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/
2,043
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Am I misinterpreting this example of convolution?
I am reading the MIT Deep Learning textbook's description of convolutional neural networks and they introduce the mathematical operation of convolution as follows:
Suppose we are tracking the ...
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What apps provide annotated steps of calculations?
I have mathematical formulas that I need to calculate and my question is:
What apps provide a calculation’s steps with explanations?
In Wolfram Alpha and ...
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Fourier series representation of a signal
Firstly, I need to write this signal as sum of two signals to find the Fourier representation, but couldn't manage to derive it. How to find it for -T/2 - T/2? It seems like the square wave added with ...
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How to define a linear operator in Maple that commutes with derivatives?
I would like to simplify an expression involving the Hilbert transform in Maple. The Hilbert transform is defined by $$ Hf(x) = \frac{1}{\pi} \ \mathrm{p.v.} \int_{-\infty}^{+\infty} \frac{f(z)}{z-x} \...
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Is there an alternative to FFT to extract spectral density within specific frequency ranges?
I am looking for an algorithm that allows me to get additional/ consistent information of the low frequency spectral components of a signal.
So far I have been using Fast fourier transforms, the issue ...
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Relationship between fourier transform and fourier series
Let $x(t) = A\sin(2 \pi f_0 t + \alpha)$ its fourier transform is given by $ X(\omega) = \frac{A \pi}{i}(e^{ia}\delta(\omega-2\pi f_0) - e^{-ia}\delta(w+2\pi f_0)) $. the fourier series complex ...
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CTFT of $t \mapsto (t − 1) e^{−2t} u(t − 1)$
Determine the CTFT and the corresponding energy of the signal $$x(t) = (t − 1) e^{−2t} u(t − 1)$$ For this same signal: find the even and odd parts of this signal and then determine there ...
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If $X\sim \text{Uniform}[-1,1]$ and $Y = X^2$, what is $E[X\mid Y]$?
Given that $X$ is $\text{Uniform}(-1,1)$ continuous, $Y = X^2$, what is the expected value of $X$ given $Y$? I was asked to calculate the MMSE of $X$ given observing $Y$, which is $E[X\mid Y]$. The ...
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Periodogram formula for sampling
The formula I know for the periodogram is
$$I\left(f\right)=\frac{1}{NT}\sum_{n=0}^{N-1}\left|x\left[n\right]\exp\left(-2\pi fjnT\right)\right|^2$$
but for this one question I have which states that $...
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Calculating fourier transform of sawtooth function
Let the following $T$-periodic signal :
then $$\begin{align}F(x(t))(\omega) =& \int_{-\infty}^\infty x(t) \exp(-i\omega t) \mathrm{d}t = \frac{A}{T} \int_{-\infty}^\infty t \exp(-i \omega t) \...
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Determining the frequency and phase spectrum
Let the following T-periodic signal :
I found that $x(t) = \frac{A \cdot t}{T} $ and its fourier series is : $S(x)(t) = \frac{A}{2} - \frac{A}{\pi} \sum_{n=1}^\infty \frac{\sin(n \omega t)}{n} $ how ...
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How can i determine the fourier series of this signal
Let the following T-periodic signal:
i think that each sub-signal is $x(t) = A \cdot tri_{\frac{T}{2}}(t)$ but i dont know how to make it periodic like in each interval of length T, we get the same ...
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Show that the solutions to differential equations are the same - decomposition property
Consider a linear and time-invariant system with differential equation $Q(D)y(t) = P(D)x(t)$
where $Q(D)$ and $P(D)$ are differential operator polynomials.
We define the zero-state response as the ...
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Sampling a cosine [closed]
If you sample a cosine of fundamental period 0.1 milliseconds with a sampling rate of 10^5 samples per second, how much phase difference is there between two consecutive samples?
I can't solve this ...
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How to find the time complexity of interpolation and addition?
I have signal p(t) with N samples, where I need to advance by t0 seconds. There are two ways to perform,
By converting into discete system like p(n+round(t0/dt)), where dt is sample time and round(t0/...
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Understanding Least Square Estimation (LSE) for vector parameter
I am trying to understand the following part from book Fundamentals of statistical signal processing by Steven Kay.
It is written that if parameter $\theta$ is vector of dimension $p \times 1$ then we ...
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Understanding the following equation
I have the following expression from Linear Least Square part of Steven M Kay book titled Fundamentals of statistical signal processing.
$\hat{\theta} = \frac{\sum_{n=0}^{N-1}x[n]h[n]}{\sum_{n=0}^{N-1}...
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Frequency bound for Hermite interpolation'
The Nyquist limit for interpolation by trig functions states that one must give at least two data points per wavelength, because data just above and just below this "folding frequency" ...
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The integral of telegraph process with GWN is a gaussian - having trouble understanding proof
let $N(t)$ be a poissonian counting process with parameter $\lambda$, we'll define $X(t)$ as a telegraph process in the following way: $$X(t) = B \cdot (-1)^{N_t}$$ where B gets values $\{-1,1\}$ with ...
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Autocorrelation of a time-limited, wide-sense stationary stochastic process
Problem
Let $\{X_t\}$ be a real-valued, wide-sense stationary, continuous stochastic process and consider the autocorrelation of samples of $\{X_t\}$ taken in time windows of width $T$:
$$R_{X; T}(\...
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Does the quantized estimate give the same MSE limit as the non-quantized estimate?
Setup: Given that we have a signal $\beta\in\{0,1\}^p$ that we want to estimate. We have an estimate $\hat{\beta}\in\mathbb{R}^p$ (entries could be continuous) such that the mean squared error
$$
\...
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Show that $\dot x = x + u, y = x$ is time-invariant.
I am given a system,
$\dot x = x + u, y = x$
where $u$ is the input, $x$ is the state, $y$ is the output.
I want to show that this system is time-invariant.
To do this, I know that $y(t) = H(u(t)) = \...
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Changing variables when $f(x)$ goes to $f(-x)$
I need to prove the property $$f(x)⊕f(x)=f(x)⊗f^*(-x)$$ That is, the autocorrelation of a function is the convolution with its time-reversed complex conjugate. I have constructed most of the proof but ...
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Best estimator of a matrix signal with binary entries
Setup: Given that we have a noisy matrix signal $\breve{B}\in\mathbb{R}^{p\times L}$ of the true signal $B\in\mathbb{R}^{p\times L}$, where the empirical distribution of the rows of $B$ converge to $\...
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Unilateral signal using DFT
I made a code for calculating DFT of any signal. I tested it with many functions but when I introduced a complex sine, my DFT give a "unilateral signal". I've never seen something like that. ...
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Does pairwise phase incoherence satisfy the triangle inequality?
Let $S$ be the unit circle in the complex plane,
$$ S = \{z \in \mathbb{C} : |z| = 1\}. $$
For values $z_1^{(1)},z_1^{(2)},z_2^{(1)},z_2^{(2)},\ldots,z_k^{(1)},z_k^{(2)} \in S$, letting
\begin{align*}...
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Solving a differential equation with time-varying coefficients
I have this differential equation for a system with input $V_\text{in}$ and output $V_o$.
$$2\ddot{V_o}(t)R_{oc} + \dot{V_o}(t)(R_{oc}^2+2R_{oc}+1)+V_o(t)(R_{oc}^2+R_{oc}) = -\dot{V_\text{in}}(t) \...
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Upper bound on "time-continuous Zadoff–Chu signals"
I'm trying to find a tight (as tight as possible) upper bound for $\lvert x_u(t) \rvert$, where $x_u(t)$ is the $T$-periodic signal defined by the Fourier sum
$$
x_u(t) = \frac{1}{N}\sum_{k=-N_0}^{N_0}...
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How the following Inverse Fourier Transform is obtained.
In one of the numerical, I came across the following power spectral density (PSD):
$S_{Y}(f) = 2; |f| < 2$ and $S_{Y}(f) = 0; \text{otherwise}$.
Its autocorrelation is given as $R_{Y}(\tau) = 8 \...
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Extrapolating 2D image pixels using second order derivative
I intend to use image pixel data prediction to improve image compression. This means that I need predict image pixels based on previous rows and columns. I cannot use the next rows and columns for the ...
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Intuition for why the fourier tranform of a polynomial is its pointwise evaluation?
Polynomials in coefficient representation can be multiplied in $O(n \, log \,n)$ time by using a fast fourier transform to convolute the coefficients. The DFT of the coefficients correspond to the ...
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Equation for Gaussian kernel's effect on a frequency's amplitude
Applying a Gaussian blur/kernel with a sigma of $\sigma_{gau}$ to a sine/cosine wave of frequency $f_{sin}$ will cause what percent reduction in the amplitude $p_{amp}$ (not power) of the sine wave?
...
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Brief references on partial coherence
A signal in additive noise is sampled by a receiver with an unstable clock.
Maybe the clock has absolute bounds on its frequency drift and sample time aperiodicity, or maybe the clock has some ...
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Definition of white process of order p
Antoni uses the following definition of white process of order p:
a process whose all cumulants up to order p are such that $$\text{Cum}\left[X(t),X(t-\tau_1)...,X(t-\tau_{r-1})\right]=C_{rX}\delta(\...
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Modern treatments of spectral estimation
I need to perform and understand parameter estimation on spectra of short random time series.
Kay's "Modern Spectral Estimation: Theory and Application" treats this topic for complex-valued ...
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Does this gradient-adjusting variant of the FFT have a name?
Consider the case of calculating an FFT of a not-quite-1-periodic function $$ t\to f(t) , t \in [0,1] , f(0) \neq f(1) $$ by transforming it into a periodic function :
$$f \to \hat f = f(t) - g(t) = f(...
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Understanding use of Identity Matrix in random vector
In my research work, I came across the following statement:
$\textbf{v}_{c,l}\in \mathbb{C}^{M \times 1}$ is noise vector (random vector) at time $l$ at receiver having $M$ antennas.
The $\textbf{v}_{...
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Understanding about summation in the given expression
In my research work, I came across the following expression:
$$ \textbf{S}_c = \sum_{k = 1}^K \textbf{H}_k \textbf{p}_c \textbf{u}_k + \textbf{V}_c$$
where dimension of $\textbf{V}_c$ is $M \times L$, ...
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Linear phase and stability of a filter
I have a question that says that a system has N poles which are placed on the plane:
$$ {{|z-0.75\,e^{j\frac{\pi}{4}}|<0.25}} $$ and I am asked if the system is stable and if it has a linear phase. ...
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Cross - correlation between two complex signals
I want to calculate the degree of similarity of two complex signals with different lengths in the time domain using python, so I’ll get a scalar value representing the degree of similarity.
I thought ...
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Decomposition of a Filter Transfer Function into Minimum-phase and All-pass filters
I'm currently studying Digital Signal Processing and I'm particularly focusing on filters' characteristics, specifically minimum-phase and all-pass filters. Understanding these properties and their ...
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Why is there an asymmetry in the scale of these filter kernels?
In this video, the kernel for a low pass filter is given to be $[\frac{1}{2},\ \frac{1}{2}]$, while the kernel for a high pass filter is given to be $[-1,\ 1]$.
Interpreting these as essentially ...
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Distribution of Gaussian noise in the frequency domain
I have $N$ samples of Gaussian noise independently drawn from a normal distribution $n(t) \sim \mathcal{N}(0, \sigma^2)$. How would the noise be distributed in the frequency domain? I.e., what is $n(f)...
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Convolution of a $2$-dimensional function with a $1$-dimensional function
Below is the schematic diagram from a very popular paper in this field of retinotopy.
In the diagram above, $g$ is a bivariate Gaussian function of the form
$$ g (x,y) = \exp \left( - \left( \frac{ (...
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Methodology to determine similarity between two signals
I'm working on the following problem:
I have an object detection model that detects bounding boxes of objects in a point cloud. I convert the point cloud to a bird-eye-view perspective, i.e. I create ...
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How to smoothly remove boundary discontinuities in functions and their derivatives at the edges of their defined domain?
If I want to Fourier transform a function
$$t \in [-1,1] \to f $$
but this function and it's first $n$ differentials are not equal at the edges : $$f^{(k)}(-1) \neq f^{(k)}(1) , k \in \{0,1,\cdots,n\}$...
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Determine $f(x)$ knowing $f(x)+f(x+\varepsilon)$
I encountered a problem at work that, in my opinion, has a fundamental mathematical reasoning to determine its solvability.
Due to an unwitting software configuration, my associate recorded the audio ...
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Is there a phase between local maxima of a sine wave and a sine wave with a linear trend? [closed]
I'm trying to better understand this simple situation: If I have two functions
$$ \begin{aligned} y_1 &= \sin(\omega t) \\
y_2 &= \sin(\omega t) + at \end{aligned} $$
Will ...
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Kalman filter) Observation matrix of measurement equation and what is a good signal?
I am trying to use a Kalman filter, but my data are somewhat deviating from the assumptions.
The noises in my measurement equation are not normally distributed. First of all, they are not zero-mean. ...
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Understanding $\operatorname{argmin}$ in the given expression
Let $B(m) \in \{-1,1\}$ be the transmitted symbols, for $m=1,2,\dots,1000$. Based on $B(m) = \pm 1$,
$$\begin{aligned} \psi_{m,0} &= h_m-\alpha f_mg_m \\ \psi_{m,1} &= h_m+\alpha f_mg_m \end{...
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