Abstract
The DTFT of a discrete-time signal is a continuous function of the frequency (\( \omega \)), and hence, the relation between \( X\left( {\text{e}^{{j}\omega } } \right) \) and \( x(n) \) is not a computationally convenient representation. However, it is possible to develop an alternative frequency representation called the discrete Fourier transform (DFT) for finite duration sequences. The DFT is a discrete-time sequence with equal spacing in frequency. We first obtain the discrete-time Fourier series (DTFS) expansion of a periodic sequence. Next, we define the DFT of a finite length sequence and consider its properties in detail. We also show that the DTFS represents the DFT of a finite length sequence. Further, evaluation of linear convolution using the DFT is discussed. Finally, some fast Fourier transform (FFT) algorithms for efficient computation of DFT are described.
References
- 1.J.W. Cooley, P.A.W. Lewis, P.D. Welch, Historical notes on the fast Fourier transform. IEEE Trans. Audio Electroacoust. 55(10), 1675–1677 (1967)Google Scholar
- 2.J.W. Cooley, P.A.W. Lewis, P.D. Welch, Historical notes on the fast Fourier transform. Proc. IEEE 55(10), 1675–1677 (1967)CrossRefGoogle Scholar
- 3.G. Goertzel, An algorithm for the evaluation of finite trignometric series. Am. Math Monthly 65, 34–35 (1958)MathSciNetCrossRefGoogle Scholar
- 4.A.V. Oppenheim, R.W. Schafer, J.R. Buck, Discrete-Time Signal Processing, 2nd edn. (Prentice Hall, 1999)Google Scholar
- 5.L. Rabiner, R. Schafer, C. Rader, IEEE Trans. Audio Electroacoust. 17(2) (1969)CrossRefGoogle Scholar