Questions tagged [estimation]

In signal processing, estimation is a technique for approximating an unobserved signal from an observed signal containing noise.

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615 views

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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4answers
227 views

Estimate Filter Coefficients from the Result of Linear Convolution with a Known Signal

If I have samples of input say x(1:500) and it passes through FIR filter with 9 taps and some unknown coefficients. The output y(1:508) is also known. The aim is to estimate the filter response in ...
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1answer
621 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 ...
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0answers
96 views

Finding the parameters and endpoints of a sinusoid

Suppose I have some signal $$ s(t) = n(t) + \left\{\begin{aligned} &0 &&: t < t_0\\ &A e^{i (2 \pi f t + \theta_0)} &&: t_0 \le t \le t_1\\ &0 &&: t > t_1 \...
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3answers
564 views

error variance of frequency and phase estimation using DFT

What is the error variance of estimating the (frequency) and (initial phase) of a sinusoid using DFT. please provide references if possible.
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2answers
90 views

Why use parametric based estimation methods - confusion regarding terms

Using the probability density function (pdf) we can estimate an unknown parameter using methods such as Maximum Likelihood estimation. If the pdf is not available, then Least Squares can be used. ...
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2answers
68 views

Is there any information we can extract from the residual

The residual in estimation is the difference between the measurement and the previous estimate. My question is when I plot this quantity, what is ,if any, information I can extract or infer from this ...
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2answers
228 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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1answer
115 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 \...
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4answers
388 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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2answers
106 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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1answer
55 views

calculating estimate of a state for a system with two observations of the state from different times

I'm struggling with a problem that I just can't seem to get a grasp of. I'm supposed to calculate an estimate for state $x(k)$ at times $k=1,2,3$ from the state-space $ \begin{align} x(k+1) &= A ...
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1answer
351 views

Direct Correlation (DC) Time Delay Estimation: Variance Keeps Decreasing for Increasing SNR?

I'm trying to reproduce the results from this paper "Discrete time techniques for Time Delay Estimation" doi:10.1109/78.193195, for both the correlation and Least Squares. I've generated a random (...
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0answers
196 views

Deterministic method to compute “Process noise covariance matrix, Q” for a Kalman filter when parameter variations of the model is known apriori

I am implementing a Kalman filter (for a linear ODE system for now). My model represents a physical device that has 6 "parameters", i.e. those values of the device do not evolve over time (within a ...
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1answer
33 views

Can you explain likelihood function in the equation?

In the paper here, just before equation 12, authors introduce likelihood function $\pi(r|\theta)$. What does $\theta$ represent here ?. What is the intuitive explanation behind equation 12?
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1answer
456 views

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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1answer
383 views

General questions on Kalman filter and difference

In the wikipedia Kalman filter link, the state variable $x_k$ takes a continuous value say a floating point number, but what if the values are integer say symbols from an alphabet set, then how does ...
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83 views

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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1answer
968 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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1answer
60 views

Is there a rule of thumb for selecting the variance of the input?

Considering an estimation problem, where I want to estimate the unknown input $x$ using the known channel parameters. This estimation problem can be solved by Least Squares. Thus, the model is $$\hat{...
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1answer
290 views

confidence interval estimate for coherence

Is there any solution for estimating Confidence interval of coherence derived by Welch method with overlapping segments? Moreover, the window length is not same for all the observations, only the ...
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1answer
342 views

How does this FLL work?

I'm trying to get my head around how band-edge frequency locked loops work. In particular, a diagram from this paper: Specifically, what does the Tanh() function and the conjugate product of the two ...
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1answer
1k views

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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135 views

Unbiased estimation of square in presence of Gaussian noise

I have a single scalar observation, $y$, of a scalar parameter $x$ in presence of additive Gaussian noise. That is $y = x + n$ where $n$ is the Gaussian noise. The variance $n$ is known to be $\sigma^...
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1answer
273 views

Is Savitzky-Golay filter a special case of an FIR lowpass filter?

The Savitzky-Golay filter is explained here. It looks to me that any Savitzky-Golay filter can equally be considered an FIR filter. Am I right about that?
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1answer
962 views

Estimate transfer function from Bode curve

I have measured the magnitude response Y_mag(f) and phase response Y_phase(f) of an unknown physical system. Is it possible to ...
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1answer
535 views

Reconstructing Signal From Its Cyclic Autocorrelation

Can a signal be reconstructed from its cyclic autocorrelation? Specifically, if we know $$ R^{\alpha}(\tau) = \int{x(t)x^{\ast}(t-\tau)e^{-j2\pi\alpha t}\mathrm{d}t}, $$ can we reconstruct $x(t)\in\...
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1answer
5k views

What would be the variance for complex number?

When $x$ is a zero mean random variable then $$\sum_{n=1}^N x_n x_n^T = N \sigma^2_x\,\text,$$ where the variance is $\sigma^2_x$. I'm considering Complex Normal Distributions where the real and ...
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1answer
183 views

Time series forecasting on dsp embedded systems

I have a time series analysis algorithm that uses arima modeling and fits a model to the time series and gets the residual values. It is more of a statistical signal processing algorithm than a ...
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1answer
454 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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1answer
245 views

What method to use for interpolation and extrapolation?

Below are discrete samples { t1, f(t1) }, { t2, f(t2) }, ... ,{ tn, f(tn) } using Mathematica syntax. {{7.0,0.354887404925574},{7.3,0.4003399403324751}, {7.6,0.5849632195845474},{7.9,0....
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1answer
124 views

Quantifying goodness of amplitude estimation

Assume you have a single sinusoid in bandlimited Gaussian noise with unknown amplitude $A$, known frequency $f_0$, and known noise spectral density $S(f)$ in $\frac{\mathrm{units}^2}{\mathrm{Hz}}$: $$...
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1answer
123 views

Histogram estimation (from sub-set of image pixels)

Problem: How to find gray-scale image histogram without iterating all pixels and binning them. Description: Suppose that I have large set of images with no common scene and of different resolutions. ...
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1answer
585 views

Estimates sub-pixel shift directly in phase region

I try to estimates shift estimation directly in phase region, by following the proposed method in this Sub-pixel Shift Estimation of Image based on the Least Squares Approximation in Phase Region by ...
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0answers
46 views

Get slopes of integer shift of an image in radians

I tried to reproduce the result from this paper, However I'm stuck at the Section III of the paper which involve integer shift. Two images which their phase correlation $\theta(k_1,k_2)$ was ...
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1answer
117 views

Correct method for up-sampling cross-power spectrum for sub-pixel motion estimation

Let's say I have two images, one reference($g$) and another shifted($f$). I tried to get subpixel accuracy of the motion. Here are my steps: Firstly I would calculate the normalized cross-power ...
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1answer
231 views

Help in understanding from book expression of variance of an estimator : PRBS vs real valued input

The question is based on Book : Fundamentals of Statistical Signal Processing by Steven Kay, Chapter 4 : Eq(4.21). The expression for the variance of the estimated coefficients when the input is PRN ...
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1answer
2k views

xcorr in MATLAB for periodic function

I have a periodic signal and I want to find it's autocorrelation function. I can calculate it exactly: $$R_{uu}(h) = \frac 1M \sum_{k=0}^{M-1} u(k)\cdot u(k-h)$$ But will ...
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1answer
339 views

Implication of Cramer Rao Lower bound and relation with SNR : non-technical overview

I need some clarifications on CRLB which has been bugging me for quite some time. Clarification: When computing the Fisher Information(FI), the mathematical expression contains $\theta$ and not $\...
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1answer
5k views

Tracking a Sine Wave with Kalman Filter - How to Account for Offset (DC Signal)?

I am attempting to create a Kalman filter to track a sine wave (I am using a linear Kalman filter example assuming I already know the frequency of the sine wave) - the example I am using is derived on ...
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1answer
75 views

Why is being jointly WSS important in signal estimation with LMMSE estimator?

Assume that we have $y_k = s_k + n_k $. We have observed $y_k$ and want to estimate $s_k$. The goal is to use LMMSE (Linear MMSE) estimator to find $y_k$ and on the other hand, we know that our ...
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1answer
53 views

Notation confusion — What is the correct operator for computation of the log-likelihood expression for complex valued data?

This question is an extension of another question of mine asked earlier here Help in understanding if the maximum likelhood estimation is working properly In that question the inputs were real valued....
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0answers
296 views

Kalman filer for estimating velocity

The goal is to use a Kalman filter to estimate velocity from noisy position measurements. I am attempting to implement a version of the filter used in the example on the Wiki page Kalman Example for ...
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2answers
212 views

Need help in understanding minimum variance estimator and CRLB concept using an example

I am reading the book, Fundamentals of Statistical Signal processing, Estimation Theory Volume 1 by Steven M. Kay. In Chapter 2, there is a Fig 2.5 which illustrates which estimator to select based ...
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1answer
101 views

Cross Correlation [closed]

I have two signals which I obtained from a State Estimator and the one the measured one "actual signal obtained from the nodes. I need to compare between the two signals how similar they are . or if ...
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1answer
1k views

How to distinguish signal and noise eigenvalues of autocorrelation matrix for MUSIC algorithm?

MUSIC (multiple signal classification) algorithm is introduced in Schmidt, Ralph. "Multiple emitter location and signal parameter estimation." IEEE transactions on antennas and propagation 34.3 (1986):...
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2answers
2k views

What are the differences between estimation and detection optimal filters (matched and Wiener)?

What is the differences between the two of them? I know that the matched filter is a detection filter because it detect what is the peak of the signal and return the time delay of reflected and ...
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1answer
153 views

Question regarding estimation of signal tone parameters (frequency, amplitude, and phase) using Macleods algorithm

I plan to use Macleods algorithm in order to estimate the parameters (frequency, amplitude, and phase) of one or more sinusoids from a block of N uniformly spaced ...
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1answer
1k views

How to estimate the autocorrelation from nonuniformly spaced data

Assume a continues-time random process $X(t)$ sampled nonuniformely in time to acquire discrete signal $x[n]$. The sampling times are known but the autocorrelation is not. Is there an accurate ...
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1answer
999 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 ...

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