I'm working with some code that does a Fourier transform (to calculate the cepstrum of an audio sample). Before it computes the Fourier transform, it applies a Hamming window to the sample:
for(int i = 0; i < SEGMENTATION_LENGTH;i++){
timeDomain[i] = (float) (( 0.53836 - ( 0.46164 * Math.cos( TWOPI * (double)i / (double)( SEGMENTATION_LENGTH - 1 ) ) ) ) * frameBuffer[i]);
}
Why is it doing this? I can't find any reason for it to do this in the code, or online.
This is an old question, but I thought the answer could be improved.
Imagine the signal you want to fourier transform is a pure sine wave. In the frequency domain, you would expect it to have a sharp spike only at the frequency of the sine. However if you took the fourier transform, your nice sharp spike would be replaced by something like this:
Why is that? Real sine waves extend to infinity in both directions. Computers can't do computations with an infinite number of data points, so all signals are "cut off" at either end. This causes the ripple on either side of the peak that you see. The hamming window reduces this ripple, giving you a more accurate idea of the original signal's frequency spectrum.
More theory, for the interested: when you cut your signal off at either end, you are implicitly multiplying your signal by a square window. The fourier transform of a square window is the image above, known as a sinc function. Whenever you do a fourier transform on a computer, like it or not, you're always choosing some window. The square window is the implicit default, but not a very good choice. There are a variety of windows that people have come up with, depending on certain characteristics you want to optimize. The hamming window is one of the standard ones.
Whenever you do a finite Fourier transform, you're implicitly applying it to an infinitely repeating signal. So, for instance, if the start and end of your finite sample don't match then that will look just like a discontinuity in the signal, and show up as lots of high-frequency nonsense in the Fourier transform, which you don't really want. And if your sample happens to be a beautiful sinusoid but an integer number of periods don't happen to fit exactly into the finite sample, your FT will show appreciable energy in all sorts of places nowhere near the real frequency. You don't want any of that.
Windowing the data makes sure that the ends match up while keeping everything reasonably smooth; this greatly reduces the sort of "spectral leakage" described in the previous paragraph.
With what I know about sound and quick research, it appears that Hamming Window is here to minimize the signal side lobe (unwanted radiation). Thus improving the quality or harmonics of the sound.
I also understand this type of window function fits good with DTFT.
You will find some good technical explanation on a stanford researcher page or wikipedia and also in a paper of Harris if you are ready for maths :D.
The FT of a finite length segment of sinusoid convolves the Fourier transform of the window against the sinusoid's frequency peak, since a property of the FFT is that vector multiplication in one domain is convolution in the other. The FT of a rectangular window (which is what any unmodified finite length of samples in an FFT implies) is the messy looking Sinc function which splatters any signal that is not exactly periodic in the window over the entire frequency spectrum.
The FT of a Hamming shaped window concentrates this "splatter" much nearer to the frequency peak after the convolution (than a Sinc function), resulting in a fatter but smoother frequency peak, but a lot less splatter across frequencies far from the frequency peak. This results in not only a cleaner looking spectrum, but also less interference from far away frequencies on any signal of interest.
This interpretation (as opposed to the "infinitely repeating" interpretation) makes it more clear why differently shaped windows than Hamming may give you better results with even less "leakage". In particular, a Hamming window will reduce the size of the first Sinc side lobe of "leakage" right next to the frequency peak in exchange for actually more "leakage" (or convolution splatter) far from the frequency of interest. Other windows may be more appropriate if you wish a different trade-off. The Harris paper (pdf here) linked in another answer above gives several examples of these different windows.
Related
I'm currently working on a program in C++ in which I am computing the time varying FFT of a wav file. I have a question regarding plotting the results of an FFT.
Say for example I have a 70 Hz signal that is produced by some instrument with certain harmonics. Even though I say this signal is 70 Hz, it's a real signal and I assume will have some randomness in which that 70Hz signal varies. Say I sample it for 1 second at a sample rate of 20kHz. I realize the sample period probably doesn't need to be 1 second, but bear with me.
Because I now have 20000 samples, when I compute the FFT. I will have 20000 or (19999) frequency bins. Let's also assume that my sample rate in conjunction some windowing techniques minimize spectral leakage.
My question then: Will the FFT still produce a relatively ideal impulse at 70Hz? Or will there 'appear to be' spectral leakage which is caused by the randomness the original signal? In otherwords, what does the FFT look like of a sinusoid whose frequency is a random variable?
Some of the more common modulation schemes will add sidebands that carry the information in the modulation. Depending on the amount and type of modulation with respect to the length of the FFT, the sidebands can either appear separate from the FFT peak, or just "fatten" a single peak.
Your spectrum will appear broadened and this happens in the real world. Look e.g for the Voight profile, which is a Lorentizan (the result of an ideal exponential decay) convolved with a Gaussian of a certain width, the width being determined by stochastic fluctuations, e.g. Doppler effect on molecules in a gas that is being probed by a narrow-band laser.
You will not get an 'ideal' frequency peak either way. The limit for the resolution of the FFT is one frequency bin, (frequency resolution being given by the inverse of the time vector length), but even that (as #xvan pointed out) is in general broadened by the window function. If your window is nonexistent, i.e. it is in fact a square window of the length of the time vector, then you'll get spectral peaks that are convolved with a sinc function, and thus broadened.
The best way to visualize this is to make a long vector and plot a spectrogram (often shown for audio signals) with enough resolution so you can see the individual variation. The FFT of the overall signal is then the projection of the moving peaks onto the vertical axis of the spectrogram. The FFT of a given time vector does not have any time resolution, but sums up all frequencies that happen during the time you FFT. So the spectrogram (often people simply use the STFT, short time fourier transform) has at any given time the 'full' resolution, i.e. narrow lineshape that you expect. The FFT of the full time vector shows the algebraic sum of all your lineshapes and therefore appears broadened.
To sum it up there are two separate effects:
a) broadening from the window function (as the commenters 1 and 2 pointed out)
b) broadening from the effect of frequency fluctuation that you are trying to simulate and that happens in real life (e.g. you sitting on a swing while receiving a radio signal).
Finally, note the significance of #xvan's comment : phi= phi(t). If the phase angle is time dependent then it has a derivative that is not zero. dphi/dt is a frequency shift, so your instantaneous frequency becomes f0 + dphi/dt.
I have some geographical trajectories sampled to analyze, and I calculated the histogram of data in spatial and temporal dimension, which yielded a time domain based feature for each spatial element. I want to perform a discrete FFT to transform the time domain based feature into frequency domain based feature (which I think maybe more robust), and then do some classification or clustering algorithms.
But I'm not sure using what descriptor as frequency domain based feature, since there are amplitude spectrum, power spectrum and phase spectrum of a signal and I've read some references but still got confused about the significance. And what distance (similarity) function should be used as measurement when performing learning algorithms on frequency domain based feature vector(Euclidean distance? Cosine distance? Gaussian function? Chi-kernel or something else?)
Hope someone give me a clue or some material that I can refer to, thanks~
Edit
Thanks to #DrKoch, I chose a spatial element with the largest L-1 norm and plotted its log power spectrum in python and it did show some prominent peaks, below is my code and the figure
import numpy as np
import matplotlib.pyplot as plt
sp = np.fft.fft(signal)
freq = np.fft.fftfreq(signal.shape[-1], d = 1.) # time sloth of histogram is 1 hour
plt.plot(freq, np.log10(np.abs(sp) ** 2))
plt.show()
And I have several trivial questions to ask to make sure I totally understand your suggestion:
In your second suggestion, you said "ignore all these values."
Do you mean the horizontal line represent the threshold and all values below it should be assigned to value zero?
"you may search for the two, three largest peaks and use their location and probably widths as 'Features' for further classification."
I'm a little bit confused about the meaning of "location" and "width", does "location" refer to the log value of power spectrum (y-axis) and "width" refer to the frequency (x-axis)? If so, how to combine them together as a feature vector and compare two feature vector of "a similar frequency and a similar widths" ?
Edit
I replaced np.fft.fft with np.fft.rfft to calculate the positive part and plot both power spectrum and log power spectrum.
code:
f, axarr = plt.subplot(2, sharex = True)
axarr[0].plot(freq, np.abs(sp) ** 2)
axarr[1].plot(freq, np.log10(np.abs(sp) ** 2))
plt.show()
figure:
Please correct me if I'm wrong:
I think I should keep the last four peaks in first figure with power = np.abs(sp) ** 2 and power[power < threshold] = 0 because the log power spectrum reduces the difference among each component. And then use the log spectrum of new power as feature vector to feed classifiers.
I also see some reference suggest applying a window function (e.g. Hamming window) before doing fft to avoid spectral leakage. My raw data is sampled every 5 ~ 15 seconds and I've applied a histogram on sampling time, is that method equivalent to apply a window function or I still need apply it on the histogram data?
Generally you should extract just a small number of "Features" out of the complete FFT spectrum.
First: Use the log power spec.
Complex numbers and Phase are useless in these circumstances, because they depend on where you start/stop your data acquisiton (among many other things)
Second: you will see a "Noise Level" e.g. most values are below a certain threshold, ignore all these values.
Third: If you are lucky, e.g. your data has some harmonic content (cycles, repetitions) you will see a few prominent Peaks.
If there are clear peaks, it is even easier to detect the noise: Everything between the peaks should be considered noise.
Now you may search for the two, three largest peaks and use their location and probably widths as "Features" for further classification.
Location is the x-value of the peak i.e. the 'frequency'. It says something how "fast" your cycles are in the input data.
If your cycles don't have constant frequency during the measuring intervall (or you use a window before caclculating the FFT), the peak will be broader than one bin. So this widths of the peak says something about the 'stability' of your cycles.
Based on this: Two patterns are similar if the biggest peaks of both hava a similar frequency and a similar widths, and so on.
EDIT
Very intersiting to see a logarithmic power spectrum of one of your examples.
Now its clear that your input contains a single harmonic (periodic, oscillating) component with a frequency (repetition rate, cycle-duration) of about f0=0.04.
(This is relative frquency, proprtional to the your sampling frequency, the inverse of the time beetween individual measurment points)
Its is not a pute sine-wave, but some "interesting" waveform. Such waveforms produce peaks at 1*f0, 2*f0, 3*f0 and so on.
(So using an FFT for further analysis turns out to be very good idea)
At this point you should produce spectra of several measurements and see what makes a similar measurement and how differ different measurements. What are the "important" features to distinguish your mesurements? Thinks to look out for:
Absolute amplitude: Height of the prominent (leftmost, highest) peaks.
Pitch (Main cycle rate, speed of changes): this is position of first peak, distance between consecutive peaks.
Exact Waveform: Relative amplitude of the first few peaks.
If your most important feature is absoulute amplitude, you're better off with calculating the RMS (root mean square) level of our input signal.
If pitch is important, you're better off with calculationg the ACF (auto-correlation function) of your input signal.
Don't focus on the leftmost peaks, these come from the high frequency components in your input and tend to vary as much as the noise floor.
Windows
For a high quality analyis it is importnat to apply a window to the input data before applying the FFT. This reduces the infulens of the "jump" between the end of your input vector ant the beginning of your input vector, because the FFT considers the input as a single cycle.
There are several popular windows which mark different choices of an unavoidable trade-off: Precision of a single peak vs. level of sidelobes:
You chose a "rectangular window" (equivalent to no window at all, just start/stop your measurement). This gives excellent precission of your peaks which now have a width of just one sample. Your sidelobes (the small peaks left and right of your main peaks) are at -21dB, very tolerable given your input data. In your case this is an excellent choice.
A Hanning window is a single cosine wave. It makes your peaks slightly broader but reduces side-lobe levels.
The Hammimg-Window (cosine-wave, slightly raised above 0.0) produces even broader peaks, but supresses side-lobes by -42 dB. This is a good choice if you expect further weak (but important) components between your main peaks or generally if you have complicated signals like speech, music and so on.
Edit: Scaling
Correct scaling of a spectrum is a complicated thing, because the values of the FFT lines depend on may things like sampling rate, lenght of FFT, window, and even implementation details of the FFT algorithm (there exist several different accepted conventions).
After all, the FFT should show the underlying conservation of energy. The RMS of the input signal should be the same as the RMS (Energy) of the spectrum.
On the other hand: if used for classification it is enough to maintain relative amplitudes. As long as the paramaters mentioned above do not change, the result can be used for classification without further scaling.
I am trying to implement FFT, and I am OK with the code etc, but the general order of things is confusing me.
Am I right in thinking that this is the correct order of things to do?
Input -> Overlap input -> Windowing -> FFT -> Phase calculations/Overlap compensation -> Output
I'm getting results close to my input frequency, but they are consistently off by some factor that I can't work out, i.e. 440Hz is always 407Hz, 430Hz is always 420Hz.
The main bit that is confusing me is the initial overlap, as I have been looking at some open source FFT code and that is the part that I can never quite work out whats going on. I seem to be getting the idea from looking at those that overlapping is supposed to happen before windowing, but to me logically, wouldn't that mess with the windowing?
Any advice would be great
Thanks
The FFT is a discrete version of the continuous Fourier Transform.
The FFT produces a 1D vector of complex numbers. This complex vector is often used to calculate a 2D matrix of Frequency Magnitude versus Frequency, and represented as a 2D graph, like this one:
A single FFT is used when you want to understand the frequency spectrum of a signal. For example, from the above FFT graph we can say that most of the energy in this female soprano's G5 note is concentrated in the 784 Hz and 1572 Hz frequencies.
STFT or "Short-Time Fourier Transform" uses a sliding-frame FFT to produce a 2D matrix of Frequency versus Time, often represented as a graph called a Spectrogram, like this one:
The STFT is used when you want to know at what time a particular frequency event occurs in the signal. For example, from the above graph we can say that a large portion of the energy in this vocal phrase occurred between 0.05 and 0.15 seconds, in the frequency range of 100 Hz to 1500 Hz.
The workflow for the FFT is:
Sample the signal -> Window the entire sample frame -> FFT -> Calculate magnitude and phase -> Output something, usually a 2D graph
If your time-domain data is available in text form and if you can post it here, we can try to help you analyze it, or you can analyze it yourself with this online FFT: Sooeet FFT calculator
If you use window for FFT, your computation will be a kind of STFT.
There are some prepared codes of STFT like 'Spectrogram' etc.
To write the code by FFT, the overlapping is inevitable,but you can use some optimization methods to minimize ghost effects.Also, the practical way for windowing may be choosing the window's bandwidth according to frequency extension. It is clear that in high frequency data's you need to select small windows which is so time consuming.
I am not good enough in Matlab to write this code adhesively:)
Good Luck
I need to know a way to make FFT (DFT) work with just n points, where n is not a power of 2.
I want to analyze an modify the sound spectrum, in particular of Wave-Files, which have in common 44100 sampling points. But my FFT does not work, it only works with points which are in shape like 2^n.
So what can I do? Beside fill up the vector with zeros to the next power of 2 ?!
Any way to modify the FFT algorithm?
Thanks!
You can use the FFTW library or the code generators of the Spiral project. They implement FFT for numbers with small prime factors, break down large prime factors p by reducing it to a FFT of size (p-1) which is even, etc.
However, just for signal analysis it is questionable why you want to analyze exactly one second of sound and not smaller units. Also, you may want to use a windowing procedure to avoid the jumps at the ends of the segment.
Aside from padding the array as you suggest, or using some other library function, you can construct a Fourier transform with arbitrary length and spacing in the frequency domain (also for non-integer sample spacings).
This is a well know result and is based on the Chirp-z transform (or Bluestein's FFT). Another good reference is given by Rabiner and can be found at the above link.
In summary, with this approach you don't have to write the FFT yourself, you can simply use an existing high-performance FFT and then apply the convolution theorem to a suitably scaled and conditioned version of your signal.
The performance will still be, O(n*log n), multiplied by some implementation-dependent scaling factor.
The FFT is just a faster method of computing the DFT for certain length vectors; and a DFT can be computed for any length of input vector. You can also zero-pad your input vector to a length supported by your FFT library, which may be faster.
If you want to modify your sound file, you may need to use the overlap-add or overlap-save fast convolution filtering after determining the length of the impulse response of your frequency domain modification.
Assume a sequence of numbers (wave-like data). I perform then the DFT (or FFT) transform. Next step I want to achieve is to find the frequencies, that correspond to the real frequencies that are included in data. As we know, DFT output has real and imaginary part a[i] and b[i]. If we look at spectrum (sqrt(a[i]^2+b[i]^2) then the maximum in it corresponds to the frequency that is included to the data. The question is how to find all frequencies from DFT? The problem arises when there are many other peaks that can be falsely selected.
I had a similar problem when doing spectral analysis processing of data when I was writing my honours thesis.
You are right: To find dominant frequencies you generally only need to look at the magnitude of the complex value in the DFT.
Unfortunately, you pretty much have to write some sort of intelligent algorithm which will identify the peaks (frequencies). The way the algorithm works is highly dependent on what the DFT looks like for your application. My DFTs all had similar characteristics, so it wasn't too difficult to put together a heuristic algorithm. If your DFT can take on any form, then you will probably get a lot of false positives and/or false negatives.
The way I did it was to identify regions in the DFT with high magnitude (peaks) which were surrounded by low magnitude (troughs). You can define the minimum difference between peaks and troughs (the sensitivity) as a constant times the standard deviation of the data. Additionally, you can say that any peaks that fall below a certain magnitude (threshold) are ignored altogether, as they are just noise.
Of course, the above technique will only really work if you have relatively well defined frequencies in your data. If your DFT is highly random, then you will need to take extra care to set the sensitivity and threshold carefully.
Don't forget that the magnitude of your data is symmetric, so you only need to look at half of it.
Once you have identified the frequencies in your DFT, don't forget to convert it into the units you want. From memory, if you have n samples taken with time discretisation dt, then if you have a peak at data point 5 (for example), where the first data point is 1, then the frequency is 1/(n*dt) radians per time unit. (I haven't done this in a while, so that formula might be off by a factor of Pi or something)