Could anyone suggest a method for computing the short time windowed fourier transform of a given signal, also known as the windowed fourier transform, of a signal with altering periodicity? Consider this example data series:
t = 1:365;
x = cos((2*pi)/12*t)+randn(size(t));
I understand the principles behind the method, but trying to implement it in matlab is another matter. I this example is not sufficient to provide an answer, any solution to a signal with varying periodicity would be much appreciated.
Related
I was using PCA for dimensionality reduction of MD (molecular dynamics) trajectory data of some protein simulations. Basically my data is xyz coordinates of protein atoms which change with time (that means I have lot of frames of this xyz coordinates). The dimension of this data is something like 20000 frames of 200x3 (atoms by coordinates). I implemented PCA using princomp command in Matlab.
I was wondering if I can do FFT on my data. I have experience of doing FFT on audio signals (1D signal). Here my data has both time and space in picture. It must be theoretically possible to implement FFT on my data and then filter it using a LPF (low pass filter). But I am unsure.
Can someone give me some direction/code snippets/references towards implementing FFT on my data?
Why people are preferring PCA more often compared to FFT and filtering. Is it because of computational efficiency of algorithm or is it because of the statistical nature of underlying data?
For the first question "Can someone give me some direction/code snippets/references towards implementing FFT on my data?":
I should say fft is implemented in matlab and you do not need to implement it by your own. Also, for your case you should use fftn (fft documentation)to transform and after applying lowpass filtering by dessignfilt (design filter in matalab), the apply ifftn (inverse fft in matlab)to inverse the transform.
For the second question "Why people are preferring PCA more often compared to FFT and filtering ...":
I should say as the filtering in fft is done in signal space, after filtering you can't generalize it in time space. You can more details about this drawback in this article.
But, Fourier analysis has also some other
serious drawbacks. One of them may be that time
information is lost in transforming to the frequency
domain. When looking at a Fourier transform of a
signal, it is impossible to tell when a particular event
has taken place. If it is a stationary signal - this
drawback isn't very important. However, most
interesting signals contain numerous non-stationary or
transitory characteristics: drift, trends, abrupt changes,
and beginnings and ends of events. These
characteristics are often the most important part of the
signal, and Fourier analysis is not suitable in detecting
them.
I have a (very) long signal in the form of vector, and I would like to apply a non-linear frequency response on that vector. For example:
v=rand(1000000,1);
nonlinFreqResponse = #(f,v) sqrt(v).*1/f;
V=wfft(v); %windowed FFt, cant do FFT on the entire signal
....?
Note that since the signal is long, running FFT on the entire signal is computationally complex and not feasible.
I believe the best solution is to apply the FFT as usual. Then apply a correction factor, or a transfer function, to your vector that is gain dependent. I'm not sure what's going on with your code, so can't help you there. Good luck!
I have audio record.
I want to detect sinusoidal pattern.
If i do regular fft i have result with bad SNR.
for example
my signal contents 4 high frequencies:
fft result:
To reduce noise i want to do Coherent integration as described in this article: http://flylib.com/books/en/2.729.1.109/1/
but i cant find any MATLAB examples how to do it. Sorry for bad english. Please help )
I look at spectra almost every day, but I never heard of 'coherent integration' as a method to calculate one. As also mentioned by Jason, coherent integration would only work when your signal has a fixed phase during every FFT you average over.
It is more likely that you want to do what the article calls 'incoherent integration'. This is more commonly known as calculating a periodogram (or Welch's method, a slightly better variant), in which you average the squared absolute value of the individual FFTs to obtain a power-spectral-density. To calculate a PSD in the correct way, you need to pay attention to some details, like applying a suitable Fourier window before doing each FFT, doing the proper normalization (so that the result is properly calibrated in i.e. Volt^2/Hz) and using half-overlapping windows to make use of all your data. All of this is implemented in Matlab's pwelch function, which is part of the signal-processing toolbox. See my answer to a similar question about how to use pwelch.
Integration or averaging of FFT frames just amounts to adding the frames up element-wise and dividing by the number of frames. Since MATLAB provides vector operations, you can just add the frames with the + operator.
coh_avg = (frame1 + frame2 + ...) / Nframes
Where frameX are the complex FFT output frames.
If you want to do non-coherent averaging, you just need to take the magnitude of the complex elements before adding the frames together.
noncoh_avg = (abs(frame1) + abs(frame2) + ...) / Nframes
Also note that in order for coherent averaging to work the best, the starting phase of the signal of interest needs to be the same for each FFT frame. Otherwise, the FFT bin with the signal may add in such a way that the amplitudes cancel out. This is usually a tough requirement to ensure without some knowledge of the signal or some external triggering so it is more common to use non-coherent averaging.
Non-coherent integration will not reduce the noise power, but it will increase signal to noise ratio (how the signal power compares to the noise power), which is probably what you really want anyway.
I think what you are looking for is the "spectrogram" function in Matlab, which computes the short time Fourier transform(STFT) of an input signal.
STFT
Spectrogram
I have some dynamic light scattering data. The machine pumps out the autocorrelation function, and a count-rate.
I can do a simple fit to the ACF
ACF = exp(-D*q^2*t)
and obtain the diffusion coefficient.
I want to obtain the same D from the power spectrum. I have been able to create a power spectrum in two ways -- from the Fourier transform of the ACF, and from the count rate. Both agree, but the power spectrum does not look like in the one in the books, so I'm not sure how to use it to work out the line width.
Attached is an image from a PDF that shows what you should get, and what I get from MATLAB. Can anyone make sense of whats going on?
I have used the code of answer #3 on this question. The resulting autocorrelation comes out exactly the same as
the machine gives me and
using MATLAB's autocorr command on the photoncount data.
Thank you for your time.
When you compute the Fourier transform from short sequences of data it often looks very noisy. There are a number of reasons for this. One reason is that the statistics of individual Fourier components are not Gaussian, and so averaging the spectra across multiple samples of data will only slowly improve the quality of the estimate.
Another causes of "noisiness" in empirical spectra behavior is that you are applying (to a finite data sample) a transform which involves a pathological sinc function and which assumes an infinite length signal. To diminish this problem, it helps to apply a "windowing-function" to your data before computing the Fourier transform. One of the more complicated but also more powerful windowing approaches is the use of so-called 'Slepian tapers'.
MATLAB conveniently implements well-known windows in functions such as hamming and hann.
I have performed an fft (fast fourier transform) on a time series waveform in Matlab, but I seem to have a weird wave actually in the fourier transform plot, although there are spikes this wave looks like something I'd expect to see only in the time domain. Is there any programming reason why this could happen?
The Fourier transform is quite similar to the Inverse Fourier Transform. A spike in one is a wave in the other. Hence, if you have one outlier datapoint in your series, you'll have a wave component in the frequency domain.
A possible programming-related issue could be an uninitialized data point, e.g. providing 1023 datapoints to a 1024-point FFT.
The fft assumes the signal is periodic so you can get some artefacts if the first and last values differ by enough to make that transition look like a step function. You are frequently better off windowing the data to avoid that phenomenon.
Note that the continuous-time Fourier transform of a finite-length signal can have things that look like "spikes" in the frequency domain. See the plots in this post of the continuous-time Fourier transform of a single period of a cosine signal and of ten periods of a cosine signal.
For example, an infinite extent cosine signal has a simple Fourier transform that's a pair of impulses at +/- the cosine frequency. But if you've only got ten periods of the cosine signal, the Fourier transform looks like this:
Steve's currently doing a nice series on Fourier transforms on his blog. He's specifically talking about 2D transforms, but you might find his discussion of windowing helpful.