ELEC 1010 HW 3

Sampling frequency = 400 Hz
4-bit quantizer
amplitude -1.0 to 1.0

period = 1/400 = 0.0025 s = 2.5 ms

Q1

a) circle at time = 0, 2.5, 5, 7.5, 10, 12.5

b)

Level	Quantization	Digital
15	0.875 → 1	1111
14	0.75 → 0.875	1110
13	0.625 → 0.75	1101
12	0.5 → 0.625	1100
11	0.375 → 0.5	1011
10	0.25 → 0.375	1010
9	0.125 → 0.25	1001
8	0 → 0.125	1000
7	-0.125 → 0	0111
6	-0.25 - > -0.125	0110
5	-0.375 → -0.25	0101
4	-0.5 → -0.375	0100
3	-0.625 → -0.5	0011
2	-0.75 → -0.625	0010
1	-0.875 → -0.75	0001
0	-1 → -0.875	0000

c)
0, 0.45 → 1011
2.5, -0.45 → 0100
5, 0.3 → 1010
7.5, -0.6 → 0011
10, 0.5 → 1011/1100
12.5, 0.2 → 1001

d)

400 * 4 = 1600 bits/sec

Q2

a)

$$\begin{array}{rl}\text{Entropy}& =\frac{9}{330}{\log }_2\frac{330}{9}\\ & +\frac{100}{330}{\log }_2\frac{330}{100}\\ & +\frac{120}{330}{\log }_2\frac{330}{120}\\ & +\frac{65}{330}{\log }_2\frac{330}{65}\\ & +\frac{30}{330}{\log }_2\frac{330}{30}\\ & +\frac{6}{330}{\log }_2\frac{330}{6}\\ & \approx 2.075679bits/students\end{array}$$

https://www.wolframalpha.com/input?i=%289%2F330%29log%282%2C330%2F9%29%2B%28100%2F330%29log%282%2C330%2F100%29%2B%28120%2F330%29log%282%2C330%2F120%29%2B%2865%2F330%29log%282%2C330%2F65%29%2B%2830%2F330%29log%282%2C330%2F30%29%2B%286%2F330%29log%282%2C330%2F6%29

$$\begin{array}{rl}\text{Minimum amount of bits required}& =330\cdot 2.075679\\ & \approx 684.97411\mathrm{bits}\end{array}$$

https://www.wolframalpha.com/input?i=%28330%29%28%289%2F330%29log%282%2C330%2F9%29%2B%28100%2F330%29log%282%2C330%2F100%29%2B%28120%2F330%29log%282%2C330%2F120%29%2B%2865%2F330%29log%282%2C330%2F65%29%2B%2830%2F330%29log%282%2C330%2F30%29%2B%286%2F330%29log%282%2C330%2F6%29%29

685 bits are required

b)

$$\begin{array}{rl}\text{Minimum amount of bits required}& =330\left(6\cdot \left(\frac{\frac{330}{6}}{330}\right){\log }_2\left(\frac{330}{\frac{330}{6}}\right)\right)\\ & =330\left(6\cdot \frac{1}{6}\cdot {\log }_2\frac{6}{1}\right)\\ & \approx 853.0376252\mathrm{bits}\end{array}$$

854 bits are required