Hi list, i'm trying to study fft in Pd and still don't understand the numbers produced by the analysis. I'm on Pd "vanilla" 0-40.2 in debian. I just patch this:
[float]
|
|
[osc~]
|
| [bang]
| /
[rfft~] / |
\ \ |
\ / \ |
\ / [print~ imaginary]
[print~ real]
and then type "0" and "1" in float box and this analysis came to terminal:
real:
-39.604 0 0 0 0 0
0 0
0 0 0 0 0 0
0 0
0 0 0 0 0 0
0 0
0 0 0 0 0 0
0 0
0 0 0 0 0 0
0 0
0 0 0 0 0 0
0 0
0 0 0 0 0 0
0 0
0 0 0 0 0 0
0 0
imaginary:
0 -0 -0 -0 -0 -0
-0 -0
-0 -0 -0 -0 -0 -0
-0 -0
-0 -0 -0 -0 -0 -0
-0 -0
-0 -0 -0 -0 -0 -0
-0 -0
0 0 0 0 0 0
0 0
0 0 0 0 0 0
0 0
0 0 0 0 0 0
0 0
0 0 0 0 0 0
0 0
real:
-39.378 -0.003371 -0.0035941 -0.0035693 -0.0035941
-0.0035851 -0.0035941 -0.0035895
-0.003594 -0.0035913 -0.0035941 -0.0035921 -0.0035932
-0.0035928 -0.003594 -0.0035931
-0.0035944 -0.0035932 -0.0035941 -0.0035934 -0.0035943
-0.0035935 -0.0035942 -0.0035938
-0.0035943 -0.0035935 -0.003594 -0.0035933 -0.0035941
-0.0035936 -0.0035942 -0.0035935
-0.0035934 0 0 0 0 0
0 0
0 0 0 0 0 0
0 0
0 0 0 0 0 0
0 0
0 0 0 0 0 0
0 0
imaginary:
0 0.07316 0.036492 0.02423 0.018069 0.014348
0.011848 0.010045
0.0086772 0.007599 0.0067242 0.0059963 0.0053794
0.0048461 0.0043795 0.0039655
0.0035944 0.0032576 0.0029496 0.0026656 0.0024016
0.0021543 0.0019211 0.0016999
0.0014889 0.0012859 0.00109 0.00090034 0.00071485
0.00053315 0.00035397 0.00017647
0 0 0 0 0 0
0 0
0 0 0 0 0 0
0 0
0 0 0 0 0 0
0 0
0 0 0 0 0 0
0 0
What this numbers represents ? Where i can study more? thanks
cristiano
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Hallo, cristiano figueirÿfffff3 hat gesagt: // cristiano figueirÿfffff3 wrote:
Hi list, i'm trying to study fft in Pd and still don't understand the numbers produced by the analysis.
You maybe want to start with my "Beginner's Guide to the FFT-objects in Pd" at http://footils.org/cms/show/60 for a very informal introduciton, and then also read the URLs linked from there. And Miller's "Techniques ..." book is good as well, but a bit more advanced.
Ciao
Frank Barknecht _ ______footils.org__
so, i don't know too much about fft, but here's what i can gather. hopefully some people who actually understand it can explain where i am wrong:
fft analyzes a 'block' of samples. a block is just a section of consecutive samples. pd's default block size is 64 samples, which is why you get 64 numbers for each FFT analysis in pd with default block size.
the maximum frequency fft analysis will give you is half of your sampling rate.
for a block size of 2, the only frequency FFT tells you about is half the samplerate (SR/2)
for a block size of 4, FFT can then tell you about SR/4 and 2xSR/4..so, for a sampling rate of 44100hz, you'd get info on 11025 and 22050 hz
for a block size of 8, you can now find out about 4 frequencies: SR/8, 2xSR/8, 3xSR/8, 4xSR/8 (ie, 5512, 11025, 16538, 22050)
for a block size of 16, you can find out about SR/16, 2xSR/16, 3xSR/16, 4xSR/16, 5xSR/16, 6xSR/16, 7xSR/16, and 8xSR/16
and so on...always until you reach half the sample rate. ..i think you also get info on 0hz, and i guess that is also somehow important when it comes to resynthesis, but i'm not too sure if or how or why.
once you get up to a block of 64 or 128 samples or more, you start to get pretty good info on the basic waves which make up that block of samples.
i'm really not sure about this next bit...but i think the reason why you get a real part and an imaginary part is because the analysis finds the angle between one sample and the next one. ..if it's a steep angle, then that's a high frequency...if it's a flatter angle, then that's a lower frequency.
and the way to find an angle is derived from pythagoras' equation....a^2+b^2=c^2
so...that works fine when the samples lie above the zero axis...but when they go below the zero axis, then to get the angle you have to take the square route of a negative number. hence the imaginary part.
i could be way off the mark here,.,,,but that's just what i could guess.
pd uber pro's.. how wrong am I?
Hallo, hard off hat gesagt: // hard off wrote:
i'm really not sure about this next bit...but i think the reason why you get a real part and an imaginary part is because the analysis finds the angle between one sample and the next one. ..if it's a steep angle, then that's a high frequency...if it's a flatter angle, then that's a lower frequency.
That's not quite correct. The data, that the FFT gives you, not only has the frequency content of a block encoded, but also the phase. In my little tutorial I compare it to a bank of [osc~] objects that are tuned in fixed integer multiples of SR/blocksize from 0 to SR/2, where FFT gives you the amplitude of each osc~ plus its phase (second inlet of [osc~]).
So we're dealing not with single numbers here, but with pairs of numbers that specify how loud and at which phase the osc~ sounds. Pairs of numbers are like coordinates into a 2-dimensional space. They could be cartesian (x and y) or polar (angle and radius). Complex numbers are just pairs of numbers as well. FFT gives out cartesian coordinates (re = x, im = y) that are connected to phase and magnitude, if you view the latter as the respective polar coordinates for that fft-bin.
To convert from cartesian to polar you can use:
magn = sqrt(re^2 + im^2) <-- cf. pythagoras here phs = arctan(im/re)
You can convert back as well using similar formula.
Why does FFT give cartesian coordinates? Well, as you can convert between both representations it could give polar directly as well. (You could make a little abstractions to get a polar [fft~] object.) For some calculations polar, for others cartesian coordinates are easier to use. To quote Miller:
The main reason we use complex numbers in electronic music is because they magically automate trigonometric calculations. We frequently have to add angles together in order to talk about the changing phase of an audio signal as time progresses (or as it is shifted in time, as in this chapter). It turns out that, if you multiply two complex numbers, the argument of the product is the sum of the arguments of the two factors.
http://crca.ucsd.edu/~msp/techniques/latest/book-html/node105.html
"Argument" is the angle here.
Ciao
Frank Barknecht _ ______footils.org__
On Nov 20, 2007 5:39 PM, Frank Barknecht fbar@footils.org wrote:
For some calculations polar, for others cartesian coordinates are easier to use. To quote Miller:
The main reason we use complex numbers in electronic music is because they magically automate trigonometric calculations. We frequently have to add angles together in order to talk about the changing phase of an audio signal as time progresses (or as it is shifted in time, as in this chapter). It turns out that, if you multiply two complex numbers, the argument of the product is the sum of the arguments of the two factors.
I still don't exactly understand why one couldn't just use (x, y) vectors; why the y value has to be multiplied by something imaginary. I mean, i/j is *defined as* the square root of -1, but it can't really *be* the square root of -1... I've accepted it and moved on to more practical questions, but that is still mysterious for me.
-Chuckk
Chuckk Hubbard wrote:
On Nov 20, 2007 5:39 PM, Frank Barknecht fbar@footils.org wrote:
For some calculations polar, for others cartesian coordinates are easier to use. To quote Miller:
The main reason we use complex numbers in electronic music is because they magically automate trigonometric calculations. We frequently have to add angles together in order to talk about the changing phase of an audio signal as time progresses (or as it is shifted in time, as in this chapter). It turns out that, if you multiply two complex numbers, the argument of the product is the sum of the arguments of the two factors.
I still don't exactly understand why one couldn't just use (x, y) vectors; why the y value has to be multiplied by something imaginary. I mean, i/j is *defined as* the square root of -1, but it can't really *be* the square root of -1... I've accepted it and moved on to more practical questions, but that is still mysterious for me.
Everything in mathematics *is* so if *definined*. You could define your own mathematics by saying "there are no two straight lines that do not subtend" (no parallels). In fact, Riemann did so, and has created a geometry different from Euklidean, which is the basis for calculation on a sphere (straight lines -> great circles).
http://en.wikipedia.org/wiki/Non-Euclidean_geometry
It does not matter, if it stresses your personal "experience", as long as it does not contradict other definitions in your mathematics. Negative numbers were a great problem to mathematicians as well:
European mathematicians however, for the most part, resisted the concept of negative numbers until the 17th century, although Fibonacci allowed negative solutions in financial problems where they could be interpreted as debits (chapter 13 of Liber Abaci, 1202) and later as losses (in Flos). At the same time, the Chinese were indicating negative numbers by drawing a diagonal stroke through the right-most non-zero digit. The first use of negative numbers in a European work was by Chuquet during the 15th century. He used them as exponents, but referred to them as “absurd numbers”.
http://en.wikipedia.org/wiki/Negative_and_non-negative_numbers#First_usage_of_negative_numbers
cu Thomas
"Prisons are needed only to provide the illusion that courts and police are effective. They're a kind of job insurance." (Leto II. in: Frank Herbert, God Emperor of Dune) http://thomas.dergrossebruder.org/
Hallo, Chuckk Hubbard hat gesagt: // Chuckk Hubbard wrote:
I still don't exactly understand why one couldn't just use (x, y) vectors;
You *are* just using (x,y) rsp. (angle, radius) vectors. You can completely ignore the meaning of i as sqrt(-1) and just view "i" as the funny name of the second axis in a 2d-coordinate system. Instead of calling one axis the x-axis and the other the y-axis you call one axis the real and the other the imaginary axis. That's all there is to it. It's nothing complex (pun intended).
Ciao
Frank Barknecht _ ______footils.org__
On Tue, 20 Nov 2007, Frank Barknecht wrote:
I still don't exactly understand why one couldn't just use (x, y) vectors;
You *are* just using (x,y) rsp. (angle, radius) vectors. You can completely ignore the meaning of i as sqrt(-1) and just view "i" as the funny name of the second axis in a 2d-coordinate system.
You can see "i" as a 90-degree rotation, that turns x-axis values into y-axis values, and y-axis values into negated x-axis values, and so on.
_ _ __ ___ _____ ________ _____________ _____________________ ... | Mathieu Bouchard - tél:+1.514.383.3801, Montréal QC Canada
On Tue, 20 Nov 2007, Chuckk Hubbard wrote:
I still don't exactly understand why one couldn't just use (x, y) vectors; why the y value has to be multiplied by something imaginary.
The complex product, (a+bi)(c+di) = (ac-bd) + (ad+bc)i, is something that can be done using vectors, and complex numbers can be thought as an extension of 2-D vectors by a product like vector*vector=vector. I believe that complex numbers are easier to use than 2-D vectors for that kind of thing, but 2-D vectors are still what we used in 12th and 13th grade physics because they were too chicken to teach us complex numbers.
I mean, i/j is *defined as* the square root of -1, but it can't really *be* the square root of -1... I've accepted it and moved on to more practical questions, but that is still mysterious for me.
This is called an algebraic extension. Another example of algebraic extension is when going from numbers to polynomials. If you take polynomials in x and force x^2 to be equal to -1 then you end up with a system of linear equations that work exactly like complex numbers.
"forcing x^2 = -1" is really a modulo, just like forcing 13=0 is called modulo 13. The complex numbers are polynomials modulo x^2+1.
A more natural way of thinking of complex numbers is that you start with real numbers and want all square roots to exist. Negative numbers started as a way to make all subtractions exist. Fractions started as a way to make all nonzero divisions exist. Why not square root, too?
_ _ __ ___ _____ ________ _____________ _____________________ ... | Mathieu Bouchard - tél:+1.514.383.3801, Montréal QC Canada
Hi, thanks for help, now i'm begining to understand :)
hard off:
"pd's default block size is 64 samples, which is why you get 64 numbers for each FFT analysis in pd with default block size."
ok, that i understand, i'll make some tests with this value...to modify the default i call the block~ object rigth?
"Complex numbers are just pairs of numbers as well. FFT gives out cartesian coordinates (re = x, im = y) that are connected to phase and magnitude, if you view the latter as the respective polar coordinates for that fft-bin."
So, each pair is one number from real and the respective from imaginary? (the first from real and first from imag. for example?).
In my example, when i first select the frequency "0"(hertz) for my osc~ (who is gonna be analyzed) why this number (-39.604) appears?
real:
-39.604 0 0 0 0 0
0 0
0 0 0 0 0 0
0 0
0 0 0 0 0 0
0 0
0 0 0 0 0 0
0 0
0 0 0 0 0 0
0 0
0 0 0 0 0 0
0 0
0 0 0 0 0 0
0 0
0 0 0 0 0 0
0 0
imaginary:
0 -0 -0 -0 -0 -0
-0 -0
-0 -0 -0 -0 -0 -0
-0 -0
-0 -0 -0 -0 -0 -0
-0 -0
-0 -0 -0 -0 -0 -0
-0 -0
0 0 0 0 0 0
0 0
0 0 0 0 0 0
0 0
0 0 0 0 0 0
0 0
0 0 0 0 0 0
0 0
thanks cristiano
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Hallo, cristiano figueirÿfffff3 hat gesagt: // cristiano figueirÿfffff3 wrote:
ok, that i understand, i'll make some tests with this value...to modify the default i call the block~ object rigth?
Yes, just use [block~ 8] or so to get less output.
"Complex numbers are just pairs of numbers as well. FFT gives out cartesian coordinates (re = x, im = y) that are connected to phase and magnitude, if you view the latter as the respective polar coordinates for that fft-bin."
So, each pair is one number from real and the respective from imaginary? (the first from real and first from imag. for example?).
In my example, when i first select the frequency "0"(hertz) for my osc~ (who is gonna be analyzed) why this number (-39.604) appears?
real: -39.604 0 0 0 0 0
0 0
0 0 0 0 0 0
0 0
...
imaginary: 0 -0 -0 -0 -0 -0
-0 -0
-0 -0 -0 -0 -0 -0
...
The number is the real part of your first "osc~" with frequency 0, that is: the "DC offset". The first "osc~" of the bank actually would be a "sig~" in Pd: it doesn't change over time because its frequency is 0.
If you would connect a [sig~ 1] to [rfft~] and print it, then you'd get all zeros everywhere except in the first real value, where you'd get the blocksize (like 8 or 64 or so). You don't get 1, because the [rfft~] isn't normalized. If you connect a [sig~ 0.5] you'd get blocksize/2 in the first real bin, and with [sig~ -1] you get -(blocksize).
Using amp = sqrt(re^2 + im^2) you can calculate the amplitude of your signal manually from re and im as well and would get e.g. 8 for [sig~ 1] at [block~ 8] (which is unnormalized again). The phase is arctan(im/re) or arctan(0/bs) = 0, which is okay for a DC signal.
Ciao
Frank Barknecht _ ______footils.org__
ah cheers frank, thanks for clearing a lot of that up. you do really good explainations.
"The main reason we use complex numbers in electronic music is because they magically automate trigonometric calculations. We frequently have to add angles together in order to talk about the changing phase of an audio signal as time progresses (or as it is shifted in time, as in this chapter). It turns out that, if you multiply two complex numbers, the argument of the product is the sum of the arguments of the two factors."
so, is that where 'fast' fourier transform comes from then?
hard off wrote:
ah cheers frank, thanks for clearing a lot of that up. you do really good explainations.
"The main reason we use complex numbers in electronic music is because they magically automate trigonometric calculations. We frequently have to add angles together in order to talk about the changing phase of an audio signal as time progresses (or as it is shifted in time, as in this chapter). It turns out that, if you multiply two complex numbers, the argument of the product is the sum of the arguments of the two factors."
so, is that where 'fast' fourier transform comes from then?
Not really, as multiplication is one mathematical operation. 'Fast' in FFT means reducing the number of mathematical operation from O(N²) to O(N log N), where N is the number of analyzed frequencies, e.g. for 64 frequencies from (a constant factor times) 64²=4096 to (another constant factor times) 64 * log(64)~116 operations.
cu Thomas
"Prisons are needed only to provide the illusion that courts and police are effective. They're a kind of job insurance." (Leto II. in: Frank Herbert, God Emperor of Dune) http://thomas.dergrossebruder.org/
On Wed, 21 Nov 2007, Thomas Mayer wrote:
Not really, as multiplication is one mathematical operation. 'Fast' in FFT means reducing the number of mathematical operation from O(N²) to O(N log N), where N is the number of analyzed frequencies, e.g. for 64 frequencies from (a constant factor times) 64²=4096 to (another constant factor times) 64 * log(64)~116 operations.
When computing orders, multiplication constants are ignored, so log bases don't matter. However, if you want to know the exact number of operations, you have to know that the FFT for powers-of-two needs 2*log2(64)*64 operations, which is 768.
Using the same trick of algorithm for non-powers of two, you can realise that log2 was only a special case: in general, the number of passes is the sum of the prime factors of the block size, so for size 63 you need 13*64 steps and for size 65 you need 18*64 steps, and various other numbers vary immensely. There are some other variants of FFT that are better for some of the worst cases of size, but not all of them.
And then those operations are just the complex multiplications: you also have to do complex additions, and those operations also have sub-operations.
_ _ __ ___ _____ ________ _____________ _____________________ ... | Mathieu Bouchard - tél:+1.514.383.3801, Montréal QC Canada
Hallo!
"The main reason we use complex numbers in electronic music is because they magically automate trigonometric calculations. We frequently have to add angles together in order to talk about the changing phase of an audio signal as time progresses (or as it is shifted in time, as in this chapter). It turns out that, if you multiply two complex numbers, the argument of the product is the sum of the arguments of the two factors."
so, is that where 'fast' fourier transform comes from then?
No - the term "fast" fourier transform comes from the special algorithm how the DFT (discrete fourier transform) is calculated (see e.g. http://en.wikipedia.org/wiki/Fft)
LG Georg
Hi, thanks for help everyone. Now it's much more clear. I downloaded the "Beginner's Guide to the FFT-objects in Pd" from Frank (http://footils.org/pkg/fft-tut.tgz). And have some more questions:
-> working on patch fft-up-close.pd: I have a blocksize=8 and a samplerate=44100 and all analysis have been printed here:
orig:
0.13004 0.26951 0.40352 0.52934 0.64446 0.74649
0.83341 0.90344
img:
0 1.0317 0.41678 0.17191 0 0
0 0
re:
4.4602 -0.58717 -0.46243 -0.44167 -0.43735 0
0 0
amp:
4.4602 1.1871 0.62253 0.47395 0.43735 0
0 0
amp-normal:
0.27876 0.074192 0.038908 0.029622 0.027334 0
0 0
after-fft:
0.13004 0.26951 0.40352 0.52934 0.64446 0.74649
0.83341 0.90344
=> Ok, with a block of 8 i have data control over 4 partials from the incoming signal rigth? The frequencies are given by multiples of Samplerate/blocksize, starting on F0= 0hz , that's rigth? So, my question is about the normalized amplitudes(amp-normal):
- Why the first is so loud? This is not the F0 (0
hz)?
- With the normalized amplitudes i can estimate wich
partials are more relevants to the resynthesis (wich partials are more louder in a particular spectrum/audio)? Is this rigth?
-> working on "specgate~.pd"
- My Pd (vanilla 0-40.2, on debian) don't recognize
the [>~] object, isn't a native object? Case not where i download this object?
thanks for all.
cristiano
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On Tue, 20 Nov 2007, cristiano figueirÿfffff3 wrote:
-39.604 0 0 0 0 0
It's -39.604 because you changed the frequency of [osc~] while it was running, so it froze in whichever position it was. If you want to get a true 0 Hz at the amplitude you want, you have to reset the phase. This behaviour is normal and wanted, as it allows to remove the clicks from changes between any pair of nonzero frequencies.
_ _ __ ___ _____ ________ _____________ _____________________ ... | Mathieu Bouchard - tél:+1.514.383.3801, Montréal QC Canada
On Wed, 21 Nov 2007, hard off wrote:
and so on...always until you reach half the sample rate. ..i think you also get info on 0hz, and i guess that is also somehow important when it comes to resynthesis, but i'm not too sure if or how or why.
it may seem like 0hz is useless, as it is inaudible, but that's only if you keep it constant. at a block size of 64 and CD quality, you can make a 344hz square wave just by toggling the value of the 0hz component.
in any decomposition in which the first harmonic is 689hz, any frequency between 0hz and 689hz needs to make waves in the history of the 0hz component to make it sound like the desired number of hz. Any frequency that is not a harmonic will also use pretty much all of the spectrum (usually at least a little of each frequency) in order to simulate whatever piece of wave happens to be in the current block.
but i think the reason why you get a real part and an imaginary part is because the analysis finds the angle between one sample and the next one. ..if it's a steep angle, then that's a high frequency...if it's a flatter angle, then that's a lower frequency.
No. "Real" stands for "Cosine" and "Imaginary" stands for "Sine", as those coefficients are those of an Euler (complex-number) exponential. Those values are made by multiplying the amplitude by the cosine and sine of the phase.
and the way to find an angle is derived from pythagoras' equation....a^2+b^2=c^2
this is actually cos^2+sin^2=1 in this context, and it helps getting back to the amplitude by removing the phase (angle).
To get the phase, you need to use atan2 or atan or a complex-number logarithm.
so...that works fine when the samples lie above the zero axis...but when they go below the zero axis, then to get the angle you have to take the square route of a negative number. hence the imaginary part.
The imaginary numbers are because the complex-number system is two-dimensional and that signals to be analysed are presumed to be two-dimensional. They are decomposable into a sum of circular trajectories. In complex-numbers, circles are made by giving an imaginary argument to an exponential (this is one of the most mind-blowing and most useful parts of calculus, at the same time).
_ _ __ ___ _____ ________ _____________ _____________________ ... | Mathieu Bouchard - tél:+1.514.383.3801, Montréal QC Canada
cos^2+sin^2=1
ah!
so, actually the FFT doesn't HAVE to use imaginary numbers....but they are just used because "the complex-number system is two-dimensional and that signals to be analysed are presumed to be two-dimensional. They are decomposable into a sum of circular trajectories"
every day a bit more of this falls into place.
Hallo, hard off hat gesagt: // hard off wrote:
cos^2+sin^2=1
ah!
so, actually the FFT doesn't HAVE to use imaginary numbers....but they are just used because "the complex-number system is two-dimensional and that signals to be analysed are presumed to be two-dimensional. They are decomposable into a sum of circular trajectories"
every day a bit more of this falls into place.
An interesting thing to note is that the rfft outputs two signals, where seemingly only one signal comes in. How can that be? Actually what comes into the rfft object are two values as well: One is the usual signal amplitude visible as a patch cord, and the other is: time!
Time is not visible in Pd, it's just there and normally always walks on. But inside of a FFT-subpatch, time is frozen for the duration of a block (or rather: the "current time" is assumed to be infinite or eternal) to allow calculating a different representation: the frequency domain representation. If you think about this freezing of time for too long this is actually quite mind-blowing as well.
Ciao
Frank Barknecht _ ______footils.org__
Frank Barknecht wrote:
hard off hat gesagt: // hard off wrote:
so, actually the FFT doesn't HAVE to use imaginary numbers....but they are just used because "the complex-number system is two-dimensional and that signals to be analysed are presumed to be two-dimensional. They are decomposable into a sum of circular trajectories"
every day a bit more of this falls into place.
An interesting thing to note is that the rfft outputs two signals, where seemingly only one signal comes in. How can that be? Actually what comes into the rfft object are two values as well: One is the usual signal amplitude visible as a patch cord, and the other is: time!
Time is not visible in Pd, it's just there and normally always walks on. But inside of a FFT-subpatch, time is frozen for the duration of a block (or rather: the "current time" is assumed to be infinite or eternal) to allow calculating a different representation: the frequency domain representation. If you think about this freezing of time for too long this is actually quite mind-blowing as well.
I think it's more like: The real part represents the energy you can measure and the imaginary part is the part you can't measure. They oscillate back and forth but combine to make a constant as long as the amplitude is constant. That way the energy of the system is conserved and doesn't periodically magically disappear and reappear. It's the same as the idea that total energy is conserved but is made up of kinetic and potential components. Only the kinetic energy can be measured directly. The potential energy can only be calculated. Say if you stand at the top of a cliff you have a potential energy that is equal to the kinetic energy you would have if you fell to the bottom. There is no real instrument that can measure the potential energy so it's imaginary in a sense, but the kinetic energy can be measured directly, so it's real.
Martin
On Wed, 21 Nov 2007, Martin Peach wrote:
Only the kinetic energy can be measured directly. The potential energy can only be calculated.
So how do you measure the kinetic energy without calculating?
There is no real instrument that can measure the potential energy
So, where is the potential energy that allows atoms to fall apart? Apparently it's stored as extra mass in the nucleus. If you snap a uranium235 with a neutron, it becomes a krypton92, a baryum141 and three neutrons. If you account the masses more precisely, which are not the above numbers, you get:
235.043930 + 1.008665 - 91.926156 - 140.914411 - 3*1.008665
Which is 0.186033... some fraction of a particle. Why is it? It's because that energy weighs something! 0.000000000000000000000000308915 gram of matter (above numbers were in nuclear units), times the square of the base lightspeed (299792458 metres/second), is about 0.00000002776 joule of energy, according to Einstein's most famous equation. This can't be an (ordinary) particle, because nucleons are all around 1 unit each, and electrons are all below 0.001 unit each.
Is throwing a ball from a cliff different? (apart from differences of scale!). I don't quite know. I know that Special Relativity says kinetic energy increases mass, but I don't know (or don't remember) what it is about gravitational energy.
_ _ __ ___ _____ ________ _____________ _____________________ ... | Mathieu Bouchard - tél:+1.514.383.3801, Montréal QC Canada
Mathieu Bouchard wrote:
On Wed, 21 Nov 2007, Martin Peach wrote:
Only the kinetic energy can be measured directly. The potential energy can only be calculated.
So how do you measure the kinetic energy without calculating?
You could measure the displacement of a spring that absorbs the collision.
There is no real instrument that can measure the potential energy
So, where is the potential energy that allows atoms to fall apart? Apparently it's stored as extra mass in the nucleus. If you snap a uranium235 with a neutron, it becomes a krypton92, a baryum141 and three neutrons. If you account the masses more precisely, which are not the above numbers, you get:
So, you can't measure the potential energy except by converting it to kinetic energy...
235.043930 + 1.008665 - 91.926156 - 140.914411 - 3*1.008665
Which is 0.186033... some fraction of a particle. Why is it? It's because that energy weighs something! 0.000000000000000000000000308915 gram of matter (above numbers were in nuclear units), times the square of the base lightspeed (299792458 metres/second), is about 0.00000002776 joule of energy, according to Einstein's most famous equation. This can't be an (ordinary) particle, because nucleons are all around 1 unit each, and electrons are all below 0.001 unit each.
You're forgetting the neutrinos which probably have a very small mass. A stationary electron has a mass equivalent to a photon of 5ll thousand electron-volts. A photon of red light is about 1eV. Photons have kinetic energy too, they can move dust about in space and lasers are used in the lab to move atoms. The equation E=mc^2 is equivalent to E=hv where h is (Planck's) constant and v is the photon frequency. Fundamentally, energy is nothing more than a rate of change; the more energy you have the faster you can change things.
Is throwing a ball from a cliff different? (apart from differences of scale!). I don't quite know. I know that Special Relativity says kinetic energy increases mass, but I don't know (or don't remember) what it is about gravitational energy.
Einstein's equivalence principle implies that gravity slows time and increases mass in the same way as velocity.
Martin
On Fri, 23 Nov 2007, Martin Peach wrote:
So how do you measure the kinetic energy without calculating?
You could measure the displacement of a spring that absorbs the collision.
By doing that, you are converting kinetic energy to potential energy in order to measure it! You are assuming that specific displacements correspond to specific amounts of potential energy that exist once there is no kinetic energy left, at one point during the action of measuring, when the object is stationary and the displacement is at its biggest.
So, where is the potential energy that allows atoms to fall apart? Apparently it's stored as extra mass in the nucleus. If you snap a uranium235 with a neutron, it becomes a krypton92, a baryum141 and three neutrons. If you account the masses more precisely, which are not the above numbers, you get:
So, you can't measure the potential energy except by converting it to kinetic energy...
No, I'm saying exactly the opposite, it can be weighted, as it's part of mass. Of course, weighting eventually involves some motion, but it doesn't have to be like throwing something at high speed.
Which is 0.186033... some fraction of a particle. Why is it?
You're forgetting the neutrinos which probably have a very small mass.
If scientists have that much trouble figuring out the mass of the neutrino, it might be because it's so small that it possibly can't account for any significant amount of this reaction. Anyway, a nuclear fission reaction is very much known to emit *plenty* of energy :) which you can't think of as being embodied by neutrinos because they very seldom interact with any matter at all. Neutrino detection equipment is all buried deep in the ground because the instruments are too sensitive, and they really need to be that sensitive, because neutrinos are so close to undetectable.
_ _ __ ___ _____ ________ _____________ _____________________ ... | Mathieu Bouchard - tél:+1.514.383.3801, Montréal QC Canada
Even more interesting to me is when my physicist friend told me that the Uncertainty Principle is deeply rooted in the trade off between time window size and frequency resolution of the FFT. Kinda demystifies a whole decade of my life...
~Kyle
On Nov 21, 2007 12:21 PM, Frank Barknecht fbar@footils.org wrote:
Time is not visible in Pd, it's just there and normally always walks on. But inside of a FFT-subpatch, time is frozen for the duration of a block (or rather: the "current time" is assumed to be infinite or eternal) to allow calculating a different representation: the frequency domain representation. If you think about this freezing of time for too long this is actually quite mind-blowing as well.
On Wed, 21 Nov 2007, Kyle Klipowicz wrote:
Even more interesting to me is when my physicist friend told me that the Uncertainty Principle is deeply rooted in the trade off between time window size and frequency resolution of the FFT. Kinda demystifies a whole decade of my life...
That's quite true. It's not about FFT in particular, it's about all of Fourier methods, and even several other signal decomposition methods than Fourier. It's about how you can only begin to know a wave by sampling it at least twice, and the lower the frequency you are trying to extract from it, the longer the delay between the two samples has to be.
Heisenberg's uncertainty principle is not specific to Quantum Theory, it's central to the sampling of all wave-like phenomena and more.
If you modify FFT somewhat, you can blow up blocks into little pieces, in which each frequency is updated exactly as fast as the uncertainty principle allows it, but I have the impression that it's quite more complicated to handle, so I never really looked into it.
_ _ __ ___ _____ ________ _____________ _____________________ ... | Mathieu Bouchard - tél:+1.514.383.3801, Montréal QC Canada
On Wed, 21 Nov 2007, Frank Barknecht wrote:
Actually what comes into the rfft object are two values as well: One is the usual signal amplitude visible as a patch cord, and the other is: time!
I don't think that you can think of the time dimension as a value of what comes into the object: I'd say that it is the index of the amplitude. Only amplitude is transmitted: time is part of the structure of the actual transmission, not the content.
Anyway, that idea would make the complex fft take three inputs: real, imaginary, and time.
What [rfft~] does is assume a zero imaginary part. There's a special version of the FFT algorithm that is only a slight optimisation because it can assume that the imaginary input is zero. [rifft~] can assume that the imaginary output is zero or maybe just not wanted, so it can make different optimisations.
_ _ __ ___ _____ ________ _____________ _____________________ ... | Mathieu Bouchard - tél:+1.514.383.3801, Montréal QC Canada
Hi, in the rfft (as opposite to the fft) the missing dataset is supposed to be 0, that means if you think of a stereo channel as input, the second channel does not contain data/is muted. the missing data it is not related to time or index. the information in both domains (time and frequency) is always exact the same. marius.
Frank Barknecht wrote:
Hallo, hard off hat gesagt: // hard off wrote:
cos^2+sin^2=1
ah!
so, actually the FFT doesn't HAVE to use imaginary numbers....but they are just used because "the complex-number system is two-dimensional and that signals to be analysed are presumed to be two-dimensional. They are decomposable into a sum of circular trajectories"
every day a bit more of this falls into place.
An interesting thing to note is that the rfft outputs two signals, where seemingly only one signal comes in. How can that be? Actually what comes into the rfft object are two values as well: One is the usual signal amplitude visible as a patch cord, and the other is: time!
Time is not visible in Pd, it's just there and normally always walks on. But inside of a FFT-subpatch, time is frozen for the duration of a block (or rather: the "current time" is assumed to be infinite or eternal) to allow calculating a different representation: the frequency domain representation. If you think about this freezing of time for too long this is actually quite mind-blowing as well.
Ciao
On Thu, 22 Nov 2007, hard off wrote:
cos^2+sin^2=1
ah! so, actually the FFT doesn't HAVE to use imaginary numbers....but they are just used because "the complex-number system is two-dimensional and that signals to be analysed are presumed to be two-dimensional. They are decomposable into a sum of circular trajectories"
FFT can also be defined to only use real numbers, but the formulas are more complicated and the data layout is slightly irregular. Pd does not support those: it supports real-to-complex hybrids, just to avoid the irregularity (I mean: DC and Nyquist are special; and you'd have to put both cosines and sines in the same block in order to really have only one signal input). Because of the irregularity, you'd need to have FFT-specific signal objects that understand the FFT blocks and can handle their values correctly.
So, in the end, complex numbers may be complex, but real numbers are complicated.
_ _ __ ___ _____ ________ _____________ _____________________ ... | Mathieu Bouchard - tél:+1.514.383.3801, Montréal QC Canada
marius schebella schrieb:
Mathieu Bouchard wrote:
this is actually cos^2+sin^2=1 in this context,
these are not unity vextors, if you square the cosine and sine part you don't get one, but a square of the amplitudes, so it is more cos^2+sin^2=amplitude^2
hm, I don't really understand what you try to say here, but cos(x)^2 + sin(x)^2 is 1 !
LG Georg
I try to say, that the results you get from an fft or rfft are the cosine and sine portion of the signal (one value per frequency) in the frequency domain. (also referred to as real and imag part. but if you square them, you will not get 1 for each frequency but the amplitude. so cos(f1)^2+sin(f1)^2=amplitude(f1)^2. marius.
Georg Holzmann wrote:
marius schebella schrieb:
Mathieu Bouchard wrote:
this is actually cos^2+sin^2=1 in this context,
these are not unity vextors, if you square the cosine and sine part you don't get one, but a square of the amplitudes, so it is more cos^2+sin^2=amplitude^2
hm, I don't really understand what you try to say here, but cos(x)^2 + sin(x)^2 is 1 !
LG Georg
On Thu, 22 Nov 2007, marius schebella wrote:
Mathieu Bouchard wrote:
this is actually cos^2+sin^2=1 in this context,
these are not unity vextors, if you square the cosine and sine part you don't get one, but a square of the amplitudes, so it is more cos^2+sin^2=amplitude^2
i mean cos and sin the functions, not the components. You get the component equation by multiplying the whole equation by amplitude^2. I said it as cos^2+sin^2=1 because that's the trigonometric identity as taught in grade 11 or so.
_ _ __ ___ _____ ________ _____________ _____________________ ... | Mathieu Bouchard - tél:+1.514.383.3801, Montréal QC Canada
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