Topic 14 - Problem Set
Do your homework on a separate piece of paper, or in a notebook of some kind. Written
work and answers on this sheet will not be counted. Show all your work including
formulas and substitutions. Minimum credit will be awarded for answers without work.
14.1.1
1. What is the most significant bit in a binary string? What is the least significant one?
2. Convert the binary number 10110 to decimal.
3. Convert the decimal number 33 to binary.
4. How many different numbers can be represented by 1 byte?
5. How many different numbers can be represented by 2 bytes?
6. Windows Notepad is a simple text program that comes with most PCs. It stores
characters as 8-bit strings. Thus the string of characters “digital technology” will take
up 18 bytes of memory as a Notepad document. Suggest why a document in
Microsoft Word which contains the exact same string “digital technology” takes more
than 18 bytes.
14.1.2
7. List three different means of storage of information in analog form.
8. List three different means of storage of information in digital form.
emf / V
9. Convert the following analog signal to non-negative voltages (whole number
voltages, use right side of interval for voltage). The sample rate is 10 s intervals.
Then convert to 3-bit binary.
4
3
2
1
0
-1
t / s
-2
-3
-4
10
20
30
40
50
60
70
80
90 100
Voltage (+): ____ ____ ____ ____ ____ ____ ____ ____ ____ ____
3-bit binary: ____ ____ ____ ____ ____ ____ ____ ____ ____ ____
10. Sketch a track of pits and bumps recording the above analog signal as a 3-bit digital
signal on a CD: Each rectangle represents 3-bits. The first two samples are done.
Then sketch the play-back signal on the graph above.
10
20
30
40
50
60
70
80
90
100
t / s
Topic 14 - Problem Set
14.1.3
11. Explain how interference of light is used to recover information stored on a CD.
14.1.4
12. Laser light having a frequency of 2.651015 Hz is used to read back
digital information from a disk. Calculate the depth of a pit on the disk.
14.1.5
13. How often should an audio signal be sampled so that its playback is high fidelity?
14. Give two ways to increase the fidelity of digitally recorded sound:
15. If the values stored digitally on a CD are 2 bytes long, how many
discrete (quantum) values can each sample have?
16. What is the minimum sample frequency for an audio CD in Hz?
17. What is the sample rate for an audio CD in ms if the sampling just meets the
requirements of problem 16?
18. How many quantum levels can be achieved with a 4-bit binary code?
19. Why is 4-bit binary code not suitable for storing music on a CD?
20. What is the purpose of the CLOCK in the Analog-to-Digital Converter?
21. A CD rotates at about 500. revolutions per second (on average) and a focused laser
beam reads the data stored as pits and bumps by measuring the intensity of the
reflected beam. Recall that if the beam is located at a bump-pit junction, the beam
undergoes destructive interference, and hence has a low intensity. The data track
begins at the center and spirals outward to the edge of the 12.0-cm diameter CD.
(a) Estimate the length of the track, given that the track width is 2.10 m.
Hint: Find the “average circumference” and multiply by the number of
track-widths that can fit in the radius of the CD.
(b) Given that a bump (bit) is 0.970 m long, how many bits can be stored on the
spiral track?
(c) To replay music from an audio CD the data must be read at a constant rate. In
order to do this the rotation rate of the disk must vary according to the radius of the
reader’s position. Explain why this is necessary, and describe how the speed changes
as the radius of the reader increases.
(d) Why is a sample rate of 40,000 Hz desired for an audio CD?
(e) Given that audio CDs record two channels (stereo left and right) and that each
sample is stored in a 16-bit binary string, estimate the number of minutes of playing
time that can be recorded on a 4 gigabit music CD.
Topic 14 - Problem Set
22. A stereo music CD can store up to 74 minutes’ worth of music at a sample rate of
40,000 Hz in 16-bit binary. Show that a CD can store just over 700 MB of data. Hint:
A byte (B) is equal to 8 bits, and stereo means two samples are stored each time, one
for the left and one for the right.
14.1.6
23. List three advantages of storage of information in digital rather than analog form.
14.1.7
24. Discuss the implications for society of ever-increasing capability of data storage.
Be sure to consider moral, ethical, social, economic, and environmental implications.
14.2.1
25. Define capacitance.
26. 175000 electrons are transferred from one plate of a capacitor to the other by a battery
having an emf of 9.25 V. Find the capacitance of the capacitor.
V
27. If the circuit shown to the right is used to charge a capacitor, then the
current falls from a maximum value to zero very quickly.
(a) Outline in terms of electron flow how current can flow for
this short time even though the capacitor is effectively an open
in the circuit (an insulator).
q+
q-
(b) Explain why the current falls from a maximum to zero.
A
14.2.2
28. Describe the structure of a charge-coupled device (CCD). What is a pixel?
29. Explain why there is a difference in shape between a CCD used in a camera, and a
CCD used in a scanner.
14.2.3
30. Explain how incident light causes charge to build up within a pixel.
14.2.4
31. Outline how the image on a CCD is digitized.
14.2.5
32. Define the quantum efficiency of a pixel.
14.2.6
33. Define magnification. Distinguish between digital magnification and optical
magnification. Which one is preferable, and why?
Topic 14 - Problem Set
14.2.7
34. Two points on an object form an image on a CCD. How far apart must the images of
the two points be so that they are resolved?
35. A 2 cm by 2 cm CCD array contains 1000 rows and 1000 columns of picture
elements.
(a) State the number of pixels in this CCD.
(b) Calculate the length of one side of a pixel.
14.2.8
36. Discuss the effects of quantum efficiency, magnification and resolution on the quality
of the processed image.
14.2.9
37. List five practical uses of a CCD, and list some advantages compared with the use of
film.
14.2.10
38. Outline how the image stored on a CCD is retrieved.
14.2.11
39. This question is about a CCD.
(a) A digital camera is used to take a photograph of a freshwater crab. The CCD in
the camera has 5.8106 square pixels. Each pixel has an area of 5.810-10 m2. A claw
of a crab has an area of 1.5 cm2. The image of the claw formed on the CCD is 0.10
cm2. Two spots on the claw are separated by 0.80 mm. Deduce that the images of the
two spots will be resolved.
(b) Light is incident on the image collection area for a time of 50 ms. The number of
photons incident on one pixel is 4.5104. Each pixel has a quantum efficiency of 85%
and a capacitance of 20 pF.
(i) State what is meant by quantum efficiency.
(ii) Estimate the change in potential difference across each pixel.
(c) Outline how the variation in potential difference across individual pixels enables a
black and white (gray scale) image to be produced by a digital camera.
40. A digital photograph is taken of a person whose eyes are 0.060 cm apart. The distance
between the eyes of the image incident on the CCD is 7.110-4 m. The square CCD
has an area of 4.010-4 m2 and contains 5.0106 pixels.
(a) Calculate the magnification of the camera.
(b) Calculate the length of one side of a pixel.
(c) By comparing the distance between two adjacent pixels and the distance between
the eyes in the image on the CCD, determine whether the eyes are resolved.