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Questions tagged [least-squares]

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4answers
250 views

Sequential Form of the Least Squares 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 ...
1
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1answer
70 views

Equivalence of ML and 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 ...
2
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1answer
444 views

What Is the Difference between RLS, LMS and Wiener Filter? When Is One Preferred Over Another?

I'm dealing with a channel equalization problem where the channel is modeled as a WSS process. I understand LMS utilises a Wiener-like approach, ie it converges to the optimal (wiener) solution. I ...
0
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1answer
74 views

Equalizers with low data rate systems

In case of using equalizers with ZigBee systems, what type of fading does the channel show, flat or frequency selective, fast or slow? Is the inter-symbol interference (ISI) problem encountered in ...
0
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1answer
226 views

Least Squares with Non Zero Mean Noise

What happens if the noise has no zero mean? I mean, if the exercise is something like: $$y(k) = A x + \eta(k)$$ When I have zero mean, I start from: $$y = A x$$ $$\Rightarrow \hat{y} = A \hat{x}$$ ...
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0answers
23 views

Non Linear Recursive/Online Least Squares

I am searching for a recursive or online non linear least squares algorithm. I want to spread the computation out as new data is sampled like in the linear Recursive Least Squares or the LMS. ...
0
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0answers
20 views

Can coefficients of a DWT be computed using least squares approximation?

I assume that it can be computed given function values for each sample as a vector and wavelet values for each scale and translation packed in a matrix. Can anyone clarify the matrix construction and ...
-1
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1answer
120 views

Properties of Optimization Techniques in Filter Design

I have this design: Can you tell me about the type of filter and the algorithm just viewing this design, or should there be other information?
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2answers
105 views

Solving a Linear Mean-Square Estimation the easy way

I have an exercise which is quite trivial. However I got stuck and I'm not sure if this the end-result. I assume there has to be a way to get this result much quicker. Given are two randomly ...
1
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1answer
96 views

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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0answers
35 views

Linear difference equation and method of Least Squares

I'm reading the book "Fault-Diagnosis Systems" by Isermann in the par. 9.2.1a. The author explains how to estimate the parameter of a linear difference equation using Least Squares. We start with a ...
2
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1answer
285 views

How to Use the DFT (FFT) to Solve a Least Squares Regularization Problem (Inverse Problem)?

Let $X$ and $K$ be an image and a Point Spread Function (PSF), respectively. The blur image $B$ is obtained as follows $$B = X * K$$ I want to solve the following general regularization problem $$...
0
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1answer
28 views

Defining Model in Kalman for Getting Position Using

I am stuck at modeling a system model, i.e. getting my state vector and input vector for navigating just using navaid and ins (tactical). My guess is that position is my only state vector and INS ...
1
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1answer
80 views

mpiir_l2(): Results are not IIR but FIR filters - why and how to avoid this?

I'm using mpiir_l2() from user Matt L's PhD thesis to design IIR filters. I set the number of numerator and denominator coefficients both to the same value (between ...
1
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1answer
56 views

Adaptive LMS Algorithim MATLAB

I'm having some trouble implementing my LMS Adaptive Filter in MATLAB to seperate wideband and narrowband signals from a voice signal. I'm using a delayed version of my input as a reference as well ...
2
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2answers
235 views

Doubt on Weighted Least Square Estimation

This is a page from the book linear algebra,geodesy and gps by Gilbert Strang.... the page explains about the justification of the inverse of the of the co variance matrix of measurement vector $b$ in ...
6
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2answers
697 views

Zero Phase Filter: Determining Initial Conditions for Forward Backward Filtering

Is anybody familiar with Gustafson's algorithm for minimizing transients in forward backward filtering [1]? I'm trying to implement it and my first guess was to check Matlab's filtfilt.m, since they ...
0
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1answer
50 views

Regularized Least Squares by Laplacian Operator - Optimal Value of the Regularization Factor (Lagrangian Multiplier)

Consider the cost function $$f(X,\lambda) = \|AX-b\|_2^2 + \alpha \|LX\|_2^2$$ $A:$Measurement matrix($R_{m\times n}$,$m \ll n$), $b:$observation vector($R_m$), $L:$Laplacian operator($R_{n \times n}...
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2answers
335 views

Why Is Non Linear Least Squares Method from MATLAB and Alglib Gives Different Results on the Same Data?

i'm trying to rewrite my Matalab prototype for some DSP to C++ and encountering a displeasing problem. I'm trying to fit data to a function $y = a * (\pi / 2 + arctg(b * x))$. In Matlab it works well ...
0
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1answer
145 views

Recursive Least Square Adaptive Linear Equalizer

For the adaptive filter to work properly, a desired signal d(n) needs to be provided. The output from the equalizer y(n) is subtracted from d(n) to produce an error signal, which is used to adjust the ...
0
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1answer
239 views

Design of FIR filters with arbitrary magnitude and phase responses

I found many information in this thesis "Algorithms for the Constrained Design of Digital Filters with Arbitrary Magnitude and Phase Responses",i want to understand it. This question is a ...
1
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3answers
130 views

Complex Least Squares Approximation

In the case of frequency domain FIR filter design, the error function given by : $$E(\omega)=H(e^{j\omega})-D(e^{j\omega}) \tag{1}$$ is a linear function with respect to the unknown filter ...
0
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1answer
55 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,......
15
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1answer
1k views

What Does Make an Error Surface Convex? Is It Determined by the Covarinace Matrix or the Hessian?

I am currently learning about least-squares (and other) estimations for regression, and from what I am also reading in some adaptive algorithm literatures, often times the phrase "... and since the ...
1
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2answers
285 views

Conceptual Question on Least Squares Estimation Method

What is meant by Least square plus Gaussian method of estimating the parameters of an unknown system? I am aware of least squares in general but some works refer it as least square+gaussian method. ...
3
votes
1answer
195 views

Least Angle Regression (LARS) without Matrix Inversion

Sorry if this is too damned long. I did what I could to abbreviate it. The question is about Least Angle Regression (LARS). I'm new to numerical work with matrices. I believe I have a way to ...
3
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1answer
1k views

Python: Least Squares Support Vector Machine (LS-SVM)

I'm looking for a Python package for a LS-SVM or a way to tune a normal SVM from scikit-learn to a Least-Squares Support Vector Machine for a classification problem. The goal of a SVM is to maximize ...
1
vote
1answer
96 views

Obtaining Correct (Least Squares Sense) Affine Transform Parameters Between Two Images

I have two images that I want to compute the affine motion model parameters. The model that I use is $$x' = a_1x+a_2y+a_3$$ $$y' = a_4x+a_5y+a_6$$ To calculate those 6 parameters, I picked 6 points (...
1
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2answers
270 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 ...
4
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6answers
336 views

The Least Norm Solution of Under Determined Linear System

Suppose I have a matrix $$A= \begin{pmatrix} 1 & 0 & 1 & 0\\ 0 & 1 & 1& 0\\ \end{pmatrix} $$ where the variables are channel information like assume $X_1$, $X_2$, $X_3$ ...
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1answer
632 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.
5
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1answer
3k views

What's the Difference Between LMS and Gradient Descent Adaptation?

I found algorithms that seems the same to me, but they are described with different names (in field of adaptive filtering). For example: LMS - least-mean-squares seems to be GD - stochastic gradient ...
0
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2answers
121 views

Estimation of Input Signal to Obtain the Desired Output Signal for an Unknown Filter

Suppose $h(n)$ is a finite impulse response which is unknown. We can feed any input signal $x(n)$ into the system and observe the corresponding output signal $y(n)$. From this, is it possible to ...
1
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2answers
56 views

What is the definition of Linear Predictive Coefficients when the optimal value is not unique?

Linear predictive coefficients of signal y are defined as the best $k$ coefficients $a_i, i = 1, \ldots, k$, that will approximate $y_n$ by $-\sum_{i=1}^k{a_iy_{n-i}}$. (Best approximation is that ...
0
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2answers
91 views

Solving LASSO ($ {L}_{1} $ Regularized Least Squares) with Gradient Descent

To the best of my knowledge, state of the art methods for optimizing the LASSO objective function include the LARS algorithm and proximal gradient methods. I was wondering however, if the LASSO ...
1
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1answer
214 views

Least Squares Linear Phase FIR Filter Design

In explication ''the geometric interpretation of least squares'' Typically, the number of frequency constraints is much greater than the number of design variables (filter coefficients). In these ...
3
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0answers
188 views

Least Squares Approximation for FIR Filter Design

Following this paper , I am trying to make a least-squares algorithm in MATLAB, but for type I (I know about firls()). ...
3
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1answer
2k views

Use MATLAB to Restore a Signal from a Given Degraded Signal Using Tikhonov Regularization

Anyone could share how to develop an application in MATLAB to restore the signal from a given degraded signal using Tikhonov regularization i.e restoring the signal $f$ via solving $$ \min || g - f ...
2
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2answers
129 views

Finding the Best Gaussian Smoothing Kernel to Minimize the Discrepancy Between Two Images

Suppose we have two grayscale images, $A$ and $B$. $A$ and $B$ very strongly resemble each other, such that the mean of the absolute difference $\lvert A - B\rvert$ is fairly small. Suppose further ...
8
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2answers
1k views

Is the Kalman Filter a Best Linear Unbiased Estimator (BLUE) for Heteroscedastic Noise?

According to the Gauss-Markov Theorem, a ordinary least squares estimator is BLUE if the noise entering a system is uncorrelated with zero mean and is homoscedastic (has a constant finite variance). I ...
0
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1answer
98 views

Received signal error vs. BER

It is my understanding that the least squares algorithm (e.g., in equalization) minimizes the received signal error. However, minimizing the received signal error does not necessarily equate to ...
22
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3answers
3k views

FIR Filter design: Window vs Parks-McClellan and Least-Squares

Are there any advantages to use a window approach over Parks-McClellan (further abbreviated here as PMcC) or Least Squares algorithms for FIR filter design of a low pass filter? Assume with today's ...
1
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1answer
145 views

RLS Algorithm (Memoryless)

I have been studying the adaptive filters lately and now when I am at RLS (Recursive Least Squar) Algorithm I came across the term used in the weighting function of the RLS called forgetting factor ($\...
0
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1answer
244 views

Estimate a transfer function from ARX models - Is ARIMAX better?

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 ...
5
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1answer
245 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
63 views

Solve for transfer function coefficients embedded in a non-linear system

Given a complex input signal $x$ and real input signal $v$, a 4th order (for simplicity) transfer function $H(z)$ is first applied to $v$ to obtain $w$, which in the time domain is represented by the ...
0
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1answer
267 views

Difference between Leaking factor and Forgetting factor

I am using a Recursive least square adaptive filter to process electromyography signals and it is working decently so far. I decided to implement an LMS adaptive filter as a noise cancellation, so ...
1
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0answers
35 views

Digital Filter Design Accuracy [duplicate]

Having only dealt with digital filter scarcely, the question dawned to me when I used the firls function in matlab to design an equalizer with a certain gain response. In general, can we prescribe ...
0
votes
1answer
60 views

1/f^2 weighting for least-squares FIR

I am currently modifying the firls.m from Octave to allow all types of FIRs, differentiators, and Hilbert transformers, while taking a peak at Matlab's. I see there ...
1
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2answers
97 views

Matlab : Stuck in implementing Least Squares estimator for this model

The formula to estimate $\mathbf{h}$ is then $$\hat{\mathbf{h}} = (X^T X)^{-1} X^T \vec{y}\tag{2}$$ I think this can be implemented in Matlab using ...