Tom Morley's Audio Page

This page contains miscellaneous calculations bearing on the accurate reproduction of music.

Fourier Transforms

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

The x axis is in units of octaves from the fundamental frequency. So for example if the fundamental frequency is 1000 Hz, then +1 is one octave up, or 2000 Hz, +3 would then be three octaves up or 8000 Hz. Negative x axis values represent octaves down, e. g., -1 would be 500 Hz and -2 would be -250 Hz.

Frequency Part of Fourier Transform of Tone Bursts
Red -- One Full cycle, Blue - two full cycles
Vertical scale is in dB

Phase Part of Fourier Transform of Tone Bursts
Red -- One Full cycle, Blue - two full cycles
Vertical scale is in degrees

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 --

Test to see if these are audibly different. Conditions -- loudspeakers (or headphones) should be as phase coherent as possible. Best would be do do experiment in a aneachoic room. These signals are presumably very sensitive to room reflections.

Time alligned means 0 degrees!!

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