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Fourier Transform over function depend on time and frequency

In my task I need to perform Inverse Fourier Transform from spectrum that depend on time and frequency arguments simultaneously. E.g., I have a discrete spectrum of some function $S(t, f)$ with $2N$ ...
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1answer
50 views

Omitting part of Frequency domain, Fourier Transform, Image Processing

In my Image and Signal Processing lecture, the Professor said that if every other column of the frequency domain of an image is zeroed out, then the reconstructed image is aliased. (along the x-axis) ...
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1answer
206 views

Why arctan equal to -90 degrees?

Can somebody show me why $$-\arctan\left(\frac{2\pi}{1-\cos(2\pi)}\right)$$ equals to $-90^\circ$ degrees? Thanks.
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1answer
78 views

symmetric window for discrete windowed Fourier transform

The discrete windowed Fourier transform of a signal $f$ of period $N$ is given by $$ Sf[m,l]=\sum_{n=0}^{N-1}f[n]g[n-m]\exp\left(\frac{-i2\pi l n}{N}\right). $$ Why is it that the window $g$ must be ...
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73 views

How to prove Fourier inverse transform worked?

$$g(t)=\int\limits_{-\infty}^{\infty}g(f)e^{i\omega t}df$$ $g(t)$ is a function of time, $g(f)$ is a function of frequency, $e^{i\omega t}$ represent wave, and $\omega = 2\pi f$, the angular ...
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64 views

Recovery of Bandlimited Signals

Let $\Omega > 0$ and denote by $\mathcal{B}_\Omega$ the subspace of $L^2(\Bbb R)$ consisting of signals that are bandlimited to $(-\Omega, \Omega)$. Denote $\mathcal{L}_{\Omega} : L^2(\Bbb R) ...
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168 views

Q: Calculating Fourier Coefficients and Inverse Fourier Transform

Let $\Omega >0$ and $x \in \mathcal{B}_{\Omega/2}$ is continuous. Define $\hat{y}(\omega) = \sum_{n \in \Bbb Z} \hat{x}(\omega - n\Omega)$. If $\hat{y}$ is expressed as \begin{equation} ...
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1answer
33 views

Elimination of complex variable in integral

I have the equation: $$\frac{1}{\tau}\intop_{0}^{\tau}A\sin\left(\Omega t\right)\cdot A\sin\left(\Omega\left(t-\lambda\right)\right)\mathrm{d}t$$ for which the attempted solution is to convert the ...
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1answer
111 views

3-bit sensors — A question about Hamming distance in signals

I came across this question on Willy Wu's riddle site You have two 3-bit sensors, A and B, that measure the same thing, whatever it is -- temperature of the room, radioactivity levels, whatever. ...
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174 views

Differential equations and Kalman filters

I have been told that every differential equation has an associated Kalman filter. How do we get the Kalman filter of a given differential equation. For example let's say we have $$my''+cy'+ky=f(x)$$ ...
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1answer
179 views

How to interpret the results of 2D Fourier Transform on an image?

I have a class where we're studying signals processing (mostly filtering of sounds and images) and while I kind of understand the results of a Fourier Transform for sounds I don't really get the ...
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1answer
99 views

Using the FFT to align two instances of the same signal

I'm working on a program that has a software oscilloscope-like viewer for audio signals. The scope basically takes in blocks of signals at a regular rate and adds them to its existing signal data. ...
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3answers
821 views

Fourier transform of $\left|\frac{\sin x}{x}\right|$

Is there a closed form (possibly, using known special functions) for the Fourier transform of the function $f(x)=\left|\frac{\sin x}{x}\right|$? $\hspace{.7in}$ I tried to find one using ...
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2answers
3k views

How can a unit step function be differentiable??

Recently, I am taking a Signal & System course at my college. In all of the signal & system textbooks I have read, we see that it is written " When we differentiate a Unit Step Function, we ...
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2answers
201 views

Mathematical explanation for image edge detection and denoising

I am trying to understand why the convolution kernel, $$\left[\begin{array}{rrr} -1&-1&-1\\ 2&2&2\\ -1&-1&-1 \end{array}\right]$$ detects the edges in an image. If anyone has a ...
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1answer
208 views

Can the measurement matrix used for compressive sensing be a sparse matrix?

I am interested in analyzing Compressive Mechanism: Utilizing Sparse Representation in Differential Privacy. In my research, the measurement matrix $A\mathbb \in R^{m \times n}$ needs to be sparse. ...
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1answer
973 views

DSP, discrete time, how to calculate the frequency

this is from Digital Signal Processing, 4th ed, Sanjit K Mitra, problem 2.39b. The question is: Determine the fundamental period of $x[n] = cos(0.6n\pi + 0.3\pi)$ Since x[n], is in square brackets, ...
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2answers
58 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
1k 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: ...
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2answers
4k 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 ...
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2answers
201 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 ...
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3answers
3k 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 ...
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1answer
42 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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843 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
63 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
61 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 ...
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1answer
7k 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
57 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]= ...
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2answers
1k 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
101 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 ...
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123 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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73 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 ...
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2answers
180 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 ...
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1answer
92 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
2k 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
62 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 ...
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5answers
375 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 ...
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1answer
119 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 ...
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1answer
109 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
402 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 ...
2
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1answer
168 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 ...
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2answers
99 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
2k 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
69 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
34 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
90 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 ...
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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 ...
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1answer
283 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
98 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 ...
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0answers
111 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 ...