Summary of Basic Concepts

Transcription

1 Transmission Summary of Basic Concepts Sender Channel Receiver Dr. Christian Rohner Encoding Modulation Demodulation Decoding Bits Symbols Noise Terminology Communications Research Group Bandwidth [Hz] - range of frequencies that pass through a medium with minimum attenuation. - this is a physical property of the medium. Symbol rate [symbols/s; samples/s; baud] - rate at which symbols (samples) are sent - at most 2B symbols/s (B: Bandwidth of the channel [Hz]) Data rate [bit/s] - amount of bits sent over the channel per second. - equal to number of symbols/s times bits/symbol. - 1 kbit/s = 1 bit/s (but: 1 kbyte = 2^1 Byte = 124 Byte) Fourier Analysis f(x) = 1 2 a + a n cos( nπ T x) + n=1 a = 1 T a n = 1 T f(t)dx n=1 f(t)cos( nπ T x)dx b n sin( nπ T x) b n = 1 T f(t)sin( nπ T x)dx

2 Channel Capacity Henry Nyquist 1924: even a perfect channel has a finite transmission capacity: B: bandwidth [Hz], V: discrete levels of signal R max = 2B log 2 V bit/s Hartley-Shannon: upper bound for a noisy channel: R max = B log 2 (1 + S/N) bit/s Noisy Channel Coding Theorem Can we send faster than the channel capacity? R < C - there exists a coding technique which allows the probability or error at the receiver to be made arbitrarily small. R > C - the probability or error at the receiver increases without bound as the rate is increased. - No useful information can be transmitted beyond the channel capacity. Capacity [bit/s] is not Bandwidth [Hz]! Block Error Rate Block Error Rate 1e- Message of S bits Block error rate p ɛ =P(bit error) P(no errors in S bits)= (1 ɛ) S P(one or more errors in S bits)= p = 1 (1 ɛ) S = 1 = 1 P(one or more errors in S bits ɛ = 1 3, S = 1 p = 1 (1 1 3 ) 1 =.63 Example block error probability 1e-1 1e-2 S=1 S=1 S=1 1e-6 1e-5 1e-4 1e-3 1e-2 1e-1 1e- bit error probability

3 Error Detection Known message Space Parity Check - XOR over all data bits, add it to the message. Block sum check - n blocks of m bits (i.e., n x m array) - parity for every row, and every column Cyclic redundancy check (CRC) - designed to detect error bursts - polynomial code - a generator polynomial of R bits will detect all single-bit errors, all double-bit errors, all odd number of bit errors, all error bursts < R, most error bursts! R. Error Detection Two-dimensional Parity Example Checksum UDP Message: Transmitted Data: s compl Check:

4 Example CRC - Computation Example CRC - Verification Generator"" G=11 Message"" " D=1111 Generator"" G=11 Message"" " D=1111 Transmitted D!2^3 : G = D!2^3 R : G = R= Check: Shared Medium Links are only seldom point-to-point... - Ethernet - Wireless (Wifi, Bluetooth, GSM, etc.) Hidden node, exposed node Multiple Access: - TDMA, FDMA, CDMA - statistical multiplexing: CSMA, CSMA/CD, ALOHA, etc. Problem 1: Collisions, Errors Problem 2: Delay to get access to the medium Network Model Sender Network Node: - Queue Receiver

5 Network Model... Problem: Congestion due to high traffic load. Result: Delay Problem: Loss due to full queues.