This page contains miscellaneous calculations bearing on the accurate reproduction of music.
The importance of time alignment.
Since the Fourier transform is complex (a + b i, remember?) we plot its absolute value as the frequency part, and its modulus (angle) as the phase part.
Here are the frequency and phase parts of the Fourier Transforms of
Interesting. The one aspect of this that suprised me What happens with the two cycle burst between about -1 (one octave down -- 500 Hz if the fundamental is 1000 Hz) and about -4 (Four octaves down, 62.5 Hz if the fundamental is 1000 Hz.) Look at that phase shift of over 360 degrees! (In the picture you see this by the jump from about 100 degrees to about-100 degrees. In actuality the jump doesn't happen, but the graph should continue above 180 degrees into the phases from 180 to 180+360.) Although the signal is -20 dB at four octaves down, maybe this says that phase alignment at the bass frequecies is perhaps more important than I thought--- It affects what appear to be mid frequency transients!
Consider two (full frequency) drivers, the second of which is further away from the listner. Assume that this distance is exactly such that there is a one cycle delay at the fundamental frequency of the tone bursts. Such a combination of drivers would play the blue (two cycle) tone burst when presented with the red (one cycle tone burst). Is this an audible distortion?
Would you put up with this kind of inaccuacy in an amplifier?
A modest poposal: Take two signals --
Time alligned means 0 degrees!!