17
votes
Accepted
What is the adjoint of a linear operator and why is it useful?
Here's the best practical information I have so far on linear operators and their adjoints. There's only one book I've come across that discusses this very practically (which I reference below); ...
- 2,064
16
votes
Understanding the Difference Between MAP Estimation and ML Estimation
Maximium A Posteriori (MAP) and Maximum Likelihood (ML) are both approaches for making decisions from some observation or evidence.
MAP takes into account the prior probability of the considered ...
- 3,040
15
votes
Accepted
What would be the variance for complex number?
I will focus on the reason of the factor $1/2$ and leave aside the estimation things.
The exact understanding should be : if a scalar Gaussian random variable (rv) is circular symmetric, its real and ...
- 6,725
10
votes
Accepted
Which Noise Reduction Algorithms Are Used in Commercial RAW Image Processors?
Common Approaches for Commercial Denoisers
Commercial denoisers are different than what you'd see on most papers. While on papers the results are mostly using objective metrics (PSNR / SSIM) and are ...
- 20.5k
9
votes
A Machine Learning Based Algorithm as an Alternative to the Matched Filter
Sure, you can learn the matched filter, as convolution with a filter is just a function applied to a signal, and e.g. Neural Networks (through the universal approximation theorem) are good function ...
- 32.5k
8
votes
Understanding the Difference Between MAP Estimation and ML Estimation
You have a set of message set $m_i$, $0 \le i \le N-1$. (For example, QPSK will be $N=4$). For the transmitted message $m_i$, the corresponding symbol vector is $\textbf{x}_i$, and the received symbol ...
- 2,303
8
votes
Accepted
Signal Estimation after detection Part 2
After signal detection, how to estimate the clean signal $s(t)$?
Matched filtering is used to detect the presence of a known signal in noise. There is no estimation part when you are talking about a ...
- 3,042
8
votes
Accepted
cramer lower bound, MAE, and MSE
The topics you bring up are at the heart of estimation theory. I highly recommend reading Steven Kay's Fundamentals of Statistical Signal Processing: Estimation Theory for a detailed background and ...
- 968
7
votes
Accepted
Sequential Form of the Least Squares Estimator for Linear Least Squares Model
Slope from all samples obtained
To summarize the question's problem, you want to calculate the slope based on all samples obtained thus far, and as new samples are obtained, update the slope without ...
- 13.7k
7
votes
Estimators for improved spectral subtraction of noise
Maximum likelihood (ML) estimator
Here will be derived a maximum-likelihood estimator of the power of the clean signal, but it doesn't seem to be improving things in terms of root mean square error, ...
- 13.7k
7
votes
Accepted
Estimate and Track the Amplitude, Frequency and Phase of a Sine Signal Using a Kalman Filter
We can build a non linear dynamic model in order to estimate the parameters of a sine signal.
Let's model the signal as $ a \sin \left( \phi \right) $ where $ \phi $ is the instantaneous phase. So the ...
- 20.5k
7
votes
Kalman Filter on Sinusoidal Signal
This isn't quite what you're asking, because it neglects the amplitude, $A$, but it's a relatively straightforward example of application of an extended Kalman filter to the frequency tracking problem....
- 26k
7
votes
Accepted
What sensors can be fused using the Kalman Filter framework
Remark: I will answer this using the Linear framework of the Kalman Filter but the idea is the same.
The Kalman Filter basically propagate and fuses Gaussian Distributions in order to calculate the ...
- 20.5k
6
votes
Accepted
Tracking a Sine Wave with Kalman Filter - How to Account for Offset (DC Signal)?
Well, in continuous time, a sinusoid with a bias can be seen as the output of the linear system
\begin{align*}
\begin{bmatrix}\dot x_1\\\dot x_2\\\dot x_3\end{bmatrix} &= \begin{bmatrix}0 & 1 ...
- 768
6
votes
How does this FLL work?
INTUITIVE EXPLANATION
This is very similar in form (as a maximum likelihood detector in both cases) to Timing Error Detectors, where instead of a "Frequency Derivative Matched Filter", A &...
- 55k
6
votes
Accepted
Why use parametric based estimation methods - confusion regarding terms
Hi: I'll try to answer as briefly as possible and only with respect to statistics. not dsp.
In statistics, if you have a nice pdf such as the normal distribution, then maximizing the likelihood is ...
- 1,117
6
votes
Accepted
Estimation of the Amplitude of a Sine / Cosine Wave and Its Independence of the SNR / Amplitude of the Wave
One may have a look at the CRLB of estimating the parameters of a sine wave.
The model for signal is given by:
$$ x \left[ n \right] = A \cos \left( \omega n + \phi \right) + w \left[ n \right], \; n =...
- 20.5k
5
votes
How Is the Formula for the Wiener Deconvolution Derived?
The Wiener Filter can also be derived by another (Easier) way.
Let's assume the following model:
$$ y = h \ast x + n $$
Namely the data is a result of a linear combination (Convolution) of $ x $ with ...
- 20.5k
5
votes
Estimators for improved spectral subtraction of noise
Update:
I'm sorry to have to say that testing shows the following argument seems to break down under heavy noise. This is not what I expected, so I have definitely learned something new. My prior ...
- 7,600
5
votes
Accepted
OFDM time vs. frequency domain channel estimation/equalization
Traditionally, OFDM became popular in WiFi and LTE because the channel model consisted of multi-path. That is, the radio signal transmitted in 1-6GHz frequencies bounced from various obstacles (walls, ...
- 2,303
5
votes
Accepted
Understanding the H1 and H2 estimators
Let's define the true transfer function $H_0=P_{xy}/P_{xx}=P_{yy}/P_{yx}$.
$H_1$: The transfer function is computed as the ratio of the cross spectrum between the input and output signals, to the ...
- 3,263
5
votes
Accepted
FFT-based phase estimation better than CRLB in MATLAB simulation
Frequency estimation is fraught with peril. I believe that one issue is that the maximum likelihood frequency estimator gets much better accuracy than just choosing the FFT bin maximizer as in the ...
- 26k
4
votes
Accepted
Why is being jointly WSS important in signal estimation with LMMSE estimator?
Recall that in the derivation of the linear MMSE estimator we force the error term $\hat{s}_k-s_k$ to be orthogonal to the signal $y_k$ i.e. $E[(\hat{s}_k-s_k)y_{k-j}] = 0$ for all $j$. The assumption ...
- 4,144
4
votes
Accepted
What is meant by optimal estimator and how to determine optimality?
So Least squares estimator is as it literally - the estimator which brings the mean square error to minimum. In the case of Gaussian white noise it has a simple and analytic solution. I recommend you ...
- 471
4
votes
Accepted
Unscented Kalman Filter Equations for Constant Turn Rate and Velocity Process Model
From a statistical point of view, the noise parameters are zero mean gaussian distribution and that does not mean that at all times the value of noise would be zero. All it says is that if you were to ...
4
votes
MMSE Estimation - Fusion of 2 Measurements
*STOP! If you only want a hint and not the complete solution please see Stanley P.'s or Peter K.'s answers. *
Since you do not specify if there is model for the temperature evolving over time $n$, I ...
- 4,144
4
votes
Amplitude Estimating Using a Windowed DFT
I have several blog articles that solve this problem exactly in a theoretical sense, and quite accurately in an implementattion.
3. DFT Pure Tone Frequency Formulas
Exact Frequency Formula for a ...
- 7,600
4
votes
Accepted
Generate signals with a particular variance and SNR
It depends on what you mean by SNR. It's a common joke in the DSP community to spell it out as "something to noise ratio", referring to the fact that there is no unique definition of SNR, so the term ...
- 2,463
4
votes
Accepted
On the Measurement Matrix Used for Compressing Sensing
It is indeed possible to formulate this setting in terms of matrix-vector products. First, let us re-formulate your $x$ (notice throughout that I use bold letters for vectors and matrices):
$$x = \...
- 1,322
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