2, i'm sure, very simple questions about signal processing

hello,

(i'm a complete newbie to this type of thing). i have 2 qusetions. 1st:

on here: http://www.bores.com/courses/intro/freq/3_ft.htm  it says:


Jean Baptiste Fourier showed that any signal or waveform could be made
up just by adding together a series of pure tones (sine waves) with
appropriate amplitude and phase.

This is a rather startling theory, if you think about it. It means, for
instance, that by simply turning on a number of sine wave generators we
could sit back and enjoy a Beethoven symphony. 

Of course we would have to use a very large number of sine wave
generators, and we would have to turn them on at the time of the Big
Bang and leave them on until the heat death of the universe.


does phase relate to time/length/cutoff? otherwise, why doesn't it
mention something about the length of the pure tones in that first
paragraph? i'm not completely sure why that first sentance is supposed
to be startling. i suppose it's the idea that any collection of sounds
can, in theory, be completely synthesized -- made up by some tones.

and also why does it say "and we would have to turn them on at the time
of the Big Bang and leave them on until the heat death of the
universe."? what's the meaning of that? (is it maybe a case of
randomness? if you let all possible occurances happen randomly
continuously for ever then you'll at some point get something
particular, a Beethoven symphony in that example. otherwise i don't see
the point behind that sentance at all).



2nd question:

here: http://www.bores.com/courses/intro/freq/3_spect.htm  is a graph
of an audio signal, then the graph of its frequency spectrum below it.
obviously the signal (the top graph) has some duration -- time is the
horizontal axis of the signal graph. whereas the frequency spectrum has
no progression of time -- it's a single snapshot of a duration of time
showing the spread of frequencies that occured within that time.
(please correct me if i'm wrong anywhere). what i'm wondering is, is it
completely arbitrary (up to you the analyser) how much time each
frequency spectrum is a snapshot of? could the pictured frequency
spectrum on that page be either, for example (a) a snapshot of one tiny
say, 5%, segment of the displayed signal or (b) a snapshot of all the
duration displayed or (c) much more of the signal than is displayed? i
mean, if you have a digital signal, could you get a frequency spectrum
from just one, (i'm not sure what the word is -- with images it'd be a
pixel), one single sample? one single number/value from your signal
that is? what's the general deal with the length of the signal (or the
number of samples) in releation to getting a spectrum frequency graph
from it?


any pointers / info much appreciated.

thanks, ben.

log fire wrote:

> hello,
> 
> (i'm a complete newbie to this type of thing). i have 2 qusetions. 1st:
> 
> on here: http://www.bores.com/courses/intro/freq/3_ft.htm  it says:
> 
> 
> Jean Baptiste Fourier showed that any signal or waveform could be made
> up just by adding together a series of pure tones (sine waves) with
> appropriate amplitude and phase.
> 
> This is a rather startling theory, if you think about it. It means, for
> instance, that by simply turning on a number of sine wave generators we
> could sit back and enjoy a Beethoven symphony. 
> 
> Of course we would have to use a very large number of sine wave
> generators, and we would have to turn them on at the time of the Big
> Bang and leave them on until the heat death of the universe.
> 
> 
> does phase relate to time/length/cutoff? otherwise, why doesn't it
> mention something about the length of the pure tones in that first
> paragraph? i'm not completely sure why that first sentance is supposed
> to be startling. i suppose it's the idea that any collection of sounds
> can, in theory, be completely synthesized -- made up by some tones.
> 
> and also why does it say "and we would have to turn them on at the time
> of the Big Bang and leave them on until the heat death of the
> universe."? what's the meaning of that? (is it maybe a case of
> randomness? if you let all possible occurances happen randomly
> continuously for ever then you'll at some point get something
> particular, a Beethoven symphony in that example. otherwise i don't see
> the point behind that sentance at all).
> 
Actually the Author is incorrect in saying that you only have to go back 
as far as the big bang.

Fourier's theory is for really, really pure tones -- sine waves that 
start up at -infinity and continue on for +infinity.  In practice (since 
this is a DSP group) you could do your average 90 minute symphony with 
it's 20kHz bandwidth with (20kHz)(90 minute)(60 seconds/minute) = 108 
million tone generators, but it would start up again as soon as it left 
off unless you actually turn the tone generators off by yourself.

When you're doing this stuff in practice you have to take the difference 
between Fourier's theory, which requires time scales into positive and 
negative infinity, and reality, in which we are interested in events of 
finite duration with no foreshadowing, into account.  This takes quite a 
bit of getting used to -- just think on it really hard, and the brain 
cells that just don't get it will eventually die of exasperation.  Your 
remaining brain cells will expand to fill in the empty spaces but you 
will find yourself unable to chat up girls in bars (or boys, depending 
on your gender and preferences).
> 
> 
> 2nd question:
> 
> here: http://www.bores.com/courses/intro/freq/3_spect.htm  is a graph
> of an audio signal, then the graph of its frequency spectrum below it.
> obviously the signal (the top graph) has some duration -- time is the
> horizontal axis of the signal graph. whereas the frequency spectrum has
> no progression of time -- it's a single snapshot of a duration of time
> showing the spread of frequencies that occured within that time.
> (please correct me if i'm wrong anywhere). 

Once again, in Fourier's theory, the time-domain sample is for all time, 
and the frequency domain sample is for all frequencies.  Limiting the 
time domain sample to a finite time so you can do transforms on it with 
finite equipment is known as "windowing", and can be an issue in 
frequency domain analysis.

> what i'm wondering is, is it
> completely arbitrary (up to you the analyser) how much time each
> frequency spectrum is a snapshot of? could the pictured frequency
> spectrum on that page be either, for example (a) a snapshot of one tiny
> say, 5%, segment of the displayed signal or (b) a snapshot of all the
> duration displayed or (c) much more of the signal than is displayed? i
> mean, if you have a digital signal, could you get a frequency spectrum
> from just one, (i'm not sure what the word is -- with images it'd be a
> pixel), one single sample? one single number/value from your signal
> that is? what's the general deal with the length of the signal (or the
> number of samples) in releation to getting a spectrum frequency graph
> from it?
> 
-- and what windowing does is smear out the frequency spectrum.  So a 
window that's 1us wide will smear the spectrum out (roughly) by 1MHz, 
where a window that's 1 second wide will smear the spectrum out by 1Hz. 
  Depending on what you need to measure you need to make your window 
larger or smaller.
> 
> any pointers / info much appreciated.
> 
> thanks, ben.

What I've told you sounds obscure.  It's mostly because the subject _is_ 
obscure, but a very small part is because I'm being a smartass.  You 
_can_ wrap your brain around this, but expect it to take some work.

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Tim Wescott <tim@wescottnospamdesign.com> writes:
> [...]
> It's mostly because the subject
> _is_ obscure, but a very small part is because I'm being a smartass.

Worthy of a beer (or a Guiness, if you prefer) in my book,
Tim. Hopefully we'll meet someday so I can pay up.
-- 
%  Randy Yates                  % "My Shangri-la has gone away, fading like 
%% Fuquay-Varina, NC            %  the Beatles on 'Hey Jude'" 
%%% 919-577-9882                %  
%%%% <yates@ieee.org>           % 'Shangri-La', *A New World Record*, ELO
http://home.earthlink.net/~yatescr

Tim Wescott wrote:
> Actually the Author is incorrect in saying that you only have to go back 
> as far as the big bang.
> 
> Fourier's theory is for really, really pure tones -- sine waves that 
> start up at -infinity and continue on for +infinity.  In practice (since 
> this is a DSP group) you could do your average 90 minute symphony with 
> it's 20kHz bandwidth with (20kHz)(90 minute)(60 seconds/minute) = 108 
> million tone generators, but it would start up again as soon as it left 
> off unless you actually turn the tone generators off by yourself.

The longest symphony in the standard repertoire is Beethoven's ninth. 
Unless it is played exceptionally slowly, it is normally well under 80 
minutes. This was Philips rationale for choosing the size of a CD - no 
standard piece should ever be split. A friend who was involved said Sony 
wanted to make it 12" across, like an LP, but couldn't actually come up 
with any plausible material to fill one up. :-\

> [...] 
> What I've told you sounds obscure.  It's mostly because the subject _is_ 
> obscure, but a very small part is because I'm being a smartass.  You 
> _can_ wrap your brain around this, but expect it to take some work.

We can all be smartasses. The difference is not everyone has the 
intellect to back up the attitude :-)

Steve

Tim Wescott wrote:
...
> Actually the Author is incorrect in saying that you only have to go
back
> as far as the big bang.
>
> Fourier's theory is for really, really pure tones -- sine waves that
> start up at -infinity and continue on for +infinity.

Tim,

if I remember relativity theory correctly, the Big Bang (and, if our
universe is on a contracting trajectory) the Big Crunch are
singularities in both space and time. I guess one can immagine the Bang
and the Crunch warping infinite time at their space-time location. One
can think of it like the bilinear transform that warps infinity to
Fs/2. In that sense, the author's statement is correct.
Talking about smart ass :-).

Regards,
Andor

Not a "very large number", but an infinite number, consider
1Hz, 1.1Hz, 1.11Hz, 1.111Hz, 1.1111Hz, ad infinitum.

Oddly enough, you wouldn't need to have them from +/-
eternity, but only for the duration of the symphony, because they
all cancel out to zero at the times before, and the times after, the
performance of the symphony.

"log fire" <1234@12.12> wrote in message
news:160120050023482899%1234@12.12...
> Jean Baptiste Fourier showed that any signal or waveform could be made
> up just by adding together a series of pure tones (sine waves) with
> appropriate amplitude and phase.
> This is a rather startling theory, if you think about it. It means, for
> instance, that by simply turning on a number of sine wave generators we
> could sit back and enjoy a Beethoven symphony.
> Of course we would have to use a very large number of sine wave
> generators, and we would have to turn them on at the time of the Big
> Bang and leave them on until the heat death of the universe.

To the purist, any sinusoid in the analysis has to exist for
+/- eternity, and you are correct in that assumption.

The result of switching on and off, (a _MODULATION_ effect)
is that there are short-term transients that we ignore, but which
do form part of the spectrum.

"log fire" <1234@12.12> wrote in message
news:160120050023482899%1234@12.12...

> 2nd question:
> here: http://www.bores.com/courses/intro/freq/3_spect.htm  is a graph
> of an audio signal, then the graph of its frequency spectrum below it.
> obviously the signal (the top graph) has some duration -- time is the
> horizontal axis of the signal graph. whereas the frequency spectrum has
> no progression of time -- it's a single snapshot of a duration of time
> showing the spread of frequencies that occured within that time.
> (please correct me if i'm wrong anywhere). what i'm wondering is, is it
> completely arbitrary (up to you the analyser) how much time each
> frequency spectrum is a snapshot of? could the pictured frequency
> spectrum on that page be either, for example (a) a snapshot of one tiny
> say, 5%, segment of the displayed signal or (b) a snapshot of all the
> duration displayed or (c) much more of the signal than is displayed? i
> mean, if you have a digital signal, could you get a frequency spectrum
> from just one, (i'm not sure what the word is -- with images it'd be a
> pixel), one single sample? one single number/value from your signal
> that is? what's the general deal with the length of the signal (or the
> number of samples) in releation to getting a spectrum frequency graph
> from it?

In article <10uje7li9krug7b@corp.supernews.com>, Tim Wescott
<tim@wescottnospamdesign.com> wrote:


> > 
> Actually the Author is incorrect in saying that you only have to go back 
> as far as the big bang.

time didn't exist before the big bang apparently according to one
scientist i heard. so if you accept that, then you wouldn't be able to
go back before the big bang in any way what so ever. and nor could
sound waves.

> Fourier's theory is for really, really pure tones -- sine waves that 
> start up at -infinity and continue on for +infinity. 

but if you have all tones, lasting for ever, that'd be a continual mass
of unchanging sound. can these never ending tones change tone? go very
quiet/silent? change basically? otherwise there wouldn't be any change
in sound ever. just a massive loud countinual din of all possible
tones. everything always.


> > 
> -- and what windowing does is smear out the frequency spectrum.  So a 
> window that's 1us wide will smear the spectrum out (roughly) by 1MHz, 
> where a window that's 1 second wide will smear the spectrum out by 1Hz. 
>   Depending on what you need to measure you need to make your window 
> larger or smaller.

what's 'us'? like microsecond? no, one single sample maybe?

so are you saying the shorter your window to analyse using a fourier
transform the more compensatory / counter action must be taken? or, the
shorter your window, the more frequencies you'll actually miss? so it's
not really possible to get a frequency spectrum from one single sample?
(which i didn't really think it could be. that'd be like a whole
picture from one pixel).

would this be slightly correct: frequencies make themselves
known/visible (at least to fourier transform analysis) intermitently
(even though they're apparently, to us - our ears and brain, happening
continuously (for a fairly short period of time though)) and taking a
snapshot of a very short period of time will miss some of the
frequencies that are occuring throughout that moment; because you've
looked inbetween as it were?

frequencies are interleaved? so the shorter your window the more
frequencies that are occuring accross the time you're looking at you'll
miss?

log fire wrote:

> In article <10uje7li9krug7b@corp.supernews.com>, Tim Wescott
> <tim@wescottnospamdesign.com> wrote:
> 
> 
> 
>>Actually the Author is incorrect in saying that you only have to go back 
>>as far as the big bang.
> 
> 
> time didn't exist before the big bang apparently according to one
> scientist i heard. so if you accept that, then you wouldn't be able to
> go back before the big bang in any way what so ever. and nor could
> sound waves.
> 
> 
>>Fourier's theory is for really, really pure tones -- sine waves that 
>>start up at -infinity and continue on for +infinity. 
> 
> 
> but if you have all tones, lasting for ever, that'd be a continual mass
> of unchanging sound. can these never ending tones change tone? go very
> quiet/silent? change basically? otherwise there wouldn't be any change
> in sound ever. just a massive loud countinual din of all possible
> tones. everything always.
> 
Not really -- the Fourier transform shows us that the tones will cancel 
each other everywhere but one spot, if they're organized correctly.

The way one is usually expected to gain insight into this is to do the 
math on a bunch of signals.  This will cough up some severely 
counter-intuitive results indicating that either your math or your 
intuition needs adjustment.  After you verify your math you're left with 
adjusting your intuition.
> 
> 
>>-- and what windowing does is smear out the frequency spectrum.  So a 
>>window that's 1us wide will smear the spectrum out (roughly) by 1MHz, 
>>where a window that's 1 second wide will smear the spectrum out by 1Hz. 
>>  Depending on what you need to measure you need to make your window 
>>larger or smaller.
> 
> 
> what's 'us'? like microsecond? no, one single sample maybe?

Microsecond -- ASCII doesn't include a "mu", so one uses u.
> 
> so are you saying the shorter your window to analyse using a fourier
> transform the more compensatory / counter action must be taken? or, the
> shorter your window, the more frequencies you'll actually miss? so it's
> not really possible to get a frequency spectrum from one single sample?
> (which i didn't really think it could be. that'd be like a whole
> picture from one pixel).

The shorter your window the less you know about low-frequency events. 
If you're sampling data then the lower your sample rate the less you 
know about high-frequency events.

And yes, the spectrum from one sample is pretty boring.
> 
> would this be slightly correct: frequencies make themselves
> known/visible (at least to fourier transform analysis) intermitently
> (even though they're apparently, to us - our ears and brain, happening
> continuously (for a fairly short period of time though)) and taking a
> snapshot of a very short period of time will miss some of the
> frequencies that are occuring throughout that moment; because you've
> looked inbetween as it were?

More or less.
> 
> frequencies are interleaved? so the shorter your window the more
> frequencies that are occuring accross the time you're looking at you'll
> miss?

More or less.


-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

log fire wrote:

> In article <10uje7li9krug7b@corp.supernews.com>, Tim Wescott
> <tim@wescottnospamdesign.com> wrote:
> 
> 
>>Actually the Author is incorrect in saying that you only have to go back 
>>as far as the big bang.
> 
> 
> time didn't exist before the big bang apparently according to one
> scientist i heard. so if you accept that, then you wouldn't be able to
> go back before the big bang in any way what so ever. and nor could
> sound waves.
> 

A cautionary note:

History, math, physics, and cosmology view *extreme* time differently ;}

For times from NOW to seconds/years/decades/??? in past, they correspond 
nicely;} Otherwise, all bets off ,[

NOTE BENE Fourier transforms/series make/presume statements about *FUTURE*