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Section 4
Orthogonal Frequency Division Multiplexing (OFDM)
Relatively new modulation scheme.
Used for DAB, ADSL & more recently wireless computer
networks (IEEE 802.11a and Hiperlan2).
It is said to be a "multi-carrier" modulation scheme
OFDM has very many, say 64 or even 1024, carrier frequencies
evenly spaced out over a range of frequencies.
A vector-modulator can take i(t) + jq(t) as a complx b-b signal.
Multiplying cos(2fCt) by i(t) & sin(2fCt) by q(t) & summing is:
Real { (i(t) +jq(t)) exp(2jfCt) }
Instead of one carrier, IEEE802.11a takes 64 carrier frequencies
over the range fC to fC + 20 MHz:
fC + 0,
fC + fD, fC + 2fD, … ,
fC +63fD
with fD = 20MHz / 64 =312.5 kHz.
We do the OFDM modulation in two stages:
(1) Apply PSK, QPSK, QAM (or other) to 64 'sub-carriers' of
frequencies: 0 , fD, 2fD , …, 63fD
(2) Vector-modulate fC with sum of modulated sub-carriers.
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There are 64 modulating signals:
X0(t) = I0(t) + jQ0(t) : modulating 0 Hz
X1(t) = I1(t) + jQ1(t) : modulating fD
X2(t) = I2(t) + jQ2(t) : modulating fD
….
X63(t) = I63(t) + jQ63(t). : modulating 63fD
With QPSK, each Xi represents 2 bits.
(IEE802.11a makes X0-X5 & X58-X63 all zero
and uses 4 others for "pilot tones", leaving 48 to use.).
Adding these together we obtain:
63
x(t) =  Xk(t) exp (2jkfD t ) : -<t<.
k=0
Sampling at 20MHz, (i.e. T = 0.05 us) this becomes:
63
x(nT) = x[n] =  Xk(nT) exp (2jk fD nT )
k=0
Make Xk[nT] =Xk : constant for 0<n<79, i.e. 4 us.
63
x[n] =  Xk exp(jk(2/N)n) :
k=0
0<n<79
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Generates a set {x[0], x[1], …, x[79]} of complx numbers.
Time-domain OFDM "symbol" lasting 4us.
Shape of pulse tell us the information.
With QPSK on 48 carriers, 248 x 2 = 4 x 10 14 different shapes.
250,000 symbols /s can be strung together.
Real part multiplies cos(2fCt) & imag pulse sin(2fCt).
Complex multiplication gives one sideband.
Expression for {x[n]} is "inverse DFT" of {X0, X1, …., X63} .
Normally generates complex sequence {x[0], x[1], …., x[63]}
With 63 samples, {x[0], x[1], …., x[63]}, no information lost.
{x[n]}0,63 contains all the information in {Xk}0,63 .
DFT of {x[n]}0,63 gets back exactly to {X0, X1, …., X63}.
OFDM demodulator is DFT followed by detectors
But we calculate {x[0], x[1], …, x[63], x[64], …, x[79]}
Inverse DFT repeats cyclically for n > 63.
So x[64] = x[0], x[65]=x[1], …, x[79]=x[15].
x[64] to x[79] is the "cyclic extension".
"Guard interval" between one symbol & the next;
Useful for carrier and symbol synchronisation at receiver.
Due to cyclic extension & cyclic nature of DFT and its inverse,
even if exact synchronisation is not achieved at the receiver,
exact data can still be recovered with a phase shift.
With 64 = 26 sub-carrier frequencies, inverse DFT can be carried
out very efficiently by an "fast Fourier transform" (FFT).
OFDM works because of orthogonality of the 64 carriers.
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Very good for channels affected by frequency selective fading
for several reasons.
First, information can be spread out across the sub-carriers in
very intelligent ways so that when some are lost due to fading,
others will compensate. This can even be done adaptively.
Second the guard-band allows for ISI, so that if one 4us OFDM
symbol rings on for a while, it only affects the beginning of the
next symbol, and this information is repeated at the end.
(Can have cyclic "prefix" instead).
So no pulse-shaping necessary!
Third (most importantly) equalisation much easier than with
single carrier systems.
Adaptive filtering is complicated
OFDM equalisation done frequency-domain after FFT.
Multiplies the FFT spectrum by a weighting function.
Multiplication much easier than adaptive filtering.
Works because of cyclic extension.
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