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Turns out that convolution and correlation are closely related. For real signals (and finite energy signals): Convolution: $\qquad y[n] \triangleq h[n]*x[n] = \sum\limits_{m=-\infty}^{\infty} h[n-m] \, x[m]$ Correlation: $\qquad R_{yx}[n] \triangleq \sum\limits_{m=-\infty}^{\infty} y[n+m] \, x[m] = y[-n]*x[m]$ Now, in metric spaces, we like to use this ...

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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 doesn't actually calculate the gradient directly, as that would require knowing: $$E(\mathbf{x}[n]e^*[n])$$ where $\mathbf{x}[n]$ is the input vector and $e[... 7 TL;DR: No, they are not necessarily the same. Gory Details Least squares is just an optimization technique. It is used in a variety of ways. For filter design it is used to select that realizable filter$H_r(e^{j\omega})$that most closely matches, in the least squares sense, the ideal required filter response$H_i(e^{j\omega})$: $$H_r(e^{j\omega}) = \... 7 The play similar role in those algorithms - the ability to forget the past and adapt to current reality. In the LMS, the classic implementation has \alpha = 1 . Namely the optimal weights at any point are function of all inputs. The Leakage factor allows to weigh the past differently in a damped manner which over times means the far past has ... 6 Note that the inverse of an FIR system is IIR, and the same is true for the inverse of an IIR system, unless it is an all-pole system, the inverse of which would be FIR. So in most cases the ideal equalizer should have an infinitely long impulse response in order to perfectly invert the channel. In practice almost all adaptive equalizers are FIR filters ... 6 That really depends on context, but generally adaptive implies that the calculations are done on-line / on the fly. In some applications, the filter is updated for a while, then the adaptation is turned off and the last lot of weights are used. 6 To expand on what Peter K. has said, if the signals being used by the filter are stationary, then the filter weights or coefficients can be determined and the filter operates as it was designed without further updates to the filter weights. However, if the signals change, or become quasi-stationary, the filter will adapt continuously. 6 These are the key differences between FIR and IIR filters, regarding the feature you wish to control are the following:$$ \begin{array}{c|lcr} \text{Feature} & \text{IIR} & \text{FIR} \\ \hline \text{Implementation} & \text{Poles & Zeros} & \text{Zeros Only} \\ \text{States} & \text{Yes} & \text{No} \\ \text{Phase Delay} & \... 6 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 language use but these might be just personal preferences. The "Subject matter" section includes comments in methodology which could take more time to address. ... 6 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 operations(convolution and matrix multiplication) same here?. The system is an FIR system, so the vector multiplication here is equivalent to convolution --- for ... 5 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 only be found through experiment. the variabe noisedB appears to be the desired SNR of the received signal. An SNR of$0\,\text{dB}$as specified by you is very poor (the noise is as strong as the ... 5 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 way to calculate the optimal solution. The LMS and RLS are sequential / on line methods to solve the same problem and given the data is stationary they all ... 5 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 frequency cutoffs for a frequency domain filter can get tricky and turn into a game of whack-a-mole because there's "high frequency" components in the heartbeat blip (... 5 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 dealing with are complex. If you write $$h(n)=\sum_{k=0}^{\infty}w_k(n)u(n-k)$$ then it isn't clear that you are dealing with complex quantities. As Robert has ... 5 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 (adaptive) filter is the numerical issues due to coefficient quantization. When the poles are closer to unit circle this will be critical. This is especially ... 5 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 each iteration and put a hard constraint to avoid poles out of the unit circle. Even within the case of the FIR filters - which are unconditionally stable- we ... 5 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 \right] = \sum_{i}^{m} {a}_{i} \sin \left[ 2 \pi \frac{ {f}_{i} }{ {f}_{s} } n + {\phi}_{i} \right] $. Input Signal -$ y \left[ n \right] $which is composed of ... 4 The adaptive filter tries to emulate the assumed filtering process between the noise reference signal and the actual noise in the noisy signal. If$n(t)$is the actual noise in the noisy signal, and$n_r(t)$is the noise reference, it is assumed that there's a linear filtering relationship between the two: $$n(t)=(n_r*h)(t)$$ where$*$denotes ... 4 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 Decomposition): $$x \left[ n \right] = \sum_{k = 0}^{\infty} {g}_{k} u \left[ n - k \right] + z \left[ n \right]$$ Where$ {\left\{ {g}_{k} \right\}}_{k = 1}^{\infty} ...

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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: $${\left( A + U C V \right)}^{-1} = {A}^{-1} - {A}^{-1} U {\left( {C}^{-1} + V {A}^{-1} U \right)}^{-1} V {A}^{-1}$$ Deriving it is by utilizing these useful identities: \begin{align} U + U C V {A}^{-1} U &... 4 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 signal component s(n-\Delta) at the input of the adaptive filter. This means that \Delta must be chosen such that the autocorrelation R_{ss}(k) of s(n) is ... 4 The Covariance Matrix is commonly defined as\mathbf Q = E\left[ (\mathbf x -\mathbf\mu_{x})(\mathbf x -\mathbf\mu_{x})^*\right]$$with \mu denoting the mean value, i.e. \mu_{x}=E\left[\mathbf x\right], and \mathbf x being column vectors. The fact that you define the covariance matrix as$$\mathbf{R}_i = E\left[\textbf{u}_i^*\textbf{u}_i \right]$$... 4 A real filter wont perfectly remove all unwanted frequencies. But then, a real X_\mathrm{anything,whatever} won't do its job perfectly. The real world isn't perfect, you only get perfection in mathematics. What's important though is that a filter can approximately remove the unwanted frequencies. For example if you have some interference at 100 kHz, then ... 4 The LMS and many of the variants of Adaptive Filters (In the Linear System context) work in the following settings (Intuitive): You have access to 2 signals. One signal is the result of the other one when a Linear System is applied. This sounds really limiting, yet in practice it is powerful and flexible. In the settings you mentioned the most known and ... 4 In that range it is guaranteed to converge. It doesn't mean it will necesseraly won't converge for higher values. If you want deeper understanding you can read about the step size in Convex Optimization context where there the step size related to the Lipschitz Constant of the function (Which matches the eigen value for Quadratic functions). If you share ... 4 I'd say there 3 approaches to do so: Properties of the LMS Filter There is an optimal step size given you know the spectrum of the correlation matrix. You may have a look at Wikipedia's Least Mean Squares Filter at Convergence and Stability in the Mean. Some other approaches related to this might be those from Variable Step Size LMS. You may have a look at ... 3 The quadratic surface is determined by the autocorrelation matrix of the data, which is always positive definite or positive semi-definite. This means that any stationary point is always a minimum. In the worst case, this minimum is not unique if the matrix is singular, but it can never be a saddle point. 3 In the most simple case, just to give intuition about the problem, it is really easy. In the Frequency Domain:$$ {Y}^{\ast} \left( \omega \right) = H \left( \omega \right) {X}^{\ast} \left( \omega \right) \Rightarrow {X}^{\ast} \left( \omega \right) = \frac{ {Y}^{\ast} \left( \omega \right) }{ H \left( \omega \right) }  Since \$ {Y}^{\ast} \left( \omega ...

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"Deep zeros" in the input spectrum are caused by a channel which strongly attenuates one or several frequencies, often due to multipath fading. This causes problems for a (linear) equalizer because it tries to compensate for these "deep zeros" by strong amplification of the corresponding frequency bands, which leads to noise enhancement. Non-linear ...

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