[Joe Bloggs]

A look at lowpass filter ringing, as occurs in "low-res" audio playback (in this case 44.1kHz):

1. An ideal impulse at high sample rate (352.8kHz) https://www.dropbox.com/s/m7jk56jj2r4o6wh/352800-impulse.wav?dl=0
Waveform plot


Frequency response plot


2. Lowpassed to 44.1kHz, with classic ringing tails caused by lowpass filter https://www.dropbox.com/s/cmwsht528pnaqke/44100.wav?dl=0
Waveform plot (linear interpolation)


Waveform plot (smoothed)

Eyeballing the plot, the oscillation frequency of the ringing is close to Nyquist, (22.05kHz, 1 cycle in 2 samples) but how to quantify this?

Frequency analysis of the impulse shows flat response up to 20500Hz or so.



3. Same as (2), but subtracted by an ideal impulse in the middle. We see that the main pulse of the lowpassed impulse is cancelled out to blend in with the ringing before and after--this lets us analyse the ringing in isolation. https://www.dropbox.com/s/ufq98h0ajblfxtk/44100-ringing%20only.wav?dl=0
Waveform plot (linear interpolation)


Waveform plot (smoothed)


Frequency analysis of the modified impulse confirms that the main impulse has been cancelled out (down to over -200dB) while the ringing that remains is confined to over 20000Hz.



4. Same as (3), but with amplitude amplified to within 0.2dB of full scale. The file from (3) may be used to gauge audibility of ringing in normal program material with your player set to the same volume at which it normally plays music; this file is "can you hear this ringing with the volume turned up to 11 in the sound file?" https://www.dropbox.com/s/6fexu9byxw2ejz2/44100-ringing%20only-amplified.wav?dl=0

 

[arnyk]

Q: Eyeballing the plot, the oscillation frequency of the ringing is close to Nyquist, (22.05kHz, 1 cycle in 2 samples) but how to quantify this?
 
A:  Eyeballing the plot on a display that shows timing fairly information accurately, one of the waves peaks at 2.499990 second and again at 2.499996 seconds. the difference corresponds to a frequency of 16.6666 KHz. The simulations or the readout may not be that accurate because the book answer is 22.1 KHz.

 

[cjl]

  Q: Eyeballing the plot, the oscillation frequency of the ringing is close to Nyquist, (22.05kHz, 1 cycle in 2 samples) but how to quantify this?
 
A:  Eyeballing the plot on a display that shows timing fairly information accurately, one of the waves peaks at 2.499990 second and again at 2.499996 seconds. the difference corresponds to a frequency of 16.6666 KHz. The simulations or the readout may not be that accurate because the book answer is 22.1 KHz.

That's clearly off a bit then, since the actual frequency content is >20kHz, as shown in the second to last image.

 

[limpidglitch]

Isn't it pretty?
It's Joe's last file which I slowed down 8x and took a few frequency snapshots, starting at the peak (the red line) and gradually moving towards the end of the ringing (the lower green line)
32768 point Hann window.
 

 

[RRod]

One thing you can try is to make a series of the rings (so subtracting out the impulse as you did above), starting at a rate where things should be audible (e.g. 11025), and then gradually going up to the full 44100. You can then let people find the last one they can actually hear. You can also make comparisons to something pathological like a maximum phase filter.