I'm applying different windows to a signal and then get the frequency responses using fft function in MATLAB. The idea is to isolate the peaks of the signal, removing the noise and reverberation.
Different windows:
My frequency responses:
Zoom in the peak:
I don't understand the reasons why there're differences, especially the peak I get using Gaussian (figure 3). I know using the Gaussian with small standard deviation I can get rid of the noise, getting a cleaner signal.
Why does this happen? Can you guys give an scientific explanition?
Thank you.
There are two relevant phenomena here:
Windows that are narrower in the time domain have a broader frequency response, and windows that are wider in the time domain have a narrower frequency response.
Multiplication of a signal with a window in the time domain is equivalent to convolution in the frequency domain
Your Gaussian window with a small standard deviation is narrower in the time domain than the other windows, so it has a wider frequency response. Convolution of this wide frequency response with the spectrum of the un-windowed signal smooths out the frequency response of the windowed signal.
Of course this smoothing comes with trade-offs. As you make your window narrower in the time domain, the spectrum of the windowed signal will become smoother, but the resolution will become increasingly coarse.
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Related
I have an acoustic waveform of a Spanish phoneme and I'd like to compute its magnitude spectrum and plot it in dB magnitude on a linear frequency scale. How would I be able to accomplish this in MATLAB?
Thanks
First a quick heads up: At stackoverflow you are expected to show some of your own efforts to solve the problem and then ask for help.
Now to your question:
You can plot the spectrogram using the "spectrogram" Matlab function.
[s,f,t] = spectrogram(x,window,noverlap,f,fs)
Check the details here: https://www.mathworks.com/help/signal/ref/spectrogram.html
For a speech signal you will want to specify the sampling frequency "fs" (you can get that when you read the file using:
[y,Fs] = audioread(filename)
You will probably want to specify the variables "window" and "noverlap" since speech signals can show distinct properties depending on the dimension of the window (fast phenomena will not be visible on big windows ). A typical values are 20ms windows with 10ms overlap (select the best value by considering your sampling frequency and the nearest 2^n value for fast Fourier calculation).
The window size and overlap are also valid when you calculate spectrum. If you apply FFT to the whole waveform then you will get the "average" spectral information for the sentence. To catch specific phenomena you must use windowing techniques and perform a short-term Fourier analysis.
use sptool
Signal Processing toolbox Show its Document
I have a set of experimental data s(t) which consists of a vector (with 81 points as a function of time t).
From the physics, this is the result of the convolution of the system response e(t) with a probe p(t), which is a Gaussian (actually a laser pulse). In terms of vector, its FWHM covers approximately 15 points in time.
I want to deconvolve this data in Matlab using the convolution theorem: FT{e(t)*p(t)}=FT{e(t)}xFT{p(t)} (where * is the convolution, x the product and FT the Fourier transform).
The procedure itself is no problem, if I suppose a Dirac function as my probe, I recover exactly the initial signal (which makes sense, measuring a system with a Dirac gives its impulse response)
However, the Gaussian case as a probe, as far as I understood turns out to be a critical one. When I divide the signal in the Fourier space by the FT of the probe, the wings of the Gaussian highly amplifies those frequencies and I completely loose my initial signal instead of having a deconvolved one.
From your experience, which method could be used here (like Hamming windows or any windowing technique, or...) ? This looks rather pretty simple but I did not find any easy way to follow in signal processing and this is not my field.
You have noise in your experimental data, do you? The problem is ill-posed then (non-uniquely solvable) and you need regularization.
If the noise is Gaussian the keywords are Tikhonov regularization or Wiener filtering.
Basically, add a positive regularization factor that acts as a lowpass filter. In your notation the estimation of the true curve o(t) then becomes:
o(t) = FT^-1(FT(e)*conj(FT(p))/(abs(FT(p))^2+l))
with a suitable l>0.
You're trying to do Deconvolution process by assuming the Filter Model is Gaussian Blur.
Few notes for doing Deconvolution:
Since your data is real (Not synthetic) data it includes some kind of Noise.
Hence it is better to use the Wiener Filter (Even with the assumption of low variance noise). Otherwise, the "Deconvolution Filter" will increase the noise significantly (As it is an High Pass basically).
When doing the division in the Fourier Domain zero pad the signals to the correct size or better yet create the Gaussian Filter in the time domain with the same number of samples as the signal.
Boundaries will create artifact, Windowing might be useful.
There are many more sophisticated methods for Deconvolution by defining a more sophisticated model on the signal and the noise. If you have more prior data about them, you should look for this kind of framework.
You can always set a threshold on the amplification level for certain frequencies, do that if needed.
Use as much samples as you can.
I hope this will assist you.
I use Matlab to calculate the fft result of a time series data. The signal has an unknown fundamental frequency (~80 MHz in this case), together with several high order harmonics (1-20th order). However, due to finite sampling frequency (500 MHz in this case), I always get the mixing frequencies from high order frequency (7-20), e.g. 7th with a peak at abs(2*500-80*7)=440 MHz, 8th with frequency 360 MHz and 13th with a peak at abs(13*80-2*500)=40 MHz. Does anyone know how to get rid of these artificial mixing frequencies? One possible way is to increase the sampling frequency to sufficient large value. However, my data set has fixed number of data and time range. So the sampling frequency is actually determined by the property of the data set. Any solutions to this problem?
(I have image for this problem but I don't have enough reputation to post a image. Sorry for bring inconvenience for understanding this question)
You are hitting on a fundamental property of sampling - when you sample data at a fixed frequency fs, you cannot tell the difference between two signals with the same amplitude but different frequencies, where one has f1=fs/2 - d and the other has f2=f2/2 + d. This effect is frequently used to advantage - for example in mixers - but at other times, it's an inconvenience.
Unless you are looking for this mixing effect (done, for example, at the digital receiver in a modern MRI scanner), you need to apply a "brick wall filter" with a cutoff frequency of fs/2. It is not uncommon to have filters with a roll-off of 24 dB / octave or higher - in other words, they let "everything through" below the cutoff, and "stop everything" above it.
Data acquisition vendors will often supply filtering solutions with their ADC boards for exactly this reason.
Long way to say: "That's how digitization works". But it's true - that is how digitization works.
Typically, one low-pass filters the signal to below half the sample rate before sampling. Otherwise, after sampling, there is usually no way to separate any aliased high frequency noise (your high order harmonics) from the more useful spectrum below (Nyquist) half the sample rate.
If you don't filter the signal before sampling it, the defect is inherent in the sample vector, not the FFT.
I am a beginner in MATLAB and I should perform a spectral analysis of an EEG signal drawing the graphs of power spectral density and spectrogram. My signal is 10 seconds long and a sampling frequency of 160 Hz, a total of 1600 samples and have some questions on how to find the parameters of the functions in MATLAB, including:
pwelch (x, window, noverlap, nfft, fs);
spectrogram (x, window, noverlap, F, fs);
My question then is where to find values for the parameters window and noverlap I do not know what they are for.
To understand window functions & their use, let's first look at what happens when you take the DFT of finite length samples. Implicit in the definition of the discrete Fourier transform, is the assumption that the finite length of signal that you're considering, is periodic.
Consider a sine wave, sampled such that a full period is captured. When the signal is replicated, you can see that it continues periodically as an uninterrupted signal. The resulting DFT has only one non-zero component and that is at the frequency of the sinusoid.
Now consider a cosine wave with a different period, sampled such that only a partial period is captured. Now if you replicate the signal, you see discontinuities in the signal, marked in red. There is no longer a smooth transition and so you'll have leakage coming in at other frequencies, as seen below
This spectral leakage occurs through the side-lobes. To understand more about this, you should also read up on the sinc function and its Fourier transform, the rectangle function. The finite sampled sequence can be viewed as an infinite sequence multiplied by the rectangular function. The leakage that occurs is related to the side lobes of the sinc function (sinc & rectangular belong to self-dual space and are F.Ts of each other). This is explained in more detail in the spectral leakage article I linked to above.
Window functions
Window functions are used in signal processing to minimize the effect of spectral leakages. Basically, what a window function does is that it tapers the finite length sequence at the ends, so that when tiled, it has a periodic structure without discontinuities, and hence less spectral leakage.
Some of the common windows are Hanning, Hamming, Blackman, Blackman-Harris, Kaiser-Bessel, etc. You can read up more on them from the wiki link and the corresponding MATLAB commands are hann, hamming,blackman, blackmanharris and kaiser. Here's a small sample of the different windows:
You might wonder why there are so many different window functions. The reason is because each of these have very different spectral properties and have different main lobe widths and side lobe amplitudes. There is no such thing as a free lunch: if you want good frequency resolution (main lobe is thin), your sidelobes become larger and vice versa. You can't have both. Often, the choice of window function is dependent on the specific needs and always boils down to making a compromise. This is a very good article that talks about using window functions, and you should definitely read through it.
Now, when you use a window function, you have less information at the tapered ends. So, one way to fix that, is to use sliding windows with an overlap as shown below. The idea is that when put together, they approximate the original sequence as best as possible (i.e., the bottom row should be as close to a flat value of 1 as possible). Typical values vary between 33% to 50%, depending on the application.
Using MATLAB's spectrogram
The syntax is spectrogram(x,window,overlap,NFFT,fs)
where
x is your entire data vector
window is your window function. If you enter just a number, say W (must be integer), then MATLAB chops up your data into chunks of W samples each and forms the spectrogram from it. This is equivalent to using a rectangular window of length W samples. If you want to use a different window, provide hann(W) or whatever window you choose.
overlap is the number of samples that you need to overlap. So, if you need 50% overlap, this value should be W/2. Use floor(W/2) or ceil(W/2) if W can take odd values. This is just an integer.
NFFT is the FFT length
fs is the sampling frequency of your data vector. You can leave this empty, and MATLAB plots the figure in terms of normalized frequencies and the time axis as simply the data chunk index. If you enter it, MATLAB scales the axis accordingly.
You can also get optional outputs such as the time vector and frequency vector and the power spectrum computed, for use in other computations or if you need to style your plot differently. Refer to the documentation for more info.
Here's an example with 1 second of a linear chirp signal from 20 Hz to 400 Hz, sampled at 1000 Hz. Two window functions are used, Hanning and Blackman-Harris, with and without overlaps. The window lengths were 50 samples, and overlap of 50%, when used. The plots are scaled to the same 80dB range in each plot.
You can notice the difference in the figures (top-bottom) due to the overlap. You get a cleaner estimate if you use overlap. You can also observe the trade-off between main lobe width and side lobe amplitude that I mentioned earlier. Hanning has a thinner main lobe (prominent line along the skew diagonal), resulting in better frequency resolution, but has leaky sidelobes, seen by the bright colors outside. Blackwell-Harris, on the other hand, has a fatter main lobe (thicker diagonal line), but less spectral leakage, evidenced by the uniformly low (blue) outer region.
Both these methods above are short-time methods of operating on signals. The non-stationarity of the signal (where statistics are a function of time, Say mean, among other statistics, is a function of time) implies that you can only assume that the statistics of the signal are constant over short periods of time. There is no way of arriving at such a period of time (for which the statistics of the signal are constant) exactly and hence it is mostly guess work and fine-tuning.
Say that the signal you mentioned above is non-stationary (which EEG signals are). Also assume that it is stationary only for about 10ms or so. To reliably measure statistics like PSD or energy, you need to measure these statistics 10ms at a time. The window-ing function is what you multiply the signal with to isolate that 10ms of a signal, on which you will be computing PSD etc.. So now you need to traverse the length of the signal. You need a shifting window (to window the entire signal 10ms at a time). Overlapping the windows gives you a more reliable estimate of the statistics.
You can imagine it like this:
1. Take the first 10ms of the signal.
2. Window it with the windowing function.
3. Compute statistic only on this 10ms portion.
4. Move the window by 5ms (assume length of overlap).
5. Window the signal again.
6. Compute statistic again.
7. Move over entire length of signal.
There are many different types of window functions - Blackman, Hanning, Hamming, Rectangular. That and the length of the window and overlap really depend on the application that you have and the frequency characteristics of the signal itself.
As an example, in speech processing (where the signals are non-stationary and windowing gets used a lot), the most popular choices for windowing functions are Hamming/Hanning of length 10ms (320 samples at 16 kHz sampling) with an overlap of 80 samples (25% of window length). This works reasonably well. You can use this as a starting point for your application and then work on fine-tuning it a little more with different values.
You may also want to take a look at the following functions in MATLAB:
1. hamming
2. hanning
I hope you know that you can call up a ton of help in MATLAB using the help command on the command line. MATLAB is one of the best documented softwares out there. Using the help command for pwelch also pulls up definitions for window size and overlap. That should help you out too.
I don't know if all this info. helped you out or not, but looking at the question, I felt you might have needed a little help with understanding what windowing and overlapping was all about.
HTH,
Sriram.
For the last parameter fs, that is the frequency rate of the raw signal, in your case X, when you extract X from audio data using function
[X,fs]=audioread('song.mp3')
You may get fs from it.
Investigate how the following parameters change the performance of the Sinc function:
The Length of the coefficients
The Following window functions:
Blackman Harris
Hanning
Bartlett
I have FFT outputs that look like this:
At 523 Hz is the maximum value. However, being a messy FFT, there are lots of little peaks that are right near the large peaks. However, they're irrelevant, whereas the peaks shown aren't. Are the any algorithms I can use to extract the maxima of this FFT that matter; I.E., aren't just random peaks cropping up near "real" peaks? Perhaps there is some sort of filter I can apply to this FFT output?
EDIT: The context of this is that I am trying to take one-hit sound samples (like someone pressing a key on a piano) and extract the loudest partials. In the image below, the peaks above 2000 Hz are important, because they are discrete partials of the given sound (which happens to be a sort of bell). However, the peaks that are scattered about right near 523 seem to be just artifacts, and I want to ignore them.
If the peak is broad, it could indicate that the peak frequency is modulated (AM, FM or both), or is actually a composite of several spectral peaks, themselves each potentially modulated.
For instance, a piano note may be the result of the hammer hitting up to 3 strings that are all tuned just a tiny fraction differently, and they all can modulate as they exchange energy between strings though the piano frame. Guitar strings can change frequency as the pluck shape distortion smooths out and decays. Bells change shape after they are hit, which can modulate their spectrum. Etc.
If the sound itself is "messy" then you need a good definition of what you mean by the "real" peak, before applying any sort of smoothing or side-band rejection filter. e.g. All that "messiness" may be part of what makes a bell sound like a real bell instead of an electronic sinewave generator.
Try convolving your FFT (treating it as a signal) with a rectangular pulse( pulse = ones(1:20)/20; ). This might get rid of some of them. Your maxima will be shifted by 10 frequency bins to teh right, to take that into account. You would basically be integrating your signal. Similar techniques are used in Pan-Tompkins algorithm for heart beat identification.
I worked on a similar problem once, and choosed to use savitsky-golay filters for smoothing the spectrum data. I could get some significant peaks, and it didn't messed too much with the overall spectrum.
But I Had a problem with what hotpaw2 is alerting you, I have lost important characteristics along with the lost of "messiness", so I truly recommend you hear him. But, if you think you won't have a problem with that, I think savitsky-golay can help.
There are non-FFT methods for creating a frequency domain representation of time domain data which are better for noisy data sets, like Max-ent recontruction.
For noisy time-series data, a max-ent reconstruction will be capable of distinguising true peaks from noise very effectively (without adding any artifacts or other modifications to suppress noise).
Max ent works by "guessing" an FFT for a time domain specturm, and then doing an IFT, and comparing the results with the "actual" time-series data, iteratively. The final output of maxent is a frequency domain spectrum (like the one you show above).
There are implementations in java i believe for 1-d spectra, but I have never used one.