Strictly speaking, yes. Somewhat like the circumference of a circle. The 
basic maths of sinusoids (and Fourier theory) presumes they are infinite 
(witness e.g. all those integrals we see all the time between ± 
infinity). As soon as we add starts and stops (aka transients) we throw 
a mathematical spanner into the works (the sinusoid is multiplied by a 
step function or rectangular window). Of course ~with care~ we can 
minimise the consequences, or conveniently ignore it; but it remains the 
case that just as the circle has no beginning nor end, a sinusoid 
doesn't either.

Richard Dobson

On 27/09/2012 23:57, peiman khosravi wrote:
> Hello,
>
> I have a physics question. I have always been told that a pure sine-wave
> can only be pure if it has no beginning and end. Is this true?
>
> Thanks
> Peiman

Yes. In strict fourier decomposition all frequencies are of infinite
duration. Any finite signal is a sum of infinite signals that
selectively cancel.

In computer land we analyze small chunks, then we start with a
frequency with a wavelength identical to the chunk size, and we only
use frequencies that are even multiples of that frequency. So if your
window length is sr/1 samples, you can detect 1hz, 2hz, 3hz, 4hz, etc.
up to nyquist. If your window length is sr/10 samples, you can detect
10hz, 20hz ... If your window is sr/12 samples long you get 12hz 24hz
36hz etc. in your analysis.

On Thu, Sep 27, 2012 at 3:57 PM, peiman khosravi
 wrote:
> Hello,
>
> I have a physics question. I have always been told that a pure sine-wave can
> only be pure if it has no beginning and end. Is this true?
>
> Thanks
> Peiman

Yes. You detect exactly one frequency bin, nominally that frequency is
sr/2 (you have looped thanks to nyquist effect from sr/2 to 0 back up
to sr/2). It tells you the sum total energy of the signal . You can
get this data without conducting an fft: since your window is one
sample long, the value of the analysis is simply the value of the
sample.

On Thu, Sep 27, 2012 at 4:22 PM, peiman khosravi
 wrote:
>
>
> On 28 September 2012 00:09, Justin Smith  wrote:
>>
>> Yes. In strict fourier decomposition all frequencies are of infinite
>> duration. Any finite signal is a sum of infinite signals that
>> selectively cancel.
>>
>> In computer land we analyze small chunks, then we start with a
>> frequency with a wavelength identical to the chunk size, and we only
>> use frequencies that are even multiples of that frequency. So if your
>> window length is sr/1 samples, you can detect 1hz, 2hz, 3hz, 4hz, etc.
>> up to nyquist. If your window length is sr/10 samples, you can detect
>> 10hz, 20hz ... If your window is sr/12 samples long you get 12hz 24hz
>> 36hz etc. in your analysis.
>>
>
> Thanks.
> So if the window is sr/sr (one sample long) then you get a broadband noise
> that spans between 0 hz up to nyquist?
>
> P
>
>>
>> On Thu, Sep 27, 2012 at 3:57 PM, peiman khosravi
>>  wrote:
>> > Hello,
>> >
>> > I have a physics question. I have always been told that a pure sine-wave
>> > can
>> > only be pure if it has no beginning and end. Is this true?
>> >
>> > Thanks
>> > Peiman
>>
>>
>> Send bugs reports to the Sourceforge bug tracker
>>             https://sourceforge.net/tracker/?group_id=81968&atid=564599
>> Discussions of bugs and features can be posted here
>> To unsubscribe, send email sympa@lists.bath.ac.uk with body "unsubscribe
>> csound"
>>
>

on 2012-09-27 at 16:09 Justin Smith wrote:

>Yes. In strict fourier decomposition all frequencies are of infinite
>duration. Any finite signal is a sum of infinite signals that
>selectively cancel.

which brings an interesting paradox: sampling is impossible, because
finite signals aren't band-limited, and band-limited signals are
infinite...

Its not the right time of the day /night to continue this but:

Fourier analysis is predicated on the assumption of periodicity - when 
it "sees" a (windowed) single cycle it does not see discontinuities but 
a pattern matching an infinite sinusoid of that frequency. So in fact it 
is incorrect even to describe it as seeing a single cycle. It assumes 
that whatever it sees is periodic over the length.  When we create a 
single cycle we are in effect convolving the single-line spectrum of 
that infinite sinusoid with that of a rectangular window; and that 
spectrum of course is anything but a single line. As soon as the 
sinusoid itself fails to fit that window length ~exactly~, as is almost 
invariably the case, the result is massive spectral leakage (comparable 
to the worst possible choice of loop points in a sampler). Which is why 
we use Hamming windows etc ion spectral analysis.

The spectrum of a single impulse is flat from DC to Nyquist (or in 
analogue terms flat over all frequencies - infinite bandwidth); which is 
why it is valuable for system testing - literally the "impulse 
response". Conversely, the spectrum of the infinite-length sinusoid is a 
single line. Moving from one situation to the other defines the 
time/frequency tradeoff. The only reason we do not use impulses when 
~listening~ to a system is that there is essentially no acoustic energy 
in a single impulse; the ear has nothing to work with. Ears need white 
or pink noise over several seconds, computers can manage with the single 
impulse.

So - we do require a certain minimum amount of time to recognise a 
frequency, and one cycle is basically nothing like enough. We simply 
perceive the rms energy (and extended bandwidth) of what is a more or 
less extended pulse. So in that sense Wishart is not entirely correct - 
the analogy between Fourier Analysis and the ear is not exact. Fourier 
extends the waveform by assuming it is periodic, whereas our ears do not 
do that- we hear the single cycle much as he describes.

Where the analogy does fit better (though still not perfectly) is with 
respect to accurate pitch detection. Given a single cycle we can't get 
enough information to converge on the pitch; we need several of them. I 
think the psycho-acoustic lower limit is around 30msecs, but at that 
duration we really are not going to be sure if it was 440Hz or 445Hz. 
And given the assumption of an anechoic environment, that threshold 
duration might in fact need to be even longer.

Fourier analysis suffers from a very similar problem, which can be 
understood simply enough if we start with a single sample - there is no 
"instantaneous frequency" information at all. As we add samples, we very 
slowly start to get an idea of what the frequency might be. We are 
unlikely to be able to give a small fraction of a cycle to an FFT and 
get a believable result. The inevitable arithmetical approximations lead 
to a definable error range. This is thus the classic time/frequency 
tradeoff or uncertainty - to get a precise value for frequency we need 
~at least~ a full cycle, preferably more to minimise the error; but then 
we cannot in principle nail down the time more accurately than the 
duration of that cycle.

And so to bed...:-)

Richard Dobson

On 28/09/2012 00:38, peiman khosravi wrote:
> OK, I see. It all becomes clear! Thanks a lot.
>
> You see I'm a little confused by something Wishart says in "On Sonic Art":
>
> "[...] if we produce a single cycle of a sine wave [...] what we will in
> fact hear is a click, a sonic impulse whose frequency is maximally
> indeterminate. This is not [...] just a limitation of the ear. It is
> intrinsic to Fourier analysis itself".
>
> Is this a limitation of computation (digital) or a limitation Fourier?
> If the latter then why isn't a complete cycle enough?
>

> So - we do require a certain minimum amount of time to recognise a
> frequency, and one cycle is basically nothing like enough. We simply
> perceive the rms energy (and extended bandwidth) of what is a more or less
> extended pulse. So in that sense Wishart is not entirely correct - the
> analogy between Fourier Analysis and the ear is not exact. Fourier extends
> the waveform by assuming it is periodic, whereas our ears do not do that- we
> hear the single cycle much as he describes.

A single cycle of a sine wave sounds like a click because it has the
frequency spectrum of a click. A single cycle of a sine wave in
isolation does not have the same spectrum as the same wave repeated
indefinitely. In fact as you are probably aware bandwidth and duration
are inversely tied. The shorter the signal the wider the bandwidth. We
hear a click from one cycle of a sine wave because the frequency
content of a single cycle of a sine wave is effectively a bandlimited
impulse, containing energy throughout the frequency spectrum.

No, the FT will indeed recognise a single cycle. For the sake of 
relative simplicity: the FFT (which is what were are usually dealing 
with; a "special case" of the more general Discrete FT) matches the 
supplied waveform against a set of "basis functions" (whicha re 
themselves implicitly periodic), which are sinusoids at integral 
harmonics of the fundamental, whose wavelength is that of the window. 
So a single cycle in the window will be successfully matched to the 
fundamental, and appear in bin 1 (bin 0 represents any DC or additive 
offset present)

But when we try to listen to a single cycle (i.e. surrounded by 
silence), we have a completely different situation, which results in our 
hearing a click.

The clue to the issue of periodicity can be seen in the conversion of 
domain, from amplitude/time to amplitude/frequency; i.e. in the latter 
case there is no temporal dimension at all. An FT is in effect timeless, 
and this translates to the principle that one cycle is simply 
representative of an infinite number of identical ones. The circle 
remains a circle of the same shape however long we look at it from a 
fixed position. Another example is the use of the FT to analyse an 
impulse response of a system such as a filter. This is "linear 
time-invariant" (LTI)  - we get the same response whenever we inject the 
impulse, so the IR itself is strictly amp/freq, not anything/time.

Put yet another way; the temporal context of a FT is something ~we~ know 
(we know which bit of a waveform we are analysing), but this is not 
information contained within the FT itself. Conversely, if we are simply 
given the FT of "some waveform", we have no idea of where or when it was 
taken. A circle drawn 200 years ago is still a circle, still with the 
same properties. There is no temporal context in a FT at all. Of course, 
in a system such as the phase vocoder, the engine keeps track of the 
temporal evolution frame by frame, and knows how to glue the frames 
together correctly. An individual analysis frame (deriving from a single 
FFT) has no sense of its place in time; it is in principle identical for 
example if the whole sequence is played backwards.

But this description hides the nasty matter of phase accumulation which 
is where pvoc gets its name. Pvoc frames are not "pure" FFTs but 
conversions to amplitude/frequency where the (unwrapped) phase is 
tracked from frame to frame. In this sense a given frame contains within 
it the whole (averaged) history of the sound up to that point.

Occasionally this can hit us in the form of smearing and reverb effects, 
which one way and another are artifacts of "time aliasing". It is in the 
end a simple summation, and while we can add up N numbers to create 
another, we cannot perform the reverse process short of knowing what all 
the original numbers were to being with. There are other sets of numbers 
which could equally give the same result.  At best we can imagine that 
the history of the sound is so to speak "implicated" in each frame in 
the sense David Bohm uses that word (the "implicate state" of the 
universe); so "infolded" into the frame that unravelling it is not possible.

We can therefore say that our single wave cycle is equally temporally 
undefined - it is simply a shape which can exist anywhere, any time. If 
we choose to place it by itself in time, we have only ourselves to blame 
for the consequences!

Richard Dobson

On 28/09/2012 10:19, peiman khosravi wrote:
>..
>
> So do you mean in terms of Fourier transform a single cycle is not
> enough to detect a sinewave even if the cycle fits the window perfectly?
> Now I'm confused!
>

Thanks you very much Victor and Richard. This has clarified many
things for me. I think I'm going to enrol on a physics course soon and
study DSP properly.

Thanks again.
Peiman

On 28 September 2012 10:58, Richard Dobson
 wrote:
> No, the FT will indeed recognise a single cycle. For the sake of relative
> simplicity: the FFT (which is what were are usually dealing with; a "special
> case" of the more general Discrete FT) matches the supplied waveform against
> a set of "basis functions" (whicha re themselves implicitly periodic), which
> are sinusoids at integral harmonics of the fundamental, whose wavelength is
> that of the window. So a single cycle in the window will be successfully
> matched to the fundamental, and appear in bin 1 (bin 0 represents any DC or
> additive offset present)
>
> But when we try to listen to a single cycle (i.e. surrounded by silence), we
> have a completely different situation, which results in our hearing a click.
>
> The clue to the issue of periodicity can be seen in the conversion of
> domain, from amplitude/time to amplitude/frequency; i.e. in the latter case
> there is no temporal dimension at all. An FT is in effect timeless, and this
> translates to the principle that one cycle is simply representative of an
> infinite number of identical ones. The circle remains a circle of the same
> shape however long we look at it from a fixed position. Another example is
> the use of the FT to analyse an impulse response of a system such as a
> filter. This is "linear time-invariant" (LTI)  - we get the same response
> whenever we inject the impulse, so the IR itself is strictly amp/freq, not
> anything/time.
>
> Put yet another way; the temporal context of a FT is something ~we~ know (we
> know which bit of a waveform we are analysing), but this is not information
> contained within the FT itself. Conversely, if we are simply given the FT of
> "some waveform", we have no idea of where or when it was taken. A circle
> drawn 200 years ago is still a circle, still with the same properties. There
> is no temporal context in a FT at all. Of course, in a system such as the
> phase vocoder, the engine keeps track of the temporal evolution frame by
> frame, and knows how to glue the frames together correctly. An individual
> analysis frame (deriving from a single FFT) has no sense of its place in
> time; it is in principle identical for example if the whole sequence is
> played backwards.
>
> But this description hides the nasty matter of phase accumulation which is
> where pvoc gets its name. Pvoc frames are not "pure" FFTs but conversions to
> amplitude/frequency where the (unwrapped) phase is tracked from frame to
> frame. In this sense a given frame contains within it the whole (averaged)
> history of the sound up to that point.
>
> Occasionally this can hit us in the form of smearing and reverb effects,
> which one way and another are artifacts of "time aliasing". It is in the end
> a simple summation, and while we can add up N numbers to create another, we
> cannot perform the reverse process short of knowing what all the original
> numbers were to being with. There are other sets of numbers which could
> equally give the same result.  At best we can imagine that the history of
> the sound is so to speak "implicated" in each frame in the sense David Bohm
> uses that word (the "implicate state" of the universe); so "infolded" into
> the frame that unravelling it is not possible.
>
> We can therefore say that our single wave cycle is equally temporally
> undefined - it is simply a shape which can exist anywhere, any time. If we
> choose to place it by itself in time, we have only ourselves to blame for
> the consequences!
>
> Richard Dobson
>
>
> On 28/09/2012 10:19, peiman khosravi wrote:
>>
>> ..
>>
>> So do you mean in terms of Fourier transform a single cycle is not
>> enough to detect a sinewave even if the cycle fits the window perfectly?
>> Now I'm confused!
>>
>
>
>
> Send bugs reports to the Sourceforge bug tracker
>            https://sourceforge.net/tracker/?group_id=81968&atid=564599
> Discussions of bugs and features can be posted here
> To unsubscribe, send email sympa@lists.bath.ac.uk with body "unsubscribe
> csound"
>

I think I forgot to attach the picture before!

P

On 28 September 2012 16:02, peiman khosravi  wrote:
> Thanks you very much Victor and Richard. This has clarified many
> things for me. I think I'm going to enrol on a physics course soon and
> study DSP properly.
>
> Thanks again.
> Peiman
>
> On 28 September 2012 10:58, Richard Dobson
>  wrote:
>> No, the FT will indeed recognise a single cycle. For the sake of relative
>> simplicity: the FFT (which is what were are usually dealing with; a "special
>> case" of the more general Discrete FT) matches the supplied waveform against
>> a set of "basis functions" (whicha re themselves implicitly periodic), which
>> are sinusoids at integral harmonics of the fundamental, whose wavelength is
>> that of the window. So a single cycle in the window will be successfully
>> matched to the fundamental, and appear in bin 1 (bin 0 represents any DC or
>> additive offset present)
>>
>> But when we try to listen to a single cycle (i.e. surrounded by silence), we
>> have a completely different situation, which results in our hearing a click.
>>
>> The clue to the issue of periodicity can be seen in the conversion of
>> domain, from amplitude/time to amplitude/frequency; i.e. in the latter case
>> there is no temporal dimension at all. An FT is in effect timeless, and this
>> translates to the principle that one cycle is simply representative of an
>> infinite number of identical ones. The circle remains a circle of the same
>> shape however long we look at it from a fixed position. Another example is
>> the use of the FT to analyse an impulse response of a system such as a
>> filter. This is "linear time-invariant" (LTI)  - we get the same response
>> whenever we inject the impulse, so the IR itself is strictly amp/freq, not
>> anything/time.
>>
>> Put yet another way; the temporal context of a FT is something ~we~ know (we
>> know which bit of a waveform we are analysing), but this is not information
>> contained within the FT itself. Conversely, if we are simply given the FT of
>> "some waveform", we have no idea of where or when it was taken. A circle
>> drawn 200 years ago is still a circle, still with the same properties. There
>> is no temporal context in a FT at all. Of course, in a system such as the
>> phase vocoder, the engine keeps track of the temporal evolution frame by
>> frame, and knows how to glue the frames together correctly. An individual
>> analysis frame (deriving from a single FFT) has no sense of its place in
>> time; it is in principle identical for example if the whole sequence is
>> played backwards.
>>
>> But this description hides the nasty matter of phase accumulation which is
>> where pvoc gets its name. Pvoc frames are not "pure" FFTs but conversions to
>> amplitude/frequency where the (unwrapped) phase is tracked from frame to
>> frame. In this sense a given frame contains within it the whole (averaged)
>> history of the sound up to that point.
>>
>> Occasionally this can hit us in the form of smearing and reverb effects,
>> which one way and another are artifacts of "time aliasing". It is in the end
>> a simple summation, and while we can add up N numbers to create another, we
>> cannot perform the reverse process short of knowing what all the original
>> numbers were to being with. There are other sets of numbers which could
>> equally give the same result.  At best we can imagine that the history of
>> the sound is so to speak "implicated" in each frame in the sense David Bohm
>> uses that word (the "implicate state" of the universe); so "infolded" into
>> the frame that unravelling it is not possible.
>>
>> We can therefore say that our single wave cycle is equally temporally
>> undefined - it is simply a shape which can exist anywhere, any time. If we
>> choose to place it by itself in time, we have only ourselves to blame for
>> the consequences!
>>
>> Richard Dobson
>>
>>
>> On 28/09/2012 10:19, peiman khosravi wrote:
>>>
>>> ..
>>>
>>> So do you mean in terms of Fourier transform a single cycle is not
>>> enough to detect a sinewave even if the cycle fits the window perfectly?
>>> Now I'm confused!
>>>
>>
>>
>>
>> Send bugs reports to the Sourceforge bug tracker
>>            https://sourceforge.net/tracker/?group_id=81968&atid=564599
>> Discussions of bugs and features can be posted here
>> To unsubscribe, send email sympa@lists.bath.ac.uk with body "unsubscribe
>> csound"
>>

I picked up my copy of 'The Audio Programming Book' and started
reading Victor's chapter on Fourier transform, DFT and STFT. It's
really great!

Best,
Peiman

On 30 September 2012 13:54, J  wrote:
> Agreed, this one's a gem, and leaves me with a lot to wrap my head around!
>
> Jeremy
>
>
> On Sun, Sep 30, 2012 at 1:45 PM, Dr. Richard Boulanger
>  wrote:
>>
>> What a fantastic discussion.  Thank you all so much!
>>
>> -dB
>>
>> ------------------------------------------
>> Dr. Richard Boulanger, Ph.D.
>> rboulanger@berklee.edu
>> Professor of Electronic Production and Design
>> Professional Writing and Music Technology Division
>> Berklee College of Music
>> 617-747-2485 (office)  774-488-9166 (cell)
>> http://csounds.com/boulanger     http://csounds.com/mathews
>> http://boulangerlabs.com    http://csoundforlive.com   http://csounds.com
>>
>> -------------------------------------------------------------------------------------------------
>>
>> On Sep 28, 2012, at 11:02 AM, peiman khosravi 
>> wrote:
>>
>> Thanks you very much Victor and Richard. This has clarified many
>> things for me. I think I'm going to enrol on a physics course soon and
>> study DSP properly.
>>
>> Thanks again.
>> Peiman
>>
>> On 28 September 2012 10:58, Richard Dobson
>>  wrote:
>>
>> No, the FT will indeed recognise a single cycle. For the sake of relative
>> simplicity: the FFT (which is what were are usually dealing with; a
>> "special
>> case" of the more general Discrete FT) matches the supplied waveform
>> against
>> a set of "basis functions" (whicha re themselves implicitly periodic),
>> which
>> are sinusoids at integral harmonics of the fundamental, whose wavelength
>> is
>> that of the window. So a single cycle in the window will be successfully
>> matched to the fundamental, and appear in bin 1 (bin 0 represents any DC
>> or
>> additive offset present)
>>
>> But when we try to listen to a single cycle (i.e. surrounded by silence),
>> we
>> have a completely different situation, which results in our hearing a
>> click.
>>
>> The clue to the issue of periodicity can be seen in the conversion of
>> domain, from amplitude/time to amplitude/frequency; i.e. in the latter
>> case
>> there is no temporal dimension at all. An FT is in effect timeless, and
>> this
>> translates to the principle that one cycle is simply representative of an
>> infinite number of identical ones. The circle remains a circle of the same
>> shape however long we look at it from a fixed position. Another example is
>> the use of the FT to analyse an impulse response of a system such as a
>> filter. This is "linear time-invariant" (LTI)  - we get the same response
>> whenever we inject the impulse, so the IR itself is strictly amp/freq, not
>> anything/time.
>>
>> Put yet another way; the temporal context of a FT is something ~we~ know
>> (we
>> know which bit of a waveform we are analysing), but this is not
>> information
>> contained within the FT itself. Conversely, if we are simply given the FT
>> of
>> "some waveform", we have no idea of where or when it was taken. A circle
>> drawn 200 years ago is still a circle, still with the same properties.
>> There
>> is no temporal context in a FT at all. Of course, in a system such as the
>> phase vocoder, the engine keeps track of the temporal evolution frame by
>> frame, and knows how to glue the frames together correctly. An individual
>> analysis frame (deriving from a single FFT) has no sense of its place in
>> time; it is in principle identical for example if the whole sequence is
>> played backwards.
>>
>> But this description hides the nasty matter of phase accumulation which is
>> where pvoc gets its name. Pvoc frames are not "pure" FFTs but conversions
>> to
>> amplitude/frequency where the (unwrapped) phase is tracked from frame to
>> frame. In this sense a given frame contains within it the whole (averaged)
>> history of the sound up to that point.
>>
>> Occasionally this can hit us in the form of smearing and reverb effects,
>> which one way and another are artifacts of "time aliasing". It is in the
>> end
>> a simple summation, and while we can add up N numbers to create another,
>> we
>> cannot perform the reverse process short of knowing what all the original
>> numbers were to being with. There are other sets of numbers which could
>> equally give the same result.  At best we can imagine that the history of
>> the sound is so to speak "implicated" in each frame in the sense David
>> Bohm
>> uses that word (the "implicate state" of the universe); so "infolded" into
>> the frame that unravelling it is not possible.
>>
>> We can therefore say that our single wave cycle is equally temporally
>> undefined - it is simply a shape which can exist anywhere, any time. If we
>> choose to place it by itself in time, we have only ourselves to blame for
>> the consequences!
>>
>> Richard Dobson
>>
>>
>> On 28/09/2012 10:19, peiman khosravi wrote:
>>
>>
>> ..
>>
>> So do you mean in terms of Fourier transform a single cycle is not
>> enough to detect a sinewave even if the cycle fits the window perfectly?
>> Now I'm confused!
>>
>>
>>
>>
>> Send bugs reports to the Sourceforge bug tracker
>>           https://sourceforge.net/tracker/?group_id=81968&atid=564599
>> Discussions of bugs and features can be posted here
>> To unsubscribe, send email sympa@lists.bath.ac.uk with body "unsubscribe
>> csound"
>>
>>
>>
>> Send bugs reports to the Sourceforge bug tracker
>>            https://sourceforge.net/tracker/?group_id=81968&atid=564599
>> Discussions of bugs and features can be posted here
>> To unsubscribe, send email sympa@lists.bath.ac.uk with body "unsubscribe
>> csound"
>>
>>
>
>
>
> --
> www.jeremykeenan.info
> www.callandresponse.org.uk
> www.fromhoneytoashes.co.uk
> www.upsidedownumbrella.info
>

It sure is!

Sent from my iPhone.

On Sep 30, 2012, at 9:28 AM, peiman khosravi  wrote:

> I picked up my copy of 'The Audio Programming Book' and started
> reading Victor's chapter on Fourier transform, DFT and STFT. It's
> really great!
> 
> Best,
> Peiman
> 
> On 30 September 2012 13:54, J  wrote:
>> Agreed, this one's a gem, and leaves me with a lot to wrap my head around!
>> 
>> Jeremy
>> 
>> 
>> On Sun, Sep 30, 2012 at 1:45 PM, Dr. Richard Boulanger
>>  wrote:
>>> 
>>> What a fantastic discussion.  Thank you all so much!
>>> 
>>> -dB
>>> 
>>> ------------------------------------------
>>> Dr. Richard Boulanger, Ph.D.
>>> rboulanger@berklee.edu
>>> Professor of Electronic Production and Design
>>> Professional Writing and Music Technology Division
>>> Berklee College of Music
>>> 617-747-2485 (office)  774-488-9166 (cell)
>>> http://csounds.com/boulanger     http://csounds.com/mathews
>>> http://boulangerlabs.com    http://csoundforlive.com   http://csounds.com
>>> 
>>> -------------------------------------------------------------------------------------------------
>>> 
>>> On Sep 28, 2012, at 11:02 AM, peiman khosravi 
>>> wrote:
>>> 
>>> Thanks you very much Victor and Richard. This has clarified many
>>> things for me. I think I'm going to enrol on a physics course soon and
>>> study DSP properly.
>>> 
>>> Thanks again.
>>> Peiman
>>> 
>>> On 28 September 2012 10:58, Richard Dobson
>>>  wrote:
>>> 
>>> No, the FT will indeed recognise a single cycle. For the sake of relative
>>> simplicity: the FFT (which is what were are usually dealing with; a
>>> "special
>>> case" of the more general Discrete FT) matches the supplied waveform
>>> against
>>> a set of "basis functions" (whicha re themselves implicitly periodic),
>>> which
>>> are sinusoids at integral harmonics of the fundamental, whose wavelength
>>> is
>>> that of the window. So a single cycle in the window will be successfully
>>> matched to the fundamental, and appear in bin 1 (bin 0 represents any DC
>>> or
>>> additive offset present)
>>> 
>>> But when we try to listen to a single cycle (i.e. surrounded by silence),
>>> we
>>> have a completely different situation, which results in our hearing a
>>> click.
>>> 
>>> The clue to the issue of periodicity can be seen in the conversion of
>>> domain, from amplitude/time to amplitude/frequency; i.e. in the latter
>>> case
>>> there is no temporal dimension at all. An FT is in effect timeless, and
>>> this
>>> translates to the principle that one cycle is simply representative of an
>>> infinite number of identical ones. The circle remains a circle of the same
>>> shape however long we look at it from a fixed position. Another example is
>>> the use of the FT to analyse an impulse response of a system such as a
>>> filter. This is "linear time-invariant" (LTI)  - we get the same response
>>> whenever we inject the impulse, so the IR itself is strictly amp/freq, not
>>> anything/time.
>>> 
>>> Put yet another way; the temporal context of a FT is something ~we~ know
>>> (we
>>> know which bit of a waveform we are analysing), but this is not
>>> information
>>> contained within the FT itself. Conversely, if we are simply given the FT
>>> of
>>> "some waveform", we have no idea of where or when it was taken. A circle
>>> drawn 200 years ago is still a circle, still with the same properties.
>>> There
>>> is no temporal context in a FT at all. Of course, in a system such as the
>>> phase vocoder, the engine keeps track of the temporal evolution frame by
>>> frame, and knows how to glue the frames together correctly. An individual
>>> analysis frame (deriving from a single FFT) has no sense of its place in
>>> time; it is in principle identical for example if the whole sequence is
>>> played backwards.
>>> 
>>> But this description hides the nasty matter of phase accumulation which is
>>> where pvoc gets its name. Pvoc frames are not "pure" FFTs but conversions
>>> to
>>> amplitude/frequency where the (unwrapped) phase is tracked from frame to
>>> frame. In this sense a given frame contains within it the whole (averaged)
>>> history of the sound up to that point.
>>> 
>>> Occasionally this can hit us in the form of smearing and reverb effects,
>>> which one way and another are artifacts of "time aliasing". It is in the
>>> end
>>> a simple summation, and while we can add up N numbers to create another,
>>> we
>>> cannot perform the reverse process short of knowing what all the original
>>> numbers were to being with. There are other sets of numbers which could
>>> equally give the same result.  At best we can imagine that the history of
>>> the sound is so to speak "implicated" in each frame in the sense David
>>> Bohm
>>> uses that word (the "implicate state" of the universe); so "infolded" into
>>> the frame that unravelling it is not possible.
>>> 
>>> We can therefore say that our single wave cycle is equally temporally
>>> undefined - it is simply a shape which can exist anywhere, any time. If we
>>> choose to place it by itself in time, we have only ourselves to blame for
>>> the consequences!
>>> 
>>> Richard Dobson
>>> 
>>> 
>>> On 28/09/2012 10:19, peiman khosravi wrote:
>>> 
>>> 
>>> ..
>>> 
>>> So do you mean in terms of Fourier transform a single cycle is not
>>> enough to detect a sinewave even if the cycle fits the window perfectly?
>>> Now I'm confused!
>>> 
>>> 
>>> 
>>> 
>>> Send bugs reports to the Sourceforge bug tracker
>>>          https://sourceforge.net/tracker/?group_id=81968&atid=564599
>>> Discussions of bugs and features can be posted here
>>> To unsubscribe, send email sympa@lists.bath.ac.uk with body "unsubscribe
>>> csound"
>>> 
>>> 
>>> 
>>> Send bugs reports to the Sourceforge bug tracker
>>>           https://sourceforge.net/tracker/?group_id=81968&atid=564599
>>> Discussions of bugs and features can be posted here
>>> To unsubscribe, send email sympa@lists.bath.ac.uk with body "unsubscribe
>>> csound"
>> 
>> 
>> 
>> --
>> www.jeremykeenan.info
>> www.callandresponse.org.uk
>> www.fromhoneytoashes.co.uk
>> www.upsidedownumbrella.info
> 
> 
> Send bugs reports to the Sourceforge bug tracker
>            https://sourceforge.net/tracker/?group_id=81968&atid=564599
> Discussions of bugs and features can be posted here
> To unsubscribe, send email sympa@lists.bath.ac.uk with body "unsubscribe csound"
> 

And I have my copy of the Csound sources... ;)

On 9/30/12, J  wrote:
> Cheers, I'll crack mine open, I've not had enough time to properly check it
> out, this is giving me some motivation to do so!
>
> Jeremy
> On Sun, Sep 30, 2012 at 2:28 PM, peiman khosravi
> wrote:
>
>> I picked up my copy of 'The Audio Programming Book' and started
>> reading Victor's chapter on Fourier transform, DFT and STFT. It's
>> really great!
>>
>> Best,
>> Peiman
>>
>> On 30 September 2012 13:54, J  wrote:
>> > Agreed, this one's a gem, and leaves me with a lot to wrap my head
>> around!
>> >
>> > Jeremy
>> >
>> >
>> > On Sun, Sep 30, 2012 at 1:45 PM, Dr. Richard Boulanger
>> >  wrote:
>> >>
>> >> What a fantastic discussion.  Thank you all so much!
>> >>
>> >> -dB
>> >>
>> >> ------------------------------------------
>> >> Dr. Richard Boulanger, Ph.D.
>> >> rboulanger@berklee.edu
>> >> Professor of Electronic Production and Design
>> >> Professional Writing and Music Technology Division
>> >> Berklee College of Music
>> >> 617-747-2485 (office)  774-488-9166 (cell)
>> >> http://csounds.com/boulanger     http://csounds.com/mathews
>> >> http://boulangerlabs.com    http://csoundforlive.com
>> http://csounds.com
>> >>
>> >>
>> -------------------------------------------------------------------------------------------------
>> >>
>> >> On Sep 28, 2012, at 11:02 AM, peiman khosravi
>> >> > >
>> >> wrote:
>> >>
>> >> Thanks you very much Victor and Richard. This has clarified many
>> >> things for me. I think I'm going to enrol on a physics course soon and
>> >> study DSP properly.
>> >>
>> >> Thanks again.
>> >> Peiman
>> >>
>> >> On 28 September 2012 10:58, Richard Dobson
>> >>  wrote:
>> >>
>> >> No, the FT will indeed recognise a single cycle. For the sake of
>> relative
>> >> simplicity: the FFT (which is what were are usually dealing with; a
>> >> "special
>> >> case" of the more general Discrete FT) matches the supplied waveform
>> >> against
>> >> a set of "basis functions" (whicha re themselves implicitly periodic),
>> >> which
>> >> are sinusoids at integral harmonics of the fundamental, whose
>> >> wavelength
>> >> is
>> >> that of the window. So a single cycle in the window will be
>> >> successfully
>> >> matched to the fundamental, and appear in bin 1 (bin 0 represents any
>> >> DC
>> >> or
>> >> additive offset present)
>> >>
>> >> But when we try to listen to a single cycle (i.e. surrounded by
>> silence),
>> >> we
>> >> have a completely different situation, which results in our hearing a
>> >> click.
>> >>
>> >> The clue to the issue of periodicity can be seen in the conversion of
>> >> domain, from amplitude/time to amplitude/frequency; i.e. in the latter
>> >> case
>> >> there is no temporal dimension at all. An FT is in effect timeless,
>> >> and
>> >> this
>> >> translates to the principle that one cycle is simply representative of
>> an
>> >> infinite number of identical ones. The circle remains a circle of the
>> same
>> >> shape however long we look at it from a fixed position. Another
>> >> example
>> is
>> >> the use of the FT to analyse an impulse response of a system such as a
>> >> filter. This is "linear time-invariant" (LTI)  - we get the same
>> response
>> >> whenever we inject the impulse, so the IR itself is strictly amp/freq,
>> not
>> >> anything/time.
>> >>
>> >> Put yet another way; the temporal context of a FT is something ~we~
>> >> know
>> >> (we
>> >> know which bit of a waveform we are analysing), but this is not
>> >> information
>> >> contained within the FT itself. Conversely, if we are simply given the
>> FT
>> >> of
>> >> "some waveform", we have no idea of where or when it was taken. A
>> >> circle
>> >> drawn 200 years ago is still a circle, still with the same properties.
>> >> There
>> >> is no temporal context in a FT at all. Of course, in a system such as
>> the
>> >> phase vocoder, the engine keeps track of the temporal evolution frame
>> >> by
>> >> frame, and knows how to glue the frames together correctly. An
>> individual
>> >> analysis frame (deriving from a single FFT) has no sense of its place
>> >> in
>> >> time; it is in principle identical for example if the whole sequence
>> >> is
>> >> played backwards.
>> >>
>> >> But this description hides the nasty matter of phase accumulation
>> >> which
>> is
>> >> where pvoc gets its name. Pvoc frames are not "pure" FFTs but
>> conversions
>> >> to
>> >> amplitude/frequency where the (unwrapped) phase is tracked from frame
>> >> to
>> >> frame. In this sense a given frame contains within it the whole
>> (averaged)
>> >> history of the sound up to that point.
>> >>
>> >> Occasionally this can hit us in the form of smearing and reverb
>> >> effects,
>> >> which one way and another are artifacts of "time aliasing". It is in
>> >> the
>> >> end
>> >> a simple summation, and while we can add up N numbers to create
>> >> another,
>> >> we
>> >> cannot perform the reverse process short of knowing what all the
>> original
>> >> numbers were to being with. There are other sets of numbers which
>> >> could
>> >> equally give the same result.  At best we can imagine that the history
>> of
>> >> the sound is so to speak "implicated" in each frame in the sense David
>> >> Bohm
>> >> uses that word (the "implicate state" of the universe); so "infolded"
>> into
>> >> the frame that unravelling it is not possible.
>> >>
>> >> We can therefore say that our single wave cycle is equally temporally
>> >> undefined - it is simply a shape which can exist anywhere, any time.
>> >> If
>> we
>> >> choose to place it by itself in time, we have only ourselves to blame
>> for
>> >> the consequences!
>> >>
>> >> Richard Dobson
>> >>
>> >>
>> >> On 28/09/2012 10:19, peiman khosravi wrote:
>> >>
>> >>
>> >> ..
>> >>
>> >> So do you mean in terms of Fourier transform a single cycle is not
>> >> enough to detect a sinewave even if the cycle fits the window
>> >> perfectly?
>> >> Now I'm confused!
>> >>
>> >>
>> >>
>> >>
>> >> Send bugs reports to the Sourceforge bug tracker
>> >>           https://sourceforge.net/tracker/?group_id=81968&atid=564599
>> >> Discussions of bugs and features can be posted here
>> >> To unsubscribe, send email sympa@lists.bath.ac.uk with body
>> "unsubscribe
>> >> csound"
>> >>
>> >>
>> >>
>> >> Send bugs reports to the Sourceforge bug tracker
>> >>            https://sourceforge.net/tracker/?group_id=81968&atid=564599
>> >> Discussions of bugs and features can be posted here
>> >> To unsubscribe, send email sympa@lists.bath.ac.uk with body
>> "unsubscribe
>> >> csound"
>> >>
>> >>
>> >
>> >
>> >
>> > --
>> > www.jeremykeenan.info
>> > www.callandresponse.org.uk
>> > www.fromhoneytoashes.co.uk
>> > www.upsidedownumbrella.info
>> >
>>
>>
>> Send bugs reports to the Sourceforge bug tracker
>>             https://sourceforge.net/tracker/?group_id=81968&atid=564599
>> Discussions of bugs and features can be posted here
>> To unsubscribe, send email sympa@lists.bath.ac.uk with body "unsubscribe
>> csound"
>>
>>
>
>
> --
> www.jeremykeenan.info
> www.callandresponse.org.uk
> www.fromhoneytoashes.co.uk
> www.upsidedownumbrella.info
>
> Send bugs reports to the Sourceforge bug tracker
>             https://sourceforge.net/tracker/?group_id=81968&atid=564599
> Discussions of bugs and features can be posted here
> To unsubscribe, send email sympa@lists.bath.ac.uk with body "unsubscribe
> csound"
>
>