Single carrierMulticarrierOFDM
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Single Carrier
- ISI, Receiver complexity  ISI, Bit rate limitation
Multi-carrier
- Negligible ISI, Approximately flat subchannels,
Less receiver complexity
- Adaptive allocation of power to each subbands
OFDM
- Overlapping spectra still separable at the receiver
- Maximum data rate in band-limited channel
- Digital implementation is possible
- No need of steep bandpass filters
Need of orthogonality in time
functions
Overlapping spectra
Synthesis of OFDM signals for multichannel
data transmission

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No need of perpendicular cutoffs and linear phases
Overall data rate  2Overall baseband bandwidth,
as N  , where N are no. of subchannels

For transmitting filters designed for arbitrary
amplitude characteristics, the received signals remain
orthogonal for all phase characteristics of the
transmission medium

The distances in signal space are independent of the
phase characteristics of the transmitting filters and
the transmitting medium
OFDM of N-data channels over one
transmission medium
To eliminate ISI and ICI  Orthogonality


 Ai2(f) H2(f) Cos 2fkT df
= 0,
k = 1, 2,…
i=1,2, … ,N,
(1)
0

 Ai(f) Aj (f) H2(f) Cos[i(f) - j(f)] . Cos 2fkT df
=0
and
0

 Ai(f) Aj (f) H2(f) Sin[i(f) - j(f)] . Sin 2fkT df
=0
0
for k = 0,1,2, …
i,j =1,2, … ,N,
ij
(2)
f1= (h + ½)fs & fi = f1+ (i-1)fs = (h + i - ½)fs, h is any +ve integer
(3)
T = 1 / 2fs seconds
(4)
Designing transmitter filter
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For given H(f),
Ai2(f)H2(f) = Ci + Qi(f) > 0,
fi - fs  f  fi + fs
=0
f  fi-fs,
f > fi + fs
(5)
where Ci is an arbitrary constant and Qi(f) is a shaping function
having odd symmetries about fi + (fs/2) and fi - (fs/2). i.e.
Qi[(fi+fs/2) + f] = -Qi[(fi+fs/2) - f],
0  f  fs/2
Qi[(fi-fs/2) + f] = -Qi[(fi-fs/2) - f],
0  f  fs/2
(6)
Furthermore, the function [Ci + Qi(f)][Ci+1 + Qi+1(f)] is an even
function about fi + (fs/2). i.e.
[Ci + Qi(fi + fs/2+f)][Ci+1 + Qi+1(fi + fs/2+f)]
= [Ci + Qi(fi + fs/2-f)][Ci+1 + Qi+1(fi + fs/2-f)]
0  f  fs/2
i = 1,2, … .N-1
(7)
The phase characteristic i(f), i = 1,2, … N, be shaped such that –
i(f) - i+1(f) =  /2 + i(f),
fi  f  fi + fs ,
i = 1,2, …, N-1
(8)
where i(f) is an arbitrary phase function with odd symmetry about
fi + (fs/2)
Examples of required filter characteristics
(i) Ci is same for all i (e.g. ½)
(ii) Qi(f), i = 1,2, …, N, is identically shaped, i.e.,
Qi+1(f) = Qi(f-fs),
i = 1, 2, …, N-1,
e.g.
Qi(f) = ½·Cos(.(f - fi)/fs), fi – fs  f  fi + fs, i=1,2, …,N,

Ai 2(f)H2(f) = Ci + Qi(f) = ½ + ½ Cos((.(f - fi)/fs)

Ai(f)H(f) = Cos((.(f - fi)/(2fs)), fi – fs  f  fi + fs, i=1,2, …,N
Examples … (continued)
Shaping of phase characteristics
i(f), i = 1,2, … ,N, are identically shaped, i.e. –
i+1(f) = i(f-fs)
i = 1,2, … ,N-1 equation (8) holds when
f - fi
f – fi
f - fi
i(f) = h  + 0 +  m Cos m  +  n Sin n 
2fs
m
fs
n
fs
m=1,2,3,4,5, …
n=2,4,6, …
fi – fs  f  fi+fs
An example with all coefficients zero, except 2 = 0.3 and h is set to –1
Shaping of phase…. (continued)
Satisfaction of 1st and 2nd requirements 
No perpendicular cutoffs and
characteristics are not required
linear
phase

Overall baseband bandwidth = (N+1)fs
As data rate/channel is 2fs,
Overall data rate = 2N fs
= [2N/(N+1)]. Overall baseband b/w
= [N/(N+1)]. Rmax
Where Rmax is 2 times overall baseband b/w, is the
Nyquist rate.
So, for large N, overall data rate approaches Rmax
Satisfaction of 3rd and 4th requirements 

Since phase chracteristic (f) of the transmission medium does not enter
into equatios (1) and (2), the received will remain orthogonal for all (f).
In the case of the fourth requirement, let
bki, k = 0, 1, 2, …; i = 1, 2, … ,N,
and
cki, k = 0, 1, 2, …; i = 1, 2, … ,N
be two arbitrary distinct sets of m-ary signal digits to be transmitted by N
channels.
The distance in signal space between these two received signal sets

d = [  [   bki ui(t - kT) -   cki ui(t - kT) ]2 dt ]½
-
i k
i k
With no ISI and ICI and applying transform domain identity –

dideal = [   (bki - cki)2  Ai2(f) H2(f) df ]½
i k
-
Thus didealis independent of the phase characteristics i(f) and (f).
Receiver structure
FFT based modulation and demodulation
Modulation using IDFT

Create N = 2N information symbols by defining
XN-k = Xk*,
k=1, ……. ,N-1
and X0 = Re{X0}, and XN = Im(X0)
Then N-point IDFT yields the real-valued sequence
1
N-1
xn =   Xk eJ2nk / N ,
n = 0,1,2,…N-1
N
k=0
where 1/N is simply a scale factor.
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The resulting baseband signal is then converted back into serial data
and undergoes the addition of the cyclic prefix (which will be
explained in the next section). In practice, the signal samples {xn} are
passed through a digital-to-analog (D/A) converter at time intervals
T/N. Next, the signal is passed through a low-pass filter to remove
any unwanted high-frequency noise. The resulting signal closely
approximates the frequency division multiplexed signal.
Cyclic prefix and demodulation using DFT

Cyclic Prefix:
-acts as a guard space,
-as cyclic convolution is performed with channel impulse response,
orthogonality of subcarriers is maintained.

Demodulation using DFT:
Demodulated sequence will beXk = HkXk + k,
k = 0,1, …, N-1
where {Xk} is the output of the N-point DFT demodulator, and k is
the additive noise corrupting the signal.
Downsides of OFDM
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Cyclic Prefix Overhead
Frequency Control
Requirement of coded or adaptive OFDM
Latency and block based processing
Synchronization
Peak-to-average power ratio (PAR)
Future research scope
Goals:
- Increase capacity, high data rate, minimum bit error rate (BER),
spectral efficiency, minimum power requirements
Problems:
- Spectral limitations, channel delay/doppler shifts,
limitation in transmission power, real time, PAR
Issues:
- coding, diversity, frame overlapping, synchronisation
techniques, adaptive estimation, w/f shaping, combined
approaches, OFDM application specific DSP architecture
AND
Any of the combination of above issues to achieve ‘Goals’ in
presence of ‘Problems’.