample values sampled waveform sampling function shift shown in Fig signal flow graph sinusoids technique term tion transform analysis truncation function truncation interval Tukey waveform h(t x(kT zero