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Some of the Existing Systems

Some of the Existing Systems. Wired Communication – Telephone Company. Dial-up – 56kbps DSL – Digital Subscriber Line ADSL: Asymmetric DSL, different upload and download bandwidth

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Some of the Existing Systems

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  1. Some of the Existing Systems

  2. Wired Communication – Telephone Company • Dial-up – 56kbps • DSL – Digital Subscriber Line • ADSL: Asymmetric DSL, different upload and download bandwidth • Available bandwidth is about 1.1MHz, divided into 256 channels, one for voice, some unused or for control, the rest divided among upstream and downstream data. My DSL at Pittsburgh was 100kbps upstream and 768kbps downstream • How ADSL is set up. Fig. 2-29. The ADSL modem is 250 QAM modems operating at different frequencies. The actual QAM depends on the noise.

  3. Wired Communications – The Cable TV Company • Cable frequency allocation. Fig. 2-48. • Downstream channel bandwidth is 6MHz, and may use QAM-64. • Upstream channel is worse so use QAM-4. • Upstream – stations contend for access (MAC layer issue, will be discusses later) • Downstream – no contention, from the head end to user • Shared medium, so some security is needed

  4. Wired communication – Optical Backbone • SONET – Synchronous Optical Network • OC-1: 51.84Mbps • OC-3: 3*51.84Mbps • OC-9: 9*51.84Mbps • … • Used for backbone switching

  5. Cellular Phone Networks • User – base station – Telephone network • FDMA – Frequency division multiplexing • TDMA – Time division multiplexing • CDMA – Code division multiplexing

  6. GSM – Global System for Mobile Communications • Second generation cell phone system (digital, first generation analog). • GSM-900 and GSM-1800 are most widely used • GSM-900 uses 890 - 915 MHz to send information from the Mobile Station to the Base Transceiver Station (uplink) and 935 - 960 MHz for the other direction (downlink). • FDMA + TDMA • Each user transmitting on a frequency and receiving on another frequency. • 124 pairs of 200 KHz channels. Each channel divided into 8 time slots for 8 users. • Each user is has a chance to transmit every 4.615 ms. Each time he can send 114 data bits – 24.7kbps.

  7. CDMA • Your 3G network uses CDMA. • A good analogy in the book – You have a group of people in a room. TDMA means they talk in turn. FDMA means that those who wants to talk sit in different corners and can’t hear other pair. CDMA means each pair talks in a different language and other people’s voice is noise to them.

  8. CDMA • The whole bandwidth is used by every user. Meaning that they can send out symbols really fast. • The trick is to make what A sent appear as 0 to B. • Because we have a fast symbol rate, for each data bit, we send out, say, 8 bits, call the “small bits” chips. • Given a bit, if 1, send out, say, -1,-1,-1,1,1,-1,1,1, and if 0, 1,1,1,-1,-1,1,-1,-1 • This is called the chip sequence. • The key is that each station has a unique chip sequence (language), and different languages are orthogonal. • Fig. 2-45.

  9. Wireless LAN Physical Layer • 802.11b,g in the 2.4G band and 802.11a in the 5G band. People now consider 802.11 as the notion of MAC layer protocol, while a, b, g, or n, are about physical layer. • 802.11b. 1, 2, 5.5, 11Mbps. • 1Mbps: BPSK modulation. 1 bit into 11 chips with Barker sequence +1 +1 +1 −1 −1 −1 +1 −1 −1 +1 −1. Why spread spectrum? Required by FCC but was later removed • 2Mbps: QPSK. • 5.5M and 11M: use some bits to select chip sequence and use two bits for QPSK • 802.11a. Up to 54Mbps. OFDM. • 802.11g. Up to 54Mbps. OFDM. • 802.11n. Up to 600Mbps. OFDM on MIMO.

  10. OFDM (Orthogonal Frequency Division Multiplexing) • In wireless communications, in addition to the bandwidth limit and additive noise, you also have multipath fading! • The faster your symbol rate is, the more badly you will be affected by multipath fading. • In effect, OFDM is like DSL: given a wideband channel, divide it into many sub channels. Each sub-channel can be modulated/demodulated independently. Because each sub-channel is of a much smaller bandwidth, multipath fading is much less severe. • In implementation, use IFFT and FFT.

  11. MIMO • Used in 802.11n. • t transmit antennas and r receive antennas. With the knowledge of channel matrix, by pre-processing the data, equivalent to min{t,r} channels.

  12. Error Control Code

  13. Error Control Code • Widely used in many areas, like communications, DVD, data storage… • In communications, because of noise, you can never be sure that a received bit is right • In physical layer, what you do is, given k data bits, add n-k redundant bits and make it into a n-bit codeword. You send the codeword to the receiver. If some bits in the codeword is wrong, the receiver should be able to do some calculation and find out • There is something wrong • Or, these things are wrong (for binary codes, this is enough) • Or, these things should be corrected as this for non-binary codes • (this is called Block Code)

  14. Error Control Codes • You want a code to • Use as few redundant bits as possible • Can detect or correct as many error bits as possible

  15. Error Control Code • Repetition code is the simplest, but requires a lot of redundant bits, and the error correction power is questionable for the amount of extra bits used • Checksum does not require a lot of redundant bits, but can only tell you “something is wrong” and cannot tell you what is wrong

  16. (7,4) Hamming Code • The best example for introductory purpose and is also used in many applications • (7,4) Hamming code. Given 4 information bits, (i0,i1,i2,i3), code it into 7 bits C=(c0,c1,c2,c3,c4,c5,c6). The first four bits are just copies of the information bits, e.g., c0=i0. Then produce three parity checking bits c4, c5, and c6 as (additions are in the binary field) • c4=i0+i1+i2 • c5=i1+i2+i3 • c6=i0+i1+i3 • For example, (1,0,1,1) coded to (1,0,1,1,0,0,0). • Is capable of correcting one bit error

  17. Generator matrix • Matrix representation. C=IG where • G is called the generator matrix

  18. Parity check matrix • It can be verified that CH=(0,0,0) for all codeword C

  19. Error Correction • Given this, suppose you receive R=C+E. You multiply R with H: S=RH=(C+E)H=CH+EH=EH. S is called the syndrome. If there is only one `1’ in E, S will be one of the rows of H. Because each row is unique, you know which bit in E is `1’. • The decoding scheme is: • Compute the syndrome • If S=(0,0,0), do nothing. If S!=(0,0,0), output one error bit.

  20. How G is chosen • How G is chosen such that it can correct one error? • The key is – Any combinations of the row vectors of G has weight at least 3 (having at least three `1’s) – and codeword has weight at least 3. • The sum of any two codeword is still a codeword, so the distance (number of bits that differ) is also at least 3. • So if one bit is wrong, won’t confuse it with other codewords

  21. The existence of H • We didn’t compare a received vector with all codewords. We used H. • The existence of H is no coincidence (need some basic linear algebra!) Let \Omega be the space of all 7-bit vectors. The codeword space is a subspace of \Omega spanned by the row vectors of G. There must be a subspace orthogonal to the codeword space spanned by 3 vectors which is the column vectors of H.

  22. Cyclic Code

  23. Linear Block Code • Hamming Code is a Linear Block Code. Linear Block Code means that the codeword is generated by multiplying the message vector with the generator matrix. • Minimum weight as large as possible. If minimum weight is 2t+1, capable of detecting 2t error bits and correcting t error bits.

  24. Cyclic Codes • Hamming code is useful but there exist codes that offers same (if not larger) error control capabilities while can be implemented much simpler. • Cyclic code is a linear code that any cyclic shift of a codeword is still a codeword. • Makes encoding/decoding much simpler, no need of matrix multiplication.

  25. Cyclic code • Polynomial representation of cyclic codes. C(x) = C_{n-1}x^{n-1} + C_{n-2}x^{n-2} + … + C_{1}x^{1} + C_{0}x^{0}, where, in this course, the coefficients belong to the binary field {0,1}. • That is, if your code is (1010011) (c6 first, c0 last), you write it as x^6+x^4+x+1. • Addition and subtraction of polynomials --- Done by doing binary addition or subtraction on each bit individually, no carry and no borrow. • Division and multiplication of polynomials. Try divide x^3+x^2+x+1 by x+1.

  26. Cyclic Code • A (n,k) cyclic code can be generated by a polynomial g(x) which has degree n-k and is a factor of x^n-1. Call it the generator polynomial. • Given message bits, (m_{k-1}, …, m_1, m_0), the code is generated simply as: • In other words, C(x) can be considered as the product of m(x) and g(x).

  27. Example • A (7,4) cyclic code g(x) = x^3+x+1. • If m(x) = x^3+1, C(x) = x^6+x^4+x+1.

  28. Cyclic Code • One way of thinking it is to write it out as the generator matrix • So, clearly, it is a linear code. Each row of the generator matrix is just a shifted version of the first row. Unlike Hamming Code. • Why is it a cyclic code?

  29. Example • The cyclic shift of C(x) = x^6+x^4+x+1 is C^1(x) = x^5+x^2+x+1. • It is still a code polynomial, because it the code polynomial if m(x) = x^2+1.

  30. Cyclic Code • Given a code polynomial • We have • C^1(x) is the cyclic shift of C(x) and (1) has a degree of no more than n-1 and (2) divides g(x) (why?) hence is a code polynomial.

  31. Cyclic Code • So, to generate a cyclic code is to find a polynomial that (1) has degree n-k and (2) is a factor of x^n-1.

  32. Generating Systematic Cyclic Code • A systematic code means that the first k bits are the data bits and the rest n-k bits are parity checking bits. • To generate it, you let where • The claim is that C(x) must divide g(x) hence is a code polynomial. 33 mod 7 = 6. Hence 33-6=28 can be divided by 7.

  33. Division Circuit • Division of polynomials can be done efficiently by the division circuit. (just to know there exists such a thing, no need to understand it)

  34. Remaining Questions for Those Really Interested • Decoding. Divide the received polynomial by g(x). If there is no error you should get a 0 (why?). Make sure that the error polynomial you have in mind does not divide g(x). • How to make sure to choose a good g(x) to make the minimum degree larger? Turns out to learn this you have to study more – it’s the BCH code.

  35. Cyclic code used in IEEE 802 • g(x) = x32 + x26 + x23 + x22 + x16 + x12 + x11 + x10 + x8 + x7 + x5 + x4 + x2 + x + 1 • all single and double bit errors • all errors with an odd number of bits • all burst errors of length 32 or less

  36. Other codes • RS code. Block code. Used in CD, DVD, HDTV transmission. • LDPC code. Also block code. Reinvented after first proposed 40 some years ago. Proposed to be used in 802.11n. Achieve close-to-Shannon bound • Trellis code. Not block code. More closely coupled with modulation. • Turbo code. Achieve close-to-Shannon bound.

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