CORE1240 HW4

Q1

(a)

(b)

Symbol	Frequency	Codeword (Huffman)
A	60	00
B	40	01
C	30	11
D	15	101
E	12	1000
F	8	10010
G	2	10011

(c)

Average codeword length = \frac{60}{167} \cdot 1 + \frac{40}{167} \cdot 3 + \frac{30}{167} \cdot 3 + \frac{15}{167} \cdot 4 + \frac{12}{167} \cdot 4 + \frac{8}{167} \cdot 4 + \frac{2}{167} \cdot 4 \approx 2.502994012 bit/symbol

(d)

Average codeword length = \frac{60}{167} \cdot 2 + \frac{40}{167} \cdot 2 + \frac{30}{167} \cdot 2 + \frac{15}{167} \cdot 3 + \frac{12}{167} \cdot 4 + \frac{8}{167} \cdot 5 + \frac{2}{167} \cdot 5 \approx 2.413173653 bit/symbol

(e)

Minimum average number of bits per symbol = \frac{60}{167} lo g_{2} \frac{167}{60} + \frac{40}{167} lo g_{2} \frac{167}{40} + \frac{30}{167} lo g_{2} \frac{167}{30} + \frac{15}{167} lo g_{2} \frac{167}{15} + \frac{12}{167} lo g_{2} \frac{167}{12} + \frac{8}{167} lo g_{2} \frac{167}{8} + \frac{2}{167} lo g_{2} \frac{167}{2} \approx 2.341068446 bits/symbol

(f)

2^{n} n n \geq 7 \geq lo g_{2} 7 \geq 2.807354922

The minimum average number of bits required is $3 bits/symbol$

Q2

(a)

A single bit error: D2

(b)

Cannot correct, multiple errors

(c)

A single bit error: P4