4 ms·

The example you've described (radio signals) is just a method of winning an argument. It doesn't actually help you, you practically do it by writing a computer

The example you've described (radio signals) is just a method of winning an argument. It doesn't actually help you, you practically do it by writing a computer program that tries out different parameters to get the right one.

klodolph 14y ago
I'll explain in more detail. Let's say you need a communications channel with a 300 Mbit/s capacity. Shannon's theorem lets you know what kind of parameters make that possible -- namely, bandwidth and signal-to-noise ratio. From there, I can make an informed decision about the entire signal chain. Once I choose the bandwidth, the theorem tells me the SNR so I can assign a noise budget to each of the components. If we assume that the noise is Gaussian, we can calculate total noise level as the root sum of squares of the noise levels of the individual components (don't forget to multiply by the gain). I can look at an amplifier IC's spec sheet and immediately say, "that's too noisy" or "that's overdesigned, I think I can get something cheaper". And thanks to theorems we got from the field of real analysis and statistics, we know that the sum of Gaussians is itself Gaussian. So rather than sticking two components together and measuring the noise, or running a computer simulation, I can simply square, sum, and square root. Each of these theorems reduces the amount of work necessary -- whether by pen and paper or by computer -- by an enormous factor. But if you can describe in similar detail how the computer program would work, I'll acknowledge your greatness.
- marshallp 14y ago
  That's just a few parameters that you avoided having to tune, a few milliseconds of computer time. I'm not great, just questioning basic assumptions that have been handed down from a time before computers were around.
  - klodolph 14y ago
    You keep on asserting that the task -- without domain knowledge of Shannon's theorem -- is solvable by computer. Can you describe how such a computer program would work? I'm unconvinced that the computer program would get a reasonable result before you run out of money paying for it.
    - marshallp 14y ago
      The Shannon theorem is basically voided by compressive sensing. But all of this (shannon+compressive) was a waste of practical people's time anyway. It's justification for pencil pushers who don't want to do real work.
      - karategeek6 14y ago
        http://www.youtube.com/watch?v=1orMXD_Ijbs&feature=related http://www.youtube.com/watch?v=1orMXD_Ijbs&feature=relat...
  - darkarmani 14y ago
    > That's just a few parameters that you avoided having to tune, a few milliseconds of computer time. Really? What happens when you need to do a similar calculation across hundreds of millions of data points?
geoka9 14y ago
How about solving differential equations numerically. Such programs are used all over the place in mathematical modeling, with applications ranging from economics to electronics. And you can't just "write a computer program that tries different parameters". You have to prove that your numerical method solves a certain class of equations first, otherwise your rocket will fly sideways, if at all.
- marshallp 14y ago
  The intelligent way to do that nowadays is to use automatic differentiation followed by optimization "plugins". And that has little to with proving theorems unless you consider all computer programs as proofs and all computer programmers as mathematicians.
  - geoka9 14y ago
    I'm not sure what automatic differentiation has to do with solving diff. equations but... I assume that all those automatic methods you mention are passed down from above in some sort of holy scriptures that we're supposed to blindly believe and use? Or maybe some "pencil pushing" mathematician came up with them first and _proved_ that they actually work?
    - marshallp 14y ago
      If you're solving DE's by hand you were probably just in a class taught by members of the Mathematician-Teaching Complex (allusion to Military-Industrial Complex).

klodolph 14y ago

I'll explain in more detail. Let's say you need a communications channel with a 300 Mbit/s capacity. Shannon's theorem lets you know what kind of parameters make that possible -- namely, bandwidth and signal-to-noise ratio. From there, I can make an informed decision about the entire signal chain. Once I choose the bandwidth, the theorem tells me the SNR so I can assign a noise budget to each of the components. If we assume that the noise is Gaussian, we can calculate total noise level as the root sum of squares of the noise levels of the individual components (don't forget to multiply by the gain). I can look at an amplifier IC's spec sheet and immediately say, "that's too noisy" or "that's overdesigned, I think I can get something cheaper". And thanks to theorems we got from the field of real analysis and statistics, we know that the sum of Gaussians is itself Gaussian. So rather than sticking two components together and measuring the noise, or running a computer simulation, I can simply square, sum, and square root. Each of these theorems reduces the amount of work necessary -- whether by pen and paper or by computer -- by an enormous factor. But if you can describe in similar detail how the computer program would work, I'll acknowledge your greatness.

marshallp 14y ago
That's just a few parameters that you avoided having to tune, a few milliseconds of computer time. I'm not great, just questioning basic assumptions that have been handed down from a time before computers were around.
- klodolph 14y ago
  You keep on asserting that the task -- without domain knowledge of Shannon's theorem -- is solvable by computer. Can you describe how such a computer program would work? I'm unconvinced that the computer program would get a reasonable result before you run out of money paying for it.
  - marshallp 14y ago
    The Shannon theorem is basically voided by compressive sensing. But all of this (shannon+compressive) was a waste of practical people's time anyway. It's justification for pencil pushers who don't want to do real work.
    - karategeek6 14y ago
      http://www.youtube.com/watch?v=1orMXD_Ijbs&feature=related http://www.youtube.com/watch?v=1orMXD_Ijbs&feature=relat...
- darkarmani 14y ago
  > That's just a few parameters that you avoided having to tune, a few milliseconds of computer time. Really? What happens when you need to do a similar calculation across hundreds of millions of data points?

geoka9 14y ago

How about solving differential equations numerically. Such programs are used all over the place in mathematical modeling, with applications ranging from economics to electronics. And you can't just "write a computer program that tries different parameters". You have to prove that your numerical method solves a certain class of equations first, otherwise your rocket will fly sideways, if at all.

marshallp 14y ago
The intelligent way to do that nowadays is to use automatic differentiation followed by optimization "plugins". And that has little to with proving theorems unless you consider all computer programs as proofs and all computer programmers as mathematicians.
- geoka9 14y ago
  I'm not sure what automatic differentiation has to do with solving diff. equations but... I assume that all those automatic methods you mention are passed down from above in some sort of holy scriptures that we're supposed to blindly believe and use? Or maybe some "pencil pushing" mathematician came up with them first and _proved_ that they actually work?
  - marshallp 14y ago
    If you're solving DE's by hand you were probably just in a class taught by members of the Mathematician-Teaching Complex (allusion to Military-Industrial Complex).