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2answers
96 views

Gaussian noise generation with Matlab

I am a Matlab beginner & need to understand what is going on this code. I know it is for Gaussian noise generation, but what do these variables mean mathematically? ...
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1answer
2k views

Regarding $x^2-a^2$ inside the argument of dirac delta

My undergraduate system textbook has this property in the appendix $$\delta(x^2-a^2)=\frac{1}{2|a|}[\delta(x-a)+\delta(x+a)]$$ and I can't seem to derive the result I tried the following: $\int_{-\...
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2answers
6k views

Is impulse response always differentiation of unit step response of a system?

I was trying to solve a question in which the transfer function of a system was asked, its unit step response being given: c(t) = 1-10exp(-t) The method that the book followed was to first find out C(...
3
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2answers
338 views

Why are these two delta function equal

In my system textbook it claims that $$\delta(x)=\delta(-x)$$ I understand the proof as follow $$\int_{-\infty}^\infty f(x)\delta(-x)\,dx$$ let $u=-x\,\:,\: du=-dx$ $$\int_{-\infty}^\infty f(x)\...
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3answers
5k views

Proofs of dirac delta property

How would I formally prove this property of dirac delta? $$\int \delta(a-x) \delta(x-b) \,dx = \delta(a-b) $$ I attempted to use the definition of a dirac delta $$\int f(x)\delta(a-x)\delta(x-b)\,dx=...
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1answer
51 views

understanding bases and frames for Gabor transform

For the 2D discrete Gabor transform, why is it that we cannot use a set of orthonormal basis for its representation, instead we have to use frames for representing it?
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0answers
1k views

Wavelets: Cone Of Influence

While reading this paper I came across the term Cone of Influence which is described as ...
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1answer
65 views

Amplitude versus time producing unexpected patterns.

I am writing a program to generate audio frequencies in multi-channel PCM format. This question may be more suited on an audio forum but I would like to know what is going on mathematically. My ...
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1answer
66 views

Periodic Fuctions - Signals -

If, in the periods, the two half's signal periodic have the same form and opposite phases, the periodic signal has symmetry of half wave. If the periodic signal $g(t)$, of period $T_0$, satisfy the ...
3
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1answer
12k views

The definition of NMSE (normalized mean square error)

Many papers use the NMSE function without ever explicitly defining it. I have always assumed that $$MSE(x,y)=\frac 1N \sum_i (x_i-y_i)^2$$ and $$ NMSE(x,y)=MSE(x,y)/MSE(x,0) = \frac{\| x-y\|_2^2}{\| ...
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1answer
61 views

Interpret convolution diagram

How do I interpret this "do convolutions" diagram? 1) How are the results computed? 2) When looking at this part: "x[n-k]" Do you interpret convolutions as delays or time reversals? $ y[n]= \sum_{k=...
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2answers
2k views

Undecimated Wavelet Transform (a trous algorithm) - how to determine 'anchor'/'center' of convolution filter

i am currently implementing the 'Undecimated Wavelet Transform' with the 'a trous' algorithm. See e.g. http://www.znu.ac.ir/data/members/fazli_saeid/DIP/Paper/ISSUE2/04060954_2.pdf, section II-A. As ...
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1answer
116 views

Given a signal in the time domain, is there a way to determine a function that produces that signal?

Disclaimer: I'm by no means an expert in any of this, and I'm just wondering whether a solution to this problem already exists. Using a raw audio waveform as an example, let's say you have a 1:00m ...
4
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0answers
148 views

Need a fast algorithm of adaptive convolution

Good morrow, gentlemen! I have to apply some kind of adaptive filter to my function $f(x).$ I present each point of my signal as a Gaussian, whose bandwidth depends on its location (not the point of ...
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0answers
84 views

Difference between Signal Processing and Filtering Theory

Here's a question. I have been reading the entries on wikipedia on signal processing and the filtering problem. It seems as both theories are conserned with the processing or estimation of some "true"...
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2answers
186 views

Detecting increasing pulse trains

I have a one dimensional point process representing the times of events which is also mixed in with lots of data that I regard as noise. The interval between the events in the point process are ...
0
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1answer
94 views

Fast evaluation of a variant of the convolution

Suppose $\{f_n\}$ and $\{g_n\}$ are finite sequences of complex numbers with $0\leq n \leq N-1$. The convolution $\{h_n\}$ of these two sequences is $$ h_n = \sum_{m = 0}^{N-1} f_m\; g_{n - m}\, . $$ ...
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2answers
3k views

Waves of differing frequency are orthogonal - help me understand

I know that sinusoidal waves of different frequencies are orthogonal to each other. For instance: ...
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1answer
79 views

Background subtraction

I have a histogram of counts which is made from ion fragmentation and noise superimposed on top of it. I also have an image of just the noise. What I want to do is to subtract the noise of the total ...
2
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5answers
605 views

Compressive sensing with non square matrices

I'm implementing the algorithm in the following paper: "Compressive sensing for wideband cognitive radios" http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=04218361 However I've run into a ...
2
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1answer
155 views

What is a norm that can measure average oscillatory amplitude?

I am numerically computing the growth of an oscillatory instability in a fluid system. Suppose for simplicity that the function $f(x)$ [defined on a finite interval] has oscillations of different ...
0
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1answer
125 views

Barlett Window - Tri function

I'm trying to understand what is meant by the following equation: Bartlett Window (Triangle): $$w(n)=TRI\left(\frac{2n}N\right)$$ Basically, it's the $TRI$ function, since, mathematically I ...
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0answers
653 views

How to measure power spectral density in matlab?

I am trying to measure the PSD of a stochastic process in matlab, but I am not sure how to do it. If this is not the right place to ask, please point me in the right direction. The stochastic process ...
3
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1answer
203 views

Detecting periodicity in point processes

I have data from a periodic one dimensional point process representing the times of events which is also mixed in with lots of data that I regard as noise. The total number of points is of order one ...
4
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2answers
104 views

Reduce formula using Euler's?

I am performing a self-study, and I am lost as to a derivation that has taken place. I basically started with this equation: $$ \Upsilon(\phi) = e^{-j\frac{N-1}{2}\phi} \ \Big[ \frac{1 - e^{j N \...
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4answers
4k views

Adjustable Sigmoid Curve (S-Curve) from $(0,0)$ to $ (1,1)$

I feel like this is such a simple question but I am at such a loss. I currently have a set of values that I would like to weigh by an S Curve. My data ranges from $0$ to $1$ and never leaves those ...
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2answers
74 views

Which Method of Convolution (If Any) Is Most Appropriate Here?

I need to convolve (or otherwise get the impulse response h(t) of) the input signal $x(t) = 2u(t)$ and $y(t) = cos(4t) + 2e^{(t-1)}$. I have tried the Fourier Transform and the Laplace Transform, but ...
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2answers
35 views

What Does The Following System Do?

I have a system $y(t) = 0.5 \int^\infty _{-\infty} x(T)[d(t-T) - d(t+T) dT] $ Where d(x) is the Dirac Delta function (couldn't find the LaTEX representation - a little rusty there, so an edit to ...
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1answer
98 views

Can one zero-pad data prior to Fourier transformation, then reverse the change afterwards?

Suppose I have a set of $n$ points $\underline{x}\in\mathbb{C}^n$ with $n \in \mathbb{P}$ ($n$ is prime), and I want to find the Fourier transform of $\underline{x}$. There are some prime-length ...
0
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1answer
47 views

Determine negligible coefficients in spectrum

Suppose I have some function $f$ that I have sampled at $N$ points and I preform a transform on it (this could be a Fourier transform, or perhaps a Hadamard, or really anything eles - I'm hoping for a ...
4
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2answers
421 views

When does Discrete Fourier analysis fail to detect a frequency?

I'm using python to learn about Discrete Fourier Analysis. What I want to understand is when does the technique fail to recover some frequency of the signal? I understand how this can occur via the ...
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0answers
129 views

Output of wavelet transforms

I am working on a time sensitive computer science and fluid dynamics project that requires me to find applications of wavelet analysis. I know that at its core, a wavelet transform simply takes a ...
3
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0answers
145 views

Exponentials of chi-squared random variables (and their sums)

Let $X_1,X_2,\ldots,X_n$ be a sequence of i.i.d. chi-squared random variables with $t$ degrees of freedom, i.e. $X_i\sim\chi^2_t$. I am wondering what is known about the distribution of $S(n,t,a)=\...
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10answers
4k views

What's the difference between $\mathbb{R}^2$ and the complex plane?

I haven't taken any complex analysis course yet, but now I have this question that relates to it. Let's have a look at a very simple example. Suppose $x,y$ and $z$ are the Cartesian coordinates and ...
0
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1answer
155 views

Continuous Hidden Markov Modeling

I am writing a speaker recognition program in Matlab using Mel Frequency Cepstral Coefficients, and although I have gotten the problem to work using discrete time wrapping, I was interested in try to ...
0
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1answer
89 views

Finding the signal $y[n]$ that results from the linear transformation of $x[n]$

I have this two signals $$x[n]=4\delta[n+1]+\delta[n-2]-2\delta[n-5]$$ $$y[n]=(3n-5)(u[n-1]-u[n-4]$$ I know that $y[n]$ results by the linear transformation of $x[n]$ and I have to find the ...
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0answers
701 views

Autocorrelation of the rectangular function

Given the autocorrelation function of the signal "s": $$r(k)=\sum_{t=-\infty}^{+\infty}{s(t)\cdot s(t+k)}$$ The autocorrelation of a rectangular function $\Pi$ (t/2) is a triangle formed by the ...
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1answer
526 views

Creating intuition about Laplace & Fourier transforms

I've been reading up a bit on control systems theory, and needed to brush up a bit on my Laplace transforms. I know how to transform and invert the transform for pretty much every reasonable function, ...
1
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1answer
3k views

Discrete Convolution

Does someone know the equation for the discrete convolution? I found here that the formula is: $$\{x*h\}[k]=\sum_{t=-\infty}^{+\infty}{x[t]\cdot h[k-t]}$$ But when using in Matlab/Octave the command ...
0
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1answer
202 views

Harmonic mean of absolute value squared discrete Fourier transform

Let $X[k] \in \mathbb{C}$ be the discrete Fourier transform of $x[n] \in \mathbb{R}$, where $k,n = 0,1,2,...,N-1$. Parseval's theorem relates the arithmetic mean (AM) of absolute value squared ...
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1answer
56 views

n-correlation function.

So I was thinking of generalization of notions in statistics, like auto-correlation or cross-correlation (auto-correlation is a specific example of cross-correlation where we take the same proccess). ...
3
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2answers
87 views

Estimating time in harmonic signal

I hope someone can help me with the following problem: Assume a periodic signal of the form $$\begin{align} s(t) &= \sum\limits_{p=1}^P \sin(p\Omega_0t)\\ &= \sum\limits_{p=1}^P \sin(\...
0
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0answers
265 views

Calculating sum of a discrete signal (explain summation)

I have to find a convolution of two signals $h[n] = 0.5^nu[n]$ $x[n] = u[n]-u[n-3]$ the final sum, which is correct is: $$\sum_{m=n-2}^n 0.5^mu[m] $$ note that i replaced $n-k$ with $m$, that is $...
0
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1answer
44 views

Frequency Change with Tape Speed

I recorded my own voice in an old tape recorder. When I put the device in fast forward mode my voice turned a little squeaky. I wonder if the fundamental frequency of the voice had changed? Is that ...
0
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1answer
76 views

Determining the Discrete Fourier Transformation of signal.

I'm studying signal analysis and now I'm at the Fourier series/transformations. I'm following an example but can't understand the last step. This problem is divided in two steps. FIRST From this ...
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1answer
93 views

Signal Processing Convolution Summation Calculation

I am learning convolution of signals, and need to do a lot of summations and math. Because y[n]=Sum(x[k]h[n-k]) from negative infinity to infinity. I am always stuck at math procedures. Also, I am ...
0
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1answer
79 views

Reconstructing signal without aliasing

$$x[n]=2\sin\left(0.2\pi n-\frac{\pi}{2}\right)+4\cos(0.5\pi n-\pi)+\cos(0.8\pi n)$$ This signal is the result of sampling $x(t)$ with a frequency of 40Hz. I have to see if this frequency can ...
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0answers
265 views

Convolution property in terms of fft (matlab)

I am working on some signal processing and I have the following data: ...
0
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1answer
95 views

Bounds on least squares and weighted least squares estimator

I was wondering if I can get some help in getting bounds on the parameters estimated by least squares (LS) and weighted least squares (WLS) methods. Suppose our observation model is: $\mathbf{y} = \...
1
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1answer
133 views

Fourier transform of a signal sequence?

Desparately, I am trying to calculate the Fourier transform of the following signal sequence. What can it be? $$f(x)=\left(\frac{\sin(\pi x)}{x\ln(T)\sin[\pi\ln(x)/\ln(T)]}\right)^2$$ while $x > ...