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 ...
Mark Borgerding's user avatar
9 votes

Estimate Sine Frequency under White Noise — simple and effective method

I assume the model to be: $$ x \left[ n \right] = \sin \left[ 2 \pi \frac{f}{ {f}_{s} } n + \phi \right] + w \left[ n \right] $$ Where $ w \left[ n \right] $ is white noise uncorrelated with the ...
Royi's user avatar
  • 19.6k
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 ...
Marcus Müller's user avatar
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 ...
jithin's user avatar
  • 2,263
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 ...
Olli Niemitalo's user avatar
7 votes

Estimate Sine Frequency under White Noise — simple and effective method

A common way to do this is to take the FFT of the input signal. Since the frequency might not be right at a FFT bin, usually a second step of interpolation is done after choosing the initial peak. A ...
Engineer's user avatar
  • 3,032
7 votes

Estimate Sine Frequency under White Noise — simple and effective method

If you have a low-noise and well-sampled signal, a quick way to estimate it is to find $\sqrt{-f''(t)/f(t)}$. For a signal $$f(t)=A \sin(\omega t+\phi)$$ the second derivative is $$-A \omega^2 \sin(\...
Nullius in Verba's user avatar
6 votes

Estimate Sine Frequency under White Noise — simple and effective method

This depends on the precision needed. If it's a pure sine wave that's noise free, you can get a very quick estimate by measuring the difference between two zero crossings. The tricky part is that most ...
Hilmar's user avatar
  • 44.7k
6 votes
Accepted

MUSIC Algorithm for Direction of Arrival (DOA) in Acoustic Signals

The steering vector basically applies the delay according to the spatial location of the Uniform Linear Array (ULA). The multiplication by the complex exponential basically applies the delay. The ...
Royi's user avatar
  • 19.6k
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 =...
Royi's user avatar
  • 19.6k
5 votes
Accepted

Relationship between information retrieval and source separation in signal processing

There are, a few discrepancies that might be making a difference here. My suggestion would be to edit the question for clarity. There are quite a few assumptions that lead to non-straightforward ...
A_A's user avatar
  • 10.7k
5 votes
Accepted

Extract Sine Phase and Amplitude - accurate and robust method

Regarding your Edit #2: It is not true that phase estimation requires that frequency be at a bin center. However if a frequency estimate requires interpolation between FFT result bins, so does any ...
hotpaw2's user avatar
  • 35.3k
5 votes

Estimate Sine Frequency under White Noise — simple and effective method

I wasn't going to answer this, since the question is stale. But I'm a little bit dissatisfied with Royi's answer and with the Kay algorithm as presented. The Kay method is, at first glance, simply ...
robert bristow-johnson's user avatar
4 votes

Extract Sine Phase and Amplitude - accurate and robust method

This is the problem we have in sinusoidal modeling. There are lots of issues, and your window function might be one of them. This paper by me is a little more general in that it not only are the ...
robert bristow-johnson's user avatar
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 ...
Cherny's user avatar
  • 471
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 ...
Atul Ingle's user avatar
  • 4,134
4 votes

Understanding the Difference Between MAP Estimation and ML Estimation

A brief, non-mathy explanation: ML assumes that all hypothesis are equally likely. MAP does not make this assumption. MAP is the optimum criterion, but under some conditions ML is optimum too. When ...
MBaz's user avatar
  • 15.3k
3 votes
Accepted

Why Is The Maximum Likelihood Estimation (MLE) Method Taken as the Benchmark for Comparing with Other Methods?

Coming up with a "good" estimator for a parameter of interest is not an easy task because it is important to define what good means. There are many ways of defining it, depending on your application. "...
Atul Ingle's user avatar
  • 4,134
3 votes
Accepted

How to apply least-squares estimation for sparse coefficient estimation?

I am not entirely sure what Matlab's LASSO routine does so I started with Ordinary Least Squares (OLS) and worked backwards. From an OLS perspective X1 as you have it won't work. You've got a ...
Chad Sexington's user avatar
3 votes

Maximum Likelihood for Colored Noise

Let's have a look on the following model: $$ y \left[ n \right] = \left( h \ast x \right) \left[ n \right] + \left( g \ast w \right) \left[ n \right] $$ Where $ x \left[ n \right] $ is the signal of ...
Royi's user avatar
  • 19.6k
3 votes

Is there an algorithm to compute the phase for a single frequecy?

You could also do this (in numpy notation): np.arctan( (signal*cos).sum() / (signal*sin).sum() )) where signal is your phase-shifted signal, cos and sin are the ...
M529's user avatar
  • 1,736
3 votes
Accepted

Is it possible to estimate variance of noise for a step answer signal?

Hi: In order to estimate the variance, you need to have an underlying model for your signal. So, suppose that the model is $y_{t+1} = y_t + \epsilon_t$ $~\forall ~ t = 1,\ldots n $. assuming that $E(...
mark leeds's user avatar
  • 1,117
3 votes

Sequential Form of the Least Squares Estimator for Linear Least Squares Model

There are really great answers. I will try to give the Sequential Least Squares approach which generalizes to any Linear Model. Sequential Least Squares Model We're after solving the Linear Least ...
Royi's user avatar
  • 19.6k
3 votes
Accepted

Estimate Parameters of Linear Combination of Harmonic Signals with Partial Known Frequencies

Bottom Line: The OP is experiencing the effects of spectral leakage and the primary motivation for windowing with the Discrete Fourier Transform. The reason the windowing approach achieves much better ...
Dan Boschen's user avatar
  • 51.4k
3 votes

Estimate Sine Frequency under White Noise — simple and effective method

As others have already mentioned in the comments, the question shows on research effort, is far too broad and needs clarification. Depending on the requirements, the MUSIC (or ESPRIT) algorithm may be ...
juliusd's user avatar
  • 31
3 votes

Algorithm for finding best matching sin wave to input signal

Well, in this noiseless case, you can just estimate the frequency directly: take the $\arccos(y(t))$ of each sample, find the difference between consecutive samples: that's how much the phase changes ...
Marcus Müller's user avatar
3 votes
Accepted

A Machine Learning Based Algorithm as an Alternative to the Matched Filter

The idea is to have a simple experiment to see if we can get, for a known signal, a better results than the Matched Filter for time delay estimation. Experiment Objective Generate, using ML (DL), a ...
Royi's user avatar
  • 19.6k
3 votes

How can I estimate the 2 parameters of this signal?

Your task is somewhat of an instance of offline change detection. Try to take a look at this and its documentation / review paper for some implemented algorithms. If you can assume your signal ...
mateC's user avatar
  • 338
2 votes

Is there an algorithm to compute the phase for a single frequecy?

This is an improvement on @Kevin McGee's suggestion to use a single frequency DFT with a fractional bin index. Kevin's algorithm doesn't yield great results: while at half bins and whole bins it's ...
mojuba's user avatar
  • 131
2 votes

Estimation of accelerating target using position measurements only

I don't have a really good answer for you, but I can see a possible problem with your second attempt: the value of $a_3$ has the potential to completely blow up the dynamics because it is multiplying ...
Peter K.'s user avatar
  • 25.7k

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