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Principles of Orthogonal Frequency Division Multiplexing
and Multiple Input Multiple Output Communications Systems
OFDM
OFDM Material
• Multicarrier communications
• Synchronization
• Issues– Synchronization– Sidelobes
• OFDMA
Intersymbol Interference
• Occurs when symbol period (Ts) is less than channel delay spread,
• ISI introduces an error floor to BER– Limits maximum throughput
• Solutions:– Equalization (high complexity)– Longer symbol periods (generally
means lower data rate)
+
x
+
+
+
+
+
x
x
x
x
x
10-210-4
10-3
10-2
10-1
10-1 100
BPSK QPSK OQPSK MSK
Modulation
Coherent Detection
Irre
du
cib
le B
ER
T
=delay spread
symbol period
BER Floor for various modulations
J. C.-I. Chuang, "The Effects of Time Delay Spread on Portable Radio Communications Channels with Digital Modulation," IEEE JSAC, June 1987
QPSK limit
Multicarrier communications: Longer period, same data rate
•Concept:–Divide original data stream at rate R into L lower rate (R/L) streams on different carriers to increase symbol time
•Long history–KINEPLEX–ANDEFT–KATHRYN
•Effects–High receiver complexity
• separate receiver chain per carrier
–Bandwidth due to sidebands–Each subcarrier experiences flat fading
Bc
B/L
H f
f
B
J. Andrews, A. Ghosh, R. Muhamed, Fundamentals of WiMAX, Prentice Hall, 2007
OFDM
• Much simpler to create multicarrier transmission using iFFT– Information carried in magnitude and phase of each bin– Then can be recovered by using FFT at receiver
• Inverse Fourier transform would be an infinite duration sine wave– Cut at Symbol duration Ts
– Rectangular windowing causes sinc spectrum in frequency domain with zeros at 1/Ts
– Orthogonal subcarriersFrequency
Magnitude
T0T0
Guard intervals and intersymbol interference
• If we space OFDM symbols by gaps at least as long as the delay spread, then there will be no intersymbol interference
• However, there will still be interference within the symbol
OFDM Symbol OFDM Symbol OFDM Symbol
Guard interval Guard interval
Delay Spread Delay Spread
Equalization and the DFT• While using longer symbol timing means OFDM can avoid
irreducible errors, still have interfering energy in band from multipath– Received signal is the (linear) convolution of channel impulse
response with transmitted signal
• DFT Circular Convolution Theorem– Circular convolution of two discrete vectors in time domain
– Is multiplication in the frequency domain
• Implication: If we can make the system behave like a circular convolution, equalization is trivial– complex multiplication per FFT bin at the receiver
*y h x
y x h
k k kY X H
Cyclic Prefix
• Adding a cyclic prefix at transmitter leads to circular convolution
• Note that misaligned timing still results in a circular convolution, just time shifted– Makes for phase
shifts in FFT bins– Correct that in a
momentJ. Andrews, A. Ghosh, R. Muhamed, Fundamentals of WiMAX, Prentice Hall, 2007
Comments on Cyclic Prefix• We’re transmitting redundant
bits (no information transfer)– Bandwidth penalty: L / (L + v)– Power penalty: L / (L + v)
• Penalty becomes negligible as L becomes large (but there are tradeoffs! – more later)
• Power penalty generally more important in practice where systems are interference limited
• Penalty can be avoided with zero prefix– Nothing transmitted in guard band
(zero prefix)– Receiver adds tail back to
beginning of symbol– Used in WiMedia
•Permits low complexity equalization for same data rates
•Single carrier tap# approximately bandwidth delay product–MAC
•OFDM, number subcarriers grows with bandwidth-delay product, so
J. Andrews, A. Ghosh, R. Muhamed, Fundamentals of WiMAX, Prentice Hall, 2007
Frequency Errors
• Primary sources of frequency errors– Doppler shift– Clock mismatches– Phase noise
• Effects– Reduction in amplitude
(missampling sinc)– Intercarrier
interference
O. Edfors, M. Sandell, J. van de Beek D. Landström, F. Sjöberg, “An Introduction to Orthogonal Frequency Division Multiplexing,” Sep 98,
Available online: http://epubl.luth.se/avslutade/0347-0881/96-16/esb96rc.pdf
Effects of Frequency Errors• Comments
– Impact greater for higher SNR signals
– Note 5% estimation error can lead to 5 dB effective degradation at 64-QAM like SNRs
– Big frequency impact is why OFDM was originally for fixed deployments
• Techniques– Data aided– Non data aided– Cyclic prefix
Fading ChannelAWGN
O. Edfors, M. Sandell, J. van de Beek D. Landström, F. Sjöberg,
“An Introduction to Orthogonal Frequency Division Multiplexing,”
Sep 98, Available online: http://epubl.luth.se/avslutade/0347
-0881/96-16/esb96rc.pdf
Solution Techniques
• Clipping– Eliminate signals above a certain
level or ratio• Peak windowing
– Filter peaks• Linear block code
– Select only those codewords with small PAPR
– Can also provide error correction• Peak Cancellation
– Subtract signals from high peaks– Need to be similar bandwidth to
limit out-of-band interfernce• Symbol Scrambling
Spectral Effects of Windowing and Clipping
Peak Cancellation, Clipping, PAPR = 4dB
Time Domain Frequency Synchronization
• Complex baseband model of passband signal
• ftx is transmitter carrier frequency, Ts is symbol period, sn is transmitted signal
• Received
Time Domain Frequency Synchronization
• Evaluate sum of products of time-delayed and conjugated repeated symbols
Estimator
• Frequency offset estimator
• Ambiguity limit
• When D = Ts
• In AWGN, this is the maximum likelihood detector with variance proportional to
Frequency domain (post-FFT)
• Similar estimator evaluated on Fourier Transformed signal
• So same performance, but much more complex as the DFT has to be calculated for both repeated symbols.
Channel Estimation
• Channel assumed static for duration of symbol, though frequency/phase varying over bandwidth
• Solution, embed pilot symbols at regular intervals in the symbol– Used closest pilot– Interpolate
H f
From IEEE Std 802.16-2004
More synchronization
• Need to detect beginning of packet– Energy detect
• Measure energy, see when it exceeds threshold
– Packet detection• Correlate with known sequence
– Delay and correlate
• Symbol timing– No problem to be off by a fraction of the guard interval from
perspective of DFT– Bad timing does get ISI though from cyclic prefixes– Better to be early (low ISI) than late
Synchronization all together
• Steps:– Detect packet beginning– Align symbol boundary– Perform coarse frequency/timing synchronization– Perform fine frequency/timing synchronization– Track changes in channel as needed
802.11a FramingIdentical symbols
Peak-to-Average Power Ratio
• Sum of large number of (somewhat) independent subcarriers leads to signal distribution that is somewhat Gaussian
• Implications– long tails for amplitude distribution– Possibly large ratios of peak-to-
power ratios• Long tails can drive amplifiers
into nonlinear region– Introduces harmonics and
significant out-of-band spectral energy
0 2 4 6 8 10 12 14 1610
-6
10-5
10-4
10-3
10-2
10-1
100
PAPR[dB]
log
(CD
F)
(a )N=16 (b) N=32 (c) N=64 (d) N=128 (e) N=256 (f) N=1024
(f)
(e)
(d)
(c)
(b)
(a)
PAPR CDF for Varying # Subcarriers
11
11
MDSMDS
P1dB,out
P1dB,in
1dBP1dB,out
P1dB,in
1dB
Noise FloorNoise Floor
Fund
amen
tal
Fund
amen
tal
Fund
amen
tal
Input Power (dBm)
Output Power (dBm)
Input Power (dBm)
Output Power (dBm)
BDRBDR
Adaptive Modulation
• Different subcarriers experience different flat fades
• Means different SINR• Adapting modulation
scheme of each subcarrier to its SINR allows the system to approach Shannon capacity
Bc
B/L
H f
f
B
B/L
J. Andrews, A. Ghosh, R. Muhamed, Fundamentals of WiMAX, Prentice Hall, 2007
OFDMA
• Multiple user access with OFDM
• Lots of flexibility possible when splitting up OFDM symbols and frames– Assign different subcarriers
to different users– Assign different time slots
to different users– Vary modulation and
coding– Vary powers– More options available with
antenna arrays
• Allocation algorithms– Maximum Sum Rate– Proportional fairness– Proportional rates
constraints
J. Andrews, A. Ghosh, R. Muhamed, Fundamentals of WiMAX, Prentice Hall, 2007
Antenna Array Algorithms and MIMO
Antenna Array Principles• The use of multiple antennas provide two forms of diversity:–Diversity gain
• Exploit multiple independent channels created by multipath diversity
• Works with uncorrelated antennas–Array gain
• Coherently combine energy from antennas• Works even with perfectly correlated antennas
as received SNR increases linearly with the number of receive antennas
• Adding additional transceiver chains is expensive (SWAP and cost), but can provide tremendous (though competing) gains
– Increase the system reliability (decrease the bit or packet error rate)– Increase the achievable data rate and hence system capacity– Increase the coverage area– Decrease the required transmit power
0 200 400 600 800 1000-10
-5
0
5
10
Samples
Fa
din
g E
nve
lop
es
[dB
]
Receive Diversity
• Oldest and simplest diversity technique
• Receiver leverages independence of fades on antennas– Selection Combining (SC)
• Choose antenna with maximum SINR• Lowest complexity
– Equal Gain Combining (EGC)• Phase align and sum signals across
antennas
– Maximum Ratio Combining (MRC)• Weight signals by SINR• Best performance (system SINR is
sum of antenna SINRs)
...
Receiver
Comparator
Short-TermAverage
PresetThreshold
Antenna
Selection Diversity
Average SNR Improvements
SN
R (
dB)
Antennas
SC
EGC
MRC
Open Loop Transmit Diversity (1/2)• Transmitter sends multiple
signals (possibly copies)– These interfere at the receiver,
but if coded properly, the receiver can recover the signal
• Simplest implementation is orthogonal space time block codes or Alamouti codes1
– Assumes flat constant channel over two symbol periods (may not be true for high mobility)
– Requires channel knowledge at receiver
– No change in rate required
• Receiver Alamouti Operation
• Output SNR 2x1 Alamouti
1. S. M. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE Journal on Selected Areas in Communications, vol 16 pp.1451–1458, Oct 1998
TX Encoder
RX Decoder
h1
h2
Open Loop Transmit Diversity (2/2)
• 2x2 STBC (same transmit encoder) SINR
– Note number of h terms maximized when Nt = Nr for a fixed number of antennas
– Also full-diversity, orthogonal STBCs exist only for certain combinations of Nt and Nr.
• Can also use space-time trellis codes for added 1-2 dB, but those have exponential complexity order
Comparison of STBC and MRC
A 4x2 Stacked Alamouti System
J. Andrews, A. Ghosh, R. Muhamed, Fundamentals of WiMAX, Prentice Hall, 2007
Space-Time Trellis Coding• Convolutional code applied to space and time domain• Each antenna output is mapped into modulation symbol• Maximum likelihood sequence estimator ( Viterbi algorithm)
Encoder structure for two antennasExample) Delay Diversity (by Wittneben [4])
Generator matrix form
Modular-4addition
(u1,u2)a1 a2 a3 a4
Output to ANT2
Output to ANT1
g11 g21 g31 g41
g12 g22 g32 g42
QPSK mapping
[a1 a2 a3 a4]
Closed Loop Systems• Transmit selection diversity
–Antenna(s) chosen which maximizes SINR–Equivalent to receiver selection diversity–Not as good as beamforming–Little bandwidth required–Makes most sense in in deployments with small bandwidths and small delay spreads (low range)
• Linear diversity precoding– Feedback channel state information to transmit encoder– Transmit encoder then attempts to fine encoding matrix which maximizes SNR
at the receiver– Higher SNR than STBC
– Typically use some sort of codebook to reduce feedback bandwidth
Beamforming Systems
0.5
1
1.5
30
210
60
240
90
270
120
300
150
330
180 0
desiredsignal
interferer
Narrowband adaptive array or linear combiner
w1
w2
wM
x1(t)
x2(t)
xM(t)
... y(t)
• The weight vector is adjusted to improve the reception of some desired signal
– Angle of arrival• MUSIC, ESPRIT
– Eigenbeamforming• No physical interpretation, but useful in multipath
environment• Minimize some cost function
• Useful for interference rejection, multipath fading mitigation, and increased antenna gain
Adaptive Beamforming• Narrowband beamforming is equivalent to spatial filtering
–By choosing appropriate sensor coefficients, it is possible to steer the beam in the desired direction
–By varying the sensor coefficients (spatial filter taps) adaptively, the interference is reduced
• Wideband beamforming requires joint space-time processing
–Phase shift at the antennas is frequency dependent
–Frequency-dependent response is required (filter)• Common algorithms
–Maximum Signal to Interference and Noise Ratio (MSINR)
–Minimum Mean Squared Error–Least Mean Squares–Minimum Variance Distortionless Response (MVDR)–Recursive Least Squares–Similar to linear precoding, but may account for interferers
Performance Comparison
• MRT refers to maximum ratio transmission – the choice of antenna
weights that maximize received SNR
• With optimal eigenbeamformer, canceling an interferer is equivalent to dropping and antenna element
3 dB
Modified from: J. Andrews, A. Ghosh, R. Muhamed, Fundamentals of WiMAX, Prentice Hall, 2007
Spatial Multiplexing• In rich scattering environments, independent data signals
transmitted from different antennas can be uniquely decoded to yield an increase in channel capacity
Source SinkModulation &
CodingDemodulation& Decoding
......
Channel
x1
xM
y1
xN
h11
h1M
hN1
hNM
Spatial Multiplexing Techniques•Open loop (Unknown channel)
–Maximum likelihood• Little gain, except at low SNR
–Zero-forcing• Evaluates pseudo-inverse of H• Can dramatically increase noise power
–MMSE• Minimizes distortion• Like Zero-forcing at high SNR, but
without the instability at low SNR–BLAST
• Layers & codes transmissions across antennas
• Effectively linear receiver with successive interference cancellation
• Receiver iterates through transmission streams using MMSE or ZF
• Works better in lab than real-world due to high SNR requirement
•Closed loop (known channel)–Singular Value Decomposition
• Computationally complex• Capacity (assuming waterfilling)
• For large SNR, capacity grows linearly with rank of H, approximately min{Nt, Nr}
–Approximations guided by• information capacity, • error probability • detection MSE• received SNR
–Can tradeoff multiplexing for diversity
Relative Capacity as function of Antenna Array Technique
• 19 BS, 3 sectors, spaced 2.8 km, mix of users
• Proportional Fair scheduling
Source: WiMAX Forum
Cooperative Antenna Arrays• Concept:
– Leverage other radios to effect an antenna array
• Applications:– Extended vehicular coverage– Backbone comm. for mesh
networks– Range extension with
cheaper devices
• Issues:– Timing, mobility– Coordination– Overhead
source
destination
Transmit Diversity
Cooperative MIMO
Source Cluster Relay cluster
First Hop Second Hop
Source Cluster Relay cluster
First Hop
Source Cluster Relay cluster
First Hop
Source Cluster Relay cluster
First Hop
Source Cluster Relay cluster
First Hop
Source Cluster Relay cluster
First Hop
Destination Cluster
Correlation/Coupling Effects
• Spacing between antennas influence correlation and coupling
• Multipath components can act like interference for beamforming which reduces antenna gain
http://www.ngwnet.ac.uk/files/wspres/mimo2.thompson.pdf[Ref. D. Figueiredo, WPMC’04]
4x4, SNR = 20 dB, 30 AS Beamforming BER
Diversity vs. BeamformingDiversity Combining
• Combine signals from different antenna elements using various algorithms
• Signal from each element is processed separately
• Signals have to be uncorrelated for maximum performance
• Mitigates fading• Increases gain• Can improve polarization match• No interference rejection
capabilities
Adaptive beamforming• Focus the antenna’s gain in the
direction of the desired signal– Achieved by manipulating the
weights associated with each element
• Antenna elements have to be separated by /2 to attain a certain phase difference in the signals– Signals are correlated
• All advantages of diversity combining
• Has interference rejection capabilities– Typically > 20 dB
OFDM/MIMO Summary
OFDM Summary
• OFDM overcomes even severe intersymbol interference through the use of the IFFT and a cyclic prefix.
• Limiting factor is frequency offset– Correctable via simple algorithm when preambles
used• Two key details of OFDM implementation are
synchronization and management of the peak-to-average ratio.
• OFDMA provides a lot of flexibility to a system’s resource allocation– Permits exploitation of multi-user diversity
MIMO Summary• Spatial diversity offers incredible
improvements in reliability, comparable to increasing the transmit power by a factor of 10–100.
• These diversity gains can be attained with multiple receive antennas, multiple transmit antennas, or a combination of both.
• Beamforming techniques are an alternative to directly increase the desired signal energy while suppressing, or nulling, interfering signals.
• In contrast to diversity and beamforming, spatial multiplexing allows multiple data streams to be simultaneously transmitted using sophisticated signal processing.
• Since multiple-antenna techniques require channel knowledge, the MIMO-OFDM channel can be estimated, and this channel knowledge can be relayed to the transmitter for even larger gains.
• It is possible to switch between diversity and multiplexing modes to find a desirable reliability-throughput operating point; multiuser MIMO strategies can be harnessed to transmit to multiple users simultaneously over parallel spatial channels.
J. Andrews, A. Ghosh, R. Muhamed, Fundamentals of WiMAX, Prentice Hall, 2007
Useful References
• O. Edfors, M. Sandell, J. van de Beek, D. Landström, F. Sjöberg, “An introduction to orthogonal frequency division multiplexing,” Sep 1996.
• A. Bahai, B. Saltzbeg, M. Ergen, Multi-Carrier Digital Communications Theory and Applications of OFDM, Springer 2nd edition, 2004.
• J. Andrews, A. Ghosh, R. Muhamed, Fundamentals of WiMAX, Prentice Hall, 2007