Discrete Fourier Transform = Correlation?

[xoxos]

perhaps there's some meaning in the idea of integration but we only talk about so much here, or it would be an open society. talk too much, you get the ass beat stick.

http://dsp.stackexchange.com/questions/ ... ect=1&lq=1

Firstly, this method is taboo in the DSP community, (and for decent reasons), although there are cases where you can get away with it, depending on your application

it's much more important to be right or wrong than to foster any kind of constructive exchange. one day, you will all come to ME.

[stratum]

Often it's interesting to see to what extent an idea can be defended because it reveals things that otherwise would stay obscured. Take the fft code above for example - the fact that it has rotation formula inside is not much revealing because with complex numbers even multiplication is rotation but we have found rotation elsewhere - in the dft matrix, even though that did not have a clear interpretation other than "just because we can do this". Useful analogies do not abruptly stop like this, they reveal a bit more. For example with the oscillation=rotation analogy that you have used to derive fast sin\cos approximation one can go further and derive a general formula for fm modulation. In the proposal dft=rotation that has not been possible yet. The fact that we can write a number of correlation operations as matrix multiplication is a nice way to visualise the redundancies in computation - apparently fft version of dft was found this way. Other than that the analogy did not serve its original purpose : to explain complex dft and no such explanation is actually necessary as the very purpose of algebraic rings such as complex number arithmetic is to reuse existing explanations. I'm still curious about it though, but assigning some useful meaning to a 2n dimensional rotation may escape humanly possible imagination, reveal new things and therefore be a useful analogy.

[aciddose]

How does something as simple as "a series of tubes" manage to escape a dog's imagination?

Again, this is the reason people are confused about the essential elements involved. It is wholly due to needlessly over-complicated explanations which in reality explain nothing at all.

It is very simple to see that the inner function is rotation of a vector by a matrix in 2d. Nothing more needs to be said!

There is absolutely no reason to wonder whether this applies as a general abstraction or not. It isn't abstract, it is concrete.

[stratum]

It isn't abstract, it is concrete

I agree about this part. The fact is that DFT has turned out to be rotation at a very elemental level. The part that I do not agree is this: consider the following example, the idea that human beings are animals. Is it correct? Yes. Would it be correct if I had said human beings are merely animals? No. A human being is something more, and that something is important. Whether that something is called intellect, spirit or merely whatever ability that is derived from language of speech, I do not know what it is, but it obviously exists as a difference.

In this very sense, DFT is not merely rotation, it is something more. In a higher dimensional space one can probably find many such 'rotations'. Any basis change can be considered to be a rotation in the sense if we rotate a room without rotating an object inside the room with respect to the room, that object would look as if it had rotated according to its surroundings. Some of such 'rotations' are interesting because they correspond to something specific. That specific thing is what you should consider to be the explanation or meaning of DFT. The idea of rotation -in this case- is not tracable to anything other than the fact that it is an higher dimensional space of vectors in which one can find many examples of such rotations.

As for what that specific thing about the 'rotation' called DFT is visible in the DFT matrix that mystran had posted, it is correlation. The title of the thread. The original poster knew it intuitively without delving into any of these details. That's an interesing property of human intellect and it is the cause of the need for looking for explanations. Interestingly math does not always give reliable hints as to what that intellect finds intuitively without having any such help. As a concrete example, the DFT code is misleading but the DFT matrix isn't and both are math. Who is to decide what should be accepted as an explanation? Not a robot following mathematical equations mindlessly, that's for sure. For a very specific argument along these lines, you can have a look at this book https://en.wikipedia.org/wiki/Shadows_of_the_Mind and that book and its predecessor are the most eleborate arguments in this direction one can probably find. Have a look at them before deciding that this kind of approach is wrong-headed.

[xoxos]

if you get mixed up and say "spit and sin" then you realise the sweeping scale and that it was in there already. like sooooooooo many other well the lot. big hands your the

[stratum]

As for what that specific thing about the 'rotation' called DFT is visible in the DFT matrix that mystran had posted, it is correlation.

This part of the text needs a correction, it would be more appropriate to say:

As for what that specific thing about the 'rotation' called DFT is visible in the DFT matrix that mystran had posted, it is correlation to a specific set of basis functions.

Somewhere earlier in the thread the information about why such a basis change is possible at all and under what conditions the transform would be bijective (i.e. reversible) is also stated, so I guess that does not need to be repeated, but I guess it may be necessary to say that correlation is a tool used to make such a change, and the tool would not work as expected if the chosen functions are not an orthonormal basis to begin with. That is the summary of this rather long thread.

[neodymDNB]

two questions about Wavelet Transform

1. can you add the resulting wavelets together and get the exact original signal like you can with sine and cosines from discrete fourier transform?

2. why isnt Wavelet used more? from what I have reed it seems like superior version of short time fourier ( windowed ),its like variable window lenght depending on freqency so its long enough to get the bass freqencies and up top it gets shorter for better time resolution

[stratum]

It's in use in image processing http://www.ni.com/white-paper/3887/en/

For audio this sentence was interesting (from
https://en.wikipedia.org/wiki/Wavelet_t ... y_analysis )

the common Morlet wavelet is mathematically identical to a short-time Fourier transform using a Gaussian window function.

So, in a way it's actually frequently in use.

That much info is easily found using just google/wiki.

Other than that I don't know. The features in viola jones object detector also look similar. https://en.wikipedia.org/wiki/Haar-like_features

As for the first question, among many possible wavelet transforms I guess one can construct one that fits that requirement.

[camsr]

2. why isnt Wavelet used more? from what I have reed it seems like superior version of short time fourier ( windowed ),its like variable window lenght depending on freqency so its long enough to get the bass freqencies and up top it gets shorter for better time resolution

The frequency resolution of wavelets follows a logarithmic slope upwards, where the STFT is a linear slope. It's intuitive to think of wavelets as a normal filter network, where each band has a spacing that is exponential in frequency. That makes the "Q" of each band the same, whereas the linear spacing has increasing "Q" with frequency.

[neodymDNB]

camsr
so the freqency resolution decreases or increases when we go higher in freqency?

edit: forgot word resolution

[Miles1981]

Decreases, as you also are using less points (you get only half of the samples for the next wavelet computation).

[neodymDNB]

Decreases, as you also are using less points (you get only half of the samples for the next wavelet computation).

ahhh,interesting... so you with short time FFT can have better freqency resolution in upper parts of freqency spectrum than wavelet?

I know that longer window lenghts give FFT better freqency resolution at the expense of time resolution,I have two questions,1. does different wavelet shapes have impact on freqency resolution? 2. at what freqency and window lenght for FFT,does wavelet get less resolution than the FFT

[camsr]

Different wavelets are used for different things. Ideally, a wavelet measures correlation perfectly when the input has a signal within it that is correlated to the wavelet signal. At that point, you see the output of the convolution spike in amplitude, unless there is too much noise on the input.

A really simple example would be, if you had a recording with a sampled bass drum in it, you could use the bass drum sample as the wavelet, and whenever the wavelet and input correlate, the convolution output goes high.

[neodymDNB]

what is the difference between DTFT and DFT? I reed in book DTFT isnt used becose it requires infinite amount of sinusoids to calculate.... it says DTFT is aperoid while DFT is periodic.

[stratum]

I have started to learn new things due to your questions:)

The answer you are looking for is here http://dsp.stackexchange.com/questions/ ... r-transfor

The discrete-time Fourier transform (DTFT) is the (conventional) Fourier transform of a discrete-time signal.

That is a theoretially significant tool, it can tell you the spectrum of a sampled signal for example, from a theoretical perpective instead of just telling you its frequently assumed content that comes out of a digital analog converter when that signal is converted to analog.
Many DSP books would tell you things like that the frequency spectrum of a sampled signal repeats itself up to infinity. This is how they have found that result.