Re: Can you recommend a good explanation of the proof of the Fourier

Rune Allnor wrote:
> On 8 Jul, 18:03, Jerry Avins <j...@[EMAIL PROTECTED]
> wrote:
>> Rune Allnor wrote:
>>> On 7 Jul, 22:41, Jerry Avins <j...@[EMAIL PROTECTED]
> wrote:
>>>> Rune Allnor wrote:
>>>>> On 7 Jul, 20:14, "maxplanck" <erik.bo...@[EMAIL PROTECTED]
> wrote:
>>>>>> Thanks very much Steve!  
>>>>>> The section of your book that you're referring to demonstrates
numerically
>>>>>> that correlation, when implemented as it is in the DFT, produces
the
>>>>>> desired results, i.e. what we want the output of the DFT to be. 
This is
>>>>>> enough proof for me to believe that it works.
>>>>> Wrong. That's smoke and mirrors that might look reasonable
>>>>> up front, but only obfuscates in the long run. Both the
>>>>> the DFT/IDFT pair and correlation is based on the inner
>>>>> product, but that's where the similarity ends.
>>> ...
>>>>> Get a good book on DSP and look for the matrix W in the
>>>>> chapter that discusses the DFT.
>>>> Rune,
>>>> What is the correlation between a sinusoid and an arbitrary function?
>>> It's the inner product between the sine and the function.
>>> Introducing the term 'correlation' is a red herring if the
>>> main objective is to study the DFT.
>> Finding the inner product is a method;
> 
> The inner product is a concept as good as any, along with
> 'product', 'sum', 'square root' etc.
> 
>> I'm after the meaning.
> 
> What's the meaning of 'difference'?
> 
>> Let me put
>> the matter differently: since basis functions are all orthogonal,
> 
> Who said they are?
> 
>> the
>> integral over a whole number of periods of the product any one of them
>> and any number of the others is necessarily zero. (Definition of
>> orthogonality.)
> 
> This is thhe definition of orthogonality. But a set of basis
> functions needs not be orthogonal.
> 
>> If the arbitrary term of the integrand includes some
>> amount of the multiplying basis function, then the amount by the which
>> result differs from zero tells us how much of the basis is "contained"
>> in the arbitrary term of the integrand.
> 
> I don't understand what you mean. The concept of 'basis' is
> the same if you talk about functions as when you talk about
> vectors in 2D. If you have an arbitrary vector x then the
> inner product c=<x,b> is computed by prijecting x onto b,
> and is thus a measure of 'similarity' between x and b.
> One can project x onto several basis vectors simultaneously,
> by collecting the basis vector in a matrix B:
> 
> c =Bx
> 
> Now c is a vector which expresses x in terms of the
> basis B. Trivial high-school linear algebra, generalizes
> straight-forward to the DFT.
> 
> The inner product is stuff I fisrt encountered at age 16,
> it is hardly revolutionary, nor controversial.
> 
>> As I see it, that's the same as
>> saying it is a measure of how well the two terms in the integrand are
>> correlated.
> 
> 'Correlation' is a term that is based on the inner product,
> not the other way around. What is the more fundamental concept
> to you: 'wheel' or 'bicycle'?

I think we're expressing different views of the same concept. Some 
people think that a sine is not a fundamental concept, but the sum of 
two exponentials whose exponents differ in sign. I don't, but recognize 
that it can be expressed that way. I also think that elementary 
trigonometry is more easily learned from my perspective. Max is seeking 
knowledge. I don't think that "inner product" is the smoother road.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
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