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# 计算机代写|自适应算法代写Cooperative and Adaptive Algorithms代考|ECE457A Ergodicity

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In the probabilistic approach, the statistical parameters of the real data are obtained through ensemble averages (or expected values). The estimation of any parameter of the stochastic process can be obtained by averaging a large number of realizations of the given process, at each instant of time. However, in many applications only a few or even a single sample of the process is available. In these situations, we need to find out in which cases the statistical parameters of the process can be estimated by using time average of a single sample (or ensemble member) of the process. This is obviously not possible if the desired parameter is time varying. The equivalence between the ensemble average and time average is called ergodicity [14,15].

The time average of a given stationary process represented by $x(k)$ is calculated by
$$\hat{m}{x_N}=\frac{1}{2 N+1} \sum{k=-N}^N x(k)$$
If
$$\sigma_{\hat{m}{x_N}}^2=\lim {N \rightarrow \infty} E\left{\left|\hat{m}{x_N}-m_x\right|^2\right}=0$$ the process is said to be mean-ergodic in the mean-square sense. Therefore, the mean-ergodic process has time average that approximates the ensemble average as $N \rightarrow \infty$. Obviously, $\hat{m}{x_N}$ is an unbiased estimate of $m_x$ since
$$E\left[\hat{m}{x_N}\right]=\frac{1}{2 N+1} \sum{k=-N}^N E[x(k)]=m_x$$

## 计算机代写|自适应算法代写Cooperative and Adaptive Algorithms代考|The Correlation Matrix

Usually, adaptive filters utilize the available input signals at instant $k$ in their updating equations. These inputs are the elements of the input signal vector denoted by
$$\mathbf{x}(k)=\left[x_0(k) x_1(k) \ldots x_N(k)\right]^T$$
The correlation matrix is defined as $\mathbf{R}=E\left[\mathbf{x}(k) \mathbf{x}^H(k)\right]$, where $\mathbf{x}^H(k)$ is the Hermitian transposition of $\mathbf{x}(k)$, that means transposition followed by complex conjugation or vice versa. As will be noted, the characteristics of the correlation matrix play a key role in the understanding of properties of most adaptive-filtering algorithms. As a consequence, it is important to examine the main properties of the matrix $\mathbf{R}$. Some properties of the correlation matrix come from the statistical nature of the adaptive-filtering problem, whereas other properties derive from the linear algebra theory.

## 自适应算法代写

$$14,15$$

$$\hat{m} x_N=\frac{1}{2 N+1} \sum k=-N^N x(k)$$

$$E\left[\hat{m} x_N\right]=\frac{1}{2 N+1} \sum k=-N^N E[x(k)]=m_x$$

## 计算机代写|自适应算法代写COOPERATIVE AND ADAPTIVE ALGORITHMS代考|THE CORRELATION MATRIX

$$\mathbf{x}(k)=\left[x_0(k) x_1(k) \ldots x_N(k)\right]^T$$

## Matlab代写

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