Help with Fourier synthesis of a sawtooth wave

Special Interest Groups Audio Processing
1

Hi there,

I’m carrying out Fourier analysis and synthesis on square and sawtooth waves as part of a university project, however, I’ve run into a problem and am unsure whether I have done something incorrectly or not. I’ve been effectively told “generate some waveforms and see what happens”, so my knowledge of Fourier is still very limited.

Using FFT in Audacity, I was successful in building a 1 KHz square wave out of sin waves. However, when I attempted the sawtooth wave, I found that the resulting shape of my waveform was inverted (see picture below).

The top waveform is a 1 KHz sawtooth wave created using the tone generator.
The bottom wave is my reconstructed wave using the first 5 harmonics. As you can see, the shape is inverted.

Have I incorrectly analyzed the waveform or is something else happening here?

2

Your analysis appears to be correct, but the starting phase for your sine waves is reversed (rotated 180 degrees).

3

Nyquist can be quite useful for playing with this sort of thing.
(also: http://wiki.audacityteam.org/wiki/Nyquist_Plug-ins_Reference)

This code can be run in the “Nyquist Prompt” effect:

(setf out (s-rest 1))
(setq f0 1000.0)
(setq k 6)

(dotimes (n k (mult (/ -2 pi) out))
  (setf out 
    (sum out 
         (mult (/ 1.0 (1+ n))
               (osc (hz-to-step (* (1+ n) f0)))))))
4

Here is the comparison between a measured sawtooth (256 Hz, FS = 8192) and the calculated values from Steve’s code (in dB).

Frequency	Level (meas.)	Level (calc.)
256	-3.94664	-3.922397541
512	-9.92532	-9.942997454
768	-13.3771	-13.46482263
1024	-15.7774	-15.96359737
1280	-17.5883	-17.90179763
1536	-19.0154	-19.48542255
1792	-20.1678	-20.82435834
2048	-21.1103	-21.98419728
2304	-21.8843	-23.00724773
2560	-22.5175	-23.92239754
2816	-23.0292	-24.75025124
3072	-23.4329	-25.50602246
3328	-23.7383	-26.20126459
3584	-23.9521	-26.84495825
3840	-24.0787	-27.44422272
4096	-30.1412	-28.00479719