## Contents

# Homework 4, ECE438, Fall 2014, Prof. Boutin

Hard copy due in class, Monday September 29, 2014.

## 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.

## Question 1

**Conversion between analog and digital frequencies**

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

**Conversion between analog and digital frequencies**

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.

## Question 3

**Downsampling and upsampling**

a) What is the relationship between the DT Fourier transform of x[n] and that of y[n]=x[4n]? (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/4],& \text{ if } n \text{ is a multiple of } 4,\\ 0, & \text{ else}. \end{array}\right. $

(Give the mathematical relation and sketch an example.)

## Question 4

**Downsampling and upsampling**

Let $ x_1[n]=x(Tn) $ be a sampling of a CT signal $ x(t) $. Let D be a positive integer.

a) Under what circumstances is the downsampling $ x_D [n]= x_1 [Dn] $ equivalent to a resampling of the signal with a new period equal to DT (i.e. $ x_D [n]= x(DT n) $)?

b) Under what circumstances is it possible to construct the sampling $ x_3[n]= x(\frac{T}{D} n) $ directly from $ x_1[n] $ (without reconstructing x(t))?

## Discussion

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