In a FFT, the resulting frequencies represent both magnitude and phase. Since each frequency element in the output array of an FFT essentially just describes the SIN wave at each frequency interval, shouldn't it just be magnitude that we need? What is the significance of the phase represented in the imaginary part of the complex number?
To clarify my question, to be able to put a meaning to the phase of a wave, I need a reference point or reference wave.
When an FFT reports the phase for each sin wave in the resulting frequency domain output, what is the reference wave with respect to which it is reporting the phase?
Because the phase of different components affects the total signal. The two functions in the plot are both summed from sine waves with periods of pi and 2pi, but the phase of the p=2pi sine waves are different. As you can see, the outputs are not the same.
Well in layman's words: magnitude tells you how much of that frequency is there, and phase tells you where it is.
FFTs (there is more than one convention) usually report phase with respect to the zero-th sample. Or if you use FFTShift, with respect to the sample at the center of an FFT window that indexes from 0 to N-1 (e.g. sample number N/2 = sin(0) for a phase of 0). The latter convention, centering phase using FFTShift, is often better, as there can be a big discontinuity at the edges of an FFT aperture, or nearly no data at the edges after using a tapered window function.
If you use FFTShift to center the phase reference, zero phase represents an even function, and a phase of pi or -pi represents an odd function in the window.
Human hearing, in general, can't discriminate the phase of a single sound source. BUT, phase is important when dealing with combined sounds, or multiple sine waves of the same frequency. Sinusoids that are in phase add or sum. Sinusoids of the opposite phase cancel. So if you have the FFT of, say, two loudspeaker responses without phase, you won't know whether they will sound great or horrible together.
Related
RSME calculates how close the predicted value is compared to the actual value, but in a point cloud, there are 2 things that I am confused about:
How do we know which point corresponds to which point, to be subtracted from?
Point clouds are 3-dimensional since it has xyz values, but how do people turn those 3 values to one RSME value?
First of all, it's RMSE, not RSME. It stands for Root Mean Square Error:
https://en.wikipedia.org/wiki/Root-mean-square_deviation
With 3D coordinates you can compare component wise, or however else you choose to define a distance measure. Then you plug this into the RMSE formula. Essentially this means comparing an expected value to your observed value.
As for the point correspondence - this depends on the algorithm of choice. Probably one of the most famous examples is ICP:
https://de.wikipedia.org/wiki/Iterative_Closest_Point_Algorithm
In a nutshell for every point of one cloud, the closest point of the other cloud is determined. Then an error measure is calculated and lastly points are transformed. This is done an arbitrary number of times, depending on the desired precision.
Since I strongly suspect that you are indeed looking for ICP, here is the description as to how they are put together:
https://en.wikipedia.org/wiki/Iterative_closest_point
Other than that you will have to do some reading yourself.
I got some problems with this homework which I have totally no idea, never got into this field before and I really need some help.
First, we have a wiener process like
Which means the probability of the process drops beneath -3 within the time interval [0,1].
Now the thing is we have to simulate the process by discretize it.
1.Suppose we first discretize the process by 100 points and simulate 10,000 process in this way.
i.e., W(0.01), W(0.02), …., W(1.00).
Note that W(t) – W(t-0.01) ~ N(0,0.01) independently.
2.Using the data obtained at 1., we approximate
by
what is the relationship between this value and the real
(larger, equal to or smaller)?
3.Repeat 1. and 2. by cutting [0,1] into 10,000 points instead. Will the
resulting probability increases or decreases?
Im really confused over here. I am a ai programmer working on a game that is designed to detect beats in songs and some more. I have no previous knowledge about audio and just reading through whatever material i can find. While i got fft working and stuff I simply don't understand the way samples are transferred to different frequencies. Question 1, what does each frequency stands for. For the algorithm i got. I can transfer for example 1024 samples into 512 outcomes. So are they a description of the strength of each spectrum at the current second? it doesn't really make sense since what i remember is that there are 20,000hz in a 44.1khz audio recording. So how does 512 spectrum samples explain what is happening in that moment? Question 2, from what i read, its a number that represent the sound wave at this moment. However i read that by squaring both left channel and right channel, and add them together and you will get the current power level. Both these seems incoherent to my understanding, and i am really buff led so please explain away.
DFT output
the output is complex representation of phasor (Re,Im,Frequency) of basis function (usually sin wave). First item is DC offset so skip it. All the others are multiples of the same fundamental frequency (sampling rate/N). The output is symmetric (if the input is real only) so use just first half of results. Often power spectrum is used
Amplitude=sqrt(Re^2+Im^2)
which is the amplitude of basis function. If phase is needed then
phase=atan2(Im,Re)
beware DFT results are strongly dependent on the input signal shape,frequency and phase shift to your basis functions. That causes the output to vibrate/oscillate around the correct value and produce wide peaks instead of sharp ones for singular frequencies not to mention aliasing.
frequencies
if you got 44100Hz then the max output frequency is half of it that means the biggest frequency present in data is 22050Hz. The DFFT however does not contain this frequency so if you ignore the mirrored second half of results then:
for 4 samples DFT outputs frequencies are { -,11025 } Hz
for 8 samples frequencies are: { -,5512.5,11025,16537.5 } Hz
The output frequency is linear to its address from start so if you got N=512 samples
do DFFT on it
obtain first N/2=256 results
i-th sample represents frequency f=i*samplerate/N Hz
where i={ 1,...,(N/2)-1} ... skipping i=0
the image shows one of mine utility apps tighted together with
2-channel sound generator (top left)
2-channel oscilloscope (top right)
2-channel spectral analyzer (bottom) ... switched to linear frequency scale to make obvious what I mean in above text
zoom the image to see the settings ... I made it as close to the real devices as I could.
Here DCT and DFT comparison:
Here the DFT output dependency on input signal frequency aliasing by sampling rate
more channels
Summing power of channels is more safe. If you just add the channels then you could miss some data. For example let left channel is playing 1 Khz sin wave and the right exact opposite so if you just sum them then the result is zero but you can hear the sound .... (if you are not exactly in the middle between speakers). If you analyze each channel independently then you need to calculate DFFT for each channel but if you use power sum of channels (or abs sum) then you can obtain the frequencies for all channels at once , of coarse you need to scale the amplitudes ...
[Notes]
Bigger the N nicer the result (less aliasing artifacts and closer to the max frequency). For specific frequencies detection are FIR filter detectors more precise and faster.
Strongly recommend to read DFT and all sublinks there and also this plotting real time Data on (qwt) Oscillocope
I have a linear regression equation from school , which gives a value between 1 and -1 indicative of whether or not a set of data points are close enough to a linear function
and the equation given here
http://people.hofstra.edu/stefan_waner/realworld/calctopic1/regression.html
under best fit of a line. I would like to use these to do simple gesture detection based on a point in 3-space (x,y,z) - forward, back, left, right, up, down. First I would see if they fall on a line in 2 of the 3 dimensions, then I would see if that line's slope approached zero or infinity.
Is this fast enough for functional gesture recognition? If not, could someone propose an alternative algorithm?
If I've understood your question correctly then (1) the calculation you describe here would probably be plenty fast enough, (2) it may not actually do what you want, and (3) the stuff that'll be slow in an actual implementation would lie elsewhere.
So, I think you're proposing to do this. (1) Identify the positions of ... something ... (the user's hand, perhaps) in three-dimensional space, at several successive times. (2) For (say) each of {x,y} and {x,z}, look at those two coordinates of each point, compute the correlation coefficient (which is what your formula describes) and see whether it's close to +-1. (3) If both correlation coefficients are close to +-1 then the points lie approximately on a straight line; calculate the gradient of that line (using a formula similar to that of the correlation coefficient). (4) If the gradients are both very close to 0 or +- infinity, then your line is approximately parallel to one axis, which is the case you're trying to recognize.
1: Is it fast enough? You might perhaps be sampling at 50 frames per second or thereabouts, and your gestures might take a second to execute. So you'll have somewhere on the order of 50 positions. So, the total number of arithmetic operations you'll need is maybe a few hundred (including a modest number of square roots). In the worst case, you might be doing this in emulated floating-point on a slow ARM processor or something; in that case, each arithmetic operation might take a couple of hundred cycles, so the whole thing might be 100k cycles, which for a really slow processor running at 100MHz would be about a millisecond. You're not going to have any problem with the time taken to do this calculation.
2: Is it the right thing? It's not clear that it's the right calculation. For instance, suppose your user's hand moves back and forth rapidly several times along the x-axis; that will give you a positive result; is that what you want? Suppose the user attempts the gesture you want but moves at slightly the wrong angle; you may get a negative result. Suppose they move exactly along the x-axis for a bit and then along the y-axis for a bit; then the projections onto the {x,y}, {x,z} and {y,z} planes will all pass your test. These all seem like results you might not want.
3: Is it where the real cost will lie? This all assumes you've already got (x,y,z) coordinates. Getting those is probably going to be more expensive than processing them. For instance, if you have a camera-based system of some kind then there'll be some nontrivial image processing for every frame. Or perhaps you're integrating up data from accelerometers (which, by the way, is likely to give nasty inaccurate position results); the chances are that you're doing some filtering and other calculations to get position data. I bet that the cost of performing a calculation like this one will be substantially less than the cost of getting the coordinates in the first place.
I'm designing a real time Audio Analyser to be embedded on a FPGA chip. The finished system will read in a live audio stream and output frequency and amplitude pairs for the X most prevalent frequencies.
I've managed to implement the FFT so far, but it's current output is just the real and imaginary parts for each window, and what I want to know is, how do I convert this into the frequency and amplitude pairs?
I've been doing some reading on the FFT, and I see how they can be turned into a magnitude and phase relationship but I need a format that someone without a knowledge of complex mathematics could read!
Thanks
Thanks for these quick responses!
The output from the FFT I'm getting at the moment is a continuous stream of real and imaginary pairs. I'm not sure whether to break these up into packets of the same size as my input packets (64 values), and treat them as an array, or deal with them individually.
The sample rate, I have no problem with. As I configured the FFT myself, I know that it's running off the global clock of 50MHz. As for the Array Index (if the output is an array of course...), I have no idea.
If we say that the output is a series of One-Dimensional arrays of 64 complex values:
1) How do I find the array index [i]?
2) Will each array return a single frequency part, or a number of them?
Thankyou so much for all your help! I'd be lost without it.
Well, the bad news is, there's no way around needing to understand complex numbers. The good news is, just because they're called complex numbers doesn't mean they're, y'know, complicated. So first, check out the wikipedia page, and for an audio application I'd say, read down to about section 3.2, maybe skipping the section on square roots: http://en.wikipedia.org/wiki/Complex_number
What that's telling you is that if you have a complex number, a + bi, you can picture it as living in the x,y plane at location (a,b). To get the magnitude and phase, all you have to do is find two quantities:
The distance from the origin of the plane, which is the magnitude, and
The angle from the x-axis, which is the phase.
The magnitude is simple enough: sqrt(a^2 + b^2).
The phase is equally simple: atan2(b,a).
The FFT result will give you an array of complex values. The twice the magnitude (square root of sum of the complex components squared) of each array element is an amplitude. Or do a log magnitude if you want a dB scale. The array index will give you the center of the frequency bin with that amplitude. You need to know the sample rate and length to get the frequency of each array element or bin.
f[i] = i * sampleRate / fftLength
for the first half of the array (the other half is just duplicate information in the form of complex conjugates for real audio input).
The frequency of each FFT result bin may be different from any actual spectral frequencies present in the audio signal, due to windowing or so-called spectral leakage. Look up frequency estimation methods for the details.