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Lab 2: Prelab (part 1)

Module by: Douglas L. Jones, Swaroop Appadwedula, Matthew Berry, Mark Haun, Jake Janovetz, Michael Kramer, Dima Moussa, Daniel Sachs, Brian Wade, Patrick Frantz, Kanglin Wang. 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: You will work through an example problem that explores the effects of sample-rate compression and expansion on the spectrum of a signal.

多频采样理论练习

考虑一个具有如 Figure 1 所示DTFT X⁢ω X ω 的采样信号。

Figure 1: 输入信号的DTFT。
Figure 1 (prelab_input.png)

假设 U=D=3 U D 3 ，使用一个信号在采样压缩和扩展之前与之后的关系 (Equation 1 和 Equation 2) 来画出该信号在通过如 Figure 2 所示的系统（无滤波器）后的DTFT响应，包括中间响应（intermediate response） W⁢ω W ω 和最终响应（final response） Y⁢ω Y ω 。注意从数字频率 ω ω 到模拟频率的转换形式将依赖于采样率。因此 X⁢ω X ω 和 W⁢ω W ω 的转换形式将会不同。

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)
Figure 2: 多频采样系统
Figure 2 (prelab_sys.png)

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