Contents
Homework 4, ECE438, Spring 2017, Prof. Boutin
Hard copy due in class, Wednesday February 8, 2017.
Do not use any other reference besides your course notes.
HOMEWORK WITHOUT A PROPER COVER SHEER OR UNSTAPLED WILL NOT BE ACCEPTED. SEE INSTRUCTIONS BELOW.
List all collaborators on your cover page. You can collaborate, but the write up must be your own. Do not copy your collaborator's solution.
The goal of this homework is to to understand how to implement a CT system as a DT system through sampling and reconstruction.
Question 1
Exercise 3, p. 3.14 of the course notes.
Question 2
Electrocardiogram signals are very susceptible to interference from the 60 Hz power present in the room where the patient is being monitored. You are going to design a high-pass digital filter to eliminate the 60 Hz interference and everything at frequencies below 60 Hz.
a) Sketch the CTFT of the (analog) high-pass filter that is needed in this case
b) You want to obtain a discrete-time representation of the electrocardiogram signal that preserves all the signal. Assuming that the highest frequency in the electroelectrocardiogram signal are at 2200 Hz, what criteria would you use to select the sampling frequency?
c) Pick a specific value of sampling frequency that satisfy the criterion you wrote in b). Then sketch the graph of the DTFT of the resulting sampling of the electrocardiogram signal. For purpose of illustration, you may assume that the CTFT of the original electrocardiogram signal has a triangular form.
d) Using the same specific value of sampling frequency as for c), sketch the graph of a digital high-pass filter that would remove the interfering frequencies (those that correspond to 60Hz or below in the analog world) from the digital signal.
Question 3
Let x(t) be a continuous-time signal and let y[n]=x(nT) be a sampling of that signal with period T>0. We would like to interpolate the samples (i.e., "connect the dots") in order to try to recover x(t).
a) Write a formula for a band-limited interpolation of the samples (i.e., an expression for a continuous signal $ x_r(t) $ in terms of the samples y[n]).
b) Prove that the formula you gave in a) yields a band-limited signal $ x_r(t) $.
c) Under what circumstances is $ x_r(kT)=x(kT) $ for all integer values of k?
d) Under what circumstances is your interpolation equal to the original signal x(t)?
Question 4
Let x(t) be a continuous-time signal and consider a sampling y[n]=x(nT) of that signal.
a) Write a formula for a zero-order hold reconstruction $ x_r(t) $ of the samples.
b) Is the interpolation you wrote in 2a) band-limited? Answer yes/no and give a mathematical proof of your answer.
c) Under what circumstances is $ x_r(kT)=x(kT) $ for all integer values of k?
d) Under what circumstances is your interpolation equal to the original signal x(t)?
Hand in a hard copy of your solutions. Pay attention to rigor!
Presentation Guidelines
- Write only on one side of the paper.
- Put all problems in order
- Use a "clean" sheet of paper (e.g., not torn out of a spiral book).
- Staple the pages together.
- Include a cover page.
- List all collaborators on your cover page. You can collaborate, but the write up must be your own. Do not copy your collaborator's solution.
- Do not use any other reference besides your course notes. Write a statement to that effect on your cover page, and sign it.
Discussion
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