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

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

One of the most widely used objective function in adaptive filtering is the MSE defined as
$$F[e(k)]=\xi(k)=E\left[e^2(k)\right]=E\left[d^2(k)-2 d(k) y(k)+y^2(k)\right]$$
where $d(k)$ is the reference signal as illustrated in Fig. 1.1.
Suppose the adaptive filter consists of a linear combiner, i.e., the output signal is composed by a linear combination of signals coming from an array as depicted in Fig. 2.1a. In this case,
$$y(k)=\sum_{i=0}^N w_i(k) x_i(k)=\mathbf{w}^T(k) \mathbf{x}(k)$$
where $\mathbf{x}(k)=\left[x_0(k) x_1(k) \ldots x_N(k)\right]^T$ and $\mathbf{w}(k)=\left[w_0(k) w_1(k) \ldots w_N(k)\right]^T$ are the input signal and the adaptive-filter coefficient vectors, respectively.

## 计算机代写|自适应算法代写Cooperative and Adaptive Algorithms代考|Linearly Constrained Wiener Filter

A deterministic discrete-time signal is characterized by a defined mathematical function of the time index $k$, ${ }^1$ with $k=0, \pm 1, \pm 2, \pm 3, \ldots$. An example of a deterministic signal (or sequence) is
$$x(k)=\mathrm{e}^{-\alpha k} \cos (\omega k)+u(k)$$
where $u(k)$ is the unit step sequence.
The response of a linear time-invariant filter to an input $x(k)$ is given by the convolution summation, as follows [7]:
\begin{aligned} y(k) & =x(k) * h(k)=\sum_{n=-\infty}^{\infty} x(n) h(k-n) \ & =\sum_{n=-\infty}^{\infty} h(n) x(k-n)=h(k) * x(k) \end{aligned}
where $h(k)$ is the impulse response of the filter. ${ }^2$
The $\mathcal{Z}$-transform of a given sequence $x(k)$ is defined as
$$\mathcal{Z}{x(k)}=X(z)=\sum_{k=-\infty}^{\infty} x(k) z^{-k}$$

In a number of applications, it is required to impose some linear constraints on the filter coefficients such that the optimal solution is the one that achieves the minimum MSE, provided the constraints are met. Typical constraints are: unity norm of the parameter vector; linear phase of the adaptive filter; prescribed gains at given frequencies.

In the particular case of an array of antennas the measured signals can be linearly combined to form a directional beam, where the signal impinging on the array in the desired direction will have higher gain. This application is called beamforming, where we specify gains at certain directions of arrival. It is clear that the array is introducing another dimension to the received data, namely spatial information. The weights in the antennas can be made adaptive leading to the so-called adaptive antenna arrays. This is the principle behind the concept of smart antennas, where a set of adaptive array processors filter the signals coming from the array, and direct the beam to several different directions where a potential communication is required. For example, in a wireless communication system we are able to form a beam for each subscriber according to its position, ultimately leading to minimization of noise from the environment and interference from other subscribers.

In order to develop the theory of linearly constrained optimal filters, let us consider the particular application of a narrowband beamformer required to pass without distortion all signals arriving at $90^{\circ}$ with respect to the array of antennas. All other sources of signals shall be treated as interferers and must be attenuated as much as possible. Figure $2.2$ illustrates the application. Note that in case the signal of interest does not impinge the array at $90^{\circ}$ with respect to the array, a steering operation in the constraint vector $\mathbf{c}$ (to be defined) has to be performed [23].

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

$$F[e(k)]=\xi(k)=E\left[e^2(k)\right]=E\left[d^2(k)-2 d(k) y(k)+y^2(k)\right]$$

$$y(k)=\sum_{i=0}^N w_i(k) x_i(k)=\mathbf{w}^T(k) \mathbf{x}(k)$$

## 计算机代写|自适应算法代写Cooperative and Adaptive Algorithms代考|Linearly Constrained Wiener Filter

$$x(k)=\mathrm{e}^{-\alpha k} \cos (\omega k)+u(k)$$

$$y(k)=x(k) * h(k)=\sum_{n=-\infty}^{\infty} x(n) h(k-n) \quad=\sum_{n=-\infty}^{\infty} h(n) x(k-n)=h(k) * x(k)$$

$$\mathcal{Z} x(k)=X(z)=\sum_{k=-\infty}^{\infty} x(k) z^{-k}$$

## MATLAB代写

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