I have a system, described by a black-box, that takes as input a signal in time (let's say something similar to a sine wave) and returns a single scalar as output.
My goal is to find the optimum signal that maximizes/minimizes the output. As constraints, the time average of the signal must be kept constant and the minimum and maximum value of the signal must be within a specific range.
The parameters to be optimized are all the points describing my signal in time.
I want to use the scipy.optimize library to minimize the output of my black-box system however, everytime the parameters are modified by the optimizer, I need to smooth the input signal in order to avoid discontinuity.
My question is: is it possible to access the input variables after they are modified by the algorithm, smooth them, and substitute them at every iteration of the algorithm?
We are analysing some signals that contains an impuls in the form of a dip in the standard signal in matlab.
Signals
As you can see on the picture, we need to find the difference between the "Zlotty" and the "Krone". The two graphs besides each other, are the graphs that needs to be analyzed.
As you can see the time of the impulse is different in when it occures and in how long the impuls is. We can not use the Time as a value of measurements because that can vary randomly.
Each graph is made by vectors containing 2.5mio datapoints.
How would you use matlab to find a difference?
You could split the problem into two parts. Ensuring the same time scale for both signals and finding a possible time shift in the alignment of the resulting signals. The first part could be achieved by using the resample function of Matlab; and the second task by using cross-correlation. Using two nested for loops, you could perform a search for the "best" stretch factor and time shift that result in the maximum correlation coefficient.
Can anybody kindly explain how the Orthogonal Frequency Division Multiple Access works and its advantages in simple English, avoiding using “Fourier”? I am totally confused about many descriptions that using “Fourier” things to explain. (Or if anyone can make “Fourier” things clearer to understand...)
In wireless it is common practice to view signal in time domain or frequency domain. For example, in time domain a simple sinusoid signal with a frequency of one Hertz (one cycle per second and hence time period of one second) shows that at the time axis sine wave will repeat itself after one second . We can view the same signal in the frequency domain as well, where it will be just a delta function at One Hertz frequency (in the frequency domain, the x-axis represent frequency and in time domain x-axis represents time). Think of the Fourier as a Tardis machine, you punch in some numbers on this machine and it will take you to frequency world to display how the same thing exist there and of course, you can get back to time world again by reversing the process which brought you in frequency world at the first place. So it is easy to see that a never ending time domain signal of time period one second is just represented by one point in the frequency domain.
What is orthogonality? To explain this, let us talk about things in frequency domain only. As usual, the signals are represented in complex numbers or complex exponentials. We can view these complex exponentials or signals in two-dimensional coordinates form as well which is called vector representation of a signal. A cosine wave of magnitude one in represented by (1 + 0 * i) or in vector form [ 1; 0 ] and Sine wave of magnitude one is represented by ( 0 + 1i ) or in vector form [0;1 ]. Two signals are said to be orthogonal if the dot product of their vectors is equal to zero. Hence cosine and sine waves are orthogonal to each other. Which simply means that if we view those two signals in time domain, there will be a moment when Cosine will be at its peak but at the same time Sine wave will be zero.
OFDM exploit this property of orthogonality and put information bits on those orthogonal signals at one time. Since signals are orthogonal, the receiver just needs to know the frequency and exact phase to retrieve all the information (by sampling process). This process provides protection against inter-symbol interference (ISI), the major advantage of OFDM technique. Also OFDM provides huge advantage in frequency selective Fading because it breaks wide band spectrum (carrier) into small chunks of spectrums (subcarriers).
Hope it might help.
Let me try to make Fourier things easier to understand. Maybe it helps. Each time-dependent signal can be written as the weighted sum of sine and cosine 'waves' (meaning sines and cosines as function of time). The correct weight of a certain sine or cosine denotes how strong that particular frequency is present in your signal.
So on one hand you can represent your signal by telling how strong it is on each time instant. In other words, it can be represented by a function of time: strength (t).
On the other hand your can also represent that same signal by telling how strong each frequency is present in it. In other words, it can be represented by a function of frequency: weight (f),
Computing weight (f) from strength (t) is called Fourier transform.
Computing strength (t) from weight (f) is called inverse Fourier transform.
While this sounds complicated, your ears are doing it all the time. You don't hear a time signal, you hear frequencies, high and low 'notes'.
Computing Fourier transforms used to be very time consuming, until something called FFT (Fast Fourier Transform) was invented. It is just a computational trick and you don't need to know anything of it to understand what a Fourier transform is.
I don't pretend this to be a full answer to your question but maybe it helps.
PS. As for OFDMA, of course radio signals are not sounds. But the principle is the same.
Without using FFT and Inverse FFT, OFDMA can be considered similar to FDMA.
I have two (or more) time series that I would like to correlate with one another to look for common changes e.g. both rising or both falling etc.
The problem is that the time series are all fairly noisy with relatively high standard deviations meaning it is difficult to see common features. The signals are sampled at a fairly low frequency (one point every 30s) but cover reasonable time periods 2hours +. It is often the case that the two signs are not the same length, for example 1x1hour & 1x1.5 hours.
Can anyone suggest some good correlation techniques, ideally using built in or bespoke matlab routines? I've tried auto correlation just to compare lags within a single signal but all I got back is a triangular shape with the max at 0 lag (I assume this means there is no obvious correlation except with itself?) . Cross correlation isn't much better.
Any thoughts would be greatly appreciated.
Start with a cross-covariance (xcov) instead of the cross-correlation. xcov removes the DC component (subtracts off the mean) of each data set and then does the cross-correlation. When you cross-correlate two square waves, you get a triangle wave. If you have small signals riding on a large offset, you get a triangle wave with small variations in it.
If you think there is a delay between the two signals, then I would use xcorr to calculate the delay. Since xcorr is doing an FFT of the signal, you should remove the means before calling xcorr, you may also want to consider adding a window (e.g. hanning) to reduce leakage if the data is not self-windowing.
If there is no delay between the signals or you have found and removed the delay, you could just average the two (or more) signals. The random noise should tend to average to zero and the common features will approach the true value.
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