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Multicarrier modulation
OFDM
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The basic idea of multicarrier modulation is to
divide the transmitted bitstream into many
different substreams and send these over many
different subchannels.
Typically the subchannels are orthogonal under
ideal propagation conditions. The data rate on each
of the subchannels is much less than the total data
rate, and the corresponding subchannel bandwidth
is much less than the total system bandwidth.
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The number of substreams is chosen to ensure
that each subchannel has a bandwidth less than
the coherence bandwidth of the channel, so the
subchannels experience relatively flat fading.
Thus, the intersymbol interference on each
subchannel is small.
The subchannels in multicarrier modulation need
not be contiguous, so a large continuous block of
spectrum is not needed for high rate multicarrier
communications. Moreover, multicarrier modulation
is efficiently implemented digitally.
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The simplest form of multicarrier
divides the data stream into multiple
to be transmitted over different
subchannels centered at different
frequencies.
modulation
substreams
orthogonal
subcarrier
The number of substreams is chosen to make
the symbol time on each substream much greater
than the delay spread of the channel or,
equivalently, to make the substream bandwidth
less than the channel coherence bandwidth.
This ensures that the substreams will not
experience significant ISI.
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Consider a linearly modulated system with data
rate R and bandwidth B.
The coherence bandwidth for the channel is
assumed to be Bc < B, so the signal experiences
frequency selective fading.
The basic premise of multicarrier modulation is
to break this wideband system into N linearly
modulated subsystems in parallel, each with
subchannel bandwidth BN = B/N and data rate RN ≈
R/N.
For N sufficiently large, the subchannel
bandwidth BN = B/N < Bc, which ensures relatively
flat fading on each subchannel.
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This can also be seen in the time domain:
the symbol time TN of the modulated signal in
each subchannel is proportional to the subchannel
bandwidth 1/BN.
So BN « Bc implies that TN » Tm, where Tm
denotes the delay spread of the channel.
Thus, if N is sufficiently large, the symbol time
is much greater than the delay spread, so each
subchannel experiences little ISI degradation.
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A multicarrier transmitter. The bit stream is divided into N
substreams via a serial-to-parallel converter. The nth substream is
linearly modulated (typically via QAM or PSK) relative to the subcarrier
frequency fn and occupies bandwidth BN. We assume coherent
demodulation of the subcarriers so the subcarrier phase is neglected in
our analysis. If we assume raised cosine pulses for g(t) we get a symbol
time TN = (1 + β)/BN for each substream, where β is the rolloff factor
of the pulse shape.
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The modulated signals associated with all the
subchannels are summed together to form the transmitted
signal, given as
(12.1)
where si is the complex symbol associated with the ith
subcarrier and Φi is the phase offset of the ith carrier.
For nonoverlapping subchannels we set fi = f0 + i(BN), i =
0,..., N -1.
The substreams then occupy orthogonal subchannels with
bandwidth BN, yielding a total bandwidth NBN = B and data
rate NRN ≈ R.
Thus, this form of multicarrier modulation does not
change the data rate or signal bandwidth relative to the
original system, but it almost completely eliminates ISI for
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BN « Bc.
Receiver for this multicarrier modulation
Each substream is passed through a narrowband filter (to remove
the other substreams), demodulated, and combined via a parallel-toserial converter to form the original data stream. Note that the ith
subchannel will be affected by flat fading corresponding to a channel
gain αi = H(fi).
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Although this simple type of multicarrier
modulation is easy to understand, it has several
significant shortcomings.
First, in a realistic implementation, subchannels
will occupy a larger bandwidth than under ideal
raised cosine pulse shaping because the pulse
shape must be time limited.
Let ε/TN denote the additional bandwidth
required due to time limiting of these pulse shapes.
The subchannels must then be separated by (1 + β
+ ε)/ TN, and since the multicarrier system has N
subchannels, the bandwidth penalty for time
limiting is εN/TN.
In particular, the total required bandwidth for
nonoverlapping subchannels is
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Thus, this form of multicarrier modulation can
be spectrally inefficient.
Additionally, nearideal (and hence expensive)
lowpass filters will be required to maintain the
orthogonality of the subcarriers at the receiver.
Perhaps most importantly, this scheme requires
N independent modulators and demodulators, which
entails significant expense, size, and power
consumption.
It is possible a modulation method that allows
subcarriers to overlap and removes the need for
tight filtering, and a discrete implementation of
multicarrier modulation, which eliminates the need
for multiple modulators and demodulators.
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We can improve on the spectral efficiency of multicarrier
modulation by overlapping the subchannels. The subcarriers must
still be orthogonal so that they can be separated out by the
demodulator in the receiver. The subcarriers
{cos(2π(f0 + i/TN )t + Φi), i = 0,1,2,...}
form a set of (approximately) orthogonal basis functions on the
interval [0, TN] for any set of subcarrier phase offsets {Φi}, since
where the approximation follows because the integral in the
third line is approximately zero for f0TN » 1
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Moreover, it is easily shown that no set of
subcarriers with a smaller frequency separation
forms an orthogonal set on [0, TN] for arbitrary
subcarrier phase offsets.
This implies that the minimum frequency
separation required for subcarriers to remain
orthogonal over the symbol interval [0, TN] is 1/
TN.
Since the carriers are orthogonal the set of
functions {g(t)cos(2π(f0 + i/TN )t + Φi), i = 0,1,2,
.. N - 1} also form a set of (approximately)
orthonormal basis functions for appropriately
chosen baseband pulse shapes g(t): the family
of raised cosine pulses are a common choice for
this pulse shape.
Given this orthonormal basis set, even if the
subchannels overlap, the modulated signals
transmitted in each subchannel can be
separated out in the receiver.
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Consider a multicarrier system where each
subchannel is modulated using raised cosine
pulse shapes with rolloff factor β.
The bandwidth of each subchannel is then BN
= (1 + β)/TN.
The ith subcarrier frequency is set to (2π(f0
+ i/TN ), i = 0,1,2, .. N - 1} , for some f0, so the
subcarriers are separated by 1/TN.
However, the bandwidth of each subchannel is
BN = (1 + β)/TN > 1/TN for β > 0, so the
subchannels overlap.
Excess bandwidth due to time windowing will
increase the subcarrier bandwidth by an
additional ε/TN.
However, β and ε do not affect the total
system
bandwidth
resulting
from
the
subchannel overlap except in the first and last
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subchannels.
The total system bandwidth with overlapping subchannels is
given by
where the approximation holds for N large. Thus, with N large,
the impact of β and ε on the total system bandwidth is negligible, in
contrast to the required bandwidth of B =N(1 + β + ε)/TN when the
subchannels do not overlap.
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In order to separate out overlapping subcarriers, a different
receiver structure is needed.
In particular, overlapping subchannels are demodulated with this
receiver structure, which demodulates the appropriate symbol
without interference from overlapping subchannels.
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The advantage of multicarrier modulation is that each
subchannel is relatively narrowband, which mitigates the effect of
delay spread. However, each subchannel experiences flat fading,
which can cause large bit error rates on some of the subchannels.
In particular, if the transmit power on subcarrier i is Pi and if the
fading on that subcarrier is αi, then the received signal-to-noise
power ratio is γi = αi2Pi/NoBN, where BN is the bandwidth of each
subchannel. If αi is small then the received SNR on the ith
subchannel is low, which can lead to a high BER on that subchannel.
Moreover, in wireless channels αi will vary over time according to a
given fading distribution, resulting in the same performance
degradation as is associated with flat fading for single-carrier
systems. Because flat fading can seriously degrade performance in
each subchannel, it is important to compensate for flat fading in
the subchannels.
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There are several techniques for doing this, including coding
with interleaving over time and frequency, frequency equalization,
precoding, and adaptive loading.
Coding with interleaving is the most common, and it has been
adopted as part of the European standards for digital audio and
video broadcasting.
Moreover, in rapidly changing channels it is difficult to estimate
the channel at the receiver and feed this information back to the
transmitter.
Without channel information at the transmitter, precoding and
adaptive loading cannot be done, so only coding with interleaving is
effective at fading mitigation.
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The basic idea in coding with interleaving over time
and frequency is to encode data bits into codewords,
interleave the resulting coded bits over both time and
frequency, and then transmit the coded bits over
different subchannels such that the coded bits within a
given codeword all experience independent fading.
If most of the subchannels have a high SNR, the
codeword will have most coded bits received correctly,
and the errors associated with the few bad subchannels
can be corrected.
Coding across subchannels basically exploits the
frequency diversity inherent in a multicarrier system to
correct for errors. This technique works well only if
there is sufficient frequency diversity across the total
system bandwidth.
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If the coherence bandwidth of the channel is large,
then the fading across subchannels will be highly
correlated, which will significantly reduce the benefits
of coding.
Most coding schemes assume channel information in
the decoder.
Channel estimates are typically obtained by a twodimensional pilot symbol transmission over both time and
frequency.
Note that coding with frequency/time interleaving
takes advantage of the fact that the data on all the
subcarriers are associated with the same user and can
therefore be jointly processed.
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The other techniques for fading mitigation are all basically flat
fading compensation techniques, which apply equally to multicarrier
systems as well as to narrowband flat fading single-carrier
systems.
In frequency equalization the flat fading αi on the ith
subchannel is basically inverted in the receiver. Specifically, the
received signal is multiplied by 1/ αi which gives a resultant signal
power αi2Pi/αi2 = Pi. While this removes the impact of flat fading on
the signal, it enhances the noise power. Hence the incoming noise
signal is also multiplied by 1/αi so the noise power becomes
NoBN/αi2 and the resultant SNR on the ith subchannel after
frequency equalization is the same as before equalization.
Therefore, frequency equalization does not really change the
performance degradation associated with subcarrier flat fading.
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Precoding uses the same idea as frequency equalization, except
that the fading is inverted at the transmitter instead of at the
receiver. This technique requires the transmitter to have
knowledge of the subchannel flat fading gains αi (i = 0, ...,N — 1),
which must be obtained through estimation. In this case, if the
desired received signal power in the ith subchannel is Pi and if the
channel introduces a flat fading gain αi in the ith subchannel, then
under precoding the power transmitted in the ith subchannel is
Pi/αi2. The subchannel signal is corrupted by flat fading with gain αi
so the received signal power is αi2Pi/αi2 = Pi as desired. Note that
the channel inversion takes place at the transmitter instead of the
receiver, so the noise power remains NoBN. Precoding is quite
common on wireline multicarrier systems like high-bit-rate digital
subscriber lines (HDSL).
The main problem with precoding is the need for accurate
channel estimates at the transmitter, which are difficult to obtain
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in a rapidly fading channel.
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Adaptive loading is based on the adaptive modulation techniques.
It is commonly used on slowly changing channels like digital
subscriber lines, where channel estimates at the transmitter can
be obtained fairly easily. The basic idea is to vary the data rate
and power assigned to each subchannel relative to that subchannel
gain. As in the case of precoding, this requires knowledge of the
subchannel fading [αi, i = 0,..., N - 1) at the transmitter. In adaptive
loading, power and rate on each subchannel are adapted to
maximize the total rate of the system using adaptive modulation
such as variable-rate variable-power MQAM.
If we apply the variable-rate variable-power MQAM modulation
scheme to the subchannels, then the total data rate is given by
where K = -1.5/ln(5Pb) for Pb the desired target BER in each
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subchannel and P the total power.
Discrete Implementation of Multicarrier Modulation
Although multicarrier modulation was invented in the 1950s, its
requirement for separate modulators and demodulators on each
subchannel was far too complex for most system implementations
at the time. However, the development of simple and cheap
implementations of the discrete Fourier transform and the inverse
DFT twenty years later - combined with the realization that
multicarrier modulation could be implemented with these
algorithms - ignited its widespread use.
The DFT and Its Properties
Let x[n], 0 ≤ n ≤ N — 1, denote a discrete time sequence. The Npoint DFT of x[n] is defined as
The DFT is the discrete-time equivalent to the continuous-time
Fourier transform, because X[i] characterizes the frequency
content of the time samples x[n] associated with the original signal
x(t). The sequence x[n] can be recovered from its DFT using the
IDFT:
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Orthogonal Frequency-Division Multiplexing (OFDM)
The input data stream is modulated by a QAM modulator,
resulting in a complex symbol stream X[0],X[1], ...,X[N — 1]. This
symbol stream is passed through a serial-to-parallel converter,
whose output is a set of N parallel QAM symbols X[0],..., X[N - 1]
corresponding to the symbols transmitted over each of the
subcarriers. Thus, the N symbols output from the serial-to-parallel
converter are the discrete frequency components of the OFDM
modulator output s(t). In order to generate s(t), the frequency
components are converted into time samples by performing an
inverse DFT on these N symbols, which is efficiently implemented
using the IFFT algorithm. The IFFT yields the OFDM symbol
consisting of the sequence x[n] = x[0], ...,x[N — 1] of length N,
where
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OFDM with IFFT/FFT implementation
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This sequence corresponds to samples of the multicarrier signal:
the multicarrier signal consists of linearly modulated subchannels,
and the right-hand side corresponds to samples of a sum of QAM
symbols X[i] each modulated by the carrier ej2πni/N, i= 0,...,N-1.
The cyclic prefix is then added to the OFDM symbol, and the
resulting time samples are ordered by the parallel-to-serial
converter and passed through a D/A converter, resulting in the
baseband OFDM signal x(t), which is then upconverted to
frequency f0.
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The transmitted signal is filtered by the channel impulse
response and corrupted by additive noise, resulting in the received
signal r(t). This signal is downconverted to baseband and filtered to
remove the high-frequency components. The A/D converter
samples the resulting signal to obtain y[n]. The prefix of y[n]
consisting of the first μ samples is then removed. This results in N
time samples whose DFT in the absence of noise is Y[i] = H[i]X[i]
(being h[n] the discrete-time equivalent lowpass impulse response
of the channel). These time samples are serial-to-parallel
converted and passed through an FFT. This results in scaled
versions of the original symbols H[i]X[i], where H[i] = H(fi) is the
flat fading channel gain associated with the ith subchannel. The
FFT output is parallel-to-serial converted and passed through a
QAM demodulator to recover the original data.
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The OFDM system effectively decomposes the wideband
channel into a set of narrowband orthogonal subchannels with a
different QAM symbol sent over each subchannel. Knowledge of
the channel gains H[i], i = 0,..., N — 1, is not needed for this
decomposition, in the same way that a continuous-time channel with
frequency response H(f) can be divided into orthogonal subchannels
without knowledge of H(f) by splitting the total signal bandwidth
into nonoverlapping subbands. The demodulator can use the channel
gains to recover the original QAM symbols by dividing out these
gains: X[i] = Y[i]/H[i]. This process is called frequency equalization.
However, as discussed for continuous-time OFDM, frequency
equalization leads to noise enhancement because the noise in the
ith subchannel is also scaled by 1/H [i]. Hence, while the effect of
flat fading on X[i] is removed by this equalization, its received SNR
is unchanged.
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Prefisso ciclico
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Precoding, adaptive loading, and coding across subchannels are
better approaches to mitigate the effects of flat fading across
subcarriers.
An alternative to using the cyclic prefix is to use a prefix
consisting of all zero symbols. In this case the OFDM symbol
consisting of x[n], 0 ≤ n ≤ N — 1, is preceded by m null samples. At
the receiver the "tail“ of the ISI associated with the end of a
given OFDM symbol is added back in to the beginning of the
symbol, which re-creates the effect of a cyclic prefix, so the rest
of the OFDM system functions as usual. This zero prefix reduces
the transmit power relative to a cyclic prefix by N/(μ + N), since
the prefix does not require any transmit power. However, the noise
from the received tail is added back into the beginning of the
symbol, which increases the noise power by N/(μ + N). Thus, the
difference in SNR is not significant for the two prefixes.
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The peak-to-average power ratio (PAR) is an important attribute
of a communication system. A low PAR allows the transmit power
amplifier to operate efficiently, whereas a high PAR forces the
transmit power amplifier to have a large backoff in order to ensure
linear amplification of the signal. Operation in the linear region of
this response is generally required to avoid signal distortion, so the
peak value is constrained to be in this region. Clearly it would be
desirable to have the average and peak values be as close together
as possible in order for the power amplifier to operate at maximum
efficiency. Additionally, a high PAR requires high resolution for the
receiver A/D converter, since the dynamic range of the signal is
much larger for high-PAR signals. High-resolution A/D conversion
places a complexity and power burden on the receiver front end.
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In general, PAR should be measured with respect to the
continuous-time signal, since the input to the amplifier is an analog
signal. The PAR is sensitive to the pulse shape g(t) used in the
modulation, and it does not generally lead to simple analytical
formulas. For illustration we will focus on the PAR associated with
the discrete-time signal, since it lends itself to a simple
characterization. However, care must be taken when interpreting
these results, since they can be quite inaccurate if the pulse shape
g(t) is not taken into account.
Consider the time-domain samples that are output from the
IFFT: if N is large then the central limit theorem is applicable, and
x[n] are zero-mean complex Gaussian random variables because the
real and imaginary parts are summed. The Gaussian approximation
for IFFT outputs is generally quite accurate for a reasonably large
number of subcarriers (N ≥ 64). For x[n] complex Gaussian, the
envelope of the OFDM signal is Rayleigh distributed, and the phase
of the signal is uniform. Since the Rayleigh distribution has
infinite support, the peak value of the signal will exceed any given
value with nonzero probability. It can then be shown that the
probability that the PAR exceeds a threshold P0 is given by
p(PAR > P0) = 1 - (1 – exp(-P0))N.
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It may be demonstrated that the maximum PAR is N for N
subcarriers. In practice, full coherent addition of all N symbols is
highly improbable and so the observed PAR is typically less than N –
usually by many decibels. Nevertheless, PAR increases
approximately linearly with the number of subcarriers. So even
though it is desirable to have N as large as possible in order to
keep the overhead associated with the cyclic prefix down, a large
PAR is an important penalty that must be paid for large N.
There are a number of ways to reduce or tolerate the PAR of
OFDM signals, including clipping the OFDM signal above some
threshold, peak cancellation with a complementary signal, allowing
nonlinear distortion from the power amplifier (and correction for
it), and special coding techniques.
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Frequency and Timing Offset
We have seen that OFDM modulation encodes the data symbols
Xi onto orthogonal subchannels, where orthogonality is assured by
the subcamer separation Δf = 1/TN. The subchannels may overlap in
the frequency domain for a rectangular pulse shape in time (sine
function in frequency). In practice, the frequency separation of the
subcarriers is imperfect and so Δf is not exactly equal to 1/TN.
This is generally caused by mismatched oscillators, Doppler
frequency shifts, or timing synchronization errors. For example, if
the carrier frequency oscillator is accurate to 1 part per million, if
f0 = 5 GHz, the carrier frequency for 802.11a WLANs, then Δf =
500 Hz, which will degrade the orthogonality of the subchannels
because now the received samples of the FFT will contain
interference from adjacent subchannels.
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Several important trends of the intercarrier interference (ICI)
can be observed. First, as TN increases, the subcarriers grow
narrower and hence more closely spaced, which then results in more
ICI. Second, the ICI predictably grows with the frequency offset
δ, and the growth is about quadratic. Another interesting
observation is that ICI does not appear to be directly affected by
N. But picking N large generally forces TN to be large also, which
then causes the subcarriers to be closer together. Along with the
larger PAR that comes with large N, the increased ICI is another
reason to pick N as low as possible, assuming the overhead budget
can be met. In order to further reduce the ICI for a given choice
of N, nonrectangular windows can also be used.
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