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

Sampling and aliasing Aliasing a sine wave:

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

Aliasing strikes!

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 Discrete Fourier Transform (DFT) representation transform Frequencies on circle

Johann Carl Friedrich Gauss (1777-1855) Heideman, Johnson, Burrus (1985)

The FFT Yields O(N log N) algorithm Divide-and-conquer algorithm for DFT Yields O(N log N) algorithm

FFT Butterfly

A once-a-minute application: JPEG Discrete Cosine Transform (DCT) Steven W. Smith's Book (1997)