1 ierg 4100 wireless communications part x: ofdm
TRANSCRIPT
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IERG 4100 Wireless Communications
Part X: OFDM
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Introduction
OFDM: Orthogonal Frequency Division Multiplexing
Converts a wideband frequency selective fading channel into a parallel collection of narrow band frequency flat sub-channels
Reduces the computational complexity associated with high data-rate transmission over frequency-selective channels
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History of OFDM
The basic principles of OFDM was proposed in several publications in the 1960’s.
Since 1966 FDM systems with overlapping spectra were proposed
The next step is a proposal to realize an FDM system with DFT
Finally, in 1971 Weinstein and Ebert proposed a complete OFDM system, which included generating the signal with an FFT and adding a guard interval in the case of multipath channels
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OFDM Applications
Broadcasting DAB (Digital Audio Broadcasting) DVB (Digital Video Broadcasting)
WLAN (Wireless local area network) IEEE 802.11a HiperLan/2
WMAN (Wireless metropolitan area network) IEEE 802.16 (WiMax)
4G LTE (Long Term Evolution)
5G ?
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Motivation
Inter-symbol interference in high-data-rate wireless communications
To avoid ISI, data rate is limited the radio environment – delay spread
Otherwise, equalizer is needed at the receiver to overcome ISI
OFDM can overcome and take advantage of multipath fading and thus eliminate inherent data rate limitations
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Time and Frequency Domain Description of Multipath
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Inter-symbol interference
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Single-Carrier Transmission vs. OFDM
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time
frequency
……
Single carrier transmission:
time
……
frequency
OFDM (Multi carrier transmission):
Each symbol sees a frequency selective fading channel
Each symbol on a subcarrier sees a frequency flat fading channel
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Single Carrier System
Sequential Transmission of WaveformsWaveforms are of short Duration T Waveforms occupy full system bandwidth 1/T
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Multi-Carrier System
Parallel Transmission of waveforms Waveforms are of long duration MT Waveforms occupy 1/Mth of system bandwidth 1/T
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Subcarriers in the Time Domain
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Subcarrier Orthogonality
In conventional FDMA The whole bandwidth is divided
into many narrow sub-channels which are spaced apart and not overlapped.
⇒ Low spectral efficiency In OFDM
By using orthogonal carriers with nulls at the center of the other carriers, the subchannels are overlapped.
⇒ Increase spectral efficiency
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In the frequency domain, the orthogonality is seen by zerosAll other subcarriers are zero when one subcarrier peaks
frequency
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OFDM Transmitter and Receiver
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Add Cyclic
Prefix & Pulse
Shaping
Serial
to
Parallel
IFFT
Parallel
to
Serial
Parallel
to
Serial
FFT
Serial
to
Parallel
Mixer
fc
Mixer &Filter
fc
FrequencyDomainSamples
TimeDomainSamples
Matched Filter
and Remove Cyclic Prefix
channel
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DFT implementation
Equivalent baseband notation
At a sample rate of Ts/N
Since
(I)DFT can be much more efficiently implemented by (I)FFT
1
0
( ) exp 2 , 0N
n n sn
s t d j f t t T
s(k) =skTsN
⎛
⎝⎜⎞
⎠⎟= dnexp j2nk⋅Δf ⋅
TsN
⎛
⎝⎜⎞
⎠⎟n=0
N−1
∑ , 0 ≤k≤N −1
1
0
( ) exp 2 , 0 1N
n nn
nks k d j IDFT d k N
N
ΔfgTs =1
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DFT implementation
Matrix representation
s=FHd
F: FFT matrix
Each dn, n=0, 1, …, N1 is a modulated
frequency domain sample
Each sn, n=0, 1, …, N1 is a sample of the
OFDM symbol, i.e., time domain sample
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OFDM Signal in the Time Domain
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Guard Interval
OFDM deals with ISI within one OFDM symbol (OFDM block)
Inter-block interference still exists Solution: Insert a guard interval that is longer than
the delay spread
Guard interval can consist of no signal. In this case, however the problem inter-carrier interference (ICI) would arise, since sub-carriers are no longer orthogonal
By cyclic prefix in OFDM symbol, ISI and ICI can be eliminated completely
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Cyclic Prefix
When the length of the cyclic prefix is larger than the delay spread, there is no inter-block interference after the cyclic prefix is removed
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Matrix representation of the ISI channel
Assume channel impulse response length is P
Matrix representation
1
0
P
t k t k tk
y h s n
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Circulant Matrix
A Circulant matrix is an n-by-n matrix whose rows are composed of cyclically shifted versions of a length-n list. For example, the circulant matrix on the list l={1, 2, 3, 4} is given by
One important property: a circulant matrix can be diagonalized by the Fourier transformation matrix
1 2 3 4
2 3 4 1
3 4 1 2
4 1 2 3
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Cyclic Prefix
In order to form a circulant matrix, instead of transmitting s, we transmit
Assume P=1, then
1 1, , , ,TT
N P N P Ns s s s s
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Cyclic Prefix
An effective circulant matrix is created using cyclic prefix
Efficiency: with ,since a vector of length will be transmitted for a length-N data vector
When N increases, efficiency increases
H
( )sN N N 1sN P
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Diagonalization of Circulant Matrix
Circulant matrix can be diagonalized aswhere
N parallel flat fading subchannels are created Note, the transmitter can diagonalize
without knowing any information about
HHFHF D
1exp 2kn
knj
NN
F
1
0
exp 2N
H knnk
knh j
N
D Gain of a sub-channel
HH
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Advantages of OFDM
With cyclic prefix, intra and inter OFDM symbol ISI can be eliminated completely
An effective circulant matrix can be created using cyclic prefix, as a result, ICI can be eliminated completely
Implementation complexity is significantly lower than that of a single carrier system with an equalizer
Provide frequency diversity Forward error correcting code such as convolutional
code with interleaver is needed as some sub-carriers will be in deep fade
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Fading Across Subcarriers
Example:
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h(τ ) =0.8δ(τ ) + 0.6δ(τ −Ts)
diag(DH ) =FFT 0.8, 0.6, 0, 0, 0, 0, 0, 0[ ]( )=[1.4, 1.22 −0.424i, 0.8 −0.6i, 0.376 −0.424i, 0.2, 0.376 + 0.424i, 0.8 + 0.6i, 1.224 + 0.424i]
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Different BERs Across Subcarriers
Compensation technique Coding across subcarriers Adaptive loading (power and rate)
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Variable-Rate Variable-Power MQAM
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γ : Channel to noise ratio | h |2 N0 B
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Adaptive Techniques
Variable-rate variable-power techniques Fixed BER, maximize average data rate Fixed data rate, minimize average BER Fixed BER and data rate, minimize
average power
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Formulation
BER in non-fading AWGN channel with MQAM (M>=4) modulation and coherent detection:
Adaptive MQAM for fixed BER
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BER ≤0.2e−1.5γPM−1
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Rate Maximization in Single-Carrier Systems
Optimal solution: Water filling
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maxP(γ)
E log2 M (γ)( )⎡⎣ ⎤⎦=maxP(γ )
E log2 1+1.5γP(γ)−ln 5BER( )
⎛
⎝⎜⎞
⎠⎟⎡
⎣⎢⎢
⎤
⎦⎥⎥
s.t. E P(γ)[ ] =P
P*(γ) =1γ0
−1γ
γ > γ0
0 otherwise
⎧
⎨⎪
⎩⎪
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Power Minimization in Single-Carrier Systems
Practical (suboptimal) solution: Fix M. Transmit at the minimum power that
meets the BER performance
Optimal solution: water filling with a carefully chosen water
level31
minM (γ)
E P γ( )⎡⎣ ⎤⎦=minM (γ )
EM (γ)−1( ) ln
15BER
1.5γ
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
s.t. E M (γ)[ ] =M
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Constellation Restriction
M is restricted to {0, …, MN} Carefully design region boundaries Power control maintains target BER
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Adaptive Loading in Multi-Carrier Systems
Pros: Smaller rate and power fluctuation Requires smaller buffer size Channel gains are known
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Rate Maximization
Concave maximization Transmit power per OFDM symbol is
fixed Constellation constraint can be imposed
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maxPk (γ)
log2 Mk(γk)( )k=1
N
∑ =maxPk(γ )
log2 1+1.5γkPk(γk)−ln 5BER( )
⎛
⎝⎜⎞
⎠⎟k=1
N
∑
s.t. Pk(γk)k=1
N
∑ =NP
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Power Minimization
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minM k (γ)
Pk γk( )k=1
N
∑ =minMk(γ )
Mk(γk)−1( ) ln1
5BER1.5γk
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥k=1
N
∑
s.t. Mk(γk)k=1
N
∑ =NM
Linear programming Data rate per OFDM symbol is fixed Constellation constraint can be imposed