discrete-time signal processing lecture 8 (dft)
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Discrete-time Signal Processing Lecture 8 (DFT)](https://reader030.vdocuments.us/reader030/viewer/2022033011/56814e4a550346895dbbceab/html5/thumbnails/1.jpg)
DISCRETE-TIME SIGNAL PROCESSINGLECTURE 8 (DFT)
Husheng Li, UTK-EECS, Fall 2012
![Page 2: Discrete-time Signal Processing Lecture 8 (DFT)](https://reader030.vdocuments.us/reader030/viewer/2022033011/56814e4a550346895dbbceab/html5/thumbnails/2.jpg)
WHY DFT
The specification of filter is usually given by the tolerance scheme.
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.
![Page 3: Discrete-time Signal Processing Lecture 8 (DFT)](https://reader030.vdocuments.us/reader030/viewer/2022033011/56814e4a550346895dbbceab/html5/thumbnails/3.jpg)
DISCRETE FOURIER SERIES (DFS)
Consider a periodic sequence x(n) with period N. We have
Usually we define . Then, we have
![Page 4: Discrete-time Signal Processing Lecture 8 (DFT)](https://reader030.vdocuments.us/reader030/viewer/2022033011/56814e4a550346895dbbceab/html5/thumbnails/4.jpg)
PROPERTIES OF DFS
![Page 5: Discrete-time Signal Processing Lecture 8 (DFT)](https://reader030.vdocuments.us/reader030/viewer/2022033011/56814e4a550346895dbbceab/html5/thumbnails/5.jpg)
MORE PROPERTIES
![Page 6: Discrete-time Signal Processing Lecture 8 (DFT)](https://reader030.vdocuments.us/reader030/viewer/2022033011/56814e4a550346895dbbceab/html5/thumbnails/6.jpg)
FOURIER TRANSFORM OF PERIODIC SIGNALS
For periodic signals, the continuous-frequency Fourier transform is given by
![Page 7: Discrete-time Signal Processing Lecture 8 (DFT)](https://reader030.vdocuments.us/reader030/viewer/2022033011/56814e4a550346895dbbceab/html5/thumbnails/7.jpg)
SAMPLING THE FOURIER TRANSFORM
Consider an aperiodic sequence x(n) with Fourier transform X(w), we can do sampling:
![Page 8: Discrete-time Signal Processing Lecture 8 (DFT)](https://reader030.vdocuments.us/reader030/viewer/2022033011/56814e4a550346895dbbceab/html5/thumbnails/8.jpg)
NEW SEQUENCE
The sequence having DFS equaling the frequency domain sampling results from the aperiodic sampling.
![Page 9: Discrete-time Signal Processing Lecture 8 (DFT)](https://reader030.vdocuments.us/reader030/viewer/2022033011/56814e4a550346895dbbceab/html5/thumbnails/9.jpg)
DFT FOR FINITE-DURATION SEQUENCES 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
![Page 10: Discrete-time Signal Processing Lecture 8 (DFT)](https://reader030.vdocuments.us/reader030/viewer/2022033011/56814e4a550346895dbbceab/html5/thumbnails/10.jpg)
EXAMPLE: DFT OF A RECTANGULAR PULSE
![Page 11: Discrete-time Signal Processing Lecture 8 (DFT)](https://reader030.vdocuments.us/reader030/viewer/2022033011/56814e4a550346895dbbceab/html5/thumbnails/11.jpg)
PROPERTIES OF DFT
![Page 12: Discrete-time Signal Processing Lecture 8 (DFT)](https://reader030.vdocuments.us/reader030/viewer/2022033011/56814e4a550346895dbbceab/html5/thumbnails/12.jpg)
COMPUTING CONVOLUTION USING DFT 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.