Digital Class Homework

ENGR 0832: Digital World 2020 SPRING 2015

Final Exam

Due Date: 04/23/2015

Student Name: ____________________

Score: __________________

Directions: Submit your report in class on Thursday, April 23 rd

, 2015. No email or late submission

will be accepted. Show all your work. No plagiarism. Good luck!

1. (40 points) What is the definition of “Nyquist rate”? What is the “Nyquist sampling theorem”?

Plot the signal 𝑠(𝑡) = sin⁡(4𝜋𝑡).

a. Assuming a sampling period 𝑇𝑠 = 0.3⁡s, plot its samples. What has happened? Have you

sampled above the Nyquist rate?

b. Assuming a bit depth of 4 bits and a sampling rate 𝑓𝑠 = 24⁡Hz, plot its samples and

convert the analog waveform defined by 𝑠(𝑡)⁡into digital.

c. Convert the digital waveform back to analog using your results in (c). The reconstructed

waveform should look like a staircase.

2. (30 points) ASCII encodes English text by using 1 byte (including parity bit) per symbol. Take

any book that you have read recently and pick a page at random. Take any full line on that page

and count the number of symbols in the line of text. Do not forget to count spaces and

punctuation marks. Next, count the number of lines on the page. Now multiply the number of

symbols per line by the number of pages in the book. Make sure to justify your work, indicate the

total number of pages in the book you picked and include a hard copy of the page in your report.

a. About how many bytes would it take to store all of the characters and punctuation in this

book, using ASCII code as a formatting method?

b. A CD-ROM holds around 700 MB of information. How many similar books could be

stored on a CD-ROM using ASCII?

c. Using the same full line on that page, find the Huffman codebook and the number of

bytes needed to store one line of text. About how many bytes would it take to store all of

the characters and punctuation in this book, using Huffman as a formatting method?

Compute the compression ratio using your results in (a).

3. (30 points) An 8 × 8 pixel image is defined as follows. The four corner pixels all have a value of

15. The two central columns and the two central rows have a value of 15. All other pixels have a

value of 0.

a. Write this image as a grid of numbers and draw it. What is the total number of pixels in

the image? How many bits are needed to represent the gray values of the pixels? What is

the total number of bits needed to represent the image?

b. If we reduce the sampling rate for the image by a factor of two along rows and columns,

what will be the size of the new image? Write this image as a grid of numbers rounded to

the nearest integer. Draw the new image.

c. Many different 8 × 8 images will look the same as the result you found in (b) when the

sampling rate is reduced by a factor of two. Specify one of these 8 × 8 images that is as

different from the image specified in (a) as you can make it. Write your image as a grid

of numbers and draw it.