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Multirate Filtering: Theory Exercise (日本語 - Japanese)

Module by: Douglas L. Jones, Swaroop Appadwedula, Matthew Berry, Mark Haun, Jake Janovetz, Michael Kramer, Dima Moussa, Daniel Sachs, Brian Wade, Patrick Frantz, Emiko Yamai, Hironori Takaryo, Yoji Yamada. E-mail the authors

Based on: Multirate Filtering: Theory Exercise by Douglas L. Jones, Swaroop Appadwedula, Matthew Berry, Mark Haun, Jake Janovetz, Michael Kramer, Dima Moussa, Daniel Sachs, Brian Wade

Summary: この演習課題を通じて、ダウンサンプルおよびアップサンプルという操作が信号のスペクトルにどのような影響を与えるかを調査・理解することができます。

マルチレート理論の演習課題

　入力信号として、図 1に示すスペクトル X⁢ω X ω を有するサンプル値信号を考えよう。ただし、 X⁢ω X ω は、入力信号のDTFT(discrete time Fourier transform)であるとします。

図 1: この課題で仮定する入力信号のスペクトル
図 1 (prelab_input.png)

　 U=D=3 U D 3 と仮定し、図 2 (帯域制限用フィルタ無し)のマルチレートシステムを通過した各部の信号のDTFT結果 W⁢ω W ω および Y⁢ω Y ω を示して下さい。 このとき、ダウンサンプルおよびアップサンプル前後の信号のスペクトルの関係式（式 1および式 2）を用いて下さい。なお、正規化角周波数 ω ωをアナログ周波数表現に変換する場合、信号のレートが異なると正規化する際に用いるサンプリング周波数が異なることに注意する必要があります。したがって、図 1における X⁢ω X ω の ω ω と以下の式における W⁢ω W ω の ω ω では、同じ記号を用いているにもかかわらず、正規化表現の際に仮定しているサンプリング周波数が異なっており、後者のサンプリング周波数は前者の 1D 1 D 倍です。

W⁢ω=1D⁢∑ k =0D−1X⁢ω−2⁢π⁢kD W ω 1 D k 0 D 1 X ω 2 k D
(1)
Y⁢ω=W⁢U⁢ω Y ω W U ω
(2)
図 2: マルチレートシステム
図 2 (prelab_sys.png)

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