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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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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15 views

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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37 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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25 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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37 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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12 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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61 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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38 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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45 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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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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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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1answer
50 views

What does $U(e^{j \lambda})$ mean?

I am reading “Signal processing for communications” by Paolo Prandoni and Martin Vetterli. In the section 4.4 The DTFT (Discrete-Time Fourier Transform) (p. 72 of 2008 ed.) they write: The somewhat ...
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Find the output $y(t)$ given the input and system response using Fourier Transform

I need to find the output $y(t)$ given the input $x(t)=\cos(t)$, and impulse response $h(t)=u(t)$ using the Fourier Transform. I know that convolution in time domain is multiplication in the frequency ...
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15 views

Autocorrelation of two functions multiplied and raised to arbitrary powers

Given a signal $A$ and a signal $B$ with autocorrelation times of $\tau_A$ and $\tau_B$, respectively, where $\tau_A > \tau_B$, is there any general statement that can be made about the ...
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57 views

Pulse train rect function

Why $ X_\delta (f) = \frac{4A}{3} ( 1 - \frac{1}{2} rect \frac{f}{B} ) $ is equal to The $ X_\delta (f) $ signal I wrote here ? my book obtained the same equation I wrote on the paper but , before ...
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50 views

Existence of low-pass filters

For $h=(h_k)_{k=0}^{n-1}\in \mathbb{R}^n$ call $\hat{h}(\omega)=\sum_k h_ke^{-j\omega k}$ and take $\gamma_h(\omega,\epsilon)\! :=\! \frac{\min_{f\in [0,\omega]} |\hat{h}(f)|}{\max_{f\in[\omega+\...
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56 views

Determining Causality and Time-Invariance for a system.

Consider the following system: $$y(t-1)=\int_{-\infty}^\infty x(𝜏)u(𝜏-t) d𝜏 $$ where $u(t)$ is the unit step function, which is zero for $t<0$ and equals $1$ for $t>0$. $(1)$ Is the system ...
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Frequency stability metrics

I'm looking for a metric (and - if possible - it's implementation in a programming language) that can measure "stability of a frequencies over time in signal". For example, if my signal is ...
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24 views

Low pass filter to remove noise for data with low frequency

I have a sensor (LM35 temp sensor) that measures the temperature every 5 seconds. This sensor worked for one year, and now I am plotting the measurements. The plot show noise in the outcome. E.g., it ...
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73 views

Inverse $z$-transform of $\frac{z_{1}^{-1}}{1 - az_{1}^{-1}z_{2}^{-2}}$

I'd like to know how to calculate the inverse $z$-transform of $$X(z_{1},z_{2}) = \frac{z_{1}^{-1}}{1 - az_{1}^{-1}z_{2}^{-2}},\quad |z_{1}|\cdot |z_{2}|^{2} > |a|.$$ I`ve tried to make the change $...
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27 views

Fourier transform and energy of a convolution with a delta comb

Hi guys i have to find the fourier transform of the convolution: $$ sinc(t/2T)*\sum\limits_{n-\infty}^{+\infty} (-1)^{n}\delta(t - nT) $$ i was thinking of express the summatory as : $$\sum\limits_{n-\...
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32 views

Period of a discrete sine function?

My teacher assigned me a problem involving a discrete periodic function and I'm a little confused on how to solve. Here is what was assigned (I paraphrased/summarized a bit): For the given function ...
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26 views

How to get coefficient of the discrete fourier Series from the fourier transform

Given $$X(v)$$ the Discrete Fourier transform of a discrete periodic signal $$x(n)$$ it's possible to arrive to the $$ c_k $$ of the fourier series $$x(n)=\sum_{k=0}^{n-1} c_k \exp(2\pi i k t) $$ ...
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17 views

Difference between x(2t-3) and x(2(t-3)).

Ofcourse x(2(t-3))=x(2t-6), but my question is in terms of operations on signals. In what order does the shifting and scaling happens in the case of x(2(t-3)) and how is it different from the ones ...
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10 views

Time reversal of odd signals

This question is about the “reflect about the $y$ axis” method of finding $x(-t)$ for a given $x(t)$. How does this method work if the signal is odd to begin with? Let’s say it is a signal whose ...
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11 views

How do we distinguish different distinct instruments in a digital sound signal?

I have a background in physics, and I've done a graduate class in harmonic analysis (although this was very theoretical class in a Hilbert space setting), plus I recently started DJing as a hobby. I'...
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26 views

Derivative of the $L2$-norm error of a linear map

Let us assume we have an error function such that $f(X):=\frac{1}{2}\left||K(X)-D|\right|_2^2$ where $X\in R^{n\times n}$ is a Gram matrix and $D\in R^{n\times n}$ is an Euclidean distance matrix and $...
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30 views

Generating Functions VS Z-Transforms as Solutions to Recurrence Relations

In a Discrete Mathematics video, recurrence relations are solved by applying generating functions to each term, doing algebra, and extracting coefficients of the result. There is no mention of Z-...
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49 views

Why does the gradient descent of square loss result in minimizing the l2 norm in compressed sensing problem

Why is the gradient descent of this problem with an extremely small step size (start at $\mathbf{x_0}=\mathbf{0}$) $$ \min_\mathbf{x} ||\mathbf{y}-\mathbf{A}\mathbf{x}||_2^2 $$ equivalent to $$ \min_\...
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22 views

decomposition of a function to piecewise functions

Is the next answer correct: $$a\left(z\right)=\sum _{\left\{k\right\}U\left\{k'\right\}:f_k\le \:z,\:z\:\in R,\:f_{k'}\ge z\:;\:z\ge 0}1-\frac{f_k}{z},\:b\left(z\right)=\sum _{\left\{k\right\}:f_k>...
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1answer
24 views

How can I perform local averaging on any manifold?

Say that I want to do local averaging on a circle. I have values for angles $\phi \in [0,1]$ where 1 "spins around" to 0. Let us call it $c[0,1]$ If I just do normal averaging $$\frac{1}{N}\...
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35 views

Is the optimizaion function convex?

I'm not sure how to approach this exercise. One idea is to derive it w.r.t z, show that there is a min-extremum at $z=f_k$ and then show that for each value from the right and the left of the loss ...
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24 views

sampling using L1 optimization

As much as I understand the F should be the interval medians (correct me if I wrong), according to the next slide, where is also the Loss function defined: What I don't understand is the next note in ...
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1answer
43 views

How to find average and power of signal expressed by fourier series

I need to find the average and the power of this signal: $$x(n)=\sum_{k=1}^{\infty}2^{-k}e^{j2{\pi}kn}$$ The problem is that the summation starts at 1 and not at 0, and a part of that how can I find ...
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1answer
26 views

Multidimensional signal synchronization

I am trying to find ways of synchronizing multidimensional waves gotten from the brain. For one dimensional signals, I used Cross correlation and was able to synchronise, but for multi dimensional ...
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12 views

Given a signal I(m,n) = a + b*m + c*n and a non linear filter e.g. median filter, find the response.

I stumbled upon this question and I can't seem to solve it correctly "Given a signal $x(n,m) = a +b{\cdot}m + c{\cdot}n $ and a filter, find the processing system response defined in a finite ...
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26 views

Computing a least-squares least-norm solution to image deconvolution

I want to deconvolve an image $h$ by a kernel $f$. More precisely, let $$G = \operatorname*{argmin}_g \|f \ast g - h\|_2$$ be the set of least-squares solutions. I want to find the least-norm solution ...
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34 views

Arithmetic entropy encoding with prime numbers.

I wonder if the following method of arithmetic entropy encoding could work for lossless compression of a binary signal: For some 24 bit signal: $Sn = \begin{pmatrix}x_{1}\\x_{2}\\ \vdots \\x_{24}\end{...
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1answer
84 views

Correlated noise in Kalman filter

Assume a standard state-space setting: \begin{align} x_{t+1} &= Ax_t + w_t\\ y_t &= Cx_t+v_t, \end{align} where the noise sequences are i.i.d. Gaussian with $w_i\sim N(0,W)$ and $v_i\sim N(0,V)...
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1answer
133 views

How to calculate the Fourier transform of the Kaiser-Bessel window?

According to Wikipedia: https://en.wikipedia.org/wiki/Kaiser_window, the Fourier transform of the Kaiser-Bessel window $w_0(x) := \left\{ \begin{array}{**lr**} \frac{I_0(\pi \alpha \sqrt{1-{(2x/L)}^2}...
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11 views

Discrete Time Convolution

I'm trying to solve a problem on convolution from Alan V.Oppenheim: Find the convolution output $y[n]$ for the following signals: $x[n]= u[n]$ and $h[n]=a^{n}u[-n-1], a>1 $ I started the evaluation:...
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17 views

Quantization and Sampling - putting it all together

So after I learned this two topic: quantization and sampling, I'm learning the way to look at both of them and try to optimize the split of a given amount of bit B to N and k, where N is the amount of ...
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45 views

DFT but using non-sinusoidal periodic waveforms.

Is there an established way of decomposing a discrete periodic (complex) signal into a sum of non-sinusoidal periodic waveforms (eg square, triangle, and sawtooth)? For my use case the input waveform ...
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9 views

Aproximately optimal high k-quatization: lagrange multiplier constraint

I'm observing the quantization topic in signal processing and there is some mathematical term, which I'm not totally understand. Here is the start of the development of the quantization for k > 1 (...
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14 views

What is the eigenvalues of $\boldsymbol{C}_1(\boldsymbol{C}_1^{-1} - \boldsymbol{C}_0^{-1} )$

Suppose that $\boldsymbol{C}_1, \boldsymbol{C}_0 \in \mathbb{C}^{KN \times KN} $ are block diagonal indefinite Hermite matrix with full rank, $\boldsymbol{C}_i = diag \lbrace \boldsymbol{D}^1_i, \...
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53 views

If you have three trigonometric functions that are defined for all real numbers, are they linearly independent as long as they have different periods?

Say, for instance, that I have trigonometric functions $f$, $g$, and $h$, where $f(x) = \cos(\pi x /2)$, $g(x) = \cos(\pi x/4)$ and $h(x) = \cos 4(\pi x)$ Since each function has a different period, ...
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8 views

Discrete Time convolution calculation error

i'm trying to find the convolution among this three signals : $$ h1= \delta(n)-\delta(n-4) $$ $$ h2= u(n) $$ $$ X= ({1}/{2})^n * u(n) $$ If i try to do $$ x*h2 $$ i get : $$ 2-2^{-n}*u(n) $$ ...
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51 views

Parseval theorem on a signal

If I have $ z(t) = x( t - \frac{ T_0 }{4 } ) $ with $ x(t) = rect ( \frac{ t - \frac{T_0}{4} } { \frac{T_0}{2} } $ I obtained that , For parseval theorem , $ Px= Pz = \frac{1}{4} sinc ^{2}( \frac{k}{2}...
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26 views

Find the values of $k$ , for which the system is stable.

I have a multiple choice question , that says , if $y(t) = (k^2 -3k -4)\log(x) + \sin(x)$ , find the values for which the system is stable ( I guess it means BIBO stable) . $$a. 1 \ \ \ \ b. 3 \ \ \ \...

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