Thursday, February 26, 2009

梳狀濾波器www.tool-tool.com

Bewise Inc. www.tool-tool.com Reference source from the internet.


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幾種梳狀濾波器的頻譜

信號處理領域中,梳狀濾波器使一個信號與它的延時信號疊加,從而產生相位抵消。梳狀濾波器的頻率響應由一系列規律分佈的峰組成,看上去與梳子類似。

離散時間系統中的梳狀濾波器滿足下式:

y[n] = ax[n] + bx[n − τ] + cy[n − τ]

其中τ 是一個表示延時的常量。梳狀濾波器也可以在連續時間系統上實現。它的頻率響應為:

H(\omega) = \frac{a + be^{-i \omega \tau}} {1 - ce^{-i \omega \tau}}

頻譜中的梳狀峰值是因為系統周期的不連續性(極點),極點的位置滿足:

\cos(\omega \tau) = \frac{1+c^2}{2c}

[編輯] 應用

NTSC制式的電視信號解碼器中以硬體(偶爾也有軟體)實現了二維和三維梳狀濾波器,以減輕雜色訊(dot crawl)等效應。梳狀濾波器也被應用在地面無線通信系統中。梳狀濾波器可以產生回聲效應,若將延時設置為幾個毫秒,則將此濾波器加在音頻信號上,就可以作為圓柱形諧振腔的模型。因為這種諧振腔能夠放大與它寬度相關的駐波對應的頻率分量。

[編輯] 頻率響應的計算

梳狀濾波器是一個線性時不變系統,因此指數函數是這一系統的特徵函數。所以當輸入信號x(n) 為指數函數的形式時

x(n) = eiωn

輸出信號y(n) 的形式為:

y(n) = H(ω)eiωn .

代入上文中梳妝濾波器頻響滿足的條件式,可得:

H(ω)eiωn = aeiωn + beiω(n − τ) + cH(ω)eiω(n − τ)
H(ω)eiωn = aeiωn + be iωτeiωn + cH(ω)e iωτeiωn

由於指數函數非零,因此有:

H(ω) = a + be iωτ + cH(ω)e iωτ

解出H(ω) 可得:

H(\omega) = \frac{a + be^{-i \omega \tau}} {1 - ce^{-i \omega \tau}}

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