Questions tagged [convolution]
Convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions.
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Proof for the energy correction factor of DFT
I am looking for a mathematical proof for the energy correction factor in conteext of windowed discrete fourier transform.
In Spectrum and spectral density estimation by the Discrete Fourier transform ...
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Prove the following [closed]
If $$x_1[n]=A$$ where $A$ is a constant, $$x_2[n]=u[n]$$ and $$x_3[n]= \left \{ \begin{array}{l} 1, \mbox{for n=0}\\-1, \mbox{for n=1}\\ 0, \mbox{otherwise} \end{array}\right.$$
Prove that $$x_3[n]\...
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Continuous-time convolution of signals with negative amplitudes
While preparing for a mid-term exam, I encountered negative amplitudes for the first time while convolving two signals.
I've already solved the problem, but my result and results from others conflict ...
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Use DFT (`fft()`) to Replicate 2D Convolution (`conv()`)
In MATLAB, I tried to convert the fft2-based multiplication using conv2.
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Intuition behind commutativity of convolution in LTI systems
Why is convolution commutative, as it seems to treat two signals in a different way in an LTI system?
If you imagine $y[n] = x[n] \star h[n]$ with $x[n]$ being an input signal and $h[n]$ being the ...
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Signal Processing of Long Term Behavior in Stochastic Systems
I am quite new to techniques of signal processing. I have a fairly generic problem and wish to find information about topics and/or techniques that may help me address this problem.
Let $x(t)$ and $y(...
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Implementing point-wise multiplication as a convolution
As the title says, I'm looking for a way to implement the point-wise multiplication of 2 signals somehow as a convolution.
Here's why:
In my system, I have an input signal $x[n]$ which has been ...
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What is the rationale behind the "same" mode of discrete convolution?
Having read up on discrete convolution and how it is implemented, I've started to think about which "mode" is applicable to which situation. For two signals ...
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Filtering in time domain vs filtering in frequency domain
The signal is an impulse repsonse. I am filtering the signal using a windowed sinc filter in both frequency and time domain. I am interested in knowing the differences between the two methods and ...
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How to Use Convolution Theorem to Apply a 2D Convolution on an Image
How do I actually apply the convolution theorem? I have my fourier transformed image matrix, and a Fourier transformed kernel, but how do I actually multiply these together to achieve the intended ...
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convolve to differentiate black and white colors
a figure for instance of size 500*500 has half above part with black and below half white should result in a white line where the
white meets the black (something like a single line at line 250 with ...
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Truncating the output of a digital filter. Which part to discard?
Suppose I have a signal $\mathbf{x}\in \mathbb{C}^{N}$ and a digital filter with impulse response $\mathbf{h}\in\mathbb{C}^L$, where $L<N$. If we pass the signal through the filter, the output will ...
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Determine Noise Level by Convolving N Signals
I using a microphone to determine (air) leakage characteristics. The signals need to be filtered to reduce the noise level and i thought of using the Spectral Subtraction Technique. Since im having 4 ...
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Fourier transform of $|x_\mathrm{a}(t)|^2$
Let $x_\mathrm{a}(t)$ be the analytic signal for real signal $x(t)$. I want to find an expression for $\mathscr{F}\{|x_\mathrm{a}(t)|^2\}(f)$ in terms of $x(t)$. The analytic signal can be written as $...
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Proving that the autocorrelation of a signal is the convolution with its time-reversed complex conjugate
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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How to perform an analytical computation of the convolution sum for a discrete-time system?
I am trying to follow some notes on "Analytical Computation of the Convolution Sum (Graphical Method)" but am getting stuck on what I am doing, or if what I am doing is correct.
There are 5 ...
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DFT - Removing window effect in spectral domain with convolution
I was thinking about the DFT windowing subject and a thought came to my mind. A DFT will yield the spectrum of a signal convoluted with spectrum of the window used, therefore having a main lobes and ...
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PyWavelets CWT : error when differentiating after convolution?
I'm trying to understand the implementation of CWT in PYWT. This topic has already helped me quite a lot but there is still a thing that is unclear to me : why is the result differentiated only after ...
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Infinite Sum of Future Signal Values of an LTI System
Let $x[n]$ be the input and $y[n]$ be the output of an LTI system. That is,
$y[n]=\displaystyle\sum_{k=-\infty}^{\infty} h[n-k]\space x[k]$
where $h[\cdot]$ is the impulse response function. We are ...
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Can time aliasing cause peaks?
For the following I use the terms “time domain signal” and “frequency domain signal” as a Fourier Transform pair. The question is for generalized cases of continuous-time signals that once sampled in ...
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Does block-based FIR filtering necessarily imply using the DFT?
Assume I want to filter a stream of data with a rather small FIR filter (in the order of 16 to 64 coefficients). In any case I have to split the input stream into chunks (blocks) of data.
Searching ...
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Correlation gives contradictory results
I am trying to find the correlation between the signals $u(t)$ and $\sin(t)[u(t)-u(t-2)]$
The correlation function $C(t) = \int^{\infty}_{-\infty} u(\tau+t)\sin(t)(u(t)-u(t-2))d\tau$
This is my ...
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What is the Fourier Transform of $\operatorname{sgn}(t) \cdot \operatorname{sgn}(t)$?
I am wondering what the Fourier Transform of $\operatorname{sgn}(t) \cdot \operatorname{sgn}(t)$ will be, where $\operatorname{sgn}(t)$ indicates the signum function. It would seem obvious that this ...
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2d convolution: What is the difference between convolution using blocked Toeplitz matrix and convolution layers?
For a given matrices $A$ of size $4\times 4$ and $B$ of size $3\times 3$ then I construct a blocked Toeplitz matrix and perform the convolution. The resulting output is of size $6 \times 6$.
I have no ...
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Why are convolutions between those functions equivalent (signal processing for theoretical neuroscience)?
I'm reading a book on theoretical neuroscience [1], in which the following definitions are given:
$\rho(t)=\sum_{i=1}^n \delta(t-t_i)$
where $\delta$ is Dirac's delta and the $t_i$ are timestamps at ...
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Convolution output signal transient correction
I'm just starting out in signal analysis and I've come across this effect. When I use convolution of a sinusoid with any other, either triangular pulse, rectangular pulse or decreasing exponential (...
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Impulse Response for an Input-Output Pair
Given an input-output pair of a LTI system
\begin{gather*}
x[ n] \ =\ 2\delta [ n+2] -\delta [ n+1] +\delta [ n-1]\\
y[ n] \ =\ 4 \delta [ n+2] +\ 4\delta [ n+1] -\delta [ n-1]
\end{gather*}
My ...
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Equipment to test my theory on time varying systems
How can I make a linearly time varying system to test my theory?
Can I ask for a PID microcontroller available from my university such that $K$ is gradually increased from 0 to $c$
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Why will the output be imprinted with delay effect or echo after convolving two direct dry signals' FFT values (convolution filtering)
I am recently working on the reproduction of the filtering effect of target play back devices, like phone, or speakers, etc using convolution techniques in MATLAB.
I firstly created a function called &...
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Fusing convolutions
I am attempting to combine two consecutive 2D convolutions into a single 2D convolution. Ideally, a convolution operation is a linear transformation, so it should be possible to merge two linear ...
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Having problems with convolution in Fourier space
First attempt:
I tried convolution in Fourier space.
step: transform grayscale image ==> (real1,imag1)
step: transform kernel (a simple box filter) ==> (real2, imag2)
step: multiply (complex) ==...
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How would the correlation coefficient between 2 images change if we rotated 1 image?
Suppose we have a image which is represented by a matrix
$
\begin{bmatrix}
a & b & c \\
d & e & f \\
g & h & i \\
\end{bmatrix}
$
and we want to find its correlation with ...
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How to handle undefined functions in a convolution?
When convolving with undefined functions, do we just treat them like variables? (eg. How do I handle $u \left( k \right)$ and $u \left( n - k \right)$ since they are just step functions with no ...
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Incorrect Power Calculation in MATLAB Convolution
I have been working on a MATLAB code to perform convolution and calculate the power of the resulting signal. However, I have encountered an issue with the power calculation in my code.
I am convolving ...
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Why does convolution give the output of a passing a signal through a filter?
I have a rudimentary understanding of Convolution, the Convolution Theorem and why the output z(t) of an LTI system can be found using the convolution of input signal x(t) and the impulse response h(t)...
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Simulating analog filter using convolution or converting with fft
My task is simple; I want to simulate analog low-pass filtering of an input signal, using Python. Note that the input signal is an array of values, not an analytical function.
My first question is if ...
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Why is the convolution of two sine waves a sinc function?
Problem/Question
After taking the convolution of two discrete sine waves of the same frequency and length the result appears to be a sinc function but I was expecting to get a sine wave since this ...
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Question about convolution of shifted function-simple
I memorized the convolution formula, all good. I started doing examples and in one of them I had to calculate the convolution of a shifted function with another one.
I know the property $$H(x)=f(x)\...
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Doubt on linear and circular convolution derivation
[I'm very new to this website and to DSP in general as well! :)]
Hello everyone!
I have a question on the derivation of the circular convolution.
As far as I understand, we can define a circular ...
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Is flipping a vector x the same as x[-n]?
I was intending to use the following convolution property in my Matlab code.
$$y[-n]=x[-n]*h[n]=x[n]*h[-n]$$
However, I am noticing that the results seems not the expected ones, for example let have
<...
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Inequality involving zero-crossings of Difference of Gaussian and Laplacian of Gaussian
I am stuck on the following problem:
Let $f$ be a smooth real function with a single inflection point and let $\sigma_1>\sigma_2>0$ be two real numbers. We denote $G_1$ and $G_2$ the associated ...
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3D convolution product with Fourier transforms, FFTW and MPI in C
My question is not really from the field of signal processing but I think you are the most suited for answering my question.
I am willing to compute the gravitational potential which can be written as ...
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Does convolution of delayed sequences produces a delayed output?
Say, I have $x[n]$ and $y[n]$ be periodic sequences of period $N$. Let us define $z[n] =x[n] \ast y[n]$ where $\ast$ denote the circular convolution operation.
What is $x[n+\delta] \ast y[n+\delta]?...
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Locate No Signal Area Between 2 Signals
I have many datasets which all looks roughly like this:
All the datasets have this in common. First there is an artifact around 0 with an unknown (but likely very high) intensity. Then there is some ...
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Reduce Signal Size to Compare Them
I have multiple experiments and each of them produce several ($k$ for example) binary signals; some artificial example next:
I have a metric to compare experiment results but I need vectors of equal ...
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How to Find the Kernel of the Convolution for Linear Interpolation?
I'm trying to solve the following exercise:
Image A was doubled by linear interpolation. The magnification was performed in two stages. In the first stage, add about zero pixel to the image between ...
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What architecture of CNN to solving a image regression problem (case study: solving Poisson equation)?
I've been working on solving Poisson problem using CNN model (you can ignore Poisson problem part if you not familiar and jump to image processing/CNN part). More specific, I am solving electric ...
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Underdetermined deconvolution of windowed output
Consider a discrete 'blurred' output $h[t]$ given by the convolution of filter $f[t]$ and signal $g[t]$. This question considers recovering $g[t]$ from a window (subset) of $h[t]$. This causes the ...
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Convolution and Fourier transform for 1D signals
My problem is the following, I have 3 curves/signals (1D) , the measure, the signal and the resolution of my detector: $\mathcal{M},\mathcal{S},\mathcal{R}$, knowing that : \begin{equation}\label{eq:...
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Why the difference of a vectorized VS non-vectorized FIR convolution differ?
I have an FIR filter that is a low pass at 11025Hz for 44100Hz, 461 taps. The reference implementation, i.e. naive convolution, works as expected. Now I've written a vectorized implementation to take ...
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