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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Matched filter ： ウィキペディア英語版 Matched filter In signal processing, a matched filter (originally known as a North filter〔After D.O. North who first introduced the concept: 〕) is obtained by correlating a known signal, or ''template'', with an unknown signal to detect the presence of the template in the unknown signal. This is equivalent to convolving the unknown signal with a conjugated time-reversed version of the template. The matched filter is the optimal linear filter for maximizing the signal to noise ratio (SNR) in the presence of additive stochastic noise. Matched filters are commonly used in radar, in which a known signal is sent out, and the reflected signal is examined for common elements of the out-going signal. Pulse compression is an example of matched filtering. It is so called because impulse response is matched to input pulse signals. Two-dimensional matched filters are commonly used in image processing, e.g., to improve SNR for X-ray. Matched filtering is a demodulation technique with LTI (Linear Time Invariant) filters to maximize SNR.〔http://cnx.org/content/m10141/latest/〕 ==Derivation of the matched filter impulse response== The following section derives the matched filter for a discrete-time system. The derivation for a continuous-time system is similar, with summations replaced with integrals. The matched filter is the linear filter, $h$, that maximizes the output signal-to-noise ratio. :$\backslash \; y()\; =\; \backslash sum\_^\; h()\; x().$ Though we most often express filters as the impulse response of convolution systems, as above (see LTI system theory), it is easiest to think of the matched filter in the context of the inner product, which we will see shortly. We can derive the linear filter that maximizes output signal-to-noise ratio by invoking a geometric argument. The intuition behind the matched filter relies on correlating the received signal (a vector) with a filter (another vector) that is parallel with the signal, maximizing the inner product. This enhances the signal. When we consider the additive stochastic noise, we have the additional challenge of minimizing the output due to noise by choosing a filter that is orthogonal to the noise. Let us formally define the problem. We seek a filter, $h$, such that we maximize the output signal-to-noise ratio, where the output is the inner product of the filter and the observed signal $x$. Our observed signal consists of the desirable signal $s$ and additive noise $v$: :$\backslash \; x=s+v.\backslash ,$ Let us define the covariance matrix of the noise, reminding ourselves that this matrix has Hermitian symmetry, a property that will become useful in the derivation: :$\backslash \; R\_v=E\backslash \backslash ,$ where $v^\backslash mathrm$ denotes the conjugate transpose of $v$, and $E$ denotes expectation. Let us call our output, $y$, the inner product of our filter and the observed signal such that :$\backslash \; y\; =\; \backslash sum\_^\; h^$ *() x() = h^\mathrmx = h^\mathrms + h^\mathrmv = y_s + y_v. We now define the signal-to-noise ratio, which is our objective function, to be the ratio of the power of the output due to the desired signal to the power of the output due to the noise: :$\backslash mathrm\; =\; \backslash frac\; =\; \backslash frac\}.$ We wish to maximize this quantity by choosing $h$. Expanding the denominator of our objective function, we have :$\backslash \; E\backslash \; =\; E\backslash v)\}^\backslash mathrm\; \backslash \}\; =\; h^\backslash mathrm\; E\backslash \; h\; =\; h^\backslash mathrmR\_vh.\backslash ,$ Now, our $\backslash mathrm$ becomes :$\backslash mathrm\; =\; \backslash frac.$ We will rewrite this expression with some matrix manipulation. The reason for this seemingly counterproductive measure will become evident shortly. Exploiting the Hermitian symmetry of the covariance matrix $R\_v$, we can write :$\backslash mathrm\; =\; \backslash frac^\backslash mathrm\; (R\_v^s)\; |^2\; \}$ ^\mathrm (R_v^h) }, We would like to find an upper bound on this expression. To do so, we first recognize a form of the Cauchy-Schwarz inequality: :$\backslash \; |a^\backslash mathrmb|^2\; \backslash leq\; (a^\backslash mathrma)(b^\backslash mathrmb),\backslash ,$ which is to say that the square of the inner product of two vectors can only be as large as the product of the individual inner products of the vectors. This concept returns to the intuition behind the matched filter: this upper bound is achieved when the two vectors $a$ and $b$ are parallel. We resume our derivation by expressing the upper bound on our $\backslash mathrm$ in light of the geometric inequality above: :$\backslash mathrm\; =\; \backslash frac^\backslash mathrm\; (R\_v^s)\; |^2\; \}$ ^\mathrm (R_v^h) } \leq \frac^\mathrm (R_v^h) 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Matched filter」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.