Questions tagged [signal-processing]

Questions on the mathematical aspects of signal processing. Please consider first if your question might be more suitable for http://dsp.stackexchange.com/

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10 views

FFT sampling does not pan out for “beautiful” data measurement

This is a half math and half electrical engineering question, I believe. So will focus on the math here. I have a BEAUTIFUL doppler radar signal here from a 24.050-24.250 GHZ sensor of an very small ...
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How can you get time related to frequency domain behavior? [FourierTransform, SignalProcessing, PowerSpectralDensity]

Context: Data shown is EEG Time domain data transformed to get an alpha brainwave plot with X= Time Units & Y= Intensity/Power of the frequency band of interest. Question: how do you get time data ...
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19 views

Relating the infinite operator to finite difference

I'm reading a book (Digital Signal Processing by Jonathan Steyn) which defines the infinite sum: $$r\equiv\sum^\infty_{m=0}z^{-m}x_n$$ This is just the sum of $x_1 + x_2 + x_3 +\dots +x_n$ to infinty. ...
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Determine the impulse response of the filter. [closed]

enter image description here Stuck on a question, mainly part 3 + 5. Could anyone point me in the right direction? ** "This image shows the structure of a digital filter in block diagram form. ...
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9 views

Measurements Numbers in Compressive Sensing

here is a question about compressive sensing. Let us denote the $k$-sparse signal $x\in \mathbb{R}^n$, measurement/sensing matrix $A\in \mathbb{R}^{m\times n}$ and the measurement $y = Ax \in \mathbb{...
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23 views

Find Fourier transform of e^-2/t sin(t) [closed]

Please solve this qstn. I need to know. Someone give explanation to this one
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1answer
41 views

What is a term-by-term explanation of Fourier Transform?

I've been self-studying Fourier Transform for a while ("vanilla" transform, not FFT, DFT or other types of that sort) but I cannot seem to understand how the mathematical expression reflects ...
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20 views

Why do we use generalized eigendecomposition for detecting signals in colored noise? [closed]

I'm trying to understand when I should use standard eigendecompositon and when generalized eigendecomposition. In particular, I'm dealing with the detection of a number of signals in colored noise. In ...
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34 views

Fourier transform of sum of delta functions

I am currently learning about fourier transform, but find it difficult to grasp in general. There is an exercise in the book, which is: Derive the continuous Fourier transform of $\delta(x−c) +\delta(...
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1answer
34 views

Why the impulse response needs to be right-sided for a system to be causal?

I have been revisiting my school notes about the definition of Causality for signal and systems. I already understand that for a system to be causal, the output at any time can only depend on past and ...
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34 views

How to derive the transfer function of the ideal differentiator?

How do I derive the transfer function, $$ H(s) = s $$ for the following ideal differentiator, $$ y(t) = \frac{d}{dt}x(t) $$ My attempt so far has been, $$ h(t) = \&'(t) $$ $$ H(s) = \int_{-\infty}^...
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Pros/Cons to using Spectral and Diffusive Graph Wavelets

As I understand, there are two major methods of constructing wavelets on graphs. Spectral wavelets, from David K Hammond et. al, and diffusive wavelets from Coifman and Maggioni. I can't quite parse ...
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15 views

Problem with exponentially modified Gaussian expression

One of the popular formulations of exponentially modified Gaussian EMG is shown below (taken from a commercial Peak fitting software called PeakFit). One point which has been bothering me for a while ...
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18 views

Time-lagged correlation between two factors

I have the following problem: I have a large dataset of a health facility where every (emergency) call was recorded. Besides this calls, I have other data about the weather, air quality and other ...
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1answer
38 views

How can I prove that the first line of a singular Toeplitz matrix is linearly dependant of the others?

Here is a Toeplitz Matrix of the form: \begin{pmatrix} a_0&a_1&a_2\cdots&a_n\\a_{-1}&a_0&a_1\cdots&a_{n-1}&\\a_{-2}&a_{-1}&a_0\cdots&a_{n-2}\\\vdots&\vdots&...
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How to find the derivative of an output in Laplace domain

I have a $Y(s)=F(s)\cdot X(s)$. Because it is a close feedback loop, I know F(s), but didn't compute the inverse Laplace transform of it because I don't know if $y(t) = f(t) \cdot x(t)$. The input x ...
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79 views

Response of a specific LTI system to an exponential

I am taking a Signals & Systems course in my Telecommunications Engineering degree, and now I am trying to put together everything I have seen. I came across this basic problem of determining the ...
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Determine the partial fraction convolution output y(ω) signal using fourier transform.

$x(t)=e^{-t}u(t)$ $H(\omega)\ =\frac{2+j\omega}{3+j4\omega-w^2}\ $ 2. $ x(t)\ =\ e^{-2t}\ u(t)\ $ $H(T)\ =\ e^{-4t}\ u(t)$ Can you guys help how to solve this two items in step-by-step. Youtube videos ...
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Minimization of $\|x\|_1$ subject to $\|Ax - y\| \leq \eta$ has an $m$-sparse solution when minimizer $x^*$ is unique

Given matrix $A \in \mathbb{R}^{m \times n}$, vector $y \in \mathbb{R}^m$, and scalar $ \eta \geq 0$, $$\begin{array}{ll} \underset{x \in \mathbb{R}^n}{\text{minimize}} & \|x\|_1\\ \text{subject ...
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Which causal LTI systems do have a state space representation?

Let $$L_{e,m}^2 = L_{\text{loc}}^2([0,\infty),\mathbb{R}^m) = \left\{f:[0,\infty) \to \mathbb{R}^m\mid \int_0^T \|f(t)\|^2 dt < \infty \text{ for all } T > 0 \right\}$$ and suppose that $G:L_{e,...
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36 views

how is this function a sum of these two functions?

So chui's book on wavelets has this diagram: I don't understand what he means by the highlight part here; aka I don't really see how the first two walsh basis functions are equal to the sums he's ...
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88 views

What is the value of a delta function?

So my understanding is that a delta function $\delta(t)$ has the value of $\infty$ at $t = 0$ and $\delta(t) = 0$ everywhere else. Essentially, it's just one big spike on a graph right? In my signal ...
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32 views

How can I find the fundamental period of the following signal? [closed]

How to find the fundamental period if it is periodic? $$ x[n] = \cos((\pi/2)n)\cos((\pi/4)n) $$ I have simplified to the following but not sure what to do next: $$ \cos((\pi/2)n)\cos((\pi/4)n) = 1/2[\...
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1answer
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Improving system identification results?

I'm performing system identification of the lateral closed-loop dynamics of a quadrotor. My model receives a setpoint and should return position and acceleration. I've proposed a second order model of ...
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3answers
97 views

What is Inverse Fourier Transform of $2𝛼^2/(𝛼^2−𝜔^2)$?

What is the inverse Fourier transform of $F(w) = \frac{4}{4+(j2\pi f)^2} $? I have two suggested solutions: Assume $j2\pi f = \omega$ and use the standard transform $e^{-\alpha |\tau |} = \frac{2\...
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1answer
33 views

Calculation for Standard Deviation Given a Gaussian Distribution

Quick background: I work as a Software Engineer in the Embedded Systems space, and my primary job function has me working with signal processing algorithms. I feel pretty good when it comes to ...
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21 views

Is there any value related to a time series pattern?

I've been digging a little into signals and I just started a networks and telecommunications class at college. We saw there are different encode techniques Amplitude, Frequency and Phase shift keying (...
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17 views

Power density via squared Fourier transform or autocorrelation for vectorial signal

Consider a set of stochastic-dependentsignals written as a vector $\underline{x}$. Those should be processed by some system with frequency dependent properties. Therefore I want to compute the power ...
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24 views

Transforming a Non-Periodic Function to a Periodic Function

We know that a non-periodic function is a function for which no $N\in\mathbb{R}$ exists such that : $$ f[n]=f[n+mN]\;\;,m\in\mathbb{Z} $$ However, we can construct a periodic extension of $f[n]$ : $$ ...
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Variance of a function of variances: Estimate SNR of measurement protocol

I'm working with a measurement technique where the variance of the signal is used to calculate the actual quantity of interest. For my thesis I want to take a detailled look into the signal-to-noise ...
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1answer
29 views

How to find the Fourier Coefficient to this discrete time signal?

I'm having trouble finding the fourier coefficients to this discrete time signal: My approach: I start by first finding the discrete period, $N$, and discrete frequency, $\Omega_0$ From the problem, ...
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2answers
70 views

Envelope of oscillator signal

How do we find envelope ODE of a modulated oscillator obeying $$y^{''}(t)+y (t)\left(\dfrac{2 \pi}{t-c}\right)^2 = 0 $$ after obtaining solution say for initial conditions $y(0) = 1, y'(0) = 0?$. Does ...
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How can we prove the correctness of the integration property of the Laplace transform?

I was going through an Electrical Engineering textbook and came across the following proof for one of the properties of the Unilateral Laplace transform. Integration property of the unilateral Laplace ...
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1answer
34 views

Integral of a shifted continuous unit step function

If the integral of a unit step function is a ramp function, then for the integration of the unit step function with a right shift of k value, is it possible that becomes a ramp function with a right ...
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15 views

Calculate the second harmonic contribution from a wave-like signal

I have a wave signal ($M_z$) computed from a simulation which has a main frequency of $\nu\approx0.158[GHz]$, and I want to extract the second harmonic contribution $M_z^{2\omega}$ of the signal. I ...
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1answer
33 views

Evaluating particular solution

I'm trying to find the solution to the following difference equation: $$y[n] - \frac{1}{4} y[n-1] - \frac{1}{8} y[n-2] =3x[n] $$ with $x[n]=(\frac{1}{2})^nu[n]$ where $u[n]$ is a step signal defined ...
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1answer
30 views

Difference in the interpretation of cross-correlation and convolution?

I'm confused on the interpretation between convolution and cross-correlation integral transforms. The convolution integral is defined as $f (t) \otimes g (t) = \int f(\tau) g(t-\tau) d\tau$ and the ...
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1answer
53 views

2D convolution of two circles

Imagine if two circles exist with definitions of $f_1(r) = circ(\frac{r}{R_1})$ and $f_2(r) = circ(\frac{r}{R_2})$ where circ is defined in a 2d dimension as: $ circ(\frac{r}{R}) = \begin{cases} 1, &...
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32 views

Can a quadruple integral be simplified through 2D autocorrelation definition?

I am trying to calculate this integral $$P(l) = \int \int \int \int \langle\Psi^*(x',y',z)\Psi(x,y)\rangle \phi^*(x,y,z)\phi(x',y',z) dx' dy' dx dy, \qquad (1)$$ which corresponds to the detection ...
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Bspline problem

I have some confusions about Bspline interpolation. In wiki and some websites, e.g. Bspline1. It is using something like repeated interpolation. But in some literature, e.g.Bspline2. It is using ...
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9 views

Should we apply filtering on points sampled from a distribution

Let us assume that there is a stationary random process defined as $\mathbf{X}_{k+1} = f\left(\mathbf{X}_k, \eta_k\right)$ , where $\eta_k$ is a Gaussian white noise and $f(.)$ is a linear/nonlinear ...
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46 views

How to find the period of this sinusoid?

I'm stuck trying to find the period of this sinusoid and would really like some pointers to different ways to approach this problem. $$x(t) = \cos(\frac{4\pi t}{5})\sin^2(\frac{8\pi t}{3})$$ I would ...
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1answer
30 views

Why is the integral of harmonically related sinusoids either $\frac{T}{2}$ or $0$?

I'm taking a signals processing class and am trying to wrap my head around why the following is true. $$\int_{t_0}^{t_0 +T}cos(\frac{2\pi}{T}kt)cos(\frac{2\pi}{T}mt) = \begin{cases} ...
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40 views

Is there a way to prove what form the fourier transform of a random variable takes?

Forgive me if any of my terminology isn't right - I come from a physics/stats background, not pure maths. I have a randomly generated time series, which is normally distributed with a mean of $0$ and ...
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13 views

Can we correctly estimate the derivative of a signal x(t) by differentiating a Gaussian process fitted to the signal x(t)?

Given measurements of a time-signal $x(t)$ at some time training points $t_i, i=1,\ldots,n$, can one estimate the time-derivative, $\frac{dx(t)}{dt}$, at the same training points using Gaussian ...
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2answers
65 views

Edge case with sampling and reconstruction.

I know I had been dabbling around this question before, here and here, but does anyone have in their bag of tricks the most simpliest and concise proof that: $$\sum_{n=-\infty}^{\infty} (-1)^n \, \...
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1answer
39 views

setting the value of a variable in such equation to have a specific output

Assume I have, for example, $x = 0.7 + 0.7i$ and the equation as below: $$\tag{1} (0.5+0.5i)y + (0.5 - 0.5i)x = 0 $$ in that case $y$ has the same magnitude of $x$. At the same time, if we swap $x$ ...
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1answer
47 views

Is there a way to get an approximation of $\det( \sigma_s^2A+B)^{1/N}$ in the GLRT?

Both $A_{N \times N}$ and $B_{N \times N}$ are symmetric Toeplitz matrix, $\sigma_s^2$ is a constant. In detection theory, the computation of $\det(\sigma_s^2A+B)^{1/N}$ arises in the computation of ...
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29 views

What's the application of matrix whose off-diagonal elements are all the same and non-zero?

For a $m$ by $m$ matrix $M$ whose off diagonal elements are all the same $\rho$, it can be re-written as $M=D+u*u^{T}$ ,where $D$ is a diagonal matrix whose disagonal elements are M's diagonal minus $\...
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29 views

Why does the period of $X_n = \cos(\omega_0 n + \phi)$ decrease as $\omega_0$ increases to $\pi$, but increase as $\omega_0$ increases to $2\pi$?

In Discrete-Time Signal Processing, Oppenheim writes For the discrete-time sinusoidal signal $x[n]=A \cos(\omega_0 n + \phi)$, as $\omega_0$ increases from $\omega_0 = 0$ toward $\omega_0 = \pi$, $x[...

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