# Questions tagged [least-squares]

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### 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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### Difference Between a 1st Order SG Filter And a Least Squares Moving Average

I have been studying SG filters and i recently found another filter which seem to be commonly used in financial data smoothing which is the least-squares moving average, this filter is also called ...
25 views

### How to find Coefficients of Autoregressive (AR) model using least means square (LMS) Algorithm without having future signal?

I want to do two things. Estimating Coefficients of AR model using LMS Using Coefficients found in step 1 and predict future samples of a signal using AR equation. I don't have a desired signal so I ...
101 views

### Range Estimation Based on Distance Differences

It is a range estimation problem, as shown in the figure. The blue circles are the anchor node' positions, which are produced by one moving anchor (the moving direction is shown with black arrow). ...
43 views

### Why Would Pre Filtering Measurement Data Affect the Least Squares Estimate?

In estimating parameters in a discrete time model I've often seen the use of filters applied to the input data, before its applied to least squares processing. I've been told that the filters are ...
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### 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 ...
612 views

### Least Squares and Auto and Cross Correlation

I am trying to understand why auto and cross correlation helps find the best fit line in least squares. I have an equation as stated here: $Ax=b$ -- I have not exact solution, so I use the least ...
90 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 ...
74 views

### Deriving the Matrix Inversion Lemma for RLS Equations vs the Woodbury Derivation

Can any one help me in deriving the matrix inversion lemma rule for RLS algorithm? I don't know how to start with. Many books have just stated but they haven't derived it.
291 views

### 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 ...
42 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 ...
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### Using MATLAB Function 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 ...
262 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 ...
62 views

### What Is the Definition of Linear Predictive Coefficients When the Optimal Value Aren't 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 ...
103 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 ...
251 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 (...
70 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 ...
269 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 ...
296 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 ...
67 views

### $\frac{1}{f^2}$ 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 ...
120 views

### Least Squares FIR for Type II

Following this paper (see the Matlab example at the end), I am trying to make a least-squares algorithm in Octave, but for type II (I know about firls). For type I ...
320 views

### How to Add Group Delay?

I was confused about the value of group delay, I am not sure whether my opinion is correct. For example, if I want to add a group delay of $0.25$ time units to a zero-phase signal, does it mean that I ...
104 views

### MATLAB: Implementing Least Squares Estimator for a Given 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 ...
127 views

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### Constant Modulus Algorithm and the Gradient Operation

CMA is a blind channel equalization algorithm with the details presented above. I am facing 3 difficulties and shall appreciate help Q1: Does $H$ and the bar over $\bar{y_k}$ represent the Transpose ...
289 views

### Motion Artifact Cancellation from ECG Signal Using 3 Axis Accelerometer Data

I have a ECG signal measured as a test subject is performing 3 different activities - sitting, walking(2mph) and jogging(5mph) and also the corresponding 3-axis accelerometer(X,Y and Z) signals. I can ...
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### How to Remove the Periodic Oscillations from a Signal

The task that I have is to remove the annual and semiannual oscillation from a set of temperature measurements, taken over several years, by means of least squares method. I found the method ...
237 views

### Questions 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 ...
160 views

### Least Square Error Estimation: Conditions for $(A^TA)^{ -1} = A^{-1}(A^T)^{-1}$?

I am new to linear algebra and have this simple question... in least sqaure estimation...the best estimation of the equation $Ax = b$ is $x_{Estimated} = A (A^t A )^{-1} A^t b$...the projection of $b$ ...
707 views

### How to Improve Least Means Squares (LMS) / NLMS Filter Performance?

Are there ways to increase computational performance of a normalized least squares (NLMS) filter? Multidelay block frequency-domain (MDF) filters have been proposed to do this, but they also take away ...
110 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 ...
242 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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### 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. ...
79 views

I'm having some trouble implementing my LMS Adaptive Filter in MATLAB to separate wideband and narrowband signals from a voice signal. I'm using a delayed version of my input as a reference as well ...
103 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,\...