life in the frequency domain jean baptiste joseph fourier (1768-1830)
Post on 21-Dec-2015
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Sampling
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Example: CD rate f = 44,100 Hz, so the highest frequency thatcan be represented is 22,050 Hz
Called Nyquist frequency = ½ cycle/sample = f/2 Hz
• One frequency can masquerade as another
• When viewed as a strobed phasor, it’s easy to see that we need to sample at least twice each period to capture the frequency unambiguously
• Nyquist’s theorem: Highest allowed signal frequency is half the sampling frequency = Nyquist frequency
• Prefiltering: avoids aliasing on A-to-D
• Oversampling: can substitute cheap digital filtering for expensive analog filtering for A-to-D or D-to-A conversion
The DFT and FFT
Think of the n samples of the signal as n points on a circle
And also think of n frequency points as n points on the circle
The Discrete Fourier Transform(DFT)
1,...,0,2
1,...,0,
1,...,0,
1
0
1
0
1
NkN
kk
Nk
Nt
tiN
ttk
tiN
kkt
k
k
exX
eXN
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representation
transform
Frequencies on circle
Heideman, Johnson, Burrus (1985)
Johann Carl Friedrich Gauss (1777-1855)
A once-a-minute application: JPEG
Steven W. Smith's Book (1997)
Discrete Cosine Transform(DCT)