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Questions tagged [maximum-likelihood-estimation]

Use this tag for any question regarding or utilizing the Maximum Likelihood (MLE / ML) method.

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ML bound for Massive MIMO

How can one estimate the ML bound for Massive Mimo detection with 128 transmitting antenna and 128 receiving antennas using 4 QAM modulation? I have seen for a 8 by 8 system it is possible to use K-...
albusSimba's user avatar
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2 answers
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Difference between Likelihood Estimation and CRLB Estimation for Cooperative Radar

I do not know if this question fits this stack, but I think it might suit here because it is something related to the radar sensing processing. The question is about the difference between the ...
Loco Citato's user avatar
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simple algorithm to calculate and update LLR for nand flash memory?

I'm workding on nand flash DSP (Digital Signal Processing) now, and encounter obstacle, which is: how to generate and update Log-Likelihood Ratio (LLR) LUT for Triple- and Quad Level Cell (TLC and QLC)...
Milin's user avatar
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Why the log likelihood is positive in some cases

I have a log likelihood that looks like the following. $$ \log(p(Z|\Theta)) = -\sum_{p = 1}^{N} \left[ L \log\left(\pi\left(A F(p, \Theta) + \sigma^2\right) \right) + \frac{\sum_{l = 1}^{L} Z_l(p) }{A ...
CfourPiO's user avatar
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How can I write the likelihood of this system

I am very confused about how to write the likelihood of the following function. I have tried one and it doesn't have a maximum at the point where it should contain a maximum. The formulation of the ...
CfourPiO's user avatar
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1 answer
33 views

Why this second part of the integral given by the gradient of the log-likelihood is zero

I am reading the book "Detection of Signals in Noise" by Robert N McDonough, A D Whalen. In chapter 10, they compute the gradient of the log-likelihood for the time of arrival. See attached ...
user144410's user avatar
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55 views

Detector for vector-valued signals

I'm trying to find the optimal probability of detection/false alarm for the following detection task. Given $N$ signal samples with $d$ dimensions (independent channels) each, assign the samples to ...
Sami's user avatar
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How is maximum log likelihood calculated for BPSK?

I am seeking to understand, exactly, how to calculate a soft MLL output from a received BPSK signal, and what information is needed. I would like to implement turbo coding in my system, but the ...
Howard's user avatar
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2 answers
339 views

Maximum Likelihood Estimation of Phase

I'm trying to solve an exercise where I need to derive the MLE of the phase parameter. I'm given the below signal with white noise $w[n] \sim \mathcal{N}(0, \sigma^2) $, amplitude $A$, angular ...
sadghi's user avatar
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3 votes
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768 views

cramer lower bound, MAE, and MSE

I am quite confused, is there any relation between MSE, CRLB and MAE. Can we test the efficiency of a Maximum likelihood estimator with the MAE, or must we use CRLB? Also is it true that CRLB is MSE?
Blobmou's user avatar
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Difference between Cramer Rao bound and mean absolute error MAE

Difference between Cramer Rao bound and mean absolute error MAE? I cannot see the difference, or where to use one and not the other. I know that Cramer and MAE are used to measure the quality of an ...
Blobmou's user avatar
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Generalized likelihood ratio to Probability estimate

I implemented a generalized likelihood ratio test (GLRT) detector/identifier. The result of the identification step results in a score for each of the objects to be identified in an ordered list. The ...
Sam Gomari 's user avatar
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116 views

Fisher Information Matrix for sinusoidal signal under multiplicative noise

Consider observations ($y$) of a sinusoidal wave with multiplicative noise ($v$) where we are estimating unknown frequency ($\omega$) and unknown initial phase ($\theta$). We can write this system ...
Ahwaq's user avatar
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303 views

Does Maximum Likelihood detector (QAM symbol detection) enhance BER?

Assuming QAM symbols (or OFDM system) with AGWN, I found that the use of ML detector does not enhance the BER but only enhances (reduces) error vector magnitude (EVM) because of symbol decision and ...
Amro Goneim's user avatar
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71 views

Maximum Likelihood for Colored Noise: Implementing using Viterbi Algorithm

I have a question regarding the ML estimation in presence of colored noise, i.e., Additive Colored Gaussian Noise. I want to implement this using a Viterbi algorithm, since I need to take the '...
Jacob's user avatar
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Question about relationship between aliasing and matched filter

while I was reading J. Barry's textbook (https://www.springer.com/gp/book/9780792375487), page 308 says the following: "... symbol-rate sampling at the matched filter output is generally at less ...
Emmmm's user avatar
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Bayesian Inference / Likelihood of Byte-stream

Introduction Let us consider the following problem; I receive a byte stream. The bytes are correlated, i.e. It is possible that values will very quite close. It is reasonable to assume that the ...
Yair M's user avatar
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1 answer
391 views

How to remove the modulation before doing frequency offset estimation?

To use DFT/FFT (or maximum likelihood) method to estimate the frequency offset introduced by the channel, we need to remove the modulation on the received data samples in the front. If the unknown ...
Linda's user avatar
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1 answer
103 views

ML estimation - solve for x

I'm trying to solve the following maximum likelihood estimation but for multiplicative noise instead of additive noise: So the goal is to do ML-estimation for a scalar constant $x$, which is ...
Phobos's user avatar
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Unable to estimate for AR model using OLS, Yule Walker and MLE

I am learning estimation methods following the book of Steven Kay, "Fundamentals of Statistical Processing, Volume I: Estimation Theory " Theory says that if the measurement noise is ...
Sm1's user avatar
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Blind Estimation of Signal Parameters for Zero mean Normal distribution

Let $\mathbb{s}$ be zero mean with a normal distribution. Let one of the observed set of signals be given by $K\mathbb{s}$ and another by $L\mathbb{s^*}$, where $K$ and $L$ both are constants, and $\...
Andrew Smith's user avatar
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2 answers
79 views

Bayesian Information criterion is independent of prior - What does 'prior' mean here?

I am trying to identify the possible number of states in my data. Each state corresponds to a different scenario. For example, I am measuring humidity and trying to identify the number of states in ...
Sashi's user avatar
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11 votes
3 answers
7k views

Understanding the Difference Between MAP Estimation and ML Estimation

There are a number of possible criteria to use in making decisions. Can someone elaborate on the difference between ML and MAP for a sequence of BPSK symbols impaired by Gaussian noise ?
Abby_DSP's user avatar
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1 answer
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Estimate Variance of Additive White Gaussian Noise (AWGN) Given Multiple Realizations with Different Mean

Given $ N $ images of the same scene, where each image is corrupted by additive white Gaussian noise of the same variance $ {\sigma}^{2} $. How can $ {\sigma}^{2} $ be estimated? So we have an Image $...
zacmar's user avatar
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4 votes
1 answer
389 views

Equivalence of Maximum Likelihood (ML) and Discrete Fourier Transfrom (DFT) Peak Finding for Single Tone Estimation

My understanding is Maximum Likelihood and DFT Peak Finding for a single tone produce the same results assuming the ML is restricted to the same frequencies as the DFT. I was wondering if there was ...
FourierFlux's user avatar
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1 answer
2k views

Maximum Likelihood Detection of Signal Vectors in Gaussian Noise

Consider a binary-input additive white Gaussian noise channel. Let $\mathbf{x}_0 = (\sqrt{Es},\sqrt{E_s},⋯,\sqrt{E_s})$ and $\mathbf{x}_1 = (-\sqrt{E_s},-\sqrt{E_s},⋯,-\sqrt{E_s})$, be two codewords ...
electronic component's user avatar
3 votes
1 answer
243 views

Why Isn't the ML Estimator (MLE) in MIMO Spatial Multiplexing Obtained by the Least Squares Solution?

In the simplest scenario of MIMO spatial multiplexing: $$\mathbf{y} = H\mathbf{s} + \mathbf{n}$$ where: $\mathbf{s}=[s_0,s_1,...s_{M-1}] \\\mathbf{y}=[y_0,y_1,...y_{N-1}]$ $\mathbf{n}=[n_0,n_1,......
user3921's user avatar
  • 267
1 vote
2 answers
264 views

MLE parameter estimation -- confusion regarding some terms in the pdf of complex normal r.v (Part 2)

This question is based on the application of the pdf which was an earlier question of mine asked here Confusion regarding pdf of circularly symmetric complex gaussian rv If $v \sim CN(0,2\sigma^2_v)$ ...
Ria George's user avatar
1 vote
2 answers
85 views

MLE formulation -- confusion regarding the terms in the equation (Part1)

If $v \sim CN(0,2\sigma^2_v)$ is a circularly complex Gaussian random variable which acts as the measurement noise in this model $$y_n = x_n + v_n \tag{1} $$ where $x \sim CN(0,2\sigma^2)$, then is ...
Ria George's user avatar
0 votes
2 answers
579 views

Optimum Filter Signal Detection for Non AWGN Channels

I have been reading this question and it confirms that the matched filter is the maximum-likelihood receiver in the presence of additive white Gaussian noise. So in the AWGN channel it maximizes the ...
VMMF's user avatar
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3 votes
1 answer
1k views

Cramer Rao Lower Bound for Cross Correlation (Time Shift Estimation)

UPDATE 2 Okay, I think I understand now why the defined CRLB is not applying to my use case. In my use case, we have very high SNR, and the the first signal $x_1$ is always the same. So the classical ...
Luca Martini's user avatar
1 vote
1 answer
136 views

Notations to Use in Formulating of Maximum Likelihood Estimation

The received noisy signal $y_n \in \mathbb{R}$ is expressed as: \begin{align} y_n = \mathbf{h}^\mathsf{T}\mathbf{u}_n + w_n. \tag{1} \end{align} $\mathbf{h} = [h_0,h_1,\ldots,h_{p-1}]^\mathsf{T} \in \...
Srishti M's user avatar
  • 616
4 votes
2 answers
182 views

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

In many research articles the performance of an estimation method is compared to that of the ML estimation performance. If the performance of the method does not achieve the ML estimation performance, ...
Srishti M's user avatar
  • 616
0 votes
1 answer
562 views

Maximum likelihood estimator of active time delay and passive time delay

A typical time delay estimation problem has the model: $$ \begin{align} x_1(t) &= s(t)+ n_1(t) \\ x_2(t) &= a s(t-D) + n_2(t) \end{align} $$ Where $n_1$ and $n_2$ are considered to be ...
Akash Maity's user avatar
0 votes
1 answer
92 views

Question about central spatial moments

Spatial moment $$ m_{pq} = \sum_{(x, y)\ \in \mathbb R} x^p y^q I(x, y) $$ The order of each moment is $p+q$. Central moment $$ \mu_{pq} = \sum_{(x, y)\ \in \mathbb R} (x - \bar x)^p (x - \...
Zhetao Zhuang's user avatar
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1 answer
1k views

How to Implement and Minimize Maximum Likelihood Expression in MATLAB

I need to check if the estimation algorithm has converged or not. I am using the Maximum Likelihood estimation method. For convergence check, we see if the log-likelihood has reached its maximum value ...
SKM's user avatar
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7 votes
3 answers
940 views

Maximum Likelihood for Colored Noise

I have the following question about the maximum likelihood (ML) in presence of inter-symbol interference and colored noise. Assume the communication system is as follows. Information source, ...
Noor's user avatar
  • 337
0 votes
1 answer
937 views

Matlab: Help in understanding if the example for Maximum Likelihood Equalizer is properly functioning

I am stuck in the part where I need to apply the MLSE Equalizer with Viterbi code. The equalizer is an optimal. I am using the Communication Toolbox http://www.mathworks.com/help/comm/ref/comm....
Ria George's user avatar
1 vote
1 answer
802 views

What Do We Expect Likelihood Function to Be Used For?

In my understanding, I let make a example. For example, you can see in the following picture. Consequently, We want to find ^sigma. and We have already known the observation data(which is random ...
gmotree's user avatar
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3 votes
1 answer
288 views

Maximum Likelihood Through a Noisy Channel

I have random variables $X_1, X_2, \cdots, X_m $, which can take $n$ values and is distributed iid according to $\Theta=(\theta_1, \theta_2, \cdots, \theta_n)$. That is $X_k$ can take values $\{1,2,\...
Devil's user avatar
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7 votes
1 answer
2k views

Maximum Likelihood Estimation in Presence of Colored Noise

I am trying to test system identification in presence of measurement noise (1) A white Gaussian noise (2) Colored noise - pink, violet. When we are estimating parameters we do so in presence of iid, ...
Ria George's user avatar
0 votes
1 answer
260 views

Maximum Likelihood Derivation

as you can see on page 314 the derivation of the ln[f etc.] equation is done via the matched filter that i have also provided below. i would like to know how this equation is used to derive the ln[f ...
user3313661's user avatar
3 votes
2 answers
407 views

Minimum SNR Requirements for Maximum Likelihood Frequency Estimation

In certain applications, you have enough SNR available to, for example, perform an FFT and identify peak location and hence the signal frequency. If my understanding is correct, parameter estimation ...
user4673's user avatar
  • 325
5 votes
2 answers
3k views

Differences Using Maximum Likelihood or Maximum a Posteriori for Deconvolution / Deblur

Are there any differences if you use Maximum Likelihood or Maximum a Posteriori to estimate the Point Spread Function for image deconvolution?
Marka's user avatar
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