Anne Reinarz Durham University

Recap: Byte Count

Recap: Byte Count

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Error Correction


Codewords


Error Bounds – Hamming distance

The number of bit positions in which two codewords differ is called the Hamming distance between these two codewords.

Error Bounds – Hamming distance

Example with 2 codewords:


Error Detection and Correction

Hamming distance

Step 1: Ensure the two strings are of equal length.


Error Detection and Correction

Hamming distance

Step 2: Compare the first two bits in each codeword (XOR them). If they are the same, record a “0” for that bit. If they are different, record a “1” for that bit. Repeat the process for the remaining bits.


Error Detection and Correction

Hamming distance

Step 3: Add all the ones and zeros in the record together to obtain the Hamming distance.


Hamming Codes

Encoding:


Hamming Codes

Decoding:


Error Correction – Hamming code

The Hamming code gives a simple way to add check bits and correct up to a single bit error:


Error Correction – Hamming code

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Error Correction – Hamming code

0 0 1 0 0 0 0 1 0 0 1 (original codeword)

0 0 1 0 1 0 0 1 0 0 1 (received codeword)


Error Correction – Hamming code

0 0 1 0 0 0 0 1 0 0 1 (original codeword)

1 2 3 4 5 6 7 8 9 10 11

0 0 1 0 1 0 0 1 0 0&nbsp&nbsp 1&nbsp (received codeword)


Error Correction – Hamming code

0 0 1 0 0 0 0 1 0 0 1 (original codeword)

p1 p2 m3 p4 m5 m6 m7 p8 m9 m10 m11

0 &nbsp 0 &nbsp 1 &nbsp &nbsp 0 &nbsp &nbsp 1 &nbsp&nbsp 0 &nbsp&nbsp 0 &nbsp&nbsp 1 &nbsp&nbsp 0 &nbsp&nbsp 0 &nbsp&nbsp&nbsp&nbsp&nbsp 1 &nbsp&nbsp&nbsp (received codeword)


Error Correction – Hamming code

0 0 1 0 0 0 0 1 0 0 1 (original codeword)

p1 p2 m3 p4 m5 m6 m7 p8 m9 m10 m11

0 &nbsp 0 &nbsp 1 &nbsp &nbsp 0 &nbsp &nbsp 1 &nbsp&nbsp 0 &nbsp&nbsp 0 &nbsp&nbsp 1 &nbsp&nbsp 0 &nbsp&nbsp 0 &nbsp&nbsp&nbsp&nbsp&nbsp 1 &nbsp&nbsp&nbsp


Error Correction – Hamming code

0 0 1 0 0 0 0 1 0 0 1 (original codeword)

p1 p2 m3 p4 m5 m6 m7 p8 m9 m10 m11

0 &nbsp 0 &nbsp 1 &nbsp &nbsp 0 &nbsp &nbsp 1 &nbsp&nbsp 0 &nbsp&nbsp 0 &nbsp&nbsp 1 &nbsp&nbsp 0 &nbsp&nbsp 0 &nbsp&nbsp&nbsp&nbsp&nbsp 1 &nbsp&nbsp&nbsp


Error Correction – Hamming code

0 0 1 0 0 0 0 1 0 0 1 (original codeword)

p1 p2 m3 p4 m5 m6 m7 p8 m9 m10 m11

0 &nbsp 0 &nbsp 1 &nbsp &nbsp 0 &nbsp &nbsp 1 &nbsp&nbsp 0 &nbsp&nbsp 0 &nbsp&nbsp 1 &nbsp&nbsp 0 &nbsp&nbsp 0 &nbsp&nbsp&nbsp&nbsp&nbsp 1 &nbsp&nbsp&nbsp


Error Correction – Hamming code

0 0 1 0 0 0 0 1 0 0 1 (original codeword)

p1 p2 m3 p4 m5 m6 m7 p8 m9 m10 m11

0 &nbsp 0 &nbsp 1 &nbsp &nbsp 0 &nbsp &nbsp 1 &nbsp&nbsp 0 &nbsp&nbsp 0 &nbsp&nbsp 1 &nbsp&nbsp 0 &nbsp&nbsp 0 &nbsp&nbsp&nbsp&nbsp&nbsp 1 &nbsp&nbsp&nbsp


Error Correction – Hamming code

0 0 1 0 0 0 0 1 0 0 1 (original codeword)

0 &nbsp 0 &nbsp 1 &nbsp &nbsp 0 &nbsp &nbsp 1 &nbsp&nbsp 0 &nbsp&nbsp 0 &nbsp&nbsp 1 &nbsp&nbsp 0 &nbsp&nbsp 0 &nbsp&nbsp&nbsp&nbsp&nbsp 1 &nbsp&nbsp&nbsp


Summary


MAC Layer


Outline


Elements of a wireless network

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Elements of a wireless network

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Elements of a wireless network

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Elements of a wireless network

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Elements of a wireless network

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Elements of a wireless network

Ad hoc mode


IEEE 802.11 Wireless LAN


Characteristics of selected wireless links

Standard Frequency Range Data Rate
802.11b 2.4 GHz Up to 11 Mbps
802.11a 5 GHz Up to 54 Mbps
802.11g 2.4 GHz Up to 54 Mbps
802.11n 2.5 and 5 GHz Up to 450 Mbps
802.11ac 5 GHz Up to 1300 Mbps

Wireless Link Characteristics


Signals

A signal is a time-varying quantity (e.g. voltage, or electromagnetic waves) that conveys information.

Properties of signals

y(t) = A sin( 2 π f t + φ )

Here:


Example: wired connection

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Wireless connection

Radio waves are a form of eletctromagnetic radiation with wave forms between 30 Hz and 300 GHz.

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Wireless network characteristics

Hidden terminal problem

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Signal Attenuation

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Exposed Terminals

Exposed terminals are senders who can sense each other but still transmit safely (to different receivers) Desirably concurrency; improves performance B ↔ A and C ↔ D are exposed terminals

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Roadmap