Deconvolution of accumulated values
$begingroup$
Introduction
I have an unknown function of time $f(t)$ that I would like to learn based on experimental observations. I have an observable $g(t)$, which, to the best of my knowledge, is given by the convolution of $f(t)$
$g(t) = int_{0}^{infty}f(t-tau)h(tau)dtau + nu(t)$
where $h(t)$ is the known filter function, and $nu(t)$ is assumed to be Gaussian noise of known variance. $h(t)$ is typically an exponentially decaying function or similar. Let us assume for simplicity that we have measured $g(t)$ in the time interval $[0,T]$ and that $f(t)=g(t)=0$ for all $t<0$
Now, my understanding is that this problem is solved in practice by discretizing it, assuming $f(t)$ to be piecewise-constant over small time intervals $Delta t$. If we define
$f(t) = f_i$ where $i = lfloor frac{t}{Delta t} rfloor$
- $g_i = g(iDelta t)$
- $nu_i = nu(iDelta t)$
- $h_i = int_{i Delta t}^{(i+1)Delta t}h(t)dt$
the convolution integral can be rewritten in its discrete form as
$g_i = sum_{j=0}^{i} f_{i-j} h_j + nu_i$
It is then possible to solve the discrete problem by matrix inversion, fourier transform, etc...
Actual question
A central point in the above approximation is that the exact value of $g(t)$ is known at discrete intervals. In practice, however, the value of $g$ is accumulated over small time intervals, as opposed to being measured instantly. Namely, the actual discrete observable one obtains from the device is
$tilde{g}_i = frac{1}{Delta t} int_{i Delta t}^{(i+1)Delta t} g(t)dt$
I have tried to repeat the procedure to rewrite $tilde{g}_i$ as a function of $f_i$. However, it appears to be a page-long tedious calculation, and I keep making mistakes. Surely, I am not the first one to come across this problem. If you can provide a link to the derivation of the above equation, or perhaps its name or keyword if such exists, I would be grateful
discrete-optimization deconvolution
asked Jan 9 at 14:42
Aleksejs FominsAleksejs Fomins
555211
$endgroup$
add a comment |
$begingroup$
Introduction
I have an unknown function of time $f(t)$ that I would like to learn based on experimental observations. I have an observable $g(t)$, which, to the best of my knowledge, is given by the convolution of $f(t)$
$g(t) = int_{0}^{infty}f(t-tau)h(tau)dtau + nu(t)$
where $h(t)$ is the known filter function, and $nu(t)$ is assumed to be Gaussian noise of known variance. $h(t)$ is typically an exponentially decaying function or similar. Let us assume for simplicity that we have measured $g(t)$ in the time interval $[0,T]$ and that $f(t)=g(t)=0$ for all $t<0$
Now, my understanding is that this problem is solved in practice by discretizing it, assuming $f(t)$ to be piecewise-constant over small time intervals $Delta t$. If we define
$f(t) = f_i$ where $i = lfloor frac{t}{Delta t} rfloor$
- $g_i = g(iDelta t)$
- $nu_i = nu(iDelta t)$
- $h_i = int_{i Delta t}^{(i+1)Delta t}h(t)dt$
the convolution integral can be rewritten in its discrete form as
$g_i = sum_{j=0}^{i} f_{i-j} h_j + nu_i$
It is then possible to solve the discrete problem by matrix inversion, fourier transform, etc...
Actual question
A central point in the above approximation is that the exact value of $g(t)$ is known at discrete intervals. In practice, however, the value of $g$ is accumulated over small time intervals, as opposed to being measured instantly. Namely, the actual discrete observable one obtains from the device is
$tilde{g}_i = frac{1}{Delta t} int_{i Delta t}^{(i+1)Delta t} g(t)dt$
I have tried to repeat the procedure to rewrite $tilde{g}_i$ as a function of $f_i$. However, it appears to be a page-long tedious calculation, and I keep making mistakes. Surely, I am not the first one to come across this problem. If you can provide a link to the derivation of the above equation, or perhaps its name or keyword if such exists, I would be grateful
discrete-optimization deconvolution
asked Jan 9 at 14:42
Aleksejs FominsAleksejs Fomins
555211
$endgroup$
add a comment |
$begingroup$
Introduction
I have an unknown function of time $f(t)$ that I would like to learn based on experimental observations. I have an observable $g(t)$, which, to the best of my knowledge, is given by the convolution of $f(t)$
$g(t) = int_{0}^{infty}f(t-tau)h(tau)dtau + nu(t)$
where $h(t)$ is the known filter function, and $nu(t)$ is assumed to be Gaussian noise of known variance. $h(t)$ is typically an exponentially decaying function or similar. Let us assume for simplicity that we have measured $g(t)$ in the time interval $[0,T]$ and that $f(t)=g(t)=0$ for all $t<0$
Now, my understanding is that this problem is solved in practice by discretizing it, assuming $f(t)$ to be piecewise-constant over small time intervals $Delta t$. If we define
$f(t) = f_i$ where $i = lfloor frac{t}{Delta t} rfloor$
- $g_i = g(iDelta t)$
- $nu_i = nu(iDelta t)$
- $h_i = int_{i Delta t}^{(i+1)Delta t}h(t)dt$
the convolution integral can be rewritten in its discrete form as
$g_i = sum_{j=0}^{i} f_{i-j} h_j + nu_i$
It is then possible to solve the discrete problem by matrix inversion, fourier transform, etc...
Actual question
A central point in the above approximation is that the exact value of $g(t)$ is known at discrete intervals. In practice, however, the value of $g$ is accumulated over small time intervals, as opposed to being measured instantly. Namely, the actual discrete observable one obtains from the device is
$tilde{g}_i = frac{1}{Delta t} int_{i Delta t}^{(i+1)Delta t} g(t)dt$
I have tried to repeat the procedure to rewrite $tilde{g}_i$ as a function of $f_i$. However, it appears to be a page-long tedious calculation, and I keep making mistakes. Surely, I am not the first one to come across this problem. If you can provide a link to the derivation of the above equation, or perhaps its name or keyword if such exists, I would be grateful
discrete-optimization deconvolution
asked Jan 9 at 14:42
Aleksejs FominsAleksejs Fomins
555211
$endgroup$
Introduction
I have an unknown function of time $f(t)$ that I would like to learn based on experimental observations. I have an observable $g(t)$, which, to the best of my knowledge, is given by the convolution of $f(t)$
$g(t) = int_{0}^{infty}f(t-tau)h(tau)dtau + nu(t)$
where $h(t)$ is the known filter function, and $nu(t)$ is assumed to be Gaussian noise of known variance. $h(t)$ is typically an exponentially decaying function or similar. Let us assume for simplicity that we have measured $g(t)$ in the time interval $[0,T]$ and that $f(t)=g(t)=0$ for all $t<0$
Now, my understanding is that this problem is solved in practice by discretizing it, assuming $f(t)$ to be piecewise-constant over small time intervals $Delta t$. If we define
$f(t) = f_i$ where $i = lfloor frac{t}{Delta t} rfloor$
- $g_i = g(iDelta t)$
- $nu_i = nu(iDelta t)$
- $h_i = int_{i Delta t}^{(i+1)Delta t}h(t)dt$
the convolution integral can be rewritten in its discrete form as
$g_i = sum_{j=0}^{i} f_{i-j} h_j + nu_i$
It is then possible to solve the discrete problem by matrix inversion, fourier transform, etc...
Actual question
A central point in the above approximation is that the exact value of $g(t)$ is known at discrete intervals. In practice, however, the value of $g$ is accumulated over small time intervals, as opposed to being measured instantly. Namely, the actual discrete observable one obtains from the device is
$tilde{g}_i = frac{1}{Delta t} int_{i Delta t}^{(i+1)Delta t} g(t)dt$
I have tried to repeat the procedure to rewrite $tilde{g}_i$ as a function of $f_i$. However, it appears to be a page-long tedious calculation, and I keep making mistakes. Surely, I am not the first one to come across this problem. If you can provide a link to the derivation of the above equation, or perhaps its name or keyword if such exists, I would be grateful
discrete-optimization deconvolution
discrete-optimization deconvolution
asked Jan 9 at 14:42
Aleksejs FominsAleksejs Fomins
555211
asked Jan 9 at 14:42
Aleksejs FominsAleksejs Fomins
555211
asked Jan 9 at 14:42
Aleksejs FominsAleksejs Fomins
555211
asked Jan 9 at 14:42
Aleksejs FominsAleksejs Fomins
555211
asked Jan 9 at 14:42
Aleksejs FominsAleksejs Fomins
555211
555211
add a comment |
add a comment |
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