1. Tim Hurley

2.  Spectrograms are used to identify and analyze sounds  Typically, x-axis represents time and y- axis represents frequency  Spectrograms are used to visually represent a Discrete Fourier Transform

4.  First way: Filters The filter takes frequencies and passes them through and rejects frequencies that are outside of the given range

5.  We already did that!! Nooooo, we found something different.  Waves are continuous sinusoidal functions. Finding the Fourier coefficients for these produced an infinite series numbers Computers don’t like infinite numbers

6.  Recall that we wanted to find Fourier coefficients in order to find an approximation for the square wave We took the integral from 0 to 2π and looked at different cases and found the form for a n and b n. This produced infinite number of values

7.  I started with a similar equation and took the summation of it from 0 to N-1 N = number of samples within a given window  Difficulties arose because I had to use multiple trig identities in order to re-write the sums/products of sinusoids so they were able to be manipulated.  After tedious trig work I was able to determine the form of the coefficients

8.  Good question!  Discrete Fourier Transform looks at a small windows of a sound signal. Breaks this window into N small fractions of seconds (ex: 0.0005 secs) Determines the discrete Fourier coefficients for this section Coefficients represent intensities of different frequencies of the wave These numbers are graphed and the process is repeated

9. Blue = higher intensity frequency Red = medium intensity frequency Yellow = low intensity frequency This will happen for every ‘window’ until entire sound is analyzed and graphed

10. “Philip–” “–s” “re–” “–e–” “–sear–” “–ch” www.seeingwithsound.com/javoice.htm

11. Questions?? By the way, this is my last assignment as an undergrad student