I use Auriotouch sample program to generate FFT data to get Frequency and magnitude. Is there any easy method available to get SFT from FFT?
If by SFT you mean DFT (the Discrete Fourier Transform of a finite vector or array of equally spaced audio sample values), it is identical to the result of an FFT computation (plus or minus some tiny numerical rounding differences) of that same sample vector.
Related
I'm trying to figure out how MATLAB does the short time Fourier transforms for its spectrogram function (and related functions like specgram, or stft in Octave). What is curious to me is that you can apparently specify the length of the window and the FFT length (number of output frequencies) independently, whereas I would have expected that these two should be equal (since the length of an FFT'd signal is the same as the length of the original signal). To illustrate what I mean, here is the function call:
[S,F,T]=spectrogram(signal,winSize,overlapSize,fftSize,rate);
winSize is the length of subintervals which are to be (individually) FFT'd, and fftSize is the number of frequency components given in the output. When these are not equal, does Matlab do interpolation to produce the required number of frequency bins?
Ultimately the reason I want to know is so that I can determine the proper units and scaling for the frequencies.
Cheers
A windowed segment of a signal can be zero-padded to a longer length vector to use a longer FFT. The frequency scaling will be determined by the length of the FFT (and the signals sample rate). The window size and window formula will determine the effective resolution, in terms of peak separation ability.
Why do this? Some FFT sizes can be computed more efficiently than others (slightly or a lot, depending on the FFT library used). Also, a longer FFT will calculate more points or bins, thus producing a higher density of interpolated points in a potentially smoother spectrum result.
I have a time series which is a linear combination of damped waves. The data is real.
Y(t) =SUM_w exp(- gamma t) sin(omega t)
There is no analytic form but this is a closest guess. I want to fourier analyze (FFT) such data and get the real frequencies and damping rates.
I am using matlab but any tool would be fine
Thanks!
Your question would be a better fit for http://math.stackexchange.com, where LaTeX rendering is available for formula. Instead, you have to use e.g. this bookmarklet for proper display:
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First of all, I assume your function is more precisely of the kind
$\sum\limits_k e^{-\gamma_k t}\sin(\omega_k t)$ for $t>0$ and $0$ for $t<0$ (otherwise the function would tend to infinity for $t\to-\infty$). Since a damped function is not periodic, you cannot use Fourier analysis but have to use the Fourier transform, which yields an amplitude for continuous frequencies instead of discrete ones. Using the complex representation $\sin(x) = \frac{e^{ix}-e^{-ix}}{2i}$, each term in the sum can be Fourier transformed individually, yielding
$\frac1{2\pi}\int_0^\infty e^{-\gamma_k t}\sin(\omega_k t)e^{-i\omega t}\,dt = \frac{\omega_k}{2\pi[(\gamma+i\omega)^2+\omega_k^2]} = \frac{\omega_k}{2\pi}\frac{(\gamma^2-\omega^2+\omega_k^2) + 2i\gamma\omega\omega_k}{[\gamma^2-\omega^2+\omega_k^2]^2+4\gamma^2\omega^2}$
Since you state the damped sine is only a guess, you have to discretize this unlimited integral somehow, though due to the damping a cutoff at some sufficiently large time $t$ should be in order. If the damping $\gamma_k$ is actually the same for all summands, your life becomes easier: Multiply the data by $e^{+\gamma t}$ to obtain a periodic signal which you can now really FFT on.
First digitize your function:
t=0:dt:T; % define sampling interval dt and duration T according to your needs
Y=sum(exp(-gamma*t).*sin(omega*t));
Then do fft and plot:
Y_f=fft(Y);
plot(abs(Y_f));
let us assume,
I have a vector t with the times in seconds of my samples. (These samples are not equally distributed on the time domain.
Also I have a vector data containing the samplevalues at the time t.
t and data have the same length.
If I plot the graph some sort of periodical signal is obtained.
now I could perform: abs(fft(data)) to get my spectrum, which is then plotted over the amount of data points on the x-axis.
How can I obtain my spectrum regarding the times in vector t and plot it?
I want to see which frequencies in 1/s or which period in s my signal contains.
Thanks for your help.
[Not the OP's intention]: FFT will give you the spectrum (global) for any number of input data points. You cannot have a specific data point (in time) associated with parts (or the full) spectrum.
What you can do instead is use spectrogram and obtain the Short-Time Fourier Transform (STFT). This will give you a NxM discrete grid of time-frequency FT values (N: FT frequency bins, M: signal time-windows).
By localizing the (overlapping) STFT windows on your data samples of interest you will get N frequency magnitude values, thus the distribution of short-term spectrum estimates as the signal changes in time.
See also the possibly relevant answer here: https://stackoverflow.com/a/12085728/651951
EDIT/UPDATE:
For unevenly spaced data you need to consider the Non-Uniform DFT (and Non-uniform FFT implementations). See the relevant question/answer here https://scicomp.stackexchange.com/q/593
The primary approaches for NFFT or NUFFT, are based on creating a uniform grid through local convolutions/interpolation, running FFT on this and undoing the convolutional effect of the interpolation filter.
You can read more:
A. Dutt and V. Rokhlin, Fast Fourier transforms for nonequispaced data, SIAM J. Sci. Comput., 14, 1993.
L. Greengard and J.-Y. Lee, Accelerating the Nonuniform Fast Fourier Transform, SIAM Review, 46 (3), 2004.
Pippig, M. und Potts, D., Particle Simulation Based on Nonequispaced Fast Fourier Transforms, in: Fast Methods for Long-Range Interactions in Complex Systems, 2011.
For an implementation (with an interface to MATLAB) try NFFT and possibly its parallelized version PNFFT. You may find a nice walk-through on how to set-up and use here.
You can resample or interpolate your sample points to get another set of sample points that are equally spaced in t. The chosen spacing or sample rate of the second set of equally spaced sample points will allow you to infer frequencies to the result of an FFT of that second set.
The results may be noisy or include aliasing unless the initial data set is bandlimited to a sufficiently low frequency to allow interpolation. If bandlimited, then you might try something like cubic splines as an interpolation method.
Although it may look like one can get a high FFT bin frequency resolution by resampling to a larger number of data points, the actual useful resolution accuracy will be more related to the original number of samples.
I am trying to prove that the white noise has constant power spectral density using matlab
but the amplitude of the spectrum looks like random amplitude.
can anyone tell me why?
here is my code.
noise = randn(1,10000);
fft_noise=fft(noise);
plot(abs(fft_noise(1:5000)))
thanks.
You need to average a bunch (law of large numbers) of FFTs of white noise to approach the average power spectral density.
If you take the FFT of an independent set of random variables from the same distribution, then you'll get an independent set of random variables from the same distribution since the inverse Fourier transform is (more or less) the same as the Fourier transform. The point is that the expected value for each frequency is the same.
you need to multiple the fft by the complex conjugate of the fft to show a flat PSD. i.e. change
fft_noise=fft(noise);
to
fft_noise=fft(noise).*conj(fft(noise));
I'm working on a control system that measures the movement of a vibrating robot arm. Because there is some deadtime, I need to look into the future of the somewhat noisy signal.
My idea was to use the frequencies in the sampled signal and produce a fourier function that could be used for extrapolation.
My question: I already have the FFT of the signal vector (containing 60-100 values e.g.) and can see the main frequencies in the amplitude spectrum. Now I want to have a function f(t) which fits to the signal, removes some noise, and can be used to predict the near future of the signal. How do I calculate the coefficients for the sine/cosine functions out of the complex FFT data?
Thank you so much!
AFAIR FFT essentially produces output as a sum of sine functions with different frequencies. The importance of each frequency is the height of each peak. So what you really want to do here is filter out some frequencies (ie. high frequencies for the arm to move gently) and then come back to the time domain.
In matlab this should be like going through the vector of what you got from fft, setting some values to 0 (or doing something more complex to it) and then use ifft to come back to time domain and make the prediction based on what you get.
There's also one thing you should consider while doing this - Nyquist frequency - this means that the highest frequency that you get on your fft is half of the sampling frequency.
If you use an FFT for data that isn't periodic within the FFT aperture length, then you may need to use a window to reduce spurious frequencies due to "spectral leakage". Frequency estimation techniques to better estimate "between bin" frequency content may also be appropriate. The phase of each cosine sinusoid, relative to the edge of the window, is usually atan2(imag[i], real[i]). The frequency depends on the sample rate and bin number versus the length of the FFT.
You might also want to look into using a Kalman filter instead of an FFT.
Added: If your signal isn't exactly integer periodic in the FFT length, then you may want to do an fftshift before the FFT to move the resulting phase measurement reference point to the center of your data vector, instead of a possibly discontinuous circular edge.