Some of the Existing Systems

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Some of the Existing Systems
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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.

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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

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Wired communication Optical Backbone

• SONET Synchronous Optical Network
• OC-1 51.84Mbps
• OC-3 351.84Mbps
• OC-9 951.84Mbps
• Used for backbone switching

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Cellular Phone Networks

• User base station Telephone network
• FDMA Frequency division multiplexing
• TDMA Time division multiplexing
• CDMA Code division multiplexing

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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.

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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 cant hear other pair. CDMA
  means each pair talks in a different language and
  other peoples voice is noise to them.

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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.

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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.

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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.

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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 mint,r
  channels.

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Error Control Code
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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)

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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

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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

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(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., c0i0.
  Then produce three parity checking bits c4, c5,
  and c6 as (additions are in the binary field)
• c4i0i1i2
• c5i1i2i3
• c6i0i1i3
• For example, (1,0,1,1) coded to (1,0,1,1,0,0,0).
• Is capable of correcting one bit error

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Generator matrix

• Matrix representation. CIG where

• G is called the generator matrix

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Parity check matrix

• It can be verified that CH(0,0,0) for all
  codeword C

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Error Correction

• Given this, suppose you receive RCE. You
  multiply R with H SRH(CE)HCHEHEH. 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.

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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
  1s) 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, wont confuse it with
  other codewords

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The existence of H

• We didnt 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.

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Cyclic Code
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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 2t1, capable of detecting 2t error
  bits and correcting t error bits.

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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.

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Cyclic code

• Polynomial representation of cyclic codes. C(x)
  C_n-1xn-1 C_n-2xn-2 C_1x1
  C_0x0, 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 x6x4x1.
• 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 x3x2x1 by x1.

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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 xn-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).

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Example

• A (7,4) cyclic code g(x) x3x1.
• If m(x) x31, C(x) x6x4x1.

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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?

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Example

• The cyclic shift of C(x) x6x4x1 is C1(x)
  x5x2x1.
• It is still a code polynomial, because it the
  code polynomial if m(x) x21.

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Cyclic Code

• Given a code polynomial

• We have

• C1(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.

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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 xn-1.

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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 33-628
  can be divided by 7.

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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)

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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 its the BCH code.

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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

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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.