Lecture 6 Mul carrier Modula on and OFDM

Wireless Communications
Lecture 6 Mul+carrier Modula+on and OFDM
Prof. Chun-Hung Liu
Dept. of Electrical and Computer Engineering
National Chiao Tung University
Fall 2014
Outline •  Multicarrier Modulation (Chapter 12 in Goldsmith’s Book)
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Data Transmission using Multiple Carriers
Multicarrier Modulation with Overlapping Subchannels
Mitigation of Subcarrier Fading
Discrete Implementation of Multicarrier
Challenges in Multicarrier Systems
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Introduction •  The basic idea of multicarrier modulation is to divide the transmitted
bitstream into many different substreams and send these over many
different subchannels
•  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.
•  The number of substreams is chosen to insure that each subchannel has a
bandwidth less than the coherence bandwidth of the channel, so the
subchannels experience relatively flat fading. (Why do we need flat
fading?)
•  Multicarrier modulation is efficiently implemented digitally. In this
discrete implementation, called orthogonal frequency division multiplexing
(OFDM), the ISI can be completely eliminated through the use of a cyclic
prefix.
•  There are some impairments of multicarrier modulation, such as
frequency offset and peak-to-average ratio, etc..
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Data Transmission Using Multiple Carriers
•  The simplest form of multicarrier modulation divides the data stream into
multiple substreams to be transmitted over different orthogonal
subchannels centered at different subcarrier frequencies.
•  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 insures that the substreams will not experience
significant ISI.
•  Consider a linearly-modulated system with data rate R and passband
bandwidth B. The coherence bandwidth for the channel is assumed to
be
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
•  For N sufficiently large, the subchannel bandwidth
,
which insures relatively flat fading on each subchannel.
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Data Transmission Using Multiple Carriers
•  Figure 12.1 illustrates a multicarrier transmitter. The bit stream is
divided into N substreams via a serial-to-parallel converter.
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Data Transmission Using Multiple Carriers
• 
• 
• 
The nth substream is linearly-modulated (typically via QAM or PSK) relative to the
subcarrier frequency
and occupies passband bandwidth
.
If we assume raised cosine pulses for
we get a symbol time TN = (1 + )BN for
each substream, where β is the rolloff factor of the pulse shape.
The modulated signals associated with all the subchannels are summed together to
form the transmitted signal, given as
(12.1)
•  For nonoverlapping subchannels, we set
•  The substreams then occupy orthogonal subchannels with passband bandwidth
yielding a total passband bandwidth
and data rate
R .
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,
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Data Transmission Using Multiple Carriers
•  The receiver for this multicarrier modulation is shown in Figure 12.2.
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Data Transmission Using Multiple Carriers
•  Although this simple type of multicarrier modulation is easy to
understand, it has several significant shortcomings.
•  In a realistic implementation, subchannels will occupy a larger
bandwidth than under ideal raised cosine pulse shaping since the
pulse shape must be time-limited.
•  Let
denote the additional bandwidth required due to timelimiting of these pulse shapes.
•  So the total required bandwidth for nonoverlapping subchannels is
(12.2)
•  Thus, this form of multicarrier modulation can be spectrally inefficient.
•  Additionally, near-ideal (and hence, expensive) low pass filters will be
required to maintain the orthogonality of the subcarriers at the receiver.
•  This scheme requires N independent modulators and demodulators,
which entails significant expense, size, and power consumption.
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Multicarrier Modulation with Overlapping Subchannels
•  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
form a set of
(approximately) orthogonal basis functions on the interval
for any
set of subcarrier phase offsets
since
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Multicarrier Modulation with Overlapping Subchannels
•  It is easily shown that no set of subcarriers with a smaller frequency
separation forms an orthogonal set on
for arbitrary subcarrier phase
offsets.
•  This implies that the minimum frequency separation required for
subcarriers to remain orthogonal over the symbol interval
is
.
•  Consider a multicarrier system where each subchannel is modulated using
raised cosine pulse shapes with rolloff factor β.
•  The passband bandwidth of each subchannel is then
•  The ith subcarrier frequency is set to
for
some , so the subcarriers are separated by
.
•  However, the passband bandwidth of each subchannel is
for β > 0, so the subchannels overlap.
•  Excess bandwidth due to time windowing will increase the subcarrier
bandwidth by an additional
.
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Multicarrier Modulation with Overlapping Subchannels
•  The total system bandwidth with overlapping subchannels is given by
(for large N)
(12.4)
•  Thus, with N large, the impact of β and on the total system bandwidth is
negligible, in contrast to the required bandwidth
when the subchannels do not overlap. 14/10/27
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Multicarrier Modulation with Overlapping Subchannels
•  In particular, overlapping subchannels are demodulated with the receiver
structure shown in Figure 12.4. 14/10/27
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Mitigation of Subcarrier Fading
•  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 BERs
on some of the subchannels.
•  In particular, if the transmit power on subcarrier i is Pi , and the fading on that
2
subcarrier is ↵i , then the received SNR is i = ↵i Pi /(N0 BN ) , where BN is the
bandwidth of each subchannel.
•  If ↵i is small then the received SNR on the ith subchannel is quite low, which can
lead to a high BER on that subchannel.
•  Moreover, in wireless channels the ↵i ’s will vary over time according to a given
fading distribution, resulting in the same performance degradation associated
with flat fading for single carrier systems.
•  Since flat fading can seriously degrade performance in each subchannel, it is
important to compensate for flat fading in the subchannels.
•  There are several techniques for doing this, including coding with interleaving
over time and frequency, frequency equalization, precoding, and adaptive
loading, etc..
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Interleaving over Time and Frequency
•  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.
•  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 to
a multicarrier system to correct for errors.
•  This technique only works well if there is sufficient frequency diversity across
the total system bandwidth.
•  If the coherence bandwidth of the channel is large, then the fading across
subchannels will be highly correlated, which will significantly reduce the effect of
coding.
•  Most coding for OFDM assumes channel information in the decoder. Channel
estimates are typically obtained by a two dimensional pilot symbol transmission
over both time and frequency.
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Frequency Equalization and Precoding •  In frequency equalization the flat fading ↵i on the ith subchannel is basically
inverted in the receiver.
•  Specifically, the received signal is multiplied by 1/↵i , which gives a resultant
signal power ↵i2 Pi /↵i2 = Pi .
•  While this removes the impact of flat fading on the signal, it enhances the
noise. Specifically, the incoming noise signal is also multiplied by 1/↵i , so the
noise power becomes N0 BN /↵i2 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.
•  Precoding uses the same idea as frequency equalization, except that the fading
is inverted at the transmitter instead of the receiver.
•  This technique requires that the transmitter have knowledge of the
subchannel flat fading gains ↵i , i = 10, . . . , N 1 , which must be obtained
through estimation.
•  In this case, if the desired received signal power in the ith subchannel is Pi ,
and the channel introduces a flat-fading gain ↵i in the ith subchannel, then
under precoding the power transmitted in the ith subchannel is Pi /↵i2 . The
subchannel signal is corrupted by flat-fading with gain ↵i , so the received
signal power is Pi ↵i2 /↵i2 = Pi , as desired.
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Adaptive Loading
•  Adaptive loading is based on the adaptive modulation techniques discussed in
Chapter 9.
•  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.
•  In adaptive loading power and rate on each subchannel is adapted to maximize
the total rate of the system using adaptive modulation such as variable-rate
variable-power MQAM.
•  Consider the capacity of the multicarrier system with N independent
subchannels of bandwidth BN and subchannel gain {↵i , i = 0, . . . , N 1} .
Assuming a total power constraint P, this capacity is given by:
(12.7)
•  The power allocation Pi that maximizes this expression is a water-filling over
frequency given by Equation (4.24):
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Adaptive Loading
(12.8)
• 
2
For some cutoff value 0 , where i = ↵i P/(N0 BN ) . The cutoff value is obtained
by substituting the power adaptation formula into the power constraint. The
capacity then becomes
(12.9)
•  Applying the variable-rate variable-power MQAM modulation scheme described in
Chapter 9 to the subchannels, the total data rate is given by
(12.10)
where K = 1.5/ ln(5Pb ) for Pb is the desired target BER in each subchannel. •  Optimizing this expression relative to the Pi ’s yields the optimal power allocation
where
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K
is a cutoff fade depth dictated by the power constraint P and K.
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Discrete Implementation of Multicarrier
•  Although multicarrier modulation was invented in the 1950’s, its
requirement for separate modulators and demodulators on each
subchannel was far too complex for most system implementations at the
time.
•  The development of simple and cheap implementations of the discrete
Fourier transform (DFT) and the inverse DFT (IDFT) twenty years later,
combined with the realization that multicarrier modulation can be
implemented with these algorithms, ignited its widespread use.
•  (Review) The DFT and its Properties:
Let x[n], 0 ≤ n ≤ N-1, denote a discrete time sequence. The N-point
DFT of x[n] is defined as
(12.13)
The sequence x[n] can be recovered from its DFT using the IDFT:
(12.14)
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Discrete Implementation of Multicarrier
•  The DFT and its inverse are typically performed in hardware using the fast
Fourier transform (FFT) and inverse FFT (IFFT).
•  When an input data stream x[n] is sent through a linear time-invariant
discrete-time channel h[n], the output y[n] is
(12.15)
The N-point circular convolution of x[n] and h[n] is defined as
(12.16)
where
denotes [n-k] modulo N. In other words,
periodic version of x[n-k] with period N.
is a
•  From the definition of the DFT, circular convolution in time leads to
multiplication in frequency:
(12.17)
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DFT and Cyclic Prefix
•  If the channel and input are circularly convoluted then if h[n] is known at
the receiver, the original data sequence x[n] can be recovered by taking the
IDFT of Y [i]/H[i], 0 ≤ i ≤ N − 1.
•  Unfortunately, the channel output is NOT a circular convolution but a
linear convolution.
•  However, the linear convolution between the channel input and impulse
response can be turned into a circular convolution by adding a special
prefix to the input called a cyclic prefix.
•  The Cyclic Prefix:
•  Consider a channel input sequence x[n]=x[0], . . . , x[N-1] of length N and a
discrete-time channel with finite impulse response (FIR)
of length
, where is the channel delay spread and
the
sampling time associated with the discrete time sequence.
•  The cyclic prefix for x[n] is defined as {x[N-µ], . . . , x[N-1]}: it consists of
the last µ values of the x[n] sequence.
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Discrete Implementation of Multicarrier
•  For each input sequence of length N, these last µ samples are appended to
the beginning of the sequence. This yields a new sequence
•  Note that with this definition,
which implies that
•  Suppose
is input to a discrete-time channel with impulse
response h[n]. The channel output y[n], 0 ≤ n ≤ N-1 is then
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Discrete Implementation of Multicarrier
•  Thus, by appending a cyclic prefix to the channel input, the linear
convolution associated with the channel impulse response y[n] for 0≤ n≤
N-1 becomes a circular convolution.
•  Taking the DFT of the channel output in the absence of noise then yields
and the input sequence x[n], 0 ≤ n ≤ N-1, can be recovered from the
channel output y[n], 0 ≤ n ≤ N-1, for known h[n] by
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Discrete Implementation of Multicarrier
•  Note that y[n], -µ ≤ n ≤ N-1, has length N+µ, yet the first µ samples y[µ], . . . , y[-1] are not needed to recover x[n], 0 ≤ n ≤ N-1, due to the
redundancy associated with the cyclic prefix.
•  Moreover, if we assume that the input x[n] is divided into data blocks of
size N with a cyclic prefix appended to each block to form
, then the
first µ samples of
in a given block are corrupted by ISI
associated with the last µ samples of x[n] in the prior block, as
illustrated in Figure 12.6.
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Discrete Implementation of Multicarrier
•  The cyclic prefix serves to eliminate ISI between the data blocks since the
first µ samples of the channel output affected by this ISI can be discarded
without any loss relative to the original information sequence.
•  In continuous time this is equivalent to using a guard band of duration
(the channel delay spread) after every block of N symbols of duration
to eliminate the ISI between these data blocks.
•  The benefits of adding a cyclic prefix come at a cost. Since µ symbols are
added to the input data blocks, there is an overhead of µ/N, resulting in a
data rate reduction of N/(µ+N).
•  The transmit power associated with sending the cyclic prefix is also wasted
since this prefix consists of redundant data.
•  The above analysis motivates the design of OFDM. In OFDM the input
data is divided into blocks of size N referred to as an OFDM symbol.
•  A cyclic prefix is added to each OFDM symbol to induce circular
convolution of the input and channel impulse response.
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Discrete Implementation of Multicarrier
Example 12.4: Consider an OFDM system with total passband bandwidth B
= 1 MHz assuming
. A single carrier system would have symbol
time
. The channel has a maximum delay spread of
µsec, so with
µsec, so there would clearly be severe ISI. Assume an
OFDM system with MQAM modulation applied to each subchannel. To keep
the overhead small, the OFDM system uses N=128 subcarriers to mitigate
ISI. So
µsec. The length of the cyclic prefix is set to
to insure no ISI between OFDM symbols. For these parameters, find the
subchannel bandwidth, the total transmission time associated with each
OFDM symbol, the overhead of the cyclic prefix, and the data rate of the
system assuming M = 16.
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Orthogonal Frequency Division Multiplexing (OFDM)
•  The OFDM implementation of multicarrier modulation is shown in
Figure 12.7.
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Orthogonal Frequency Division Multiplexing (OFDM)
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Orthogonal Frequency Division Multiplexing (OFDM)
•  The IFFT yields the OFDM symbol consisting of the sequence x[n] =
x[0], . . . , x[N-1] of length N, where
•  This sequence corresponds to samples of the multicarrier signal: i.e. 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 carrier frequency
.
•  The transmitted signal is filtered by the channel impulse response h(t)
and corrupted by additive noise, so that the received signal is
•  This signal is downconverted to baseband and filtered to remove the high
frequency components.
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Orthogonal Frequency Division Multiplexing (OFDM)
•  The A/D converter samples the resulting signal to obtain
−µ ≤ n ≤ N-1. 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
•  The OFDM system effectively decomposes the wideband channel into a
set of narrowband orthogonal subchannels with a different QAM symbol
sent over each subchannel.
•  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 proceeded by µ null samples, as illustrated in Figure 12.8.
•  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
recreates the effect of a cyclic prefix, so the rest of the OFDM system
functions as usual.
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Orthogonal Frequency Division Multiplexing (OFDM)
•  This zero prefix reduces the transmit power relative to a cyclic prefix
by
, 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
. Thus, the
difference in SNR is not significant for the two prefixes.
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Challenges in Multicarrier Systems
•  Peak to Average Power Ratio: •  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. This
is demonstrated in Figure 12.11
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Challenges in Multicarrier Systems
•  The PAR of a continuous-time signal is given by
and for a discrete-time signal it is given by
•  Any constant amplitude signal, e.g. a square wave, has PAR=0 dB. A
sine wave has PAR=3 dB since
and
so PAR=1/0.5=2.
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Challenges in Multicarrier Systems
•  Frequency and Timing Offset: OFDM modulation encodes the data
symbols
onto orthogonal subchannels, where orthogonality is assured
by the subcarrier separation
.
•  The subchannels may overlap in the frequency domain, as shown in
Figure 12.12 for a rectangular pulse shape in time.
•  In practice, the frequency separation of the subcarriers is imperfect: so
Δf is not exactly equal to
. So intercarrier interference (ICI) could
happen.
•  This is generally caused by mismatched oscillators, Doppler frequency
shifts, or timing synchronization errors.
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Challenges in Multicarrier Systems
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