1. Life in the frequency domain
Jean Baptiste Joseph Fourier ( )

2. Spectrogram, Northern Cardinal

3. Sampling

4. 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

5. Sampling and aliasing
Aliasing a sine wave:

6. 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

7. Aliasing strikes!

9. 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

11. The Discrete Fourier Transform (DFT)
representation
transform
Frequencies on circle

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

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

15. FFT Butterfly

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