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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How to convert measured insertion loss to an impulse response through ifft? Question around factor of 2
Specifically I'm unsure about a factor of 2 scale factor when converting the measured insertion loss to a double sided spectrum, necessary for the ifft.
In the past I have always believed that ...
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Question regarding CA-CFAR, specifying using the FFT within the algorithm and zero-padding. Or really just dealing with the edge samples
To compute the value of each CUT fast, I have been using the FFT to convolve/sum the test cell together. But I have seen that the zero-padding needed for linear convolution is being included in the ...
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Which domain (time or frequency) does filtering actually occur in real-life?
In introductory course or textbook on signal processing, there seems to be an underlying mantra of
"Filtering only happens in time domain, while filter design only happens in frequency domain&...
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Autocorrelation via FFT
I read through a lot of the already existing posts about autocorrelation via FFT on this forum and elsewhere but I am still confused. Also, I am still not sure whether this is the right place to ask ...
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How to solve the convolution of two signals when one of them isn't explicitly given and also reconstruct it?
I'm given the following:
Where in the last box it says "reconstruct".
As you can see $x(t)$ is multiplied by the impulse train $p(t)$ and then passed through this LPF and then to the ...
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Why does the window function appear in the linear chirp spectrum?
The Fourier Transform of a chirp is another chirp
$$
\mathscr{F}\{\,e^{j\pi k t^2}\}(f)
\;=\; \sqrt{\frac{1}{|k|}}\;e^{\,j\,\operatorname{sgn}(k)\,\frac{\pi}{4}}\;
\exp\!\Bigl(-\,j\pi\,\frac{f^2}{k}\...
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How to calculate the periodic system response of an LTI system?
Given $h (t) = e^{-t} u(t)$, where $u$ denotes the Heaviside step function, find $h_{\text{per}}(t)$, the periodization of $h$ with period $T$.
I found $H(\omega) = \dfrac{1}{1 + j\omega}$, and now I'...
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LTI convolution integral $\int _{-\infty}^{\infty} L\{ x(\tau) \delta(t - \tau)\} \ d\tau$ why $L$ act on $t$ rather than on $\tau$?
In the usual derivation of the LTI convolution formula, we often start with this identity:
$$x(t) = (x*\delta)(t) = \int _{-\infty}^{\infty} x(\tau) \delta(t - \tau) \ d\tau$$
Then for a linear system ...
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Why does the output of LTI systems involve convolution rather than correlation?
For a LTI system, suppose it output $h[n]$ for input $\delta[n]$ ("impulse response"). We can decompose any input signal $x[n]$ using linear combination of the time dilations of $\delta[n]$:
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On the definition of circular convolution
Let $x[n]$ and $y[n]$ be two periodic signals of period $N$. Can the circular convolution of $x$ and $y$ be defined as follows:
$$ (x \circledast y)[n] = \sum_{k=0}^{N-1} x[k] y[n-k] \qquad (n \in \...
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DFT and convolution for periodic signals
Let $F_x[k]$ denotes the DFT of the periodic signal $x[n]$. If $x$ and $y$ are periodic with period $N$, then their (usual) convolution $x*y$ is also periodic with period $N$. Is it true that
$$
F_{x*...
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CMSIS multiply and accumulate FIR filtering algorithm - concern about length of output
The zero-state response of a filter with impulse response $h[k]$ to an input signal $x[k]$ can be calculated by performing the convolution
$$y[k] = x[k] \ast h[k] \tag1$$
If $x[k]$ has length $m$ and $...
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How to do time-domain oversampling of the received OFDM signal
Following the approaches in [1], [2], and other related references, it is well established that the time-domain OFDM signal can be oversampled without increasing the effective length of the channel ...
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Is a window function the same as a blur?
I have become confused about a subtlety about window functions.
Context
My application is that I start with an interferogram (which is basically an image of a cosine function). The pipeline is:
I ...
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Is multiplication in time always convolution in frequency?
Say there are two signals $x_1(t)$ and $x_2(t)$ that are not equal except for over the domain $t=-0.5$ to $t=0.5$. Therefore $rect(t)x_1(t)=rect(t)x_2(t)$. Taking the Fourier transform of both sides ...
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