How to interpret Fourier Transform result?

Can anybody tell me what result of discrete fourier transform means? I know all theoretical stuff and pretty graphs, that it is a change of domain from time to frequency and so on.

But I want to know what result means.

Example 1. Let's say I have a vector of 256 values. Every each of them is equal to 9.81. Result of FFT is 256 numbers equal to 0. Why?

Example 2. Again I have a vector of 256 values. 255 of them are equal to 9.81, and one of them (last one) is equal to 1.00

As a result of FFT transform I get a vector of 256 values, with first 255 ranging between 8.810 and 8.91, and last one equal to 0. What does it mean?

EXAMPLE 3 - REAL LIFE SCENARIO - KEY QUESTION I am recording data from an accelerometer. I record 256 values. Depending on whether I record walking up the stairs, or sitting, I get different result from Fourier Transform ran on this data. What is the significance of each number in the vector of 256 values? The two vectors from fourier transform will be different from each other, by looking at them, what do they tell me about samples from which they originated?

The resultant vector of 256 values represents decomposition of acceleration values into what?

My wrong understanding

I am told that domain becomes frequency instead of time. So numbers along x axis represent frequency. Frequency 1, frequency 2, ... frequency 256. For a sample of 256 numbers.

So my undersanding was:

If I have such sample:

4,8,15,16

then each number has a frequency of 1. So if I plot a graph with x axis that represents, at tick labeled 1 I would put value of 4, and leave tick 2, tick 3 and tick 4 at 0.

If sample was:

4,4,4,4

Then I would have {0,0,0,1}

If sample was:

4,4,8,8

Then I would have {0,2,0,0}

But this is not fourier transform, to me it looks like frequency distribution. Can anybody tell me then what do actual VALUES that I get in return vector mean?

EDIT: To be clear, I am not asking how we arrive at these values. I want to know how to interpret these values, what do they tell me about input.

Solution 1:

Let me suggest you a game to make it clear: Create the inverse Fourier that means construct a function $Y(t)$ that describes your sequence.

Point all your data versus time. Then start using the following function (in Mathematica or any other math program) (instead of complex expression this is easier to understand the connect to roots of unity):

$$y_i(t,k)= \sum_{n=1}^k \frac{1}{k} \cos\left(\frac{2 \pi (n-1) t}{k}\right)$$

(Discrete)Plot it and test! You may need to add a couple of different such functions $y_i(t,k)$ together (superposition) while every function with different $k$ and also multiplied perhaps with different amplitudes $a_i$ to get your function $Y(t)$ (like playing LEGO):

$$Y(t)=\sum_i a_i\; y_i(t,k)$$

This equation would as good as possible mimic (inverse Fourier) your sequence. If you do so you will understand that the transform is exactly the opposit way, and what its details value for you.

I did this quite often and works great to have sequences comprehensive.

Your first example and a period of $k=2$ for instance trivially:

$$Y(t)=y_1(t,2)= 9.81 \sum_{n=1}^2 \frac{1}{2} \cos\left(\frac{2 \pi (n-1) t}{2}\right)$$

PS: By the way $y_i(t,k)$ converges.

Additional explanation:

I hope I got the set up of your question correctly: You have discrete sequences of data like $\{9.81,...\}$ or $\{9.81,...,1.00,9.81...\}$ on the time line and transform these via Fourier to frequency domain.

Perhaps I understand what is your issue, and this is probably deeper going (frequency/amplitude uncertainty principle). I brought above an exercise (a game that should help) but lets try to answer your questions more directly regarding the values.

Can anybody tell me what result of discrete fourier transform means?

The result of the Fourier Transform as you will exercise from my above description will bring you only knowledge about the frequency composition of your data sequences. That means for example 1 the zero 0 of the Fourier transform tells you trivially that there is no superposition of any fundamental (eigenmode) periodic sequences with corresponding frequencies. This is why I wrote you $Y=y_1$. Your sequence is a periodic eigenmode of itself. Trivially your numbers of value 9.81, how many they might be (256), do not provide any additional information than they have in the sequence itself, because they are just the amplitude of your preiodic sequence which is its own eigenmode. So the Fourier transform tells you: there is a trivial case of frequency of eigenmode sequence and your numbers tell you there is trivial case of amplitudes - i.o.w NOTHING CHANGES in your sequence the behaviour is of no change. In other words you will learn here that the spectrum contains only one eigenmode and that is the sequence itself with a ground amplitude 9.81. Nothing more than this!

The second example is drammatic different. The amplitude $(1.00-9.81)$ (actually the difference) is carrying great information (as for example the location of a particle in physics, certain behaivourial change in the sequence/ accelleration...). However in order to precisely point this out you need a Fourier that contains a spectrum of frequencies. So remind the case of wave and particle in physics, when you want to locate the particle at a certain time/location (analogue to your $(1.00-9.81)$ signal at a certain time) that means precise amplitude, you would need all fundamental frquencies and waves/modes of the spectrum to sum them up and obtain the $(1.00-9.81)$ at a percise certain time point (this is why in my previous message I wrote as good as possible). In an abstract interpretation this superposition and sum up is implicitly done by your measurement instrument, and it points precisely to $(1.00-9.81)$ at a percise certain time point i a sequence. The problem however is that when you have obtained this way a precise $(1.00-9.81)$ at a certain time then you (your device) can not observe the frequencies, because you summed them up (implicitly) in the waves; in reverse if you perform a Fourier transform on the data sequence recorded, as you did, you obtain all the frequencies of the fundamental waves/modes composing your signal $(1.00-9.81)$ but at the cost that you can not more precisely fix the amplitude $(1.00-9.81)$ (of course in process of your decomposition the result is decomposed and scattered).

This is the uncertainty principle of any spectrum/Fourier. Unfortunately what happens in your thought scenario is that you want to perform a Fourier transform but ask for the meaning of the amplitudes ($(1.00-9.81)$). The answer is clear: The Fourier transform / spectrum of frequencies does not give you any information about the amplitude of the superposition.

Side note: At the same time keep into mind that the first example is the buttom line or callibration of the second example.

Annecdotic: The first example is an orchestra of one drum and itself repeating at a certain frquency and 9.81 pitch. The second could be an orchestra of the same drum playing at the same frequency and 9.81 pitch but in addition many instrumental players who are scilent all time except of precisely one instance of time, when playing one singular shot of 1.00 pitch.

For example 3 we learned already that the Fourier transform and the spectrum of frequencies do not give you any information about your absolute values in time domain (pitch)! But it gives you information on how often in a sequence you will change to sit position or vice versa. So if the spectrum shows more high frequencies bandwidth, that means you change often your position and if low frquencies bandwidth it shows you less change position. The Fourier transform will tell you how you behave, whether you accellerate more often up and down or less. It does not give you any information about the amplitudes (uncertainty paradox). Trivially the information about the amplitudes is already what you see in the time domain.

So before interpreting your data decide whether you observe in time domain (amplitudes and absolute value of your data) or frquency domain (frequency, behaviour, change).

The resultant vector of 256 values represents decomposition of acceleration values into what?

Into a spectrum of frequencies that is a characteristic fingerprint of your accellaration behaviour brought in that sequence of data record.

Hope this helps you to better understanding and is Answer to your Question. I still suggest you to do the exercise I proposed. Nothing is better than own experience.

Solution 2:

It is completely normal to get 0 if the values are equal. This comes from an equality of the roots of unity of powers of 2, and for DFT you always extends to powers of two (esp if you do programming, in order to get from O(N^2) to O(nlogn).

You can see on wikipedia.

What DFT does has to do with polynomials. You can transform your list into a polynomial and then compute the values of that polynomial at the roots of unity of the closest power of 2 which is bigger than the number of items in your list.

Eg. 4,3,4,1 can be represented as $4 x^3 + 3x^2 + 4x+ 1$.

In your 4,4,4,4 example, you would get $4 x^3 + 4x^2 + 4x+ 4$. Because of the fact that DFT evaluates the polynomial at the roots of unity, in this case you will get 0 (you can check this yourself, the root of unity of order 4 is $i$.)

In short, from a list of numbers -> polynomial $p$. let the degree of the polynomial be $k$ with $k$ power of 2. Then if $w_k$ is the basic root of unity of order k, the result of the dft at index $t$ will be $p(w_k)^t$., where $t$ ranges betweeen 0 and $k-1$.

Solution 3:

You're looking at the frequency of a certain number occuring, but instead it's actually supposed to give you the frequency of the sequence as if it were a wave. For example, if you'd drawn a sine wave on an Etch-a-Sketch, you could get a sequence of discrete values approximating it. That is, $(x,x,x,x)$ would not have a frequency at 4. $(x,y,x,y)$ would have a frequency at 2, but $(x,x,y,y)$ would have a frequency at 4. Ignoring alias frequencies, anyway.