Latest revision as of 08:13, 27 September 2013

Homework 6, ECE438, Fall 2013, Prof. Boutin

Harcopy of your solution due in class, Friday October 4, 2013 (That's the day of the test! )

Presentation Guidelines

• Write only on one side of the paper.
• Use a "clean" sheet of paper (e.g., not torn out of a spiral book).
• Staple the pages together.
• Include a cover page.
• Do not let your dog play with your homework.

Conversion between analog and digital frequencies

Question 1 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. Assume that the highest frequencies of interest in the electrocardiogram signal are at 2500 Hz. Choose an appropriate sampling frequency for your A/D convertor, and sketch the desired frequency response of the digital filter. Be sure to show how you calculated the cutoff frequency for the digital filter.

Question 2 Long term climate change is a topic of great interest at this time. To see if there has been a significant long-term trend in temperatures in Lafayette, IN, you have downloaded temperature data from the U.S. Weather Service. The file contains the average monthly temperature at the Purdue airport for the past 100 years. Thus it consists of 1200 samples. In order to see if there is a long-term trend, you will need to remove the annual cycle from the data. Sketch the desired frequency response of an ideal low-pass digital filter that will accomplish this. Be sure to show how you calculated the cutoff frequency of the digital filter.

Downsampling and upsampling

Question 3

a) What is the relationship between the DT Fourier transform of x[n] and that of y[n]=x[5n]? (Give the mathematical relation and sketch an example.)

b) What is the relationship between the DT Fourier transform of x[n] and that of

$ z[n]=\left\{ \begin{array}{ll} x[n/5],& \text{ if } n \text{ is a multiple of } 5,\\ 0, & \text{ else}. \end{array}\right. $

(Give the mathematical relation and sketch an example.)

Discrete Fourier Transform

Question 4 Compute the DFT of the following signals

a) $ x_1[n] = \left\{ \begin{array}{ll} 1, & n \text{ multiple of } N\\ 0, & \text{ else}. \end{array} \right. $

b) $ x_2[n]= e^{j \frac{\pi}{3} n } \cos ( \frac{\pi}{6} n ) $

c) $ x_3[n] =(\frac{1}{\sqrt{2}}+j \frac{1}{\sqrt{2}})^n $

Question 5 Compute the inverse DFT of $ X[k]= e^{j \pi k }+e^{-j \frac{\pi}{2} k} $.

Discussion

Please discuss the homework below.

• Comment/question here
  • answer here

Back to ECE438, Fall 2013, Prof. Boutin