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Discrete-time Signal Processing Lecture 8 (DFT)

Husheng Li, UTK-EECS, Fall 2012. Discrete-time Signal Processing Lecture 8 (DFT). Discrete Fourier Transform (DFT) has both discrete time and discrete frequency. DTFT has a continuous frequency, which is difficult to process using digital processors.

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Discrete-time Signal Processing Lecture 8 (DFT)

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  1. Husheng Li, UTK-EECS, Fall 2012 Discrete-time Signal ProcessingLecture 8 (DFT)

  2. Discrete Fourier Transform (DFT) has both discrete time and discrete frequency. DTFT has a continuous frequency, which is difficult to process using digital processors. DFT has a fast computation algorithm: FFT. Why DFT The specification of filter is usually given by the tolerance scheme.

  3. Consider a periodic sequence x(n) with period N. We have Usually we define . Then, we have Discrete Fourier series (DFS)

  4. Properties of DFS

  5. More properties

  6. For periodic signals, the continuous-frequency Fourier transform is given by Fourier transform of periodic signals

  7. Consider an aperiodic sequence x(n) with Fourier transform X(w), we can do sampling: Sampling the Fourier transform

  8. The sequence having DFS equaling the frequency domain sampling results from the aperiodic sampling. New sequence

  9. Consider a finite-duration sequence x(n) with length N. We can define its DFT as the DFS of the periodic sequence where . The DFT is given by DFT for finite-duration sequences

  10. Example: DFT of a rectangular pulse

  11. Properties of DFT

  12. Since there is a fast computation algorithm in DFT, we can compute convolution via DFT: Compute the DFTs of both sequences Compute the product of both DFTs. Compute the output using IDFT. The length of DFT should be properly chosen; otherwise we will see aliasing. Computing convolution using DFT

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