Life in the frequency domain
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Life in the frequency domain. Jean Baptiste Joseph Fourier (1768-1830). Spectrogram, Northern Cardinal. Sampling. Sampling. Example: CD rate f = 44,100 Hz , so the highest frequency that can be represented is 22,050 Hz. Called Nyquist frequency = ½ cycle/sample = f/ 2 Hz. - PowerPoint PPT Presentation
Transcript of Life in the frequency domain
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Life in the frequency domainJean Baptiste Joseph Fourier (1768-1830)
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Spectrogram, Northern Cardinal
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Sampling
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SamplingExample: CD rate f = 44,100 Hz, so the highest frequency thatcan be represented is 22,050 Hz Called Nyquist frequency = cycle/sample = f/2 Hz
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Sampling and aliasingAliasing a sine wave:
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One frequency can masquerade as anotherWhen viewed as a strobed phasor, its easy to see that we need to sample at least twice each period to capture the frequency unambiguouslyNyquists theorem: Highest allowed signal frequency is half the sampling frequency = Nyquist frequency
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Aliasing strikes!
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Prefiltering: avoids aliasing on A-to-DOversampling: can substitute cheap digital filtering for expensive analog filtering for A-to-D or D-to-A conversion
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The Discrete Fourier Transform(DFT)representationtransformFrequencies on circle
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Heideman, Johnson, Burrus (1985)Johann Carl Friedrich Gauss (1777-1855)
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The FFTDivide-and-conquer algorithm for DFTYields O(N log N) algorithm
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FFT Butterfly
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A once-a-minute application: JPEG Steven W. Smith's Book (1997)Discrete Cosine Transform(DCT)
