Lecture 8: Discrete Fourier Transform Svetoslav Nikolov Ørsted•DTU, Building 349
Today
DFT
Relations to DTFT
FFT
Applications
DFT
In Matlab:
Other places:
Frequencies
Finite-energy signals
Spectrum of analog signal
Spectrum of discrete-time signal
Spectrum calculated using DFT
Example – finite energy signals
-10 -5 0 5 10 0 1 2 |X a (f)| [V/Hz] -10 -5 0 5 10 0 1 2 |X d (f)/f s | [V/Hz] -10 -5 0 5 10 0 1 2 |X fft (m)/f s | [V/Hz] Frequency [Hz] Spectrum of analog signal Spectrum of discrete-time signal DFT
Example - zoomed-in
Magnitude [V] 0 2 4 6 8 10 12 0 5 10 15 20 25 Frequency [Hz] Magnitude [V] 22 V dft_pad_demo
Periodic signals
Relation b/w a single pulse xae(t) and a periodic signal xpa(t) formed by repeating xae with a period T:
If xpa(t) is sampled correctly, then all frequency components are present in the sampled signal xpd(n).
If an integer number of periods is used in the DFT, then the spectrum of the can be found as:
Example
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -10 -5 0 5 10 t [s] x(t) [v] 0 1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 2 Frequency [Hz] Ampl. [v]
Example …
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -10 -5 0 5 10 t [s] x(t) [v] 0 2 4 6 8 10 12 14 16 18 20 0 0.5 1 1.5 2 2.5 Frequency [Hz] Ampl. [v]
DFT as complex filtration
0 2 4 6 8 -1 0 1 0 2 4 6 8 -1 0 1 0 2 4 6 8 -1 0 1 0 2 4 6 8 -1 0 1 0 2 4 6 8 -1 0 1 0 2 4 6 8 -1 0 1 0 2 4 6 8 0 0.5 1 0 2 4 6 8 -1 0 1
DFT as Complex Filtration
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 1 2 3 4 5 6 7 8 f/fs dft_demo
Oscilloscope display
demo_spek_ana Signal: Triggering: 0.2 V
Fast Convolution using FFT
FFT FFT X IFFT y(n) X1(n) X2(n)
Description:
Lecture 8: Discrete Fourier Transform Svetoslav Nikolov Ørsted•DTU, Building 349
Tags:
signal | dft | fft | spectrum | frequenc | exampl | period | discret
Created:
5/4/2006 1:06:50 PM
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