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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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How to speed up convolution for symmetrical property of a half-band FIR filter?
I've discovered that such convolution can be further sped because of symmetric coefficients but I'm not able to get it right.
The filter is a low pass filter at 11025Hz for 44100Hz, with 461 taps.
The ...
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Multiplying a finite signal with a unity window
I have a simple question to ask:
Suppose $x[n]$ is finite and has a support for $|n|<L$
Let's denote its DTFT by $\operatorname{DTFT}\big\{x[n]\big\}(e^{j\omega})=X(\omega)$
Let's consider a unity ...
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Convolution of step function with exponentials
I have a problem in signals and systems to solve which is basically math
So suppose we have two functions x(t) = H(t) which is the Heaviside unit step function as input and an impulse response
$$h(t)=...
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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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Confusion about convolution [duplicate]
I'm facing confusion about the definition of the convolution between two discrete periodic signals.
Basically, the definition of convolution between s and ...
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Convolution error when using DFT for non-periodic functions
I need help understanding why FFT-based computation of convolution on a finite domain, between two non-periodic functions often gets it wrong. Specific questions:
(1) why does FFT-based computation ...
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Comparing methods of blind source separation
I am currently working on blind source separation using sparse hypotheses and convolutive mixtures. For my project, I have compared three different methods and calculated the Signal-to-Distortion ...
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How can Haar basis be written as the impulse response of an LTI system?
Haar wavelets are defined as:
$$\phi_{0,0}(t) = \begin{cases}
1, & \text{ for } 0<t< 1/2\\
-1, & \text{ for } 1/2<t<1 \\
0, & \text{ ...
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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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Effect of BIBO-Instability on the frequency response of a ideal LPF
I recently came across this post stating that the continuous ideal LPF is BIBO-unstable since the impulse response is not absolutely integrable, and this post stating some examples.
I have been trying ...
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Is downsampling LTI for bandlimited inputs?
This question stems from the discussion we had on a previous question of mine.
The point of contention is whether the downsampling operator is time invariant or not. Which gives us a new condition to ...
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Is it possible to consider circular convolution in case of appending the guard interval as a suffix instead of a prefix
In OFDM system, the cyclic prefix is added into the head of time-domain signal to enable the circular convolution with the channel, and then perform one-tap frequency-domain equalization at the ...
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Discrete convolution of a causal signal with a delayed unit step
I am computing the discrete convolution of a signal $x[n] = a^nu[n]$ with a delayed unit step, say $h[n] = u[n-5]$.
If I place both directly into the discrete convolution formula I end up with:
$\frac{...
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What is the reason my DFT of a convolution of signals with disjoint spectrums is not zero everywhere?
Assuming the convolution in the time domain produces a signal which spectrum is the product of individual spectrums, when the two convoluted signals are sinusoids with different frequencies, the DFT ...
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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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