Re: Test report MBP built-in audio device
Re: Test report MBP built-in audio device
- Subject: Re: Test report MBP built-in audio device
- From: Richard Dobson <email@hidden>
- Date: Tue, 26 Aug 2008 12:34:31 +0100
Mikael Hakman wrote:
..
Wow, that is just ... wrong.
Wow, it is wrong because . . . it is just wrong, right? Are you a priest?
It is more a case of wondering just where to start in explaining why it
is wrong. It takes a long time to write explanations! And the DFT is not
something that lends itself to short explanations. A lot easier just to
point you to a book or something.
Do you understand why windows are used?
Sure I do, and this is why I don't want to use them. Do you really
understand what DFT does? Or do you still believe that it computes
frequencies present in a signal?
It depends! We are waiting to hear what your understanding of the DFT
is. My impression is that you are using the DFT while thinking it is the
DTFT. Do you make use of zero-padding at all?
This is an attempt at a potted non-mathematical summary of the DFT:
The output of a DFT is a ~sampled~ spectrum which is periodic at
intervals of +-2PI all the way to +- infinity; each bin corresponds to
the central value of a frequency-domain sinc function. Each 'bin" is a
harmonic of the "fundamental of analysis" determined by the number of
points relative to the sample rate. Described non-mathematically, each
bin is the result of a form of pattern matching (a form of ring
modulation) between the internal representation of that (complex)
harmonic inside the DFT (aka basis functions) and the input sequence.
When a source component exactly fits the basis function only the central
sinc peak is non-zero, so we get the single spectral line.
There is a bandwidth associated with each bin (the filter-bank model of
the DFT), such that source components not aligned exactly with a bin
will still register strongly in that bin, but will also register in all
the other bins (remembering how the sinc function works) - this is the
dreaded spectral leakage, a direct consequence of using a short-time and
of course ~finite~ window (rectangular or otherwise). The closer the
components, the more samples we need to distinguish them. This is of
course intuitively obvious - we can only recognise a 1Hz beat when we
hear 1secs (at least) worth of sound. If we only inspect 20msecs of it,
we can be forgiven for thinking only one component is present. Any
time-varying element really messes up DFT outputs!
One of the functions that generates the most sidelobes is the step
function; which corresponds in principle to the "instrument startup"
that seems to give so many problems for you. Clearly, anything with the
steep almost-vertical slope of a transient must have a complex spectral
profile - even if the signal being started in this way is a mere
sinusoid. Worst-case it is a click - which as we know is broadband in
nature (and may be a source of aliasing). In the sonogram it will be a
vertical band potentially all the way to Nyquist.
Hann or similar windowing reduces the level of the sidelobes generated
by the rectangular window, so in that sense it increases accuracy in
that it unmasks lower-level frequency components, at the price of
widening the main lobe. In almost all applications, the gains offered by
windowing vastly outnumber the "advantages' of the rectangular window.
Of course a proper account of the DFT involves lots of maths and ideally
some pictures too, e.g.
http://www.complextoreal.com/fft3.htm
Richard Dobson
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