# mobc4s ```Mobile Computing
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07/03/16 BMGC
Section 4
Orthogonal Frequency Division Multiplexing (OFDM)
Relatively new modulation scheme.
Used for DAB, ADSL &amp; more recently wireless computer
networks (IEEE 802.11a and Hiperlan2).
It is said to be a &quot;multi-carrier&quot; 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) &amp; sin(2fCt) by q(t) &amp; 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.
Mobile Computing
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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 &amp; X58-X63 all zero
and uses 4 others for &quot;pilot tones&quot;, leaving 48 to use.).
Adding these together we obtain:
63
x(t) =  Xk(t) exp (2jkfD t ) : -&lt;t&lt;.
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&lt;n&lt;79, i.e. 4 us.
63
x[n] =  Xk exp(jk(2/N)n) :
k=0
0&lt;n&lt;79
Mobile Computing
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Generates a set {x, x, …, x} of complx numbers.
Time-domain OFDM &quot;symbol&quot; 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) &amp; imag pulse sin(2fCt).
Complex multiplication gives one sideband.
Expression for {x[n]} is &quot;inverse DFT&quot; of {X0, X1, …., X63} .
Normally generates complex sequence {x, x, …., x}
With 63 samples, {x, x, …., x}, 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, x, …, x, x, …, x}
Inverse DFT repeats cyclically for n &gt; 63.
So x = x, x=x, …, x=x.
x to x is the &quot;cyclic extension&quot;.
&quot;Guard interval&quot; between one symbol &amp; the next;
Useful for carrier and symbol synchronisation at receiver.
Due to cyclic extension &amp; 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 &quot;fast Fourier transform&quot; (FFT).
OFDM works because of orthogonality of the 64 carriers.
Mobile Computing
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07/03/16 BMGC
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 &quot;prefix&quot; 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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