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

You have to look at the autocorrelation function of $y(t)$: $$R_y(\tau)=E\{y(t)y(t+\tau)\}\tag{1}$$ with $$y(t)=(h_A\star a)(t) - (h_B\star b)(t)\tag{2}$$ where $\star$ denotes convolution. If you write out $(2)$ with integrals and plug it into $(1)$ then, with the given assumptions on $a(t)$ and $b(t)$, you'll see that the mixed terms with the negative sign ...


2

You have a typo in your definition of $R_y(k)$, and an error in the time indices when developing the equation. The proper definition of autocorrelation for complex signals (in the case of wide-sense stationary processes) is $$R_y(k)=\mathbb{E}\left[y(n)\overline{y(n+k)}\right],$$ and setting $k=0$, we obtain $$\begin{align} R_y(0)&=\mathbb{E}\left[y(n)\...


2

The OP's updated working is incorrect. Following up what Hilmar suggested gives \begin{align} Y(t) &= a\left(X(t)\right)^2\\ &= a\left(S(t) + N(t)\right)^2\\ &= a\left(S(t)\right)^2 + 2aS(t)N(t) + a\left(N(t)\right)^2\\ &{\large\Downarrow}\\ E[Y(t)]&= aE\left[\left(S(t)\right)^2 \right] + 2aE\left[S(t)N(t)\right] + aE\left[\left(S(t)\...


2

Put your second equations into your first equation, express $Y(t)$ as a function of $S(t)$ and $N(t)$ Apply the definition for mean and autocorrelation. Simplify and solve


2

Any Deconvolution method which takes into account the noise in the image is basically a stochastic approach. Usually, the model for Deconvolution is: So having the noise in there makes it a problem with stochastic properties. Remark If by stochastic you meant sampling from the Posterior Distribution then you may have a look at Stochastic Image Denoising by ...


1

You have a input signal $x(n)$ (the colored noise in your case) and a desired output signal $d(n)$. Assuming that the FIR coefficients are real. Define the filter weight vector $\mathbf{w}$ and the input vector $\mathbf{x}(n)$ $$ \mathbf{w} = [w_0, w_1, \ldots, w_{N-1}]^T $$ $$ \mathbf{x}(n) = [x(n), x(n-1), \ldots, x(n-N+1)]^T $$ The output of the filter is ...


1

"The PSD is defined only for stationary processes" is a common misconception. The Wiener–Khinchin theorem states that for a wide-sense stationary process, the PSD is the Fourier transform of the autocorrelation function. This is sometimes (wrongly) taken as the "definition" of PSD. Actually, the PSD is defined as $$S_y(f)=\lim_{T\to\infty}...


1

Hi: I don't know whether white noise can have a non-zero mean ( Dilipe can tell us that ) but, if it can't, then just call it whatever you call white noise with a non-zero mean. But, assuming it's zero, this is how you can do it. ( If it's not zero, then the solution is similar but the last step would be a little different. Use $$\operatorname{Var}\left\{X\...


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