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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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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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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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39 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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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 ...
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67 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
30 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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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) = ...
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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? $w_1=(5/2)=2.5 \rightarrow T_1 = 2\pi/w_1 = 2\pi/2.5 =2.513$ $w_2=(1/4)=0.25 \rightarrow T_2 = 2\pi/w_2 = 2\pi/0.25=25.13$ ...
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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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142 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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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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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
30 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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27 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. ...
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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
54 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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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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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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37 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: $$ ...
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1answer
49 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 ...
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1answer
36 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
35 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 = ...
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1answer
34 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} ...
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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 $$ ...
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1answer
36 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 ...
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1answer
45 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 ...
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How to create obtain aliased version of $f(t)$ by upsampling whenever $f(t)$ at every $t$ is available

Suppose there is original complex-valued $f(t)$ with $t$ ranging from $-\infty$ to $\infty$. It is possible obtain samples from original $f(t)$ at every $t$ with some negligible error. If one ...
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Eigenvalues for correlation matrix which have the form of an harmonic function

As a continuation to this question, I took the matrix $C_{2 \times 2}$ which is: $$C=\left[ \begin{array}{} a& ace^{-\frac{|\phi_1-\phi_2|}{2}}\\ ace^{-\frac{|\phi_1-\phi_2|}{2}} ...
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1answer
11 views

Unit of the second derivative of the power spectral density

To characterize a subtle oscillation embedded in a time varying voltage signal measured in microvolts, I took the second derivative of the PSD (which I computed as the fft of the autocorrelation) ...
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Is it possible to use regularization to minimize the (expected) number of non-zero digits in a number?

This question may be slightly related to this question on length of the representation of a number in a certain basis. Introduction / Background In image and video coding, particularly the ...
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How do I validate my ARMAX model?

Say I have some ouput $y_1, y_2, \ldots, y_N$ and inputs $x_1, x_2, \ldots, x_N$ which, by various time series methods, I've found to match an ARMAX(2,2,1) model. So I've found the estimations for ...
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Variance of $\hat{b} =\underset{b}{\mathrm{argmin}} \sum_{t=1}^N [y_t - bu_t]^2 $

We have $y_t = bu_t + e_t$ where $u_t$ is the input signal and I'm trying to find an expression for the variance of the estimate for b that is determined to be $$\hat{b} = ...
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116 views

How can I recover a sequence of numbers given a corrupted version of it?

I have an unknown sequence of real numbers $x_i$ and a known sequence of real numbers $y_i$; $y_i$ is a corrupted version of $x_i$, i.e., $$y_i=x_i+n_i$$ where $n_i$ is a random number distributed ...
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What is the output $y(t)$ when you have input $x(t) = \cos(2 \pi t) $ and frequency response response $h(t) = u(t) - u(t - 1/2)$?

The output $y(t)$ is the convolution of input $x(t)$ with impulse response $h(t)$: $$ y(t) = h(t) * x(t) $$ This is a linear, time invariant system. What is the output $y(t)$ in real form when you ...
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Solving a non-linear parametric equation

I am interested in solving a parametric equation where the unknown function is a function of time, and there is also an input. For example: $ y^{2}(t) + y(t) = \sin(t)$ I am coming from a signal ...
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Single Sideband LSB-SC Demodulation

The problem is how the phase φ effects the outcome when the input(message signal) is the DSB-SC LSB. It's : message: $m(t)=A_{m}cos(ω_{m}t)$ carrier: $c(t)=A_{c}cos(w_{c}t)$ I found that the LSB ...
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Can downsampling create energy at the Nyquist frequency?

I am a bit surprised by the following and would like to share it with you. I expect I am mistaken somewhere and will be happy to be corrected. I have searched StackExchange not only in Mathematics ...
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How to find out the Power of $x(t)$?

I am studying signals and system. I learned that \begin{align} P&=\lim_{L\to\infty} \frac 1{2L} \int_{-L}^{L} |x(t)|^2 dt\\ P&=\frac 1{T} \int_{<T>} |x(t)|^2 dt ~~~\mbox{, P could be ...
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1answer
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In signals processing why is the discrete sequence x[n] undefined (as opposed to 0) when n is not an integer?

In Oppenheim & Schafer's "Discrete Time Signals Processing" it's written that: ... it is important to recognize that x[n] is defined only for integer values of n. It is not correct to think of ...
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Why is this defined as $u[n]$?

In LTI systems there are two famous functions the unit step function and the unit impulse functions. And they are defined as follows. $$ u[n] = \begin{cases} 1, & \text{if $n$ $\ge$ 0} \\ 0, ...
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Is it possible to do effectively irrational-interval sampling of a continuous signal?

Suppose there is a real-valued $f(t)$, with $t$ being time. And one wishes to sample at interval of $\pi$, for example. Perfect irrational-interval sampling is not possible, but is there a way to do ...
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1answer
33 views

is there any good way to figure out number of fourier series frequencies of some signal?

Suppose you have $f(t)$, but you do not know the exact function and can only measure $f(t)$ at certain time. Assume $f(t)$ is complex-valued with $t$ being "time." One wishes to find out the number ...
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1answer
17 views

Relationship between short-time and large-frequency asymptotics in Fourier transform

I am trying to understand how the short-time behaviour of a function $f(t)$ influences the large-frequency asymptotics of its Fourier transform $g(\omega)=\mathcal{F}[f(t)](\omega)\equiv ...
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Signal processing and algebraic geometry

Signal processing is a pretty huge branch of what I would (maybe wrongly) call electrical engineering. I have heard here and there whispers of interesting connections between signal processing - in ...
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How to sketch frequency response obtained from H(z)?

How to sketch frequency response obtained from H(z)? I'm adding an example and its solution below. I did not understand some of the things for question 7.4, part D. Any help is appreciated! There ...