12 votes
Accepted

For complex values, why use complex conjugate in convolution?

Turns out that convolution and correlation are closely related. For real signals (and finite energy signals): Convolution: $\qquad y[n] \triangleq h[n]*x[n] = \sum\limits_{m=-\infty}^{\infty} h[n-m] ...
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8 votes
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What's the Difference Between LMS and Gradient Descent Adaptation?

The LMS algorithm is based on the idea of gradient descent to search for the optimal (minimum error) condition, with a cost function equal to the mean squared error at the filter output. However, it ...
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  • 23.7k
8 votes
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Difference between Leaking Factor and Forgetting Factor

The play similar role in those algorithms - the ability to forget the past and adapt to current reality. In the LMS, the classic implementation has $ \alpha = 1 $. Namely the optimal weights at any ...
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  • 41.3k
7 votes

What's the advantage of adaptive IIR filter against FIR?

These are the key differences between FIR and IIR filters, regarding the feature you wish to control are the following: $$ \begin{array}{c|lcr} \text{Feature} & \text{IIR} & \text{FIR} \\ \...
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  • 1,357
7 votes
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Are all least square filters adaptive?

TL;DR: No, they are not necessarily the same. Gory Details Least squares is just an optimization technique. It is used in a variety of ways. For filter design it is used to select that realizable ...
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  • 22.9k
6 votes
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Channel Estimation / Equalization - Estimate Channel Inverse Using White Noise Statistics Only

A Regular Random Process is is the the result of White Noise Going through a Minimum Phase LTI System. A Non Perfectly Predictable Random Process can be defined as (See Wold Theorem / Wold ...
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  • 41.3k
6 votes
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MATLAB : Proper estimation of weights and how to calculate MSE for QPSK signal for Constant Modulus Algorithm

Your code reveals many misconception about what the CMA is supposed to achieve: your step size mu is much too small; note, however, that the optimal step size can ...
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  • 80.4k
6 votes

Adaptive equalizer for IIR channel?

Note that the inverse of an FIR system is IIR, and the same is true for the inverse of an IIR system, unless it is an all-pole system, the inverse of which would be FIR. So in most cases the ideal ...
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  • 80.4k
6 votes

Least Mean Squares (LMS) Filter Weight Update

That really depends on context, but generally adaptive implies that the calculations are done on-line / on the fly. In some applications, the filter is updated for a while, then the adaptation is ...
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  • 22.9k
6 votes

Least Mean Squares (LMS) Filter Weight Update

To expand on what Peter K. has said, if the signals being used by the filter are stationary, then the filter weights or coefficients can be determined and the filter operates as it was designed ...
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6 votes
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Covariance matrix of an adaptive filter input

The Covariance Matrix is commonly defined as $$\mathbf Q = E\left[ (\mathbf x -\mathbf\mu_{x})(\mathbf x -\mathbf\mu_{x})^*\right]$$ with $\mu$ denoting the mean value, i.e. $\mu_{x}=E\left[\mathbf ...
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6 votes

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

All three are Estimators / Predictors. All of them try to estimate the coefficients of Linear Filter which minimizes an MMSE Cost Function. The Wiener filter assumes all data is given and sets the ...
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  • 41.3k
6 votes
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Can I have some feedback on this research paper?

Comments provided here are in two broad categories: Presentation and Subject matter. The "Presentation" section is the easiest to address. There are some things that could be rephrased in terms of ...
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  • 10.1k
6 votes
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Doubts on LMS derivation

Here I expected $y(n)$ is to be computed by convolving $x(n)$ with $h(n)$, but in the equation given by Wikipedia it is shown as a matrix multiplication $y(n) = h^H(n).x(n)$. Are these two ...
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  • 22.9k
6 votes
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Recommended Resources / Literature Search Terms for a Solutions to a Specific Kind of Multi Harmonic Signal Structure

If I understand this problem correctly you have access to 2 signals: Noise Signal - $ w \left[ n \right] $. It is composed of a linear combination of harmonic signals. Something like $ w \left[ n \...
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  • 41.3k
5 votes
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Deriving the Matrix Inversion Lemma for RLS Equations vs the Woodbury Derivation

It is not clear what are you asking but I will try answer both things. Deriving the Matrix Inversion Lemma The Matrix Inversion Lemma goes as: $$ {\left( A + U C V \right)}^{-1} = {A}^{-1} - {A}^{-...
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  • 41.3k
5 votes

Removing low frequencies from a signal

You want to remove the heart beat signal and keep the "noise". We can solve this problem by using a denoising algorithm, and subtracting the denoised signal from the original signal. Setting ...
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  • 4,014
5 votes
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Modelling Unwanted Signal in a LMS Adaptive Filter

The LMS and many of the variants of Adaptive Filters (In the Linear System context) work in the following settings (Intuitive): You have access to 2 signals. One signal is the result of the other one ...
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  • 41.3k
5 votes

For complex values, why use complex conjugate in convolution?

The use of the conjugate in the formation of the adaptive filter isn't necessary. However, if you do not write the output using a conjugate then it is quite easy to forget that the variables you are ...
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  • 2,741
5 votes
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Why can adaptive IIR filters result in unstable solutions?

The IIR filter doesn't have to be unstable, but it has the potential of being so; unlike the FIR case which doesn't have even the potential. One reason for the (potential) unstability of an IIR (...
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  • 26.8k
5 votes

Why can adaptive IIR filters result in unstable solutions?

Although what @Fat32 wrote is correct, I think the potential instability of IIR filters is not the main reason for the instability of an adaptive IIR filter. After all, we can calculate the poles in ...
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  • 291
5 votes

How to derive the "well-known" solution to Unconstrained Array Gain?

A common way is to make use of the Schwarz inequality. First note that: $$\frac{|w^Hd|^2}{w^HQw} = \frac{|w^HQ^{1/2}Q^{-1/2}d|^2}{w^HQw}$$ Using the Schwarz inequality on the numerator: $$\frac{|w^HQ^{...
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  • 598
5 votes

LMS Convergence and the Step Size ($ \mu $) Parameter

In that range it is guaranteed to converge. It doesn't mean it will necesseraly won't converge for higher values. If you want deeper understanding you can read about the step size in Convex ...
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  • 41.3k
5 votes
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How Could One Accelerate the Convergence of the Least Mean Squares (LMS) Filter?

I'd say there 3 approaches to do so: Properties of the LMS Filter There is an optimal step size given you know the spectrum of the correlation matrix. You may have a look at Wikipedia's Least Mean ...
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  • 41.3k
4 votes

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

In the most simple case, just to give intuition about the problem, it is really easy. In the Frequency Domain: $$ {Y}^{\ast} \left( \omega \right) = H \left( \omega \right) {X}^{\ast} \left( \omega \...
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  • 41.3k
4 votes
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LMS adaptive filter relatively delayed signal and reference inputs

The adaptive filter tries to emulate the assumed filtering process between the noise reference signal and the actual noise in the noisy signal. If $n(t)$ is the actual noise in the noisy signal, and ...
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  • 80.4k
4 votes

Adaptive filtering: Optimum filter length and delay

In order to be able to choose an optimal value for the delay $\Delta$ it's important to understand how the system works. The purpose of the delay is to decorrelate the desired signal $s(n)$ and the ...
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  • 80.4k
4 votes

Usefulness of Practical filter ouptput

A real filter wont perfectly remove all unwanted frequencies. But then, a real $X_\mathrm{anything,whatever}$ won't do its job perfectly. The real world isn't perfect, you only get perfection in ...
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4 votes

Learning the Coefficients of Auto Regressive (AR) Model Using Least Mean Squares (LMS) Filter for Signal Prediction

In order to use the LMS to learn an AR Model one should use the predictor variant of the Least Mean Squares (LMS) filter. Basically we predict the $ x \left[ n \right] $ sample using past samples: $ \...
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  • 41.3k

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