Questions tagged [parameter-estimation]

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How do I prove or show that adding an extra time series channel provides an additional information about the parameter?

I am trying to understand how adding an additional channel of time series helps/hurts an estimation problem. For example, let's say we have a single channel time series, and we are trying to estimate ...
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8 votes
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MUSIC Algorithm for Direction of Arrival (DOA) in Acoustic Signals

I have seen multiple reviews of the MUSIC algorithm for the estimation of DOA. Most of those reviews consider a complex signal input with a complex steering vector. So, how do I implement this ...
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7 votes
2 answers
557 views

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

Consider we have to detect a known signal added with Gaussian noise. In this scenario, the matched filter is known to be an optimal filter for SNR. The question: is there any machine learning ...
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When is it necessary to use a Kalman Filter, and not a simple estimation method?

I know its a stupid question, but I got understand this very fundemental point. Say we have a sinousodial signal, which we want to extract from a noisy (known variance and mean) measurement. It is ...
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How to determine the CRLB for FFT based estimator with MATLAB?

I am trying to determine the CRLB for FFT-based frequency estimation in MATLAB. For this, I am simulating a single sinusoid in white noise with different amplitudes and constant noise power. Here is ...
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2 answers
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Algorithm for finding best matching sin wave to input signal

Question Suppose I take a signal $y(t) = \cos(2\pi \cdot 4.1t + 2)$ and I sample it uniformly up to 2 seconds. Given only these time samples, how would I design an algorithm that finds the pair of ...
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1 answer
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converting complex sum of exponential signal to real signal

I have the following Signal model, that generates a discrete-time complex signal, where $a_k$ - amplitudes, $\phi_k$ - phases, $\alpha_k$ - damping factors and $f_k$ - frequencies are the parameters ...
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5 answers
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Simple and Effective Method to Estimate the Frequency of a Single Sine Signal in White Noise

Given a sinusoidal signal, how can we efficiently determine its frequency?
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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 ...
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Transfer function estimation from frequency response

Let's assume that we know that we are dealing with a SISO second order system for which we have the frequency response (magnitude and phase for a known frequency range ω). What methods would people ...
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2 answers
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A Fast Power Grid Frequency Estimation Approach Using Frequency Shift Filtering

I have read the paper A Fast Power Grid Frequency Estimation Approach Using Frequency Shift Filtering. I want to prove it using a numerical example, I have problem how to calculate the ...
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11 votes
2 answers
143 views

Does the Wiener Filter Achieve the Cramer Rao Lower Bound (CRLB)?

I have been told (Wikipedia agrees) that the Wiener filter is optimal when signal and (additive) noise are WSS. Optimal in the sense that it minimizes the mean-square error. The Cramér–Rao bound is ...
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Parameter tracking using Augmented state vector approach and unscented Kalman filter

I'm trying to reproduce and extend figure 9 results in Nonlinear dynamical system identification from uncertain and indirect measurements"HU Voss, J Timmer, J Kurths - International Journal of ...
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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 ...
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7 votes
2 answers
121 views

Determine the Signal Curve from Parameters of a Power Curve by Noisy Measurement

I have a class of signals described by function: $$ f(inc,d,t)=inc\cdot t^d $$ where inc and d have a finite set of values like 1, 2, 3, i.e. $$ inc, d\in \left \{1,2,3 \right \} $$ and $$ 0\leq t<...
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Estimate Parameters of Linear Combination of Harmonic Signals with Partial Known Frequencies

Consider real signal of the form: $$ \begin{align*} x(n) = &\sum_{m=1}^{M} \left( c_m \cos(2\pi f_m n) + s_m \sin(2\pi f_m n) \right) =\\ &\sum_{m=1}^{M} a_m \cos(2\pi f_m n + \phi_m) \end{...
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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 ...
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Is my implementation of Generlized Pencil-of-Function correct?

I have a time-series data that should be of the form $$ f(t) = \sum_nA_n(e^{\alpha_n t}+e^{-\alpha_n(T-t)});\;\alpha_n=\eta_n+i\omega_n;\;i=\sqrt{-1} $$ and I want to find parameters $A_n$, $\eta_n$, ...
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How can I calculate the height at which the mobile phone is located?

I need to calculate Hm - the height at which the mobile phone is located. For calculating I can use: Hb - the BTS height, m L - the rssi-parameter, dB, f - frequency, MHz. When I use an Okuruma-...
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1 vote
1 answer
210 views

Blind Estimation of Signal Parameter and Noise Variance

Let $y[n]= h*x[n] + w[n]$, where $h$ is an unknown but deterministic parameter, $x[n]$ is a BPSK random variable with equal probability of +1 and -1, $w[n]$ are i.i.d. Gaussian with zero mean and ...
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3 answers
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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 ?
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4 votes
1 answer
104 views

Damped spring mass system - parameter estimation

I've tried to calculate the parameters of a damped spring mass system of the form $m~ y''(t)+d~y'(t)+c~y(t)=F(t)$ but I have some problems determining the mass m of the system. The damped ...
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1 vote
1 answer
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Neural networks in system identification - What type of activation functions?

I made a free software for all operative systems, even Android. It's called Deeplearning2C. It can train a neural network and generate C code and MATLAB-code. C-code for embedded systems and MATLAB-...
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6 votes
1 answer
550 views

How to Linearly Combine Two Unbiased Estimators of One Parameter without Knowledge of Their CoVariance?

I have two unbiased estimators of one parameter, $\tau$. The first estimator, $r_1$, is the better estimator with lower variance than the second estimator, $r_2$. I also have: $ \mathbb{E} \left[ {r}...
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5 votes
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 $...
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5 votes
2 answers
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Dominant Frequency Peak Decreases with Increasing Window Size

I have a signal that looks like this. I analyse it using fast Fourier transforms to identify the frequency with the largest peak, which is always close to zero. (There are no other clear peaks.) If I ...
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1 vote
1 answer
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Why is the concept of a "state covariance matrix" necessary in estimation?

I'm currently taking a course in optimal estimation (and it's still very early in the course). Much of our work is based around the idea of a measurement model $y=Hx + v$ This model assumes our ...
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2 votes
2 answers
462 views

Tracking a sine wave with noise

This will probably be an extremely simple question for some one with any background in signal processing(not my background) Let say I have signal $$x(t)=A\sin(\omega t)$$ where A is known and $\omega$...
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How sensitive is parameter estimation to uncertainty in time?

Suppose I have the following deterministic system that is a function of time: $y = k*t + b$ Now let's say I have the ability to measure this system but there is a zero-mean noise component in the ...
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1 answer
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Does there exist a simple regression method to fit a single sinusoid period to a data set?

I'm currently working on a project where I want to fit a single sinusoid period to a data set. Essentially I have very good control/knowledge of the signal's dominant frequency, so I'm only sampling ...
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1 vote
0 answers
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Complex Spectral Phase Evolution (CSPE) Performance depending on signal windowing?

I am look into CSPE. "Signal Analysis Using the Complex Spectral Phase Evolution (CSPE) Method" The method is simple. It compares the original signal's FFT and shifted signal FFT in phase domain so ...
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1 vote
1 answer
267 views

Cramer-Rao Lower Bound

In estimation problems, we may use Cramer-Rao Lower Bound (CRLB) to evaluate the best performance. But if there is no unbiased estimator can attain CRLB, what is the meaning of CRLB? To clarify the ...
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1 answer
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Demonstrate the mean of the sample variance

Let's suposse I have a vector of elements $x(n) = \{x(0), x(2), \cdots ,x(N-1)\}$ from a random process X of mean $\mu_x$ and variance $\sigma_x^2$. I want to see if I can estimate the mean and ...
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7 votes
4 answers
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Sequential Form of the Least Squares Estimator for Linear Least Squares Model

I'm currently working on a project in which I need to find the tilt of a surface. Let's assume I'm only concerned with a single dimension tilt (i.e. slope) to begin. I currently have the ability to ...
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2 votes
0 answers
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How to calculate the Fourier Transform of a solvable chaos waveform?

Recently I am stucking in frequency estimation of a solvable chaos waveform. Its local analytic expression in time domain is $$ z(t)=s_m(u_m-s_m)e^{\beta(t-mT)}\cos(\omega_0 t+\varphi),mT\leq t<(m+...
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4 votes
1 answer
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Sequential Non Linear Least Squares Problem

I have the the following non-linear function, $$s(x;A_k,\mu_k,\sigma_k)=\sum_{k=1}^2 A_k \exp\left(\frac{-(x-\mu_k)^2}{\sigma_k^2}\right)$$ with unknown (but deterministic) parameters $A_k,\mu_k,\...
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1 vote
2 answers
3k views

Least Squares Fitting to Inverse Exponential Function

I have a time series of measurements that resembles the shape of an exponential function. The samples are a bit noisy and sometimes there is a weak sine like ripple signal ontop of it. Simplified the ...
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1 vote
0 answers
251 views

Parseval's Theorm and Effective Bandwidth

This question says that RMS bandwidth (effective bandwidth) is defined based on the carrier frequency of a signal. This makes intuitive sense to me that the carrier frequency shouldn't determine the ...
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1 vote
2 answers
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Is it possible to estimate variance of noise for a step answer signal?

I know there is not possible to find the true noise of a measured signal. The only way to "find" the noise is to estimate the noise. Noise has the mean 0, but the variance varies. So assume that we ...
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1 vote
2 answers
302 views

MMSE - How to minimize a complex error with respect to a set of real parameters

Suppose there's a complex signal $X(k)$ (where $k \in \{0, 1, 2,...,N - 1\}$) corrupted by additive complex noise. Its estimate $\hat{X}(k)$ is a linear combination of a set of real parameters $A_r$ ($...
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8 votes
4 answers
522 views

MMSE Estimation - Fusion of 2 Measurements

Let's say I have 2 measurements of the same phenomenon (for example current temperature) and I want to find the MMSE (minimum mean square error) estimator, i.e to minimize the MSE (mean square error). ...
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1 vote
1 answer
157 views

Period of a quasiperiodic digital signal

I have a signal made of a perturbed square wave, sampled so that there are at least six samples per period, for a total of 15 to 50 periods. The sampling frequency and the signal frequency are ...
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5 votes
2 answers
151 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, ...
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1 answer
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Estimate a Transfer Function from ARX Models vs. ARIMAX?

There is diffrent models which can be used to create a dynamical model by using least squares. Those models are following: ARX ARMAX ARIMAX OE BJ But if my goal with creating a dynamical model is to ...
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0 answers
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Parameter estimation of mixture of damped sinusoids

What are the most known methods (algorithms) for the parameter estimation of the superimposed (mixture) of damped (complex) sinusoids? The noiseless signal model (with $P$ sinusoids) is as follows: $...
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3 votes
1 answer
1k views

What is meant by optimal estimator and how to determine optimality?

Considering an estimation problem of estimating a scalar deterministic parameter $a$ from the observations $y$ which are corrupted by randomvariable $w$. The observations are $y[n] = a + w[n]$ Least ...
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5 votes
1 answer
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Maximum Likelihood Estimator (MLE), MMSE and LS - Are All of Them Regressor, Estimator and Predictor?

Can all three criteria ML, MMSE, and LS be called regressor, estimator, and predictor ? If not, an intuitive explanation of why they can't be, would be good.
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1 vote
1 answer
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How to Apply Least Squares Estimation for Sparse Coefficient Estimation?

The model is expressed as, $$y(n) = \sum_{i=0}^{p-1} r(i) x(n-i) + v(n) \tag{1}$$ where $\mathbf{r} = [r_1,r_2,\ldots,r_p]^T$ is the sparse channel coefficients of length $p$, $\mathbf{x} = [x_1,x_2,....
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0 votes
1 answer
487 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 ...
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0 votes
2 answers
107 views

Help in proper notations and mathematical formulation

For a given series, $S = \{-1,0,-2,1,etc\}$. The number of elements in $S$ is $N = 100$. Each $s_i$ belongs to a alphabet from a finite alphabet set $\mathcal{A} = (a_1=0, a_2=1, a_3=2, a_4=3, a_5=-1,...
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