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Pick a signal x(t) representing a note of the middle scale of a piano (but not the middle C we did in class) and obtain its CTFT <math>X(f)</math>. Then pick a sampling period <math>T_1</math> for which no aliasing occurs and obtain the DTFT of the sampling <math>x_1[n]=x(n T_1)</math>. More precisely, write a mathematical expression for <math>X_1(\omega)</math> and sketch its graph. Finally, pick a sampling frequency <math>T_2</math> for which aliasing occurs and obtain the DTFT of the sampling <math>x_2[n]=x(n T_2)</math> (i.e., write a mathematical expression for <math>X_2(f)</math> and sketch its graph.) Note the difference and similarities between <math>X(f)</math> and <math>X_1(\omega)</math>. Note the differences and similarities between <math>X_1(\omega)</math> and <math>X_2(\omega)</math>. | Pick a signal x(t) representing a note of the middle scale of a piano (but not the middle C we did in class) and obtain its CTFT <math>X(f)</math>. Then pick a sampling period <math>T_1</math> for which no aliasing occurs and obtain the DTFT of the sampling <math>x_1[n]=x(n T_1)</math>. More precisely, write a mathematical expression for <math>X_1(\omega)</math> and sketch its graph. Finally, pick a sampling frequency <math>T_2</math> for which aliasing occurs and obtain the DTFT of the sampling <math>x_2[n]=x(n T_2)</math> (i.e., write a mathematical expression for <math>X_2(f)</math> and sketch its graph.) Note the difference and similarities between <math>X(f)</math> and <math>X_1(\omega)</math>. Note the differences and similarities between <math>X_1(\omega)</math> and <math>X_2(\omega)</math>. | ||
| − | You may post your answers on [[ | + | You may post your answers on [[Sampling pure frequencies ECE438F10|this page]] for collective discussion/comments (but this is optional). |
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== Question 2== | == Question 2== | ||
Latest revision as of 04:59, 7 September 2010
Homework 2, ECE438, Fall 2010, Prof. Boutin
Due Wednesday September 8, 2010. Hard copy due by 4:20pm in class, electronic copy in Prof. Boutin's dropbox (the ECE438 HW2 Assignment box) by 6pm.
Question 1
Pick a signal x(t) representing a note of the middle scale of a piano (but not the middle C we did in class) and obtain its CTFT $ X(f) $. Then pick a sampling period $ T_1 $ for which no aliasing occurs and obtain the DTFT of the sampling $ x_1[n]=x(n T_1) $. More precisely, write a mathematical expression for $ X_1(\omega) $ and sketch its graph. Finally, pick a sampling frequency $ T_2 $ for which aliasing occurs and obtain the DTFT of the sampling $ x_2[n]=x(n T_2) $ (i.e., write a mathematical expression for $ X_2(f) $ and sketch its graph.) Note the difference and similarities between $ X(f) $ and $ X_1(\omega) $. Note the differences and similarities between $ X_1(\omega) $ and $ X_2(\omega) $.
You may post your answers on this page for collective discussion/comments (but this is optional).
Question 2
Pick five different DT signals and compute their z-transform. Then take the five z-transforms you obtained and compute their inverse z-transform.
You may post your answers on this page for collective discussion/comments (but this is optional).
I just realized that there is no class Monday so we will not be able to cover the inverse z-transform before Wednesday (when the homework is due). Therefore, I am changing the homework: the second part of the question (compute the inverse z-transforms) will be part of Homework 3 instead. Sorry about the confusion. Have a great labor day weekend! --Mboutin 19:54, 3 September 2010 (UTC)
Instructions:
- Hand in a hard copy of your homework on September 8 in class.
- hand in an anonymous scan of your solution (e.g., write out your name before scanning, or replace it by a pseudo-name) and drop it in Prof. Boutin's dropbox (in the ECE438 HW2 Assignment box).
We will then do a double-blind peer of the homework.
