SpaceTime Coding for FrequencySelective Fading Channels

1
Space-Time Coding for Frequency-Selective Fading
Channels

• Anupama Lakshmanan
• Kwok Wong
• April 30, 2003

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Outline

• Flat-fading
• Diversity
• Space-time codes
• Orthogonal constructions and easy decoding
• Frequency-selective fading
• OFDM
• Diversity
• OFDM atop space-time codes?

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Flat- vs. Frequency-selective

• Flat-fading Symbol period gt Delay spread
• Frequency-selective Symbol period lt Delay spread

Flat-fading illustration

• - Multi-paths within symbol period
• - When they combine, constructive or destructive
  interference
• - Some symbols can get deeply faded, while some
  can get amplified
• - System design for worst case deep fades
• - Diversity

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Diversity

• Two receive antennas
• Maximal ratio combining
• Fundamental principle its less likely that
  both receive antennas suffer deep fades
• Two transmit and One receive antenna
• Alamouti scheme
• Transfers complexity to base station
• With double the transmit power, performance
  identical to 1 Tx antenna, 2 Rx antenna system
  (above)
• Assumes fades are nearly same over two symbols
• Quasi-static assumption (highly correlated or
  slow fading)
• Other techniques
• Space-time trellis codes
• Orthogonal design Space-time block codes
  (ODSTBC) Tarokh et al.

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Channel Model
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Transmission Matrix

• Average energy of symbols transmitted from each
  antenna normalized to 1/M.
• One receive antenna (multiple receive antenna
  handled similarly)

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• Received signal at time t
• nt are i.i.d and CN(0,1/(2SNR))
• In matrix notation,

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• Maximum likelihood decoding
• map received point to closest constellation point
  (closest S to received vector)

Cross terms, i.e., coupling of symbols makes
decoding computationally intensive
9

• Can we decouple (diagonalize) SS for
  transmission matrix S?
• Yes, make the transmissions in different antennas
  orthogonal,
• SS
• SS diagonal matrix
• Orthogonal design STBC

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• SS D diagonal matrix

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Example Alamoutis scheme

• Two transmitters and one receiver
• S
• Received values at times 0 and 1 form vector y
• SS

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Alamouti scheme decoding
Linear in ci
Linear in ci2

• Optimization over A2 possibilities (exponential)
  reduced to optimization over 2A possibilities
  (linear)

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• Maximum likelihood decoding for Alamouti scheme
  revisited

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Coding schemes for arbitrary number of transmit
antennas using orthogonal designs

• A real orthogonal design of size n an n x n
  orthogonal matrix with entries the indeterminates
  ?x1,?x2,,?xn.
• Theorem Tarokh, et al.
• An orthogonal design of size n exists if and
  only if n 2, 4 or 8.

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Complex orthogonal designs

• A complex orthogonal design of size n an n x n
  orthogonal matrix with entries the indeterminates
  ?x1,?x2,,?xn, their conjugates ?x1,?x2,,?xn,
  or multiples of these indeterminates by ?i where
  i ?-1.
• Theorem Tarokh, et al.
• A complex orthogonal design of size n exists only
  if n2.
• Alamouti scheme is in a sense unique
• Real orthogonal designs exist only for n 2, 4
  or 8.

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Flat- vs. Frequency-selective

• Flat-fading Symbol period gt Delay spread
• Frequency-selective Symbol period lt Delay spread
• Can we turn a frequency-selective fading channel
  into a flat-fading channel?
• Yes, increase the symbol period Tnew KT
• One big symbol lasts K smaller symbol periods
• Lose throughput 1/(KT) symbols/s instead of 1/T
  symbols/s

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Throughput and Multi-path Resistance

• To gain back throughput
• Different codes carry independent bits
• CDMA Codes are of the following type
• Inputs are orthogonal
• But channel responses make the received signals
  no longer orthogonal
• OFDM Codes are of sinusoids
• Inputs are orthogonal
• Channel does not distort shape of input
  (eigenfunctions)
• Outputs are also sinusoids (complex) scaled
  versions of input sinusoids
• Outputs remain orthogonal different sinusoids
  act as separate non-interacting pipes carrying
  data
• Over to Kwok

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Review of OFDM

• Orthogonality (of sub-carriers) Frequency
  spacing between sub-carriers multiple of symbol
  rate (symbol-period/carrier-period integer)

N sub-carriers
Windowed sinusoids
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Review of OFDM (cont.)

• Maps a block of scalar symbols (complex numbers)
  into a vector (with each component modulated on
  orthogonal sub-carries (bases))
• Transform serial (high rate) stream to N parallel
  (low rate) streams
• Symbol period gtgt delay spread of channel
  transform wideband frequency-selective channel
  into N parallel, narrowband channels with
  flat-fading
• Each OFDM symbol can be considered as a vector.
• Maintenance of orthogonality needed for
  demodulation (decoding)
• Integration (matched filtering) over one complete
  symbol period
• Amplitude and phase of the sub-carrier must
  remain constant over integration period

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Effects of Multipath and Frequency-Selective
Fading

• Frequency-selective fading can cause groups of
  neighboring sub-carriers to be heavily attenuated
  (need diversity)
• Multipath
• Overlap ( Td) of adjacent symbols (ISI) causes
  transient fluctuation in amplitude and phase of
  sub-carriers (breakdown of orthogonality causes
  ICI) -- this region must be excluded from the
  integration (matched filtering) interval
• Delayed copies cause amplitude scaling and phase
  rotation in non-overlapped region

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Symbol 3
Symbol 1
Symbol 2
Path 1
Symbol 1
Symbol 2
Symbol 3
Constant amplitude scaling and phase rotation
Path 2
ISI
ISI
Non-constant amplitude scaling and phase rotation
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Guard-Band (Cyclic-Prefix)

• DFT ? periodicity of symbol waveform
  cyclic-prefix periodically extends symbol
  waveform beyond one period
• Tolerant of timing offset
• Integration need not start exactly at beginning
  of each symbol period (i.e., beginning within the
  guard-band OK, so long as it is one complete
  period)
• Timing offset introduces phase rotation of all
  sub-carriers (proportional sub-carrier
  frequencies) taken care of by equalization
• ISI reduces effective length of
  guard-band/timing-offset
• ISI is localized within guard-band transforms
  frequency-selective channel into N parallel
  flat-fading sub-channels
• The remaining effects caused by multipath
  (amplitude scaling and phase rotation) are
  corrected by channel equalization

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Diversity Needed Deep Fading of Sub-carriers

• Error Correction Coding (time diversity)
• Works most effectively if the errors are randomly
  (uniformly) distributed.
• Interleave (scramble) serial data prior to
  sequential assignment to sub-carriers destroys
  correlation between burst errors of consecutive
  symbols (neighboring sub-carriers) caused by deep
  fading
• Multiple Rx-antennas (receive diversity
  maximum-ratio combining)
• Multiple Tx and Rx antennas Space-Time Coding

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Incorporation of Space-Time Coding for Added
Diversity
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OFDM with S.T.C. (cont.)

• Received signals at two consecutive time steps
  are given by
• Use Alamoutis scheme to code the transmitted
  sequence of symbol-vectors
• Giving

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OFDM with S.T.C. (cont.)

• Generalizing Alamoutis decoding scheme
• Decoding scheme decouples codewords
• and
• Benefit of transmit diversity

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OFDM with S.T.C. (cont.)

• Since the channel ? is diagonal, we can apply the
  decoding scheme to each component of the received
  symbol-vector separately
• Yields N decoupled sets of Alamoutis 2Tx-Rx
  system.

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Summary

• S.T. coding assumes flat fading.
• OFDM transforms a frequency-selective channel
  into N parallel, non-interfering, flat
  sub-channels.
• S.T. coding is used with OFDM to provide transmit
  diversity for each sub-channel.

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References

• 1 S.M.ALamouti, A simple transmit diversity
  technique for wireless communications, Journal
  of Selective Communications, vol. 16, no. 8, pp.
  1451-1458, October 1998.
• 2 V.Tarokh, R. Jafarkhani, and A.R. Calderbank,
  Space-time block codes from orthogonal designs,
  IEEE Trans. Inform. Theory, vol. 45, pp.
  1456-1467, July 1999.