Headphone CSD waterfall plots - Page 56

Earbuds.

 

AKG K 314 P

 

Some visual seam issues in the graphs, but nothing major - I'll sort it out some day. Colors remain unchanged for now.

 

What I mean is, if a tone stops abruptly, that's a broadband phenomenon.  I looked at the FFT of a single cycle of a 1000Hz tone with silence before and after it, and it looks nothing like the FFT of a continuous 1000Hz tone.  So a tone that stops abruptly would be testing a phone's decay in all frequencies not just the frequency the tone was at.

If you've got a single cycle of an x kHz sine wave and you run it through FFT, you'll either have very poor resolution (1 millisecond for one 1 kHz cycle) or better (but not perfect) resolution with additional silent samples but still only 1 ms with actual sound data. But look at the amplitude plot with your eyes and you can easily determine the exact frequency of the sine wave + where it starts and stops. (I'm not at all sure what you mean, still.)

There are a couple of problems with this, two obvious ones being that a 1 kHz sine wave sampled at 44.1 kHz requires 44.1 samples for a single cycle and another being that if you zero-pad, which you have in order to get a usable result, the FFT won't like the discontinuity. The result will be anything but a single peak at 1 kHz.

Edited by xnor - 10/14/12 at 3:18pm

Two vintage on-ears.

It's not a matter of the FFT not 'liking' the discontinuity--it's just that a single cycle of a sine wave with silence before and after is a very different thing from a continuous sine tone and the difference in the FFT plot reflects that. But then you don't need me to tell you that lol

Xnor, Joebloggs: well put!

 

Case 1) Tone corresponds to an FFT bin and window is achieved by zero padding in the time domain:

 

If one applies a rectangular window to a tone in the time domain (meaning the tone abruptly starts and stops) then that corresponds, in the frequency domain, to convolving the FFT of a rectangular window (a sinc function) with the tone (a delta/single spike if the tone corresponds to an FFT bin.) Since the tone in this case is a delta, the result in the frequency domain is a sinc function whose peak occurs at the tone frequency. The lobe's width of the sinc is proportional to the size of the window. The larger the window (the more tone cycles in the time domain), the narrower the sinc...

 

Case 2) Tone does not correspond to an FFT bin:

 

One will get side lobes (instead of an delta in the frequency domain) when calculating the FFT of a tone, if the tone does not correspond to any of the FFT bin frequencies. Worst case scenario is if the tone frequency is half way between bins...

 

Hope this helps.

Edited by ultrabike - 10/14/12 at 9:43pm

FFT is kinda like the effect they use in certain films to blur out certain parts. Lower resolution = can't make out the fine details. I can't see how FFT having a weakness of resolution shows a fundamental difference between one cycle and many cycles in this case... Unless it's in the context of measurement or something.

Vid, it does not matter if we look at a discrete fourier transform or not. The fact remains that a single sine period is composed of non only that frequency but also a bunch of others that describe the discontinuity at the end of the period. Just like a square wave and saw tooth shape can be decomposed into fundamental plus harmonics...

I had a quick google on that and apparently we're talking about artifacts of digital sampling?

DFT is an invertible, discrete, linear mapping. There is no blurring. Zero-padding doesn't increase resolution, it just causes a higher interpolation-density in the other domain. You have to use more data (more cycles) to increase resolution.

 

So the resolution will be quite bad with a single cycle and windowing a single cycle will probably make the result unusable unless you use a very low dynamic range window function. It is the discontinuity (abrupt start and stop) that causes the undesirable effects we see. Like ultrabike wrote, it's like using a rectangular window.

 

 

What you could do is record a x kHz sine wave that stops abruptly and look at the last cycle and post-ringing in the time domain.

Edited by xnor - 10/15/12 at 5:35am

The topic can be seen as such because windowing is an inescapable fact of fft analysis. But fundamentally, xnor's definition is valid regardless of sampling or note: a single period sine + silence is effectively the convolution of a continuous pure tone with a rectangular window (1's during a single sine period, zeros after that) which frequency content is broad. Doesn't matter is we discretize this and take an fft, you can analytically derive the frequency content and get to a similar conclusion.

Just like a square wave, a rectangular window is composed of many frequencies. As was already mentioned, the extreme of a rectangular window is a pulse (a Dirac function with only zeroes except a single one) and it has effectively all frequencies present (flat line in the frequency domain).

When I used the word blur, I referred (clumsily) to the effect of pixelation, where you decrease the resolution of an image, increase it back up again and end up with a low-resolution version of the original picture - and it's akin to this that you get as output from FFT. I.e. frequency bins that span some given range of frequencies, dependent on the sampling size, and thus you get a low-resolution 'pixelated' 'image' of the frequency distribution, akin to a view of a crotch in a bashful film.

 

If you get an abrupt start and stop, surely that's a result of aliasing in sampling and not a physical property of a sine wave? I couldn't quite unpack the jargon in arnaud's explanation...

vid, have a read here if not already done, in particular the explanation on how Square Waves are constructed: http://www.innerfidelity.com/content/headphone-measurements-explained-square-wave-response . The same reasoning can be used to understand the broad frequency content of any abrupt transition, including the rectangular window necessary to stop the pure tone from ringing after 1 cycle.

Edited by arnaud - 10/15/12 at 7:26am