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Help in understanding a coding technique based on inverse mapping of a dynamical system

Based on paper titled : Simultaneous Arithmetic Coding and Encryption Using Chaotic Maps by Kwok-Wo Wong et. al The Authors use a non-linear dynamical system for generating keys to be used in ...
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2answers
47 views

$\cos(2\pi f nT +2\pi N_n) =\ cos(2\pi f nT)$ and more, why?

I am studying signals and systems and this came up? Could someone explain why is $\cos(2\pi f nT +2\pi N_n)$ equal to $\ cos(2\pi f nT)$? The book says: "because $N_n$ is an integer" I am ...
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2answers
38 views

Prove Differentiator is Linear and Time-Invariant

The differentiator gives an output equal to the derivative of its input. Show that the differentiator is a linear time invariant system. Consider the input $f(t)=\sin(t^2).$ Attempt For time-...
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1answer
20 views

Quick Fourier Series help?

I was given a graph (shown above) and was asked to represent this as a Fourier Series. I was able to solve $a_0$ with no problem. However, when I was integrating for $a_n$ and $b_n$, I was having a ...
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1answer
25 views

Signal whose Laplace transform contains derived Dirac-deltas: How do I find the inverse transform?

I must reconstruct the input signal to a system, knowing the output signal and the system transfer function. At the end, I found that the Laplace-Transform of the input signal is something like: $$ s^...
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23 views

Find the inverse z transform of $H(z)=\frac{0.2685}{1-0.146z^{-1}}-\frac{0.2685}{1-6.8493z^{-1}}$

I need to find the inverse z transform of: $H(z)=\frac{0.2685}{1-0.146z^{-1}}-\frac{0.2685}{1-6.8493z^{-1}}$ My initial attempt gave: $h(n)=0.2685(0.146)^nu(n)+0.2685(6.8493)^nu(-n-1)$ by using the ...
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9 views

Calculate system output of 2nd Order discrete LTI with cosine input

Consider this time-discrete LTI: $$ H(z) = \frac{z^{-1} - 0.25z^{-2}}{1 - 0.5z^{-1} +0.4z^{-2}}$$ $$ = -0.625 + \frac{0.7907 e^{j1.1645}}{1-0.6324e^{-j1.1645}z^{-1}} + \frac{0.7907 e^{-j1.1645}}{1-0....
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2answers
47 views

How to show that the signal $x_n = A\cos(\omega n)$ can be fully predicted by a system with two weights $w_1,w_2$

I am trying to solve the following exercise: Show that the signal $x_n = A\cos(\omega n)$ can be fully predicted by a system with two weights $w_1,w_2$ (i.e. $x_n = w_1 x_{n-1} + w_2 x_{n-2}$). ...
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0answers
32 views

Polar form of the Fourier transform of $\sin(t)$

I'm studying signal processing, and I came across the Fourier transform of sin(t). It ends up being a purely imaginary (dirac delta) impulse pair. But when considering the frequency domain ...
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1answer
24 views

What is the Permutation Matrix in FFT DFT Factorization?

Given: $$F_N = \frac{1}{\sqrt{N}} \begin{bmatrix} 1&1&1&1&\cdots &1 \\ 1&\omega&\omega^2&\omega^3&\cdots&\omega^{N-1} \\ 1&\omega^2&\omega^4&\omega^...
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1answer
27 views

Why is the following system is not time invariant?

The system is as follows: $y[n] = x[2n]$ Shouldnt the system be time invariant because $y[n-n_0] = x[2n-2n_0]$ and $T(x[n-n_0]) = x[2n-2n_0]$ These are both equal, therefore why is the system not ...
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2answers
33 views

Find an function that oscillates between a given upper and lower envelope

Suppose I'm given two real, continuous functions $f(x)$ and $g(x)$ such that $f(x)\ge g(x)$ for all real $x$. I'd like to determine an oscillating function $h(x)$ that has $f(x)$ as its upper-envelope ...
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39 views

Prove that taking the inverse Fourier transform of frequency returns time.

If we evaluate the inverse Fourier transform of X(w) how do we know we get x(t) back? Link to X(w) and x(t) equations I know that integrating in the frequency domain results in getting information ...
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33 views

Creating a balanced bidirectional pulse pair, also called the Lilly Wave and change it's sample amounts in octave / matlab

I was told this should be in the mathematical stackexchange Original question link below http://stackoverflow.com/questions/36111340/creating-a-balanced-bidirectional-pulse-pair-also-called-the-lilly-...
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1answer
43 views

Shape of Impulse Responses of $ARMA(p,q)$ Processes

Suppose that $x_t$ is an $ARMA(p,q)$ stochastic process, $$ \phi(L)x_t = \theta(L)\varepsilon_t ,$$ where $\varepsilon_t \sim N(0,\sigma^2)$, and $\phi(L)$ and $\theta(L)$ are lag-polynomials given ...
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20 views

I downloaded the SPARCO package (MATLAB)and when setting it up a problem has occured

I downloaded SPARCO1.2 (MATLAB) from http://www.cs.ubc.ca/labs/scl/sparco/ and when I use the command 'sparcoSetup' it says everything is successful. But then when I use the command 'checkProblems' ...
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2answers
27 views

Signal operation, shifting and scaling

so I have a question regarding this continuous time signal: $$y(t) = \int_{-\infty}^t x(2\tau) \, d\tau$$ Now the question was to find if this function was causal, so i proceeded to check the impulse ...
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1answer
21 views

Is this system invertible?

$y(t) = \int\limits_{-\infty}^{\infty} \frac {x(t)^2}{x(t-1)} dt\\$ I was trying to prove or disprove the invertibility of this function. The only thing I could think of was differentiating it. But ...
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16 views

How to define this test signal?

I generate any plots in any parts of the interval $[0,100]$ (let it be set $A=V^{n}$ where $n \in \mathbf Z$), I get a test signal. I would like to understand how you can write this mathematically for ...
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37 views

Equation involving fractions of integrals

In the context of a signal processing problem, let's say we have the following angles that are functions of time $\tilde{\tau}\in[0,1]$ $ \phi_i(\tilde{\tau}) = \left\{\begin{array}{ll} \...
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14 views

Pearson correlation of neural responses with it's linear estimation

I am trying to anderstand the following fact: Suppose I have a linear estimation of a stimulus: $ \hat{s} = \mathbf{w}^T(\mathbf{r} - \mathbf{f}(s_0)) + s_0$ where $\mathbf{w}$ is a vector of ...
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0answers
50 views

Fourier Transforms and Sums

Suppose I have the following sum: $$ \sum_{x = -\infty}^{\infty} \int_{-\pi}^{\pi} f(j) \; e^{i j x} dj $$ Assuming that everything is sufficiently smooth and convergent, then exchanging the sum with ...
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40 views

Calculating a function from its auto-correlation

How do I calculate a function if I know its auto-correlation? To be more specific, I have a function of one variable, let's call it $g(x)$, and I know it's the cross-correlation of a function $f(x)$ ...
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0answers
45 views

How can we use theory from $L^2(\mathbb{R})$ on a sequence of numbers (discrete signal)

In have problems understanding connection between theory that is done in $L^2(\mathbb{R})$ and its application on discrete signal. look at this paper http://home.ustc.edu.cn/~zhanghan/cs/...
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1answer
71 views

Fourier transform of a 2D image, and noise cancelation

I'm a statistics grad student, and I just started getting into Digital-Image-Processing (an analogy for processing super-large contingency tables). In the book "Digital Image Processing" by Gonzalez ...
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1answer
35 views

Infinite sum of discrete unit-step signals

Trying to sketch the following signal: $$\sum_{k=-\infty}^\infty (u[k]-u[k-3])(u[n-k]-u[n-k-3])$$ Where $u[n]$ is the unit step signal (the Heaviside function, $1$ when $n\ge 0$ and $0$ otherwise). ...
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39 views

Multi dimensional multiresolution analysis, designing biorthogonal wavelets.

With inspiration from this question, I'm wondering if biorthogonal bases for arbitrary dimensions are possible to construct with the same mechanism. I am thinking a subsampling of a factor of $N$ in ...
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wavlete transform vs (scaled) Gabor transform

I've read about the scaled Gabor transform $$(G_\Psi f)(b,a)(\omega) = \frac{1}{\sqrt{a}} \int_\mathbb{R} f(x)\Psi(\frac{x-b}{a})e^{-i\omega x}dx$$ and the wavlete transform $$(L_\Psi f)(b,a) = \frac{...
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2answers
72 views

Is $x(t)=\sin(5t/2)+\cos(2t/8)+\sin(3t/6)$ periodic or aperiodic? Find the fundamental period and frequency of the signal.

Is $x(t)=\sin(5t/2)+\cos(2t/8)+\sin(3t/6)$ periodic or aperiodic? $w_1=(5/2)=2.5 \rightarrow T_1 = 2\pi/w_1 = 2\pi/2.5 =2.513$ $w_2=(1/4)=0.25 \rightarrow T_2 = 2\pi/w_2 = 2\pi/0.25=25.13$ $w_3=(1/...
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0answers
23 views

spectral density of Guassian random matrix

I am interested in the spectral properties of Gaussian random matrix. I can see the constant dominance (mostly by the two most extreme ones-largest and smallest-) of the extreme eigenvalues in the ...
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1answer
151 views

Smooth sawtooth wave $y(x)=\cos(x-\cos(x-\cos(x-\dots)))$

Consider an infinite recursive function $$y(x)=\cos(x-\cos(x-\cos(x-\dots)))$$ $$y=\cos(x-y)$$ Plotting the function $y(x)$ implicitly we get a smooth sawtooth-like wave: Was this function ...
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1answer
37 views

A sign in a book that I can understand what it mean: $1_{x}$, $1_{y}$.

In a book that I read about mapping, It said: *Any mapping $f:X\rightarrow Y$ satisfies: $f1_{x}=1_{y}f=f$ *$g:Y\rightarrow X$ is a reverse mapping with $f:X\rightarrow Y$ only when $gf=1_{x}$ and $...
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0answers
34 views

Finding convolution of two functions?

1. Continuous Functions $x_1(t)$ and $x_2(t)$ definitions' link How to evaluate $(x_1∗x_2)(t)$ at $t = −T, 0, +T$ in terms of $T$ 2. Discrete Functions $x_1[n]$ and $x_2[n]$ definitions' link ...
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29 views

Help in plotting the Z-transform and Fourier Transform of the following Sequences.

I'm taking Digital Signal Processing class at the moment and while I believe I understand the theory behind the z-transform and fourier transform, in this case DFT and DTFT, I'm stuck as to how to ...
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1answer
31 views

Determining if a function is linear, time invariant, both or not

I have the function $y(t)=t^2x(t-1)$ and I need to figure out if it is linear or not and time invariant or not. By the looks of it I guessed it to be not linear but the answer is linear but not time ...
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1answer
57 views

Difficult problem involving a percentage of the period of a sinusoid

Im having difficulty intuitively understanding how to solve this problem: $x(t) = A\cos(\omega t + \phi)$ $A > 0$ $\phi\in(−\pi,\pi]$. $x(t) ≥ 2.4$ for $18$% of each period takes $0.123$ ...
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28 views

Implementing Normalized Cross-Correlation using FFT - How to?

Is there any way to calculate the normalized cross correlation between 2 signals by using the FFT? (I managed to implement it already for standard cross correlation equation). Thanks in advance,
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Is constant system a Causal System?

Is y(t) = 1 a causal system? From the definition of causal systems , a causal system is a system where the output depends on past and current inputs. Here the system doesn't depend on any input. So,...
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Is this system Causal?

The output system is: $x(t)$ -->(S)--> $y(t) = \int_{-\infty}^{t}x(\tau) d\tau$ Recall that the system is causal if the output at $t$ depends only on input before $t$, or if the impulse response $...
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1answer
60 views

building time signal after inverse FFT

I have managed to implement both forward and inverse FFT transforms in C#. And, i tested them by taking signal both ways and got the real part after iFFT equals the original signal. Now , I have ...
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20 views

When studying 2D gabor functions why is a gaussian called elliptical?

Consider $$G(x,y)=\frac{1}{2\pi\sigma\beta}e^{-\pi\left[\frac{(x-x_0)^2}{\sigma^2}+\frac{(y-y_0)^2}{\beta^2}\right]}e^{i[\xi_0x+\nu_0y]}.$$ This is the product of a complex plane wave and what this ...
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1answer
25 views

Specific question about downsampling in frequency domain

I'm confused about why Equation 4.74 can be expressed as Equation 4.76 after the summation index is expressed as in Equation 4.75. Could somebody please explain? Thanks! Downsampling DTFT
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40 views

What is the requirement for separable parameters in an LSQ fit?

I am trying to determine the amplitude of an amplitude modulated sinus as accurate as possible. My sampling frequency is sufficently high. The entire model looks as follows: $$ A*sin(2*pi*f_1*t+p_1)*...
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1answer
51 views

Sampling the Sine Function

Consider the sampled sine function, $f(n)=\sin(\omega n)$, where $n$ is an integer. If $\omega_2 = 3\pi/2$, does there exist an $0 \leq \omega_1 \leq \pi$ such that $\sin(\omega_1 n)=\sin(\...
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1answer
44 views

Why is autocorrelation used without normalization in signal processing field?

According to the wikipedia(Link), autocorrelation has two definition. Oh my god! In statistics, the definition of the autocorrelation between times $s$ and $t$ is like the following: $$\displaystyle ...
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1answer
41 views

Is interpolating well-sampled data (Nyquist-Shannon theorem) a cheat?

Suppose to sample a signal $s(t)$ with bandwidth $B$ with a sampling frequency $f_c$. Suppose also that the number of sample collected is $N$ (the duration of the signal acquisition is then $T = \frac{...
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1answer
36 views

Orthogonality of Two Signals.

My last question's link: Reconciling different definitions of orthogonality However, I failed to understand why they are equivalent. If $f$ and $g$ are real, \begin{align} \int_{<T>}f(t)g(t)~...
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1answer
48 views

Reconciling different definitions of orthogonality

I want to establish about orthogonality in my mind. I knew the orthogonality of two functions $f$ and $g$ in interval T like the following: $$ \int_{<T>}f(t)g^*(t)~dt=0 \tag{1} $$ where $$ g^*(...
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1answer
37 views

Application of Residue Theorem to inverse Fourier transform

I'm reading through a derivation in a book and am having trouble understanding a step. Here's a screenshot 3.46 is the equation in $(k,\omega)$ space. They're doing an inverse Fourier transform back ...
1
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1answer
48 views

Fast Fourier Transform as Matrix Factorization

I'm given a vector of length 4 and three matrices that correspond to a Fast Fourier Transform, I'm not exactly sure which one, but I guess it's supposed to be the Cooley-Tukey algorithm. Here is the ...