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Lab 2: Theory

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: Introduction by Douglas L. Jones, Swaroop Appadwedula, Matthew Berry, Mark Haun, Jake Janovetz, Michael Kramer, Dima Moussa, Daniel Sachs, Brian Wade

Summary: ダウンサンプル比Dのダウンサンプラは、DサンプルにつきD-1個のデータを間引き、信号のレートを減少する。アップサンプル比Uのアップサンプラは、サンプル値データ間にU-1個のゼロ値データを挿入し、信号のレートを増加する効果を有している。これらは時変要素を含むため、線形時不変システムの条件を緩和し、信号処理の自由度を飛躍的に向上することを可能としている。ダウンサンプラとアップサンプラは単純な処理ではあるが、非常に深い意味を有している。

Introduction

 以下の演習では、図 1に示すmultirate processing(マルチレート処理)が信号に対してどのような影響を与えるかを考察する。下図のsample-rate compressor(ダウンサンプラ) ( D D )は、ダウンサンプラに入力されるサンプル値のうち D Dサンプルにつき D1 D 1 サンプルのサンプル値を間引く処理を行う。一方、sample-rate expander(アップサンプラ) ( U U )は、隣接する入力サンプル値間に U1 U 1 サンプルのゼロ値のデータを挿入し、信号のレートを増加する処理を行う。図 1において、ダウンサンプル比とアップサンプル比が同じ値( D=U D U )であると仮定すると, フィルタFIR1とフィルタFIR3は同じサンプルレート F s F s で動作し、フィルタFIR2はより低いサンプルレート F s D F s D で動作する。

図 1: 簡単なマルチレートシステムの構成図
図 1 (multirate_sys.png)

 後述の実験・演習では、図 1に示したシステムを実装し、このシステムの実行時にダウンサンプル比とアップサンプル比を変化して、その挙動を確認する。実行中にこれらの値を変更するためのインタフェースとして、すでに準備したツールを提供する。また、マルチレート処理が信号に与える影響を検討するために、FIR1からFIR3までのそれぞれのフィルタを一時的に取り除いてその影響を調べることも可能である。FIR1とFIR3は、どのような目的を果たしているのか?また、これらのフィルタが存在しない場合には何が起きるのだろうか?

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