Discrete Signal Processing - Science topic

Dear Mihir Malladi,

Please check those links i hope that are on subject.

Discrete mathematics, sometimes called finite mathematics, is the study of fundamentally discrete mathematical structures, as opposed to continuous structures. Unlike real numbers, which have the property of varying "smoothly", objects studied in discrete mathematics (such as relative integers, simple graphs, and statements in logic1) do not vary in this way, but have distinct values. separated. Discrete mathematics therefore excludes subjects in "continuous mathematics" such as calculus and analysis. Discrete objects can often be enumerated by integers. More formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets.

Also please check those links i hope that are on subject.

Best regards

output layer are 0 element , matlab bug

Dear Pezhman,

I sincerely appreciate your help. Thank you.

I did an optimal energy based window using prolate spheroidal wave functions. For Mainlobe width / sidelobe level Check the 2016 paper by Sarkar and Khan, "Simulation Based Design Analysis of an Adjustable Window Function", http://file.scirp.org/Html/3-3400472_71777.htm which cites the Samad paper and implements an improved window.

Hi,

Using MATLAB software, the maximum number of levels Jmax with which a signal can be decomposed, can be determined using the following equation :

Jmax= fix(log₂[(N/N_w)-1])

N is the length of the signal, Nw gives the length of the decomposition filter associated with the chosen mother wavelet and fix is to round the value in the brackets to its nearest integer. 

Best regards,

The stability of a system with feedback does NOT depend on any input signal. When a circuit is able to self-excitement, it wil be self-excited (caused by noise and/or power switch-on transients)

...just a guess.. maybe what will help is Groebner basis .

see e.g   https://reference.wolfram.com/language/ref/GroebnerBasis.html

" Basic Examples  (1)

Compute a Gröbner basis:

In[1]:=GroebnerBasis[{x^2 - 2 y^2, x y - 3}, {x, y}]

Out[1]  = ~9+2y^4 ,3x-2y^3  "

hope it helps

Cheers

 

If you ensure that the optimum is within your factorial experimental range and your results are accurate and the experimental error low, a face centered design is a good choice if your experimental range cannot be extended. The variance is not far from isotropic (symmetric). Indeed, the goddness of the quadratic model can be evaluated by ANOVA to get reliable and sound results.

As Luis Miguel Gato Díaz well said above, the envelope is the magnitude of the analytical signal made up of the two quadrature components (Q is the signal you have and I is the Hilbert transform of Q). The envelope is the hypotenuse of the square triangle whose sides are Q and I. The following MATLAB program illustrates this. NB: The MATLAB instruction 'hilbert' (used as 'y = hilbert(x);' in my code below) finds the analytical signal associated with x, not just the Hilbert transform, i.e., it creates a complex signal whose real part is the signal you have (x) and whose imaginary part is its Hilbert transform). The Hilbert transform is the imaginary part of the analytical signal ( imag(y)), which I plot in green). Enjoy.

t = 0:1e-4:1;

x = [1+cos(2*pi*25*t)].*cos(2*pi*100*t);   % create a modulated signal

clf;

y = hilbert(x);       % y is the analytic signal

plot(t,x,'b')         % plot original signal in blue

xlabel('Seconds')

hold on

plot(t,imag(y),'g')   % plot Hilbert transform part in green

legend('Original','Hilbert')

X=[0,1];

Y=[0,0];

plot(X,Y)             % plot the x axis

plot(t,abs(y),'r','LineWidth',2)  % Plot only +ve envelope

xlim([0 0.1])                                   % plot just part of it

xlabel('Seconds')

title('The absolute value of the Analytic Signal is the envelope; the zero crossings of the HT are the peaks', 'FontSize',16);

Let me understand if I got it well: you sampled the spectrum of a given signal in the range 18-26 GHz and you want to recover the time domain signal.

If the problem is stated this form, IFFT is the only straight approach to time domain extraction as the spectrum is not in the standard wavelet decomposition form able to reconstruct the original signal from its representation. Ofcourse WT is an algorithm and you can apply it any time you want, but the obtained results of the time domain signal depend seriously on the choices of the IWT filters you have chosen.

Once you have your spectrum in the frequency domain you should operate reconstruction: this means you have to guess somehow:

first you need to know how many steps levels should be present in the signal because the division in the frequency domain sould be described by the graphical form as reported hereafter:

LLLL;LLLH;;LLH;LH;H for a 4 level subdivision. Everi letter L and H represents a frequency span of half width with respect to the higher level.

Not very easy to explain here.

So step are:

- upsample LLLL and LLLH by a factor of 2 interleaving zeros in the frequency domain;

 -filter each frequency domain signal by the proper reconstruction filter of the chisen wavelet and then add them (take care of the delay introduced by the chosen filters)

repeat this step considering now the obtained signal coming from LLLL and LLLH upsampled and filtered , after the sum as they where the level 3 low pass signal in the frequency domain, LLL

Hope I was clear enough. 

Forgot to mention that I measured the output voltage with a (too cheap) voltmeter.

Actually the incident (infra)red light creates electron-hole pairs in the pn-layer, therefore modelling the PIN diiode as a current sources is a common way to go. Then one usually connects this current to an Operational Transconductance Amplifier.

For my own means it was enough to measure the voltage that the photocurrent induced over a resistor.

If you add a 4:1 mux only for several signals in the LVDS bus this will mean a delay mismatch between the signals which go through the mux and the signals which don't.  The delay depends on a specific 4:1 mux partnumber and might achive several ns. You should either consider if your device can afford such a delay mismatch or you should compensate this delay mismatch by adding the same 4:1 mux (or a logic device with similar delay) to the rest of signals in the LVDS bus.

Send me a mail to mdo@fct.unl.pt and I'll send you a copy of the version of the algorithm that I and my colleagues elaborated with the introduction of the concept of resolution.

The envelop of that signal is constant hence the papr  is one.

We have published TF adaptive filtering methods based on the short time Fourier transform in several venues.  I am attaching two papers, published in IEEE Trans Sig Proc and in Signal Processing.  The basis of the method is covered in section VI of the first paper and is described in more detail in the second paper. I am also attaching a MATLAB script that computes the STFT with the correct group delay, using a two tap recursion. 

Oh, I see what you mean now. I was talking about a shift in time domain, for example delaying the wavelet two or three samples. My mistake.

I wrote the following code in MATLAB to answer your question. There is a plot attached too.

Best,

Wanderson

%Ricker wavelet analysis

t = (-(0.1):0.001:0.1)'; %time series

Fc = 30; %central frequency

p = Fc.*Fc.*t.*t;

amplitude = (1-2*pi*pi.*p).*exp(-pi*pi.*p);

plot(t,amplitude)

ylabel('Amplitude')

xlabel('Timeseries (s)')

grid on

%wavelet phase rotated

phase = 90; %degrees

phase = phase*pi/180; %radians

x = hilbert(amplitude); %hilbert transform

x2 = cos(phase)*real(x) - sin(phase)*imag(x);

hold on

plot(t,x2,'-r')

In 2D images, there are (at least) two types of curvature. One describing the intensity landscape (e.g., cup, cap, saddle, etc.) and the other describing the shape of the isophotes (curves of equal intensity).

The first is described by the principal curvatures, which are the Eigenvectors of the Hessian matrix. 

k1 = (Lxx + Lyy - sqrt(4*Lxy^2 + (Lxx-Lyy)^2))/2

k2 = (Lxx + Lyy + sqrt(4*Lxy^2 + (Lxx-Lyy)^2))/2

The second is the isophote curvature.

k = - (-2*Lx*Lxy*Ly + Lxx*Ly^2 + Lyy*Lx^2 ) / ((Lx^2 + Ly^2)^(3/2))

More details are provided in the book of B.M. ter Haar Romeny (Front end vision...).

In this notation, Lx is the first order derivative to x, Lxx is the second order derivative to x, Lxy is the partial derivative to x and to y, etc..

For curvature estimation, I recommend a derivative that is rotation invariant and (at least) twice differentiable. Otherwise, you measure the shape of the pixel grid, which is undesirable. So, a higher-order B-spline or a Gaussian derivative could be appropriate. The small derivative kernels [-1, 0, +1] are not recommended. More information about the Gaussian or B-spline derivatives can be found in [Bouma e.a., Fast and Accurate Gaussian Derivatives based on B-Splines, LNCS, 2007]. A full text PDF version is available in the following link:

my data is discrete time and with Markov processes.

P(x_3,t_3|x_1,t_1)=\int P(x_3, t_3|x_2,t_2)p(x_2,t_2|x_1,t_1)dx_2

t_3-t_2=t_2-t_1=\tau

i calculate that in matlab but i think it's output is wrong.

i want to plot right and left side this formula for different tau and for on tau they fit exactly.

but i could not find this tau. may my program is wrong.

Approach Number one: Think about it the other way around: A line spectrum represents in the time domain continuous sinusoidals with the periods defined by the spectral lines. Now you can use one of the characteristics of the Fourier Transform- its symmetry: What was frequency domain is now time domain and vice versa..... That way you see, that the "lines in the time domain" are continuous waves in the frequency. (see my Screen Shot 1.

Apporach number 2, closer to your desired "proof" (ScreenShot 2):

A time discrete signal consists of a series of amplitude modulated, time shifted dirac pulses: See the "Sum over s(nT)*delta(t-nT)"

If you now Fourier-Transform the whole term, you must not forget, that the FT of a shifted dirac delta in time itself is already a continuous wave in the frequency. Like the dirac at t=0 produces the "White Noise Spectrum". Now all those continuous waves from that many shifted diracs will sum up to a continuous, periodically repeated  spectrum- et voila! (You may try to read the second line from the right....) 

This is not a stringent proof, but I hope it helps!

uli

PS: There may be typos in the figs ;-) 

Basically I would agree with Paul and Juehui. However, it depends a lot on why you want to remove the diurnal part. One can suspect that it will be mainly low frequencies for instance, so that maybe you can applied a multiscale analysis. Then you can expect to see the details you are looking for in the highest details signals.

Mohan, this answer is for the antennas in electromagnetics. The electromagnetic field associated with an antenna is the Fourier Transform of the current aperature distribution of the antenna. This depends greatly on the shape and feed design of the antenna. An added complication is that the time variations of the current introduce solutions to the math that involve a far field and a very complicated near field.

Think of it this way.  The electromagnetic radiation that propagates away from the antenna, never to return, is governed by the far field and obeys the inverse square law for propagation distance. The near field is much more complicated and involves higher order inverses and the fact that the energy in the near field does not propagate. It expands and contracts in response to the frequencies of the excitation current. However the energy is exchanged by the electric and magnetic field as is the normal case of propagation.

I forget the formulas for how far the near field extends from the antenna but I know that the higher the effective gain, the further the near field extends. In some very large gain antennas, the near field may extend 800 meters! Good luck.

In our recently supported project, we will observe sun radiations as a natural source of random samples, similar to www.random.org or others. But, we would like to share our data for your randomness tests. Those data will have (almost) exact statistics like, mean, variance and higher order moments. Best.

.. I might be wrong Alfonso, but I think "unequally spaced" refers to the samples, not the features in the signal... The traditional FFT algorithm is designed for equally spaced points so here it is just requested if a computationally efficient one is available for unevenly spaced data. IMHO the answer depends on the spacing of the data. If it is just slightly uneven you can simply smear your signal (e.g. by a gaussian kernel) and use the FFT on a narrowly resampled data. I did not know the link proposed by Herbert Homeier so thanks for sharing.

In my archive I had this http://www.lanl.gov/DLDSTP/fast/

I think you can find further information online if you look for NUFFT, or the NFFT and PFFT libraries

Yes you can use it

only change you are creating is instead of filter coefficients adapted over time( normal time domain adaptive filters), you can adapt wavelet coefficients to adapt over time( transform domain adaptive filters)

It is really hard to say. Let me explain why:

-Traditionally discrete mathematics is tied very well and is one in the same with what you refer to as "computer theory". The very foundations of computing is in pure mathematics... so it would be nonsensical to unlink them.

-The material world tends to work well with real values and error. That being said usually floating point and real valued mathematical problems can be of interest. In particular some realms of real analysis can fall into this picture. This holds for a lot of geometric problems and especially when it comes to probabilities which the medical field needs to use constantly.

Now, with this being said, your secondary question is slightly different than your title question. Pure mathematics should not be segregated from discrete or real mathematics. Its kind of like saying "what is more important to use chemistry or physics", you need one for the other so it doesn't make as much sense to put it that way. If you want to model things on computers, it is discrete mathematics all the way but the results and heuristic based solutions we often see useful in practice require a lot of the time continuous mathematics to properly explain and use them. As for pure mathematics, some of the most important results in "computer theory" are from this domain and are of great importance to knowing what we can and cannot solve in finite time if our assumptions about computers are true.

Dear Sree,

Check this video: http://www.youtube.com/watch?v=1mTU0H0-Ank

The code is available at: https://code.google.com/p/oflute/

The documentation (in spanish) is available at: https://oflute.googlecode.com/svn/doc/memoria/memoria.pdf

It is an student final master project.

I hope this helps.