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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18 views

Double Fourier transform of a single variable function

So what I'd like to achieve is the following: suppose we have a time dependent signal $f(t)$ - e.g. the wave of a song. This song contains a drumbeat that (for the sake of simplicity) has a single ...
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A generalized differential equation for a convolved differential operator.

The solution to perhaps the world's very first differential equation $$f'(x) = f(x)$$ is the well known exponential functions $$f(x) = k\exp\left[x\right]$$ But what if we consider another ...
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35 views

Sine wave - Calculate x(t) on 20KHz where t = 1

I'm trying to calculate a sine wave using the formula... $$ x(t) = A\sin{(2 \cdot \pi \cdot f_q \cdot t)} $$ Where $t$ is time (seconds), $A$ is amplitude and $f_q$ is frequency (Hz) When I calculate ...
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Unstable behavior of Discrete Fourier Transform and Fourier Transform alternatives

I have come across a "weird" behavior of the Discrete Fourier Transform (which I will refer to from now as DFT). suppose $$x=(1,-1,1,-1)$$ The real part of the DFT of $x$ is $$Real(DFT(x))=(0,0,4,...
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How can I pair spectral densities and covariance from graph?

I'm asked to pair the following spectral densities $R(f)$ with the correct covariance function $r(\tau).$ I know that $c$ belongs to $e$ and $a$ belongs to $g$. How can I determine that $b$ belongs to ...
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what is the co-variance of system output cooresponding to a non-iid random variable

Consider the following system. $$ \begin{aligned} &\dot{e_1}=c_1e_1+c_2e_2+\eta\\ &\dot{e_2}=c_3e_1+c_4e_2-c_5 sign(e_1)+a \eta \end{aligned} $$ which $\eta$ is a gaussian random variable and {...
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19 views

How to find correlation between two values and remove noise?

I have two curves A and B like that are highly correlated as shown in the figure below where ...
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55 views

How to understand multi-resolution analysis and wavelet transform?

I just started learning multi-resolution analysis. I know that given a scaling function, through dilation and translation, a sequence of spaces can be generated: $\cdots V_{-1} \subset V_0 \subset V_1 ...
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22 views

Recursive Least Squares for PID gains Java algorithm troubleshooting

I am attempting to use a recursive least squares algorithm to control the gains of a PID controller using model reference adaptive control. I have attempted to run it on my controller, but the ...
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25 views

Minimum size to get an accuracy spectral density using Fourier transformation

To get an accuracy result of spectral densities using Fourier transformation for a domain with the size $n$ by $n$. The minimum size of the input domain is $2n-1$ by $2n - 1$. Is it true? Sorry, ...
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32 views

Why must the length of a sequence under the discrete Fourier transform be equal to the input sequence length?

Let's consider a continuous signal, $f(t)$, which has been sampled $N$ times, with spacing $T$ between samples. We denote the $N$ samples $f[0], f[1], ..., f[N-1]$. The Fourier transform of the ...
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49 views

Graph Fourier transform: the adjoint notation for the eigenbasis matrix

It is well-known that for a real symmetric matrix $L$ (here, graph Laplacian) one can write the eigenvalue decomposition as $$ L = U \Lambda U^{\mathsf T}, $$ where $U$ is a real eigenvector ...
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70 views

Good kernels that exist in the real world.

A Good kernel (bounded approximate identity) in $L^1(\mathbb{T})$ is a sequence of functions $\{g_n\}$ satisfying: (i) $\frac{1}{2\pi}\int_0^{2\pi}g_n(t)dt=1~~$ (ii) $||g_n||_1=O(1)~~$ (iii) $~~\...
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27 views

A piecewise function as the output signal of an LTI system

In an LTI system, consider the following: The input signal: $$ x(t)= \begin{cases} 16 \quad & ; -7<t<0 \\ 0 & ; \text{otherwise} \\ \end{cases} $$ And the unit impulse ...
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44 views

Fourier transforms and the angular frequency

In the context of Fourier transforms, is the following true? $$ \mathcal{F} \left( F(t) \right) =2\pi \, f(-\omega ) $$ If it is not, what small changes can I make to the equation to make it true?...
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31 views

Unit impulse response of a discrete-time LTI system

The problem: Consider a discrete-time LTI system. If the output signal is: $$ y[n]=5 \left( \frac{1}{5} \right) ^n u[n] -2^{-n} u[n] $$ , then the input signal will be: $$ x[n]=\left( \frac{...
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Energy spectral density in an LTI system

Consider the LTI system: $$x\left( t \right) \to LTI\, system \to y\left( t \right) $$ The graph of the $x(t) $ signal is the following: Remark: The graph when $t>0$ is a quarter of a circle. Q1:...
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38 views

What does tilde on top of functions mean in fourier series and fourier transforms?

For an example these in these equations for continuous time fourier series: Does it denote that the function is periodic? If it denotes that the function is periodic then shouldn't there be a tilde ...
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71 views

Frequency spectrum and unit impulse response

In the context of signal processing, consider the following system: $$ x\left( t \right) \to System\, A \to z\left( t \right) \to System \, B \to y\left( t \right) .$$ Let the global system be ...
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24 views

What's the minimal time to distinguish 1 Hz difference in frequency?

Given a fragment of a single-frequency wave, what's the minimal length of it to calculate its frequency with error less than 1 Hz? Or if there are two fragments with frequency difference of 1 Hz, how ...
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36 views

Orthogonal functions definition

I know the condition for two functions to be orthogonal. It is $\int_{a}^{b}f_1(x)f_2(x)dx=0$. Also, on YouTube and websites this kind of formula is used, no other. I read from my book Signals and ...
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Calculation of alarm level for a signal detection in Poisson distributed noise

I am trying out my luck with the first post here... Well, to start off I have a problem, and I am not really sure about the solution. I hope that someone might chime in with suggestions.. We have an ...
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Why does a shift over the $x$ axis of the sine/cosine function has not only a high value for the DC coefficient but also for $0$?

I can't understand why does a shift over the $x$ axis of the sine/cosine function has not only a high value/spike for the DC coefficient but also for zero? I saw this in a video but I couldn't get it. ...
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52 views

Sequence of polynomials approximating sines and cosines

Suppose that you create a sequence of polynomials by first stating that $$ P_1(x) = \pi x $$ and then describing successive polynomials by: $P_{n+1}(x)$ is the indefinite integral of $2\pi P_n(x)$, ...
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26 views

Boundary of a Function from Bessel's Inequality

From Bessel's inequality, we know that for a partial sum of Fourier Series represented by $$[a_0, \mathbf{a}, \mathbf{b}], \mathbf{a}, \mathbf{b} \in \mathbb{R}^N$$ of a periodic function that is ...
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26 views

Approaching the Fourier transform of a damped cosine with a linear frequency chirp

I want to take the Fourier transform of the following transient signal, $$f(t) = e^{-t/\tau} \cos((\omega_0 + m t)t)$$, where $m$ is some gradient parameter in units of $\rm{Hz}/s$. I thought this ...
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36 views

Is there any condition for matched filter to be optimal?

The question is related to the optimallity of matched filter. The definition of matched filter is given in the link here: https://en.wikipedia.org/wiki/Matched_filter Cosider that the signal is ...
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1answer
29 views

The system $y(n) = c\cdot x(n) +d$ is given. Is the system linear, time-invariant, stable and causal?

The system $y(n) = c\cdot x(n) +d $ is given. (Unfortunately, nothing more is given.) Is the system linear, time-invariant, stable and/or causal? I assumed, that $c,d\in\mathbb{R}$ Linearity We ...
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1answer
50 views

Posterior of a corrupted transformation

Let $z\sim\mathcal{N}(z|0,1)$ and let $p(x|z)=\mathcal{N}(x|f(z),\gamma)$ where $f:\mathbb{R}\to\mathbb{R}$ is a differentiable bijection. I am trying to approximate the posterior $p(z|x)$ with a ...
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14 views

if $x[n] = (-1)^n$ and $y[n] = (-2)^n$, why the system is not LTI?

I am a student and I answered this question "it can be LTI" but my answer was incorrect and the right answer is that this system can't be LTI (Linear time-invariant system). I don't understand why. ...
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27 views

Calculate energy of signals

Consider the following two signals $$\begin{aligned} x(t) &= e^{-kt} u(t)\\\\ y(t) &= x(t) \,\frac{\sin(\theta t)}{\pi t}\end{aligned}$$ Determine the relation between $k$ and $\...
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34 views

Self-inverting basis of “skewed derivative” kernels. Is it useful?

Continuing my hobby from here of trying to find simple convolutional kernels which build families with interesting properties, today I found a new one. Consider the matrix: $$T=\begin{bmatrix}1&2&...
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33 views

Linear deconvolution using FFT

I want to deconvolve a filtered signal with a known input to recover the filter used using FFTs. Let $x$ be a vector of length $N$ and $h$ a filter of length $K$ where $N > K$. Let $x \ast h = y$,...
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34 views

Calculation of the integral (triangular function)

The following function f(t) could be used as a window function in Fourier transform. I would like to ask your help to see the intermediate steps(where the ???? are) of calculation of the following ...
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32 views

Chebyshev Filter Low Pass Conceptual Question

If I have a 2D signal (say a 2D image) thats defined on $[-1,1]^2$. I sample the 2D signal on discrete Chebyshev Points (Chebyshev-Legendre Points), say there is 60 grid points per side. The ...
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14 views

3D plot of a residue image in MATLAB

I'm pretty new to MATLAB and am working on image denoting functions and algorithms. I'd like to see the effect of 2 parameters $h$ and $\gamma$ on a residue image which I have with me. I want to ...
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25 views

Simplifying unit step functions

I'm having quite a hard time trying to simplify the following equations: $sinc(t)\delta(t) \\ u(t)u(t) $ (by u(t) I mean the unit step function) also, the integral $ \int^{+\infty}_{-\infty} cos(...
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31 views

How to calculate real pixel color from a blurred image using $n$ equations in $n$ unknowns?

I've been dealing with a big image de-blurrying issue for past months and now I'm stuck with this issue that I want to get original sharp image from a blurred image by using some extra data and math. ...
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25 views

Derive optimal vector of non-negative loadings with a given output vector and mixing matrix

I have posted this on crossvalidate but so far noone could answer. I am tyring to decompose a noisy vector into a sum of signals. Say we have $s$ independent signals and $c$ channels. Each signal can ...
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1answer
35 views

Differentiation in the Fourier Domain

If we have a exponential modified Gaussian function G(t) and we add the first derivative from G(t) to get a new function X(t). X (t) = G + K*d/dt (G) ----(1) where K is a constant real number ...
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1answer
47 views

Finding $y(t)$ in a causal system given an input-output relationship

Find $y(t)$ in a causal system with input-output relationship $$\frac{dy(t)}{dt} + 3y(t) = x(t)$$ where $$x(t) = e^{2t} \cdot u(-t).$$ and $u$ is the Heaviside function. To try ...
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32 views

Taking the Laplace transform of the derivative of some function

I need to find the Laplace transform of $\dfrac{dh(t)}{dt}$ where $h(t) = e^{-100t} \cdot u(t)$ and $u(t)$ is the Heaviside function. I have two ideas for solving this problem but I am unsure which ...
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39 views

How can I evaluate the following involving dirac delta functions?

\begin{align} \delta(x)\delta(x) = ? \end{align} \begin{align} \delta(x)\delta(x-a) = ? \end{align} \begin{align} \delta(x)/x = ? \end{align} In the above problems, a is a real number and a $\ne$...
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28 views

confusion in partial fraction

I have to find partial fraction of given signal $X=[s^3-2s^2-5s+6 ] / [s^2-s+2] $ for this i am working with following MATLAB code: ...
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1answer
15 views

How to determine divisor for composite sine wave

Suppose I added two sine waves $f(x) = sin(2\pi x)$ and $g(x) = sin(2\pi x)$ together and then divided by 2 like this $\frac {f(x) + g(x)}{2}$. If I plot this I get a graph where all three sine waves ...
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29 views

meaning of double headed arrow

I am studying properties of discrete time Fourier transform and I encountered a notation shown highlighted in attached photo What is meant by this notation? Is it meaning equality? screenshot of ...
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32 views

Mean of a random process

I have this problem that I can't solve: let $ x(t) = A cos(2 \pi ft + \theta ) $ a random process, where $ \theta $ is a variable that can take only the values $ -\pi/2 $ or $\pi/2$ with equal ...
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36 views

Comparing energy and power signal

I am trying to define and graph the signal to noise ratio of my system in the frequency domain. The system is LTI, excited by a pulse and subjected to filtered ZMWN. The only relevant portion of my ...
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12 views

Simplify 2D convolution

A 2-D function is given as \begin{equation} R_{w}(b',f')=\int \int_\mathbb{R^2} M_{w}(b,f)\delta(b'-\tilde{b}(b,f))\delta(f'-\tilde{f}(b,f)); \end{equation} where \begin{equation} \tilde{b}(b,f)=b-\...
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Derivation of the reassigned STFT magnitude of a complex exponential

I have a complex exponential defined as $x(t)=Ae^{i2\pi f_{0}t}$ and its short time Fourier transform with respect to the analysis window, $g(t)$ is defined as, $$X_{g}(b,f) = \int_\mathbb{R} x(t+b)g(...