Yeah, "bad shit" happens, like black eyes from a drunken fistfight, or a cock-and-balls drawing in permanent marker on someone's forehead.
But "everyone in the room agrees gang-raping an unconscious woman is fun" is a bit beyond "bad shit" that any drunken teen would accidentally do without their parents around. This should be seen as no more normal/OK than a gang of drunken teens deciding to go out and knife a gay/black/jew/nerd to death just for laughs, "because boys will be boys." Rape apologists like you are missing the actual societal problems (a total disregard for, if not gleeful violation of, the personal boundaries of women's bodies), trying to assign blame on "big bad government regulations against partying" instead of pervasive misogyny.
Right, because once parents stopped being allowed to get their kids likker'd up, they lost the ability to teach them that gang-raping a comatose woman is un-cool.
Wait, no, that makes no sense.
I'll believe that overly-restrictive alcohol prohibition has some culpability for specifically alcohol-related problems (like incidences of binge drinking and alcohol poisoning among young drinkers). But gang-raping a drugged woman, then tweeting it to your pals to show how cool you are, isn't something that "just happens" whenever you get drunk --- when you're drunk, you're just uninhibited in carrying out actions that you'd somewhat like to do cold sober (but realize you wouldn't get away with). This incident indicates a cold-sober culture of misogyny (revealed by "vero in vino"), which is an entirely separate problem from alcohol-prohibition-specific harms.
Presumably, the relative "lack of flexibility" of the European-style options ends up built into their market price. Thus, the price of a more restrictive option will be lower than a more flexible one (according to how much investors think the risk of inflexibility will cost). If your strategy works out right (so the stock price is high at the expiration time), you can make a lot more profit buying cheaper, inflexible options than "wasting" the higher cost of unused flexibility. The European options hence aren't "inferior," they're just a different kind of gamble which will find its own price competitive with the US-style ones.
I'm not muddled --- of course businesses want to keep wages down for *everyone*; starting with half the population willing to work for lower wages is very helpful for achieving that end (and a long-term strategic investor class will want to keep it that way, not kill the golden-egg-laying goose).
Your analysis specifically assumes that labor is a limited supply pool that can drive up competitive prices. China has over a billion person population, largely employed in rural situations (e.g. a huge pool of potential technology workers, with just a little selection/training to skim 10,000,000+ top-1%-intelligence candidates from the rice paddies). Likewise, in the USA, unemployment is ~8% among people actively looking (and much higher counting the underemployed and given-up-trying). "Competition" in this situation isn't to drive up employee wages by offering just enough more to get your competitor's workers to switch: instead, the competition is to offer as much less as you can, while telling your employees how grateful they should be to get anything in this tough economy (despite record-breaking profits for their overlords).
Note a key distinction: I don't think "the salary differential is due to misogyny" at its root --- misogyny is an effect (useful to those benefiting), rather than a cause. The love of money (the profit motive) is the root cause, that makes maintaining institutionalized misogyny a multi-trillion-dollar prize.
As to "you're free to start your own business with egalitarian hiring practices," no --- that's a completely empty freedom for me and the overwhelming majority of working-class stiffs in this country who don't have a couple hundred thousand spare dollars lying around to start up ventures. I'm equally "free to buy the Empire State Building" (if I only had a zillion dollars) --- but, even if my buying the Empire State Building would somehow usher in heaven on earth, it ain't happening. The Capitalist system selects for viciously amoral profiteering sociopaths to end up with most of the wealth (i.e. resources for shaping future change) in the world, e.g. the sort of people who are overjoyed to continue exploiting gender wage differences. Even if I was a freak millionaire with a social conscience, my one-man-crusade for an equal wage workplace wouldn't make a dent; I'd just run myself bankrupt competing against the more profitable louts.
"this, largely, is why the salary differential is declining.": No, the salary differential is largely declining because male workers are getting lower and lower pay (thanks to stagnant or declining wages despite growing labor productivity and profitability); not because working women are being raised up. Yay!??
Capitalism isn't "responsible for all that's wrong with the world" --- but I am going to hold it responsible for the wrongs in the large portions of the world under its thrall. Remember, "greater efficiency" in Capitalism only directly means "greater efficiency at stuffing the pockets of the already wealthy," which is undoubtedly happening. You've been conned into assuming this is equivalent to, or at least necessarily causative of, improvements for everyone else. The real world record indicates otherwise.
Or, based on the example of more advanced capitalist societies (e.g. the U.S.A.), capitalism helps lock sexism in. Now, instead of simple crude ignorant misogyny by the ruling male class (which can slowly be fixed by education and living proof that women, too, have fully functional human minds), the managerial class has a *profit motive* for intentionally keeping women marginalized and unequal. With 20-30% wage cuts at stake, the "enlightened-self-interest" profiteer will gladly hire obedient, compliant (desperate) women --- so long as they make no attempt to rock the boa(r|t) and fight against discriminatory wages. That's an extra 20-30% from each workers wage now placed in management's pocket, available to buy government support and propaganda campaigns to teach everyone how grateful women should be for their new positions (and how men should lower their wage expectations if they expect to compete).
The 20-30% wage differential for equal work is actually not just in China --- approximately the same level exists in the USA. Of course, in the USA, managers know better than to openly talk about their discriminatory practices. But, so long as everything is done with a proper level of plausible deniability, and the occasional sacrifice of a low-level scapegoat for high-level problems, the real practice of gender wage discrimination contributes significantly the country's ever-widening gap between the working class and management/investment elites. After all, the USA is a "post-feminism" country, where "women already have equal rights," so "there's no real problems for whiny bitches to complain about"; people complaining about gender inequality "are just man-hating feminazis," who earn lower wages "because they just aren't as good workers."
Here's just one recent example from a few days ago.
While President Obama was being the first US president to be awarded the Israeli Medal Of Distinction (Israel's highest civilian honor) by Israeli president Simon Peres, most news organizations were reporting live from the event (and Peres' gushing admiration of Obama's "unforgettable contribution to [the people of Israel's] security").
At the exact same time, guess what FOX news was showing? They were reporting on repealing Obamacare, only interrupted by an ad for their own show... about how "with a friend like Obama are Israel’s enemies gaining strength. Sean gets expert insight on a special Hannity."
What framing are FOX's viewers getting? Would a FOX viewer ever learn that, in the opinion of Israel's president, Obama is a reliably steadfast supporter of the Israeli government's positions? Or, would they be presented with a consistently counter-factual frame that Obama is hostile to Israel, and prefers to pal around with his Kenyan Muslim Commie friends?
The downside of a true-sinc stereo: it takes a heck of a long time to "warm up" before it can play.
Anyways, yes, the Fourier basis representation for the signal (designed for periodic functions) is indeed problematic when you try and shoehorn it into representing a non-periodic signal. Of the infinite number of curves that one could draw through your samples "x" and "y", the sum-of-sines-and-cosines one is awkward if you want to suddenly start/stop the signal without a "click" from spurious frequency components introduced by zero-padding (instead of periodic repetition). I have a personal quirk for liking to think about the intermediate frequencies "not existing" in a short time window, because it's a cute analogue to the Heisenberg Uncertainty Principle --- with a finite measurement window, you're fundamentally unable to uniquely distinguish tones more finely than one wave number (inversely proportional to the window width) apart.
While the "true" sinc stereo is a joke, the basic concept is not. Of the infinite number of curves that match your "x" samples, you can interpret them with sinc interpolation (truncated/approximated when you finally need to calculate final results). To filter x->y to remove high frequencies for downsampling, convolve this signal with a wider sinc (or other bandpass window, possibly with slightly less steep edges in Fourier space to cut down on real-space ringing). The efficiency/accuracy of this bandpass indeed won't be perfectly sharp (as you originally asserted) without infinite computer resources to calculate an infinite-length filter kernel, but it can converge pretty quickly, past the limits of quantization noise in a 16 or 24-bit signal, with finite resources --- with cleaner results than the large spurious frequency tails created by zero-padding a sample sequence interpreted as a Fourier sum. And again, on the playback side, your stereo's DAC may already be using some truncated-sinc-like interpolation filter to generate continuous interpolated output.
I agree that our only actual point of disagreement is interpreting what time window the filter designer is "responsible" for --- indeed, one can't control the frequency spectrum over an infinite (or at least lifetime-of-the-universe) window when producing a soundwave with finite support.
On the other hand, maybe you should upgrade your stereo to one that does "true" sinc interpolation (starting infinitely before you hit the "play" button, and stopping infinitely after) --- then I can give you a sampled signal which, when properly interpolated to an infinite time window, has a frequency spectrum with finite support:)
Tacking on 248 samples of "0" to an 8-sample signal isn't the same as "doing nothing"; from a mathematical standpoint, it's little different from tacking on 248 samples of "airplane full of nails crashing into a chalkboard factory". It's true, there's no digital filter I can apply to my 8-sample-long window to control the frequency spectrum if you expect to listen to my 8 samples, then tack on 248 samples of your own noises (including silence) and FFT the whole stretch. But, I can filter the 8-sample segment so that, measured within that 8-sample window I was responsible for filtering, the frequency spectrum has a "perfect" cutoff (not my problem what you do before or after).
x = [1 0 0 0 0 0 0 0]; % x is only defined on 8 samples over the interval. There are an infinite number of continuous signals that could be sampled this way.
Following your procedure through to y: octave:5] y = ifft(Y); octave:6] y y =
0.87500 0.12500 -0.12500 0.12500 -0.12500 0.12500 -0.12500 0.12500
so y is also defined at 8 sample points; as for x, there are an infinite number of curves that could fit these. One of these curves is the sum of frequencies indicated by Y. But what does fft(y,256); mean? From the Matlab documentation,
"Y = fft(X,n) returns the n-point DFT. fft(X) is equivalent to fft(X, n) where n is the size of X in the first nonsingleton dimension. If the length of X is less than n, X is padded with trailing zeros to length n."
So, now you have y defined in a larger window (y = 0.87500 0.12500 -0.12500 0.12500 -0.12500 0.12500 -0.12500 0.12500 0 0 0 0 0.... 0). See my response above to another poster's question: when you enlarge the sampling window, you "create" a lot of possible "intermediate" frequencies that "don't exist" (i.e. are indistinguishable from sums of integral frequencies in the shorter window). By padding y with zeros to a larger window, you're looking at a *different* signal from the un-padded y alone; consequently, you need the "extra frequencies" that you ascribe to the "non-sharp-cutoff" to correctly describe the different "y+0,0,0,0,...,0" signal (which is distinct from y). But that doesn't mean the cutoff isn't perfect as defined on the original signal x->y. In fact, if you periodically *repeat* y (y->y,y,y...,y instead of y->y,0,0,0...) you'll see the "sharp cutoff" still applies since the periodic signal is still the sum of the original frequencies in y.
Same response as I gave to the AC post above making the same point: this indicates a non-specious importance for the initial >>44.1kHz sampling, even if the on-chip digital circuitry immediately downsamples to properly-filtered 44.1kHz. It's a semantic distinction whether you should really call an internally-higher-than-44.1kHz-sampling DAC a 44.1kHz DAC, given that it is relying on information outside the 44.1kHz sampling range to produce the output rendered down to 44.1kHz. But yes, I agree that in practice dropping a good "44.1kHz" DAC (which is "really" a 96kHz DAC + internal downsampling) into your 24/96 signal chain won't result in audible problems.
In a finite window, *any* signal can be represented as a sum of elements with frequencies corresponding to n=0 (DC offset), 1, 2, 3,...., infinity integral cycles in the window. A signal corresponding to a non-integral number of cycles, e.g. 100.5, is indistinguishable over the window from some (infinite) combination of integral cycle waves. If you measured in a window twice as long, the 100.5-cycle signal would now be a unique, identifiable 201-cycle component. So, in an important sense, in a finite window the "intermediate" frequencies "don't exist" --- they can't do anything different from the (infinite series) of integral frequencies. Thus, you can create a cutoff that is as "perfect" as is meaningful in a finite window.
1) Digitally, yes you can. Take the DFT of the data; zero out all components above your frequency cutoff; reconstruct the signal as the sum of below-cutoff frequencies. Voila, a perfect sharp cutoff. The only subtlety is that you can only choose an exact cutoff corresponding to some integral number of cycles in your sampling window, so you can't cutoff at exactly sqrt(e*pi)kHz --- but you do have plenty of wave numbers from which to select a perfect cutoff (increasing with the size of your DFT window).
2) Untrue: a 44kHz *sampling rate* has a 44/2=22kHz Nyquist cutoff. Frequencies f>22kHz Nyquist limit "wrap around" to f-22kHz difference frequencies.
But yes, I agree, on the playback side there's no audible difference between a (sufficiently well made) 44.1kHz and 96kHz DAC.
Right, which shows why 96kHz digital sampling *is* critical, even if you immediately digitally downsample on-chip before passing it along to the next device in the processing chain.
You sure can hear the difference if you stick a 44.1kHz DAQ in a 96kHz signal chain before filtering out ultrasonic high frequency components (if there are enough to make a difference). The advantage of 96kHz recording isn't that it can capture any more human-audible frequencies than 44kHz can, but that you have a lot more leeway to prevent aliasing of signals above the Nyquist limit down into the audible range (a 25kHz tone sampled at 44kHz results in a spurious, highly audible (25-44/2)=3kHz aliasing signal).
It's pretty much impossible to build analog frequency filters with a sharp cutoff (e.g. everything below 20kHz and below gets through, everything above 22kHz is -60dB attenuated), so recording at 44.1kHz sampling requires either being absolutely certain the original sound source has minimal high-frequency harmonics, or heavy analog filtering that cuts well into the audible high frequency range. With 96kHz sampling, it's much easier to build an analog filter that gradually rolls off high frequencies between 20kHz and 40kHz (...producing a >40kHz sound is tricky in the first place), preventing aliasing without the filter cutting into the audible range. Once digitized, it's trivial to make a *digital* filter with a perfect frequency cutoff to downsample the 96kHz to aliasing-free 44.1kHz.
Maybe the smartphone-equivalent of the watch chain will come back into style to fix the dropping issue. I'm sure manufacturers would love a chance to sell another high-profit-margin fashion accessory to go with every phone.
"Free energy" --- the "useful" part, that monotonically decreases with the increase of entropy, of the overall conserved quantity of energy in the universe --- is indeed "lost." Energy can "change states" irreversibly, hence non-renewably.
Or, this topic of research is reaching maturity where understanding of the fine details will increase, but no major upheavals in the basic parameters.
From the Wikipedia page on "age of the universe":
The first reasonably accurate measurement of the rate of expansion of the universe, a numerical value now known as the Hubble constant, was made in 1958 by astronomer Allan Sandage.[23] His measured value for the Hubble constant yielded the first good estimate of the age of the universe, coming very close to the value range generally accepted today.
Between then and now, there was a fair amount of work put into searching for reasons why the universe wasn't about twice as old (coming from older models/observations), but the cutting-edge predictions pretty much settled down to where they are now (the speed of light or mass of the electron haven't changed radically in 50 years, either). Now, we have an experiment agreeing on the age of the universe to within ~0.4% of the previous best experiment.
Yes, there's always a nonzero possibility of big scientific upheavals --- but, past a certain point of maturity (once a field moves from pure wild-ass speculation to multiple repeatable experiments), scientific revolutions have historically always occurred out at the Nth decimal place.
More modern cosmological views tend to assume that there wasn't actually a singularity. There's a singularity in our current mathematical models of it --- but that's a problem with the models not having the right parts to describe the very early universe, not an indication that the universe was singular or even asymptotically approaching singular from positive time. The general "mental image" of the early universe as described by modern cosmologists like Stephen Hawking involves a transition from a region where the time dimension is no longer "special" in having a "forward-moving" direction --- in this part of the universe (which forms a smooth non-singular boundary edge to our flowing-time universe), the question "what came before?" no longer makes any sense, because there is no time direction for "before."
That should provide you with even more noodle shredding than an asymptotically infinite universe:)
Is Slashdot's sarcasm meter broken again today? I'm pretty certain the parent poster was trying to (snarkily) make the point that interpretation of religious texts *has* changed a whole lot (in opposition to the grandparent post indicating religious rigidity back to a "bunch of bronze age shepards"), despite all the replies taking him overly literally.
Slashdot Mirror
Yeah, "bad shit" happens, like black eyes from a drunken fistfight, or a cock-and-balls drawing in permanent marker on someone's forehead.
But "everyone in the room agrees gang-raping an unconscious woman is fun" is a bit beyond "bad shit" that any drunken teen would accidentally do without their parents around. This should be seen as no more normal/OK than a gang of drunken teens deciding to go out and knife a gay/black/jew/nerd to death just for laughs, "because boys will be boys." Rape apologists like you are missing the actual societal problems (a total disregard for, if not gleeful violation of, the personal boundaries of women's bodies), trying to assign blame on "big bad government regulations against partying" instead of pervasive misogyny.
Right, because once parents stopped being allowed to get their kids likker'd up, they lost the ability to teach them that gang-raping a comatose woman is un-cool.
Wait, no, that makes no sense.
I'll believe that overly-restrictive alcohol prohibition has some culpability for specifically alcohol-related problems (like incidences of binge drinking and alcohol poisoning among young drinkers). But gang-raping a drugged woman, then tweeting it to your pals to show how cool you are, isn't something that "just happens" whenever you get drunk --- when you're drunk, you're just uninhibited in carrying out actions that you'd somewhat like to do cold sober (but realize you wouldn't get away with). This incident indicates a cold-sober culture of misogyny (revealed by "vero in vino"), which is an entirely separate problem from alcohol-prohibition-specific harms.
Presumably, the relative "lack of flexibility" of the European-style options ends up built into their market price. Thus, the price of a more restrictive option will be lower than a more flexible one (according to how much investors think the risk of inflexibility will cost). If your strategy works out right (so the stock price is high at the expiration time), you can make a lot more profit buying cheaper, inflexible options than "wasting" the higher cost of unused flexibility. The European options hence aren't "inferior," they're just a different kind of gamble which will find its own price competitive with the US-style ones.
A cat is a mammal. A dog is a mammal. Therefore, a cat is a dog, QED.
I'm not muddled --- of course businesses want to keep wages down for *everyone*; starting with half the population willing to work for lower wages is very helpful for achieving that end (and a long-term strategic investor class will want to keep it that way, not kill the golden-egg-laying goose).
Your analysis specifically assumes that labor is a limited supply pool that can drive up competitive prices. China has over a billion person population, largely employed in rural situations (e.g. a huge pool of potential technology workers, with just a little selection/training to skim 10,000,000+ top-1%-intelligence candidates from the rice paddies). Likewise, in the USA, unemployment is ~8% among people actively looking (and much higher counting the underemployed and given-up-trying). "Competition" in this situation isn't to drive up employee wages by offering just enough more to get your competitor's workers to switch: instead, the competition is to offer as much less as you can, while telling your employees how grateful they should be to get anything in this tough economy (despite record-breaking profits for their overlords).
Note a key distinction: I don't think "the salary differential is due to misogyny" at its root --- misogyny is an effect (useful to those benefiting), rather than a cause. The love of money (the profit motive) is the root cause, that makes maintaining institutionalized misogyny a multi-trillion-dollar prize.
As to "you're free to start your own business with egalitarian hiring practices," no --- that's a completely empty freedom for me and the overwhelming majority of working-class stiffs in this country who don't have a couple hundred thousand spare dollars lying around to start up ventures. I'm equally "free to buy the Empire State Building" (if I only had a zillion dollars) --- but, even if my buying the Empire State Building would somehow usher in heaven on earth, it ain't happening. The Capitalist system selects for viciously amoral profiteering sociopaths to end up with most of the wealth (i.e. resources for shaping future change) in the world, e.g. the sort of people who are overjoyed to continue exploiting gender wage differences. Even if I was a freak millionaire with a social conscience, my one-man-crusade for an equal wage workplace wouldn't make a dent; I'd just run myself bankrupt competing against the more profitable louts.
"this, largely, is why the salary differential is declining.": No, the salary differential is largely declining because male workers are getting lower and lower pay (thanks to stagnant or declining wages despite growing labor productivity and profitability); not because working women are being raised up. Yay!??
Capitalism isn't "responsible for all that's wrong with the world" --- but I am going to hold it responsible for the wrongs in the large portions of the world under its thrall. Remember, "greater efficiency" in Capitalism only directly means "greater efficiency at stuffing the pockets of the already wealthy," which is undoubtedly happening. You've been conned into assuming this is equivalent to, or at least necessarily causative of, improvements for everyone else. The real world record indicates otherwise.
Or, based on the example of more advanced capitalist societies (e.g. the U.S.A.), capitalism helps lock sexism in. Now, instead of simple crude ignorant misogyny by the ruling male class (which can slowly be fixed by education and living proof that women, too, have fully functional human minds), the managerial class has a *profit motive* for intentionally keeping women marginalized and unequal. With 20-30% wage cuts at stake, the "enlightened-self-interest" profiteer will gladly hire obedient, compliant (desperate) women --- so long as they make no attempt to rock the boa(r|t) and fight against discriminatory wages. That's an extra 20-30% from each workers wage now placed in management's pocket, available to buy government support and propaganda campaigns to teach everyone how grateful women should be for their new positions (and how men should lower their wage expectations if they expect to compete).
The 20-30% wage differential for equal work is actually not just in China --- approximately the same level exists in the USA. Of course, in the USA, managers know better than to openly talk about their discriminatory practices. But, so long as everything is done with a proper level of plausible deniability, and the occasional sacrifice of a low-level scapegoat for high-level problems, the real practice of gender wage discrimination contributes significantly the country's ever-widening gap between the working class and management/investment elites. After all, the USA is a "post-feminism" country, where "women already have equal rights," so "there's no real problems for whiny bitches to complain about"; people complaining about gender inequality "are just man-hating feminazis," who earn lower wages "because they just aren't as good workers."
Here's just one recent example from a few days ago.
While President Obama was being the first US president to be awarded the Israeli Medal Of Distinction (Israel's highest civilian honor) by Israeli president Simon Peres, most news organizations were reporting live from the event (and Peres' gushing admiration of Obama's "unforgettable contribution to [the people of Israel's] security").
At the exact same time, guess what FOX news was showing? They were reporting on repealing Obamacare, only interrupted by an ad for their own show... about how "with a friend like Obama are Israel’s enemies gaining strength. Sean gets expert insight on a special Hannity."
What framing are FOX's viewers getting? Would a FOX viewer ever learn that, in the opinion of Israel's president, Obama is a reliably steadfast supporter of the Israeli government's positions? Or, would they be presented with a consistently counter-factual frame that Obama is hostile to Israel, and prefers to pal around with his Kenyan Muslim Commie friends?
You're deeply concerned about the careers of women, who you like to hire... because you can pay them less for doing more reliable quality work.
What a moving tribute to China's acceptance of the modern Western Capitalist workplace values.
The downside of a true-sinc stereo: it takes a heck of a long time to "warm up" before it can play.
Anyways, yes, the Fourier basis representation for the signal (designed for periodic functions) is indeed problematic when you try and shoehorn it into representing a non-periodic signal. Of the infinite number of curves that one could draw through your samples "x" and "y", the sum-of-sines-and-cosines one is awkward if you want to suddenly start/stop the signal without a "click" from spurious frequency components introduced by zero-padding (instead of periodic repetition). I have a personal quirk for liking to think about the intermediate frequencies "not existing" in a short time window, because it's a cute analogue to the Heisenberg Uncertainty Principle --- with a finite measurement window, you're fundamentally unable to uniquely distinguish tones more finely than one wave number (inversely proportional to the window width) apart.
While the "true" sinc stereo is a joke, the basic concept is not. Of the infinite number of curves that match your "x" samples, you can interpret them with sinc interpolation (truncated/approximated when you finally need to calculate final results). To filter x->y to remove high frequencies for downsampling, convolve this signal with a wider sinc (or other bandpass window, possibly with slightly less steep edges in Fourier space to cut down on real-space ringing). The efficiency/accuracy of this bandpass indeed won't be perfectly sharp (as you originally asserted) without infinite computer resources to calculate an infinite-length filter kernel, but it can converge pretty quickly, past the limits of quantization noise in a 16 or 24-bit signal, with finite resources --- with cleaner results than the large spurious frequency tails created by zero-padding a sample sequence interpreted as a Fourier sum. And again, on the playback side, your stereo's DAC may already be using some truncated-sinc-like interpolation filter to generate continuous interpolated output.
I agree that our only actual point of disagreement is interpreting what time window the filter designer is "responsible" for --- indeed, one can't control the frequency spectrum over an infinite (or at least lifetime-of-the-universe) window when producing a soundwave with finite support.
On the other hand, maybe you should upgrade your stereo to one that does "true" sinc interpolation (starting infinitely before you hit the "play" button, and stopping infinitely after) --- then I can give you a sampled signal which, when properly interpolated to an infinite time window, has a frequency spectrum with finite support :)
TL;DR version of my verbose answer above:
Tacking on 248 samples of "0" to an 8-sample signal isn't the same as "doing nothing"; from a mathematical standpoint, it's little different from tacking on 248 samples of "airplane full of nails crashing into a chalkboard factory". It's true, there's no digital filter I can apply to my 8-sample-long window to control the frequency spectrum if you expect to listen to my 8 samples, then tack on 248 samples of your own noises (including silence) and FFT the whole stretch. But, I can filter the 8-sample segment so that, measured within that 8-sample window I was responsible for filtering, the frequency spectrum has a "perfect" cutoff (not my problem what you do before or after).
The trick you're playing on yourself here is:
x = [1 0 0 0 0 0 0 0]; % x is only defined on 8 samples over the interval. There are an infinite number of continuous signals that could be sampled this way.
Following your procedure through to y:
octave:5] y = ifft(Y);
octave:6] y
y =
0.87500 0.12500 -0.12500 0.12500 -0.12500 0.12500 -0.12500 0.12500
so y is also defined at 8 sample points; as for x, there are an infinite number of curves that could fit these. One of these curves is the sum of frequencies indicated by Y. But what does fft(y,256); mean? From the Matlab documentation,
"Y = fft(X,n) returns the n-point DFT. fft(X) is equivalent to fft(X, n) where n is the size of X in the first nonsingleton dimension. If the length of X is less than n, X is padded with trailing zeros to length n."
So, now you have y defined in a larger window (y = 0.87500 0.12500 -0.12500 0.12500 -0.12500 0.12500 -0.12500 0.12500 0 0 0 0 0 .... 0). See my response above to another poster's question: when you enlarge the sampling window, you "create" a lot of possible "intermediate" frequencies that "don't exist" (i.e. are indistinguishable from sums of integral frequencies in the shorter window). By padding y with zeros to a larger window, you're looking at a *different* signal from the un-padded y alone; consequently, you need the "extra frequencies" that you ascribe to the "non-sharp-cutoff" to correctly describe the different "y+0,0,0,0,...,0" signal (which is distinct from y). But that doesn't mean the cutoff isn't perfect as defined on the original signal x->y. In fact, if you periodically *repeat* y (y->y,y,y...,y instead of y->y,0,0,0...) you'll see the "sharp cutoff" still applies since the periodic signal is still the sum of the original frequencies in y.
Same response as I gave to the AC post above making the same point: this indicates a non-specious importance for the initial >>44.1kHz sampling, even if the on-chip digital circuitry immediately downsamples to properly-filtered 44.1kHz. It's a semantic distinction whether you should really call an internally-higher-than-44.1kHz-sampling DAC a 44.1kHz DAC, given that it is relying on information outside the 44.1kHz sampling range to produce the output rendered down to 44.1kHz. But yes, I agree that in practice dropping a good "44.1kHz" DAC (which is "really" a 96kHz DAC + internal downsampling) into your 24/96 signal chain won't result in audible problems.
In a finite window, *any* signal can be represented as a sum of elements with frequencies corresponding to n=0 (DC offset), 1, 2, 3, ...., infinity integral cycles in the window. A signal corresponding to a non-integral number of cycles, e.g. 100.5, is indistinguishable over the window from some (infinite) combination of integral cycle waves. If you measured in a window twice as long, the 100.5-cycle signal would now be a unique, identifiable 201-cycle component. So, in an important sense, in a finite window the "intermediate" frequencies "don't exist" --- they can't do anything different from the (infinite series) of integral frequencies. Thus, you can create a cutoff that is as "perfect" as is meaningful in a finite window.
1) Digitally, yes you can. Take the DFT of the data; zero out all components above your frequency cutoff; reconstruct the signal as the sum of below-cutoff frequencies. Voila, a perfect sharp cutoff. The only subtlety is that you can only choose an exact cutoff corresponding to some integral number of cycles in your sampling window, so you can't cutoff at exactly sqrt(e*pi)kHz --- but you do have plenty of wave numbers from which to select a perfect cutoff (increasing with the size of your DFT window).
2) Untrue: a 44kHz *sampling rate* has a 44/2=22kHz Nyquist cutoff. Frequencies f>22kHz Nyquist limit "wrap around" to f-22kHz difference frequencies.
But yes, I agree, on the playback side there's no audible difference between a (sufficiently well made) 44.1kHz and 96kHz DAC.
Right, which shows why 96kHz digital sampling *is* critical, even if you immediately digitally downsample on-chip before passing it along to the next device in the processing chain.
You sure can hear the difference if you stick a 44.1kHz DAQ in a 96kHz signal chain before filtering out ultrasonic high frequency components (if there are enough to make a difference). The advantage of 96kHz recording isn't that it can capture any more human-audible frequencies than 44kHz can, but that you have a lot more leeway to prevent aliasing of signals above the Nyquist limit down into the audible range (a 25kHz tone sampled at 44kHz results in a spurious, highly audible (25-44/2)=3kHz aliasing signal).
It's pretty much impossible to build analog frequency filters with a sharp cutoff (e.g. everything below 20kHz and below gets through, everything above 22kHz is -60dB attenuated), so recording at 44.1kHz sampling requires either being absolutely certain the original sound source has minimal high-frequency harmonics, or heavy analog filtering that cuts well into the audible high frequency range. With 96kHz sampling, it's much easier to build an analog filter that gradually rolls off high frequencies between 20kHz and 40kHz (...producing a >40kHz sound is tricky in the first place), preventing aliasing without the filter cutting into the audible range. Once digitized, it's trivial to make a *digital* filter with a perfect frequency cutoff to downsample the 96kHz to aliasing-free 44.1kHz.
Maybe the smartphone-equivalent of the watch chain will come back into style to fix the dropping issue. I'm sure manufacturers would love a chance to sell another high-profit-margin fashion accessory to go with every phone.
"Free energy" --- the "useful" part, that monotonically decreases with the increase of entropy, of the overall conserved quantity of energy in the universe --- is indeed "lost." Energy can "change states" irreversibly, hence non-renewably.
Or, this topic of research is reaching maturity where understanding of the fine details will increase, but no major upheavals in the basic parameters.
From the Wikipedia page on "age of the universe":
The first reasonably accurate measurement of the rate of expansion of the universe, a numerical value now known as the Hubble constant, was made in 1958 by astronomer Allan Sandage.[23] His measured value for the Hubble constant yielded the first good estimate of the age of the universe, coming very close to the value range generally accepted today.
Between then and now, there was a fair amount of work put into searching for reasons why the universe wasn't about twice as old (coming from older models/observations), but the cutting-edge predictions pretty much settled down to where they are now (the speed of light or mass of the electron haven't changed radically in 50 years, either). Now, we have an experiment agreeing on the age of the universe to within ~0.4% of the previous best experiment.
Yes, there's always a nonzero possibility of big scientific upheavals --- but, past a certain point of maturity (once a field moves from pure wild-ass speculation to multiple repeatable experiments), scientific revolutions have historically always occurred out at the Nth decimal place.
More modern cosmological views tend to assume that there wasn't actually a singularity. There's a singularity in our current mathematical models of it --- but that's a problem with the models not having the right parts to describe the very early universe, not an indication that the universe was singular or even asymptotically approaching singular from positive time. The general "mental image" of the early universe as described by modern cosmologists like Stephen Hawking involves a transition from a region where the time dimension is no longer "special" in having a "forward-moving" direction --- in this part of the universe (which forms a smooth non-singular boundary edge to our flowing-time universe), the question "what came before?" no longer makes any sense, because there is no time direction for "before."
That should provide you with even more noodle shredding than an asymptotically infinite universe :)
Read the references --- it consists of a bigger fraction of visible and dark matter than previously estimated, with less (dark) energy.
Is Slashdot's sarcasm meter broken again today? I'm pretty certain the parent poster was trying to (snarkily) make the point that interpretation of religious texts *has* changed a whole lot (in opposition to the grandparent post indicating religious rigidity back to a "bunch of bronze age shepards"), despite all the replies taking him overly literally.
Well, with some virtual machines some JIT compiling, it's still chugging away as the back-end code on some more modern platforms...
Apparently, the way to determine the universe's real age is to closely scrutinize its wrinkles and stretch marks.