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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, Kamolchanok Kriengchaipruck. 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.

Multirate Theory Exercise

พิจารณา sampled signal ที่มี DTFT X⁢ω X ω แสดงใน รูป 1.

รูป 1: DTFT ของ input signal.
รูป 1 (prelab_input.png)

สมมติว่าAssuming U=D=3 U D 3 , ใช้ความสัมพันธ์ดังกล่าว ระหว่าง DTFT ของ signal ก่อนและหลัง sample-rate compression และ expansion (สมการ 1 and สมการ 2) เพื่อร่างค่า DTFT response ของ signal ที่ผ่านเข้าระบบ multirate ของ รูป 2 (โดยปราศจากการ filtering). รวมทั้ง intermediate response W⁢ω W ω และ final response Y⁢ω Y ω . ให้ระวังว่าการแปลงค่าจาก digital frequency ω ω เป็น analog frequency ขึ้นอยู่กับค่า sampling rate. ฉะนั้นการแปลงค่า จึงแตกต่างกันสำหรั X⁢ω X ω และ W⁢ω W ω .

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)
รูป 2: Multirate System
รูป 2 (prelab_sys.png)

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