1. Life in the frequency domain Jean Baptiste Joseph Fourier (1768-1830)

2. Spectrogram, Northern Cardinal

3. Sampling

4. 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 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

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

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

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

16. FFT Butterfly

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