Chapter 4: Orthogonal Frequency Division

Multiplexing (OFDM)
4.1. Introduction:
Wireless communications continue to grow rapidly as the need for
reaching data anywhere at any time rises. The increasing demand for
high rate data services along with the requirement for reliable
connectivity requires novel technologies. Orthogonal frequency division
multiplexing (OFDM) is a modulation scheme that allows digital data to
be efficiently and reliably transmitted over a radio channel, even in
multipath environments. OFDM transmits data by using a large number
of narrow bandwidth carriers. These carriers are regularly spaced in
frequency, forming a block of spectrum. The frequency spacing and time
synchronization of the carriers is chosen in such a way that the carriers
are orthogonal, meaning that they do not cause interference to each
other. This is despite the carriers overlapping each other in the frequency
domain, figure 4.1 shows the different between OFDM and FDM. The
name ‘OFDM’ is derived from the fact that the digital data is sent using
many carriers, each of a different frequency (Frequency Division
Multiplexing) and these carriers are orthogonal to each other, hence
Orthogonal Frequency Division Multiplexing.

Frequenc
y
(a) Eight sub-channels spectrum using FDM.

Saving of the bandwidth Frequenc

y
(b) Eight sub-channels spectrum using OFDM.

Fig.4.1.Comparison between the bandwidth utilization for FDM and OFDM.

(a) FDM, (b) OFDM.
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 OFDM is currently under significant investigation due to various
advantages such as high spectral efficiency. Robustness to channel
fading, immunity to impulse interference, and capability of handling
very strong multipath fading and frequency selective fading without
having to provide powerful channel equalization. OFDM is known as an
effective technique for high bit rate applications, where the delay spread
of the channel extends over many symbol periods. Despite these
advantages, OFDM techniques also face several challenges. First, there
is the problem associated with OFDM signals having a high peak-to-
average-power ratio (PAPR) that causes nonlinearities and clipping
distortion. This can lead to power inefficiencies that need to be
countered. Second, OFDM signals are very susceptible to phase noise
and frequency dispersion, and the design must mitigate these
imperfections. This also makes it critical to have accurate frequency
synchronization.

4.1.1 Applications of OFDM:

Orthogonal Frequency Division Multiplexing (OFDM) has been

recognized as an excellent method for high speed data communication.
Its history dates back to the 1960s, but it has become popular because
economical integrated circuits that can perform the necessary high speed
digital operations have become available. Today, OFDM is used in such
systems as Asymmetric Digital Subscriber Line (ADSL) as well as
wireless systems such as IEEE 802.11a/g (Wireless Fidelity (Wi-Fi)) and
IEEE 802.16 (Worldwide interoperability for Microwave Access (Wi-
MAX)). It is also used for wireless Digital Audio/Video Broadcasting
(DAB/DVB). Furthermore, OFDM is one of the prime technologies,
which considered for use in the 4G cellular mobile networks and Long-
Term Evolution (LTE). The high-speed short-range technology known as
Ultra-Wideband (UWB) uses an OFDM standard set by the WiMedia

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Alliance. OFDM is also used in wired communications like power-line
networking technology.
4.1.2 OFDM implementation:
OFDM is accomplished with digital signal processing (DSP). We can
program the IFFT and FFT math functions on any fast PC, but it is
usually done with a DSP IC or an appropriately programmed FPGA or
some hardwired digital logic. With today’s super-fast chips, even
complex math routines like FFT are relatively easy to implement. In
brief, we can put it all on a single chip. Recent advances in the Very
Large Scale Integration (VLSI) technology enable making of
high-speed chips that can perform large size FFT at an
affordable price
4.2. Orthogonality in OFDM System:
Orthogonality is a property that allows multiple information signals to
be transmitted perfectly over a common channel and detected without
interference. Loss of orthogonality results in interference between these
information signals and degradation in the communication system
performance. There are several ways of looking at what makes the
subcarriers in an OFDM signal orthogonal, and why this prevents
interference between them. OFDM signals are made up from a sum of
sinusoids, with each corresponding to a subcarrier. The baseband
frequency of each subcarrier is chosen to be an integer multiple of the
inverse of the symbol time, resulting in all subcarriers having an integer
number of cycles per symbol. As a consequence, the subcarriers are
orthogonal to each other. Figure 4.2 shows the construction of an OFDM
signal with five subcarriers, the phase of all these subcarriers is zero, and
T is the symbol time. Mathematically, suppose we have a set of signals
S, where Si is the ith element in the set. The signals are orthogonal if they
match the condition in Eq. 4.1. If any two different orthogonal functions
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within the set are multiplied, and integrated over a symbol period, the
result is zero.
 k i j

a

b si (t ) s j (t )dt  
*
, (4.1)
 0 i j


Where, * indicates the complex conjugate, the interval [b,a] is a symbol

period, and k is a constant value. A matched receiver for one of the
orthogonal functions, a subcarrier in the case of OFDM, will only see the
result for that function, while the results from all other functions in the
set integrate to zero and thus have no effect. Equation 4.2 shows a set of
orthogonal sinusoids, which represent the subcarriers for an un-
modulated real OFDM signal.

 sin( 2 if t ) 0t T


si (t )   , (4.2)
 0 otherwise.


∆f = I / T Hz, where I is an integer

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 Fig. 4.2. An OFDM signal with five subcarriers.
(a) time domain, (b) frequency domain.
(a.1-5) show individual subcarriers, with 1-5 cycles per symbol respectively.
(a.6) shows the result for the summation of the 5 subcarriers.
(b.1-6) show the frequency domain of the time waveforms in (a.1-6) respectively.

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 Another way to view the orthogonality property of OFDM signals is to
look at its spectrum. In the frequency domain, each OFDM subcarrier
has a sinc (sin(x)/x) frequency response, because the receiver is
concerned each subcarrier transmitted for a fixed time symbol time (T),
with no tapering at the ends of the subcarrier]. The reason of
orthogonality in frequency domain is that, each subcarrier has an integer
number of cycles over a symbol period. Because of previous reason, the
spectrum of each subcarrier has a null at the center frequency of each of
the other subcarriers in the system. This result in no interference between
the subcarriers, Fig. 4.3. shows the frequency response of the five
subcarriers shown in Fig. 4.2.a. Each subcarrier has a sinc frequency
response with a peak at the center frequency and nulls at the peaks of the
other subcarriers. When the receiver samples at the center frequency of
each subcarrier, the present energy is that of the desired signal, plus
noise. To maintain orthogonality between subcarriers tones, it is
necessary to ensure that the symbol time contains one or multiple cycles
of each sinusoidal tone waveform. This is normally the case since tone
frequencies are integer multiples of the symbol period (∆f = 1/T). As
long as orthogonality is maintained, it is still possible to recover the
individual subcarriers signals despite their overlapping spectrums.

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 Frequency

Frequency

Fig.4.3. Frequency response of five subcarriers in an OFDM symbol.

(a) The spectrum of each subcarrier, (b) The over all combined response of the five
subcarriers.

4.2.1. DFT/IDFT Orthogonality:

IDFT and DFT are used respectively for modulating and demodulating
the parallel symbols constellations on the orthogonal subcarriers. These
signal processing algorithms replace the banks of subcarrier oscillators
and coherent modulators/demodulators that would otherwise be required.
IDFT is used at the OFDM transmitter to map an input signal onto a set
of orthogonal subcarriers. Similarly, the transform is used again at the
OFDM receiver to process the received subcarriers. This separation of
signal energy is the reason that the OFDM subcarriers spectrums can
overlap without causing interferenc]. The most common way to
implement IDFT and DFT is by IFFT and FFT algorithms respectively.
IFFT/FFT performs the same operations as an IDFT/DFT, except that it
is much more computationally efficient.

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4.3. OFDM Basic System:
Figure 4.4 shows the basic block diagram of an OFDM system. The
transmitter section takes a serial data bit stream and splits it into N
parallel bit streams, where N is the number of the IFFT points. Each bit
stream is mapped into output symbols constellations (complex data) to
be modulated onto a unique orthogonal subcarrier and combined together
using IDFT (IFFT) yielding the time domain output samples. IFFT
converts each N parallel input symbols to N parallel output samples,
each input symbol or output sample has a period of Ts seconds, each N
parallel output samples forms one OFDM symbol (in time domain) with
a period of Ts seconds, because the N samples are in parallel.
A Guard Interval (GI) is then added to each OFDM symbol in time
domain, in order to avoid problems caused by the mixing of subsequent
symbols in the receiver. Each OFDM symbol consists of (N+G) samples,
where N is the number of the effective OFDM symbol samples (N
parallel output sample of the IDFT), and G is the number of the guard
period samples. After that, parallel to serial converter converts the signal
from parallel samples to serial to form the baseband OFDM signal. The
baseband OFDM signal is then mixed up to the required Radio
Frequency (RF), to be transmitted through a wireless channel.
The receiver section performs the inverse of the transmitter function,
first the RF received signal is stepped down to get the baseband OFDM
signal. This signal is converted from a serial to parallel form then, the
guard period is removed and the DFT converts the time domain samples
back to frequency domain representations (output symbols
constellations). Finally, the de-mapping signal block recovers the N
parallel bit stream, which is converted back to the original serial data
stream by the parallel to serial converter.

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 Frequency Time
Domain Domain

Baseband
Serial Serial Symbol Guard Parallel OFDM
Bit to Mapping Period to Signal
OFDM
Stream Parallel (Modulation) Insertion Serial
IFFT
Converter (Cyclic Converter
Prefix)

N Bit Streams N Input N Output

Symbols Samples

N+G Samples

(a) OFDM Transmitter

Time Frequency
Domain Domain

Serial
Serial Guard Symbol Parallel Bit
to Period Demapping To Stream
Baseband (Demodul-
OFDM Parallel Removal FFT Serial
Converter (Cyclic ation) Converter
Baseband
Signal
Prefix)

N Input N Output N Bit Streams

Samples Symbols

N+G Samples

(b) OFDM Receiver

Bit Stream

Complex signal

Fig. 4.4. The basic OFDM system. (a) Transmitter, (b) Receiver

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4.3.1Serial to Parallel/Parallel to Serial Converter:
At the transmitter serial to parallel converter is needed to convert the
original serial data bit stream to N parallel bit streams, each with a rate
of 1/N of the original data rate. Each parallel bit stream is mapped into
symbols constellations and then applied to a point of the IFFT to be
modulated at a unique orthogonal frequency. Serial to parallel converter
creates slower parallel bit streams so, the bandwidth of the modulated
symbol is decreased by a factor of N, or equivalently, the duration of the
modulation symbol is increased by a factor of N. Beside these slower bit
stream a Proper selection of system parameters, such as number of
carriers and carrier spacing, can greatly reduce ISI. Parallel to serial
converter do the opposite after converting the signal to the time domain
by the IFFT and adding the GI samples to produce a serial time domain
samples (baseband OFDM signal).

At the receiver first, serial to parallel converter is used to convert the

serial samples of the baseband OFDM signal to a parallel form to be
transformed by the FFT (after removing the GI samples) from the time
domain to the frequency domain and demapped. Finally, parallel to serial
converter is used to reconvert parallel bit streams to the original serial
data bit stream.

4.3.2. Symbol Mapping/De-Mapping:

The symbol mapping block maps each parallel bit stream to a symbol
constellation stream (input symbols), these constellations can be taken
according to any digital modulation schemes such as M-PSK or M-
QAM. After mapping, each symbol is represented by a complex in-phase
and quadrature-phase (I-Q) vector.

The input symbol rate (fs) can be calculated from Eq. 4.3, where Ts is
the input symbol time in second, fpb is the parallel bit stream rate in

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bit/second, fb is the serial (original) bit stream rate in bit/second, N is the
number of the FFT/IFFT points and m is depend on the used modulation
scheme, where any modulation scheme takes m bits to produce one
symbol constellation.

1 f fb
fs   pb symbol / s , f pb  bit / s ,
Ts m N

1 f
fs   b symbol / s . (4.3)
Ts mN

An example of 16-QAM modulation scheme, m=4 then, the input

symbol rate (symbol/second) is 1/4 of the parallel bit stream, and the
parallel bit stream (bit/second) is 1/N of the original serial bit stream
rate.

f pb fb
fs   symbol/second
4 4N

In the receiver, symbol de-mapping maps the received parallel I-Q

vectors (input symbols constellations) back to parallel data bits. During
transmission, noise and distortion becomes added to the signal due to
thermal noise, signal power reduction and imperfect channel
equalization.

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4.3.3. IFFT/FFT:
The ability to define the signal in the frequency domain, in software
on VLSI processors, and to generate the signal using the IFT is the key
to its current popularity. The use of the reverse process in the receiver is
essential if cheap and reliable receivers are to be readily available. At
OFDM transmitter, the IFFT treats the input symbols (symbols
constellations) as though they are in the frequency-domain, and brings
them into the time domain. The IFFT takes N parallel input symbols at a
time, each input symbol acts like a complex weight for the corresponding
subcarrier to be modulated. Since the input symbols are complex; the
value of each input symbol determines both the amplitude and the phase
of a sinusoid for one subcarrier. each input symbol or output sample has
a period of Ts seconds, each N parallel output samples form one OFDM
symbol (in time domain) with a period of Ts seconds, because the N
samples are in parallel (O/P sampling rate (fa)= N/Ts).

In practical OFDM system, most of the subcarriers are modulated with

data (not all subcarriers). The outer subcarriers are unmodulated and set
to zero amplitude and phase (zeros insertion). At the receiver, a FFT
block is used to process the received signal (OFDM symbols) and bring
it into the frequency domain.

4.3.4. Zeros Insertion:

Because of the low-pass filters required for the analog-to-digital and
digital to-analog conversion (ADC and DAC) of the transmitted and
received signals (baseband OFDM signal), not all N subcarriers can be
used, if an N-point IFFT is applied for modulation. The subcarriers close
to the Nyquist frequency fa/2, where fa = 1/Ta, is attenuated by these
filters and thus cannot be used for data transmission. As shown in Fig.
4.5, these zero subcarriers provide a frequency guardband before the

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Nyquist frequency also, effectively act as an interpolation of the signal,
and allows for a realistic roll off in the analog anti-aliasing
reconstruction filters. Also the DC-subcarrier might be heavily distorted
by DC offsets of the ADCs and DACs and should thus be avoided for
data.
Figure 4.6 shows a simple block diagram of OFDM transmitter with
zeros insertion and guard period addition. At the receiver after
demodulating the signal by the FFT zeros are removed before the output
of the FFT is applied on the signal de-mapping.

Fig. 4.5. The effect of the anti-aliasing filter on the outer subcarriers.

Fig. 4.6. Zeros insertion and guard period addition in OFDM transmitter.

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4.3.5. Guard Period Insertion:
One of the most important properties of OFDM transmissions is the
robustness against multipath delay spread. This is achieved by having a
long symbol period, which minimizes the ISI. The level of robustness
can in fact be increased even more by the addition of a guard period
between transmitted symbols. The guard period allows time for
multipath signals from the pervious symbol to die away before the
information from the current symbol is gathered. If the delay spread is
longer than the guard period then they begin to cause ISI. However,
provided the echoes are sufficiently small they do not cause significant
problems. This is true most of the time as multipath echoes delayed
longer than the guard period will have been reflected of very distant
objects. The ratio of the guard period to the useful symbol duration
(effective OFDM symbol) is application dependent. If this ratio is large,
then the overhead will increase causing a decrease in the system
throughput. Usually, the guard period is selected to have a length of one
tenth to a quarter of the symbol period. The only drawback of this
principle is a slight loss of effective transmitted power, as the redundant
guard period must be transmitted.
The guard period could be a section of all zero samples transmitted
in front of each OFDM symbol. Since it does not contain any useful
information, the guard period would be discarded at the receiver. This
guard period is not used in practical systems, because a long silence can
cause the receiver time synchronization to be lost. The most effective
guard period to be used is a cyclic extension of the symbol, by replicate
part of the OFDM time-domain symbol (a few samples) from back to the
front to create a guard period called cyclic prefix, as shown in Fig. 4.7,
this effectively extends the length of the symbol, while maintaining the
orthogonality of the waveform.

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 Using a cyclic prefix generates a continuous signal, with no
discontinuities at the joint between the original OFDM symbol and the
guard period. As shown in Fig. 4.7, because the subcarrier frequencies
are chosen to be integer multiples of the inverse of the input symbol
time, resulting in all subcarriers having an integer number of cycles per
symbol. Also, this make the OFDM symbol appear periodic over the
OFDM symbol time.

Fig 4.7. Cyclic prefix addition generates a continuous periodic signal.

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4.4. Adaptive Modulation
Adaptive modulation is a powerful technique for maximizing the data
throughput of subcarriers allocated to a user. Adaptive modulation
involves measuring SNR of each subcarrier in the transmission, then
selecting a modulation scheme that will maximize the spectral efficiency,
while maintaining an acceptable Bit Error Rate (BER). This technique
has been used in ADSL, to maximize the system throughput. Using
adaptive modulation in a wireless environment is difficult as the radio
channel response and SNR can change very rapidly, requiring frequent
updates to track these changes. Any errors in channel estimation can
result in large increase in the BER, due to the small link margin used.
The modulation scheme in an OFDM system can be selected based on
the requirement of power or spectrum efficiency.
An important advantage of OFDM is that different modulation
schemes can be used on different sub-channels for layered services. Most
OFDM systems use a fixed modulation scheme over all carriers for
simplicity. However each carrier in multiuser OFDM system can
potentially have a different modulation scheme depending on the channel
conditions. Any coherent or differential, phase or amplitude modulation
scheme can be used including Binary Phase Shift Keying (BPSK),
Quadrature Phase Shift Keying (QPSK), 8-Phase Shift Keying (8-PSK),
16-Quadrature Amplitude Modulation (16-QAM), 64-QAM, .… Each
modulation scheme provides a tradeoff between spectral efficiency and
the BER. The spectral efficiency can be maximized by choosing the
highest order modulation scheme that will give an acceptable BER.
Figure 4.8 shows, an example of applying adaptive modulation to an
individual subcarrier as the channel SNR varies with time. The SNR
must be greater than a specific threshold to maintain a maximum BER.

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 Fig. 4.8. An adaptive modulation scheme based on the SNR of the
channel.
Adaptive modulation has a number of key advantages over using static
modulation. In systems that use a fixed modulation scheme the carrier
modulation must be designed to provide an acceptable BER under the
worst channel conditions. This results in most systems using BPSK or
QPSK. These give a poor spectral efficiency (1-2 bits/s/Hz) and provide
an excess link margin most of the time. Using adaptive modulation, the
remote stations can use a much higher order modulation scheme when
the radio channel conditions are suitable. Thus as a remote station
approaches the base station the modulation can be increased from 1
bits/s/Hz (BPSK) up to 4-6 bits/s/Hz (16-QAM – 64-QAM),
significantly increasing the spectral efficiency of the overall system.
Figure. 4.9 illustrates how higher order modulations like 64-QAM are
used closer to the base station, while lower order modulations like QPSK
are used to extend the range of the base station. Adaptive modulation can
effectively control the BER of the transmission.

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 Fig. 4.9. Adaptive modulation scheme for each station.
There are several limitations with adaptive modulation. Overhead
information needs to be transferred, as both the transmitter and receiver
must know what modulation is currently being used. In addition, as the
mobility of the remote station is increased, the adaptive modulation
process requires regular updates, further increasing the overhead. There
is a tradeoff between power control and adaptive modulation. If a remote
station has a good channel path, the transmitted power can be maintained
at a high modulation scheme (e.g. 64 QAM), or the power can be
reduced and the modulation scheme reduced accordingly (e.g. QPSK).
4.5. Multipath Effects
A major problem in most wireless systems is the presence of a
multipath channel. In a multipath environment, the transmitted signal
reflects from several objects such as trees, hills, buildings, vehicles, or
walls. The multiple versions of the signal cause the received signal to be
distorted. Many wired systems also have a similar problem where
reflections occur due to impedance mismatches in the transmission line.
One of the most important reasons to use OFDM is the efficient way it
deals with multipath delay spread. By dividing the input data stream into
N subcarriers, the symbol duration is increased N times. This also
reduces the relative multipath delay spread, relative to the symbol time,
by the same factor. To eliminate Inter Symbol Interference (ISI) almost
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completely, a guard period is introduced for each OFDM symbol. The
guard period is chosen larger than the expected delay spread, such that
multipath components from one symbol cannot interfere with the next
symbol. As shown in Fig. 4.10, the baseband OFDM symbol consists of
the effective OFDM symbol (N samples) and the guard period part (G
samples). The total length of the OFDM symbol in seconds is Ttotal= Tg
+ Ts, where Tg is the length of the guard period in seconds, and Ts is the
length of the effective OFDM symbol in seconds. As long as maximum
excess delay (Tmax) is smaller than the length of the guard period (Tg),
the distorted part of the signal will stay within the guard period, which
will be removed later at the receiver. Therefore, ISI due to multipath
components of one symbol and the next symbol will be prevented.

OFDM symbol time

Ttotal

Previous symbol current symbol next symbol

Guard FFT interval

Period Ts
Tg

Previous symbol Symbol from path A next symbol

Previous symbol Symbol from path B next symbol

Previous symbol Symbol from path C next symbol

Tmax

Fig. 4.10. Effect of ISI in the presence of the guard period.

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 In an OFDM signal, the amplitude and phase of the subcarrier must
remain constant over the period of the symbol in order for the subcarriers
to maintain orthogonality. At the symbol boundary, the amplitude and
phase change suddenly to the new value required for the next data
symbol. In multipath environments, ISI causes spreading of the energy
between the symbols, resulting in transient changes in the amplitude and
phase of the subcarrier at the start of the symbol. The length of these
transient effects corresponds to the delay spread of the radio channel.
The transient signal is a result of each multipath component arriving at
slightly different times, changing the received subcarrier vector. Figure
4.11 shows this effect. Adding a guard period allows time for the
transient part of the signal to decay, so that the FFT is taken from a
steady state portion of the symbol. The remaining effects caused by the
multipath, such as amplitude scaling and phase rotation are corrected by
channel equalization.

(a) No multipath.

(b) With multipath.

Fig. 4.11 Function of the guard period for protecting against ISI.
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 The guard period could consist of no signal at all. In this case,
however, the problem of Inter-Carrier Interference (ICI) would arise. ICI
is crosstalk between different subcarriers, which means they are no
longer orthogonal. This effect is illustrated in Fig. 4.12. In this example,
a subcarrier 1 and a delayed subcarrier 2 are shown. When an OFDM
receiver tries to demodulate the first subcarrier, it will encounter some
interference from the second subcarrier, because within the FFT interval,
there is no integer number of different cycles between subcarrier 1 and 2.
At the same time, there will be crosstalk from the first to the second
subcarrier for the same reason. To eliminate ICI, the OFDM symbol is
cyclically extended in the guard period, as shown in Fig. 4.13. This
ensures that delayed replicas of the OFDM symbol always have an
integer number of cycles within the FFT interval, as long as the delay is
smaller than the guard time. As a result, multipath signals with delays
smaller than the guard time cannot cause ICI.

As an example of how multipath affects OFDM, Fig. 4.14 shows

received signal from a two-ray channel, where the dotted curve is a
delayed replica of the solid curve. Three separate subcarriers are shown
during three symbol intervals. In reality, an OFDM receiver only sees the
sum of all these signals, but showing the separate components makes it
more clear what the effect of multipath is. Figure 4.14 shows that the
OFDM subcarriers are BSK modulated, which means that there can be
180 degree phase jumps at the symbol boundaries. For the dotted curve,
these phase jumps occur at a certain delay after the first path. In this
particular example, this multipath delay is smaller than the guard time,
which means there are no phase transitions during the FFT interval.
Hence, an OFDM receiver sees the sum of pure sine waves with some
phase offsets. This summation does not destroy the orthogonality

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between the subcarriers, it only introduces a different phase shift for each
subcarrier. The orthogonality is lost if the multipath delay is larger than
the guard time. In that case, the phase transitions of the delayed path fall
within the FFT interval of the receiver. The summation of the sine waves
of the first path with the phase modulated waves of the delayed path no
longer gives a set of orthogonal pure sine waves, resulting in a certain
level of interference.

Fig. 4.12. Effects of multipath with zero signal in the guard time; the
delayed subcarrier 2 causes ICI on subcarrier 1 and vice versa.

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Fig. 4.13. Effects of multipath with a cyclic prefix as a guard period; the
delayed subcarrier 2 and subcarrier 1 still orthogonal in the FFT time.

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Fig. 4.14. Example of an OFDM signal, with three subcarriers in a two-
ray multipath channel.

4.6. Frequency Selective Fading

Frequency selective fading causes deep fading at certain frequencies.
This is due to the phase response of the multipath components varying
with frequency. The received phase, relative to the transmitter, of a
multipath component corresponds to the number of wavelengths the
signal has traveled from the transmitter. The wavelength is inversely
proportional to the frequency and so for a fixed transmission path the
phase will change with frequency. The path distances of each of the

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multipath component is different and so results in a different phase
change.
Figure 4.15 shows an example of two-path transmission. Path 1 is a
direct signal and has a transmission distance of 10 m, while the second
path is a reflection with a longer transmission distance of 25 m. For a
wavelength of 1 m each path is an integer number of wavelengths hence
the phase change from transmitter to receiver will be 0° for each path. At
this frequency, the two paths will reinforce each other. If we change the
frequency to have a wavelength of 0.9 m then path 1 will be 10/0.9 =
11.111λ, or a phase of 0.111 × 360° = 40°, while second path will be
25/0.9 = 27.778 λ, a phase of 0.778 × 360° = 280°. This makes the two
paths out of phase, which results in a reduction in the signal amplitude at
this frequency. For environments with a large number of multipath
components, complex variations in the fading versus frequency will
occur.

Fig. 4.15. Two-path transmission to demonstrate frequency selective

fading.
Figure 4.16 shows that because the OFDM waveform is composed of
multiple narrowband orthogonal carriers, frequency selective fading is
localized to a subset of carriers that are relatively easy to equalize. On
the other hand, frequency selective fading is more harmful in case of
single carrier system. In OFDM system, the distribution of the data over
many carriers means that frequency selective fading will cause some bits
to be received in error while others are received correctly. By using a

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forward error correcting code, which adds extra bits at the transmitter, it
is possible to correct many or all of the bits that were incorrectly
received.
Single carrier system OFDM system

Frequency Frequency
S0
S1
S2 S0 S1 S2 S3 S4 S5
Time
S3
S4
S5
Serial symbol stream used to Each of the symbols is
modulate a single wide band used to modulate a
carrier separate carrier

(a) Symbols modulation in a single carrier system and an OFDM system.

(b) The effect of frequency selective fading in case of a single carrier

system and an OFDM system. (The dotted area represent the transmitted
spectrum, the solid area is the receiver input).
Fig. 4.16. OFDM is more resistant to frequency selective fading when
compared to a single carrier system.
4.7. Synchronization
Before an OFDM receiver demodulates the subcarriers, it has to
perform at least two synchronization tasks. First, it has to find out where
the symbol boundaries are and what the optimal timing instants are to
minimize the effects of ICI and ISI. Second, it has to estimate and
correct the carrier frequency offset of the received signal, because any
offset introduces ICI.

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1. Sensitivity to Frequency Offset
One significant problem with OFDM is its sensitivity to frequency
offsets affecting the performance. The demodulation of an OFDM signal
with an offset in the frequency can lead to a high bit error rate. As
mentioned before, all OFDM subcarriers are orthogonal if they all have a
different integer number of cycles within the FFT interval. If there is a
frequency offset, then the number of cycles in the FFT interval is not an
integer anymore, this resulting in ICI, and a lack of correction for phase
rotation of the received data vectors. The FFT output for each subcarrier
will contain interfering terms from all other subcarriers. The
characteristics of ICI are similar to Gaussian noise, hence it leads to
degradation of the SNR. The amount of degradation is proportional to
the fractional frequency offset, which is equal to the ratio of frequency
offset to the frequency spacing. The amount of ICI for subcarriers in the
middle of the OFDM spectrum is approximately twice as large as that for
subcarriers at the band edges, because the subcarriers in the middle have
interfering subcarriers on both sides, so there are more interferes within a
certain frequency distance.
Frequency errors will tend to occur from two main sources. These are
local oscillator errors and Doppler spread. Any difference between
transmitter and receiver local oscillators will result in a frequency offset.
This offset is usually compensated for, by using frequency tracking,
however any residual errors result in a degraded system performance.
Movement of the transmitter or receiver results in Doppler shift in the
signal. This appears as a frequency offset for free space propagation.
This offset is usually corrected for as part of the local oscillator
compensation. A much more serious problem is that of Doppler spread,
which is caused by movement of the transmitter or receiver in a
multipath environment. Doppler spread is caused by the different relative
velocity of each of the reflected multipath components, resulting in the
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signal being frequency modulated. This frequency modulation on the
subcarriers tends to be random due to the large number of multipath
reflections that occur in typical environments. This Doppler spread is
typically poorly compensated for and results in degradation of the signal.
The use of pilots and reference symbols (preamble) are efficient
methods for carrier recovery and channel equalization. A pilot can be a
sine wave or a known binary sequence. The two-dimensional
(time/frequency) signal feature in COFDM makes pilot and reference
symbol insertion very flexible. Pilots can be inserted in frequency
domain (fixed carriers) and reference symbols in the time domain (fixed
data packets). Because they are transmitted at the predetermined
positions in the signal frame structure, it can be captured in the receiver
whenever the frame synchronization is recovered. The number of pilots
and reference symbols used in a COFDM system determines the tradeoff
between payload capacity and transmission robustness.
2. Sensitivity to Timing Errors
OFDM is relatively tolerant to timing errors, due to the inclusion of
the guard period between symbols. For a channel with no multipath
delay spread, the time offset error can be as much as the length of the
guard period with no loss of orthogonality results. Because of the cyclic
nature of the guard period, changing the time offset (less than the guard
period length) simply results in a phase rotation of all the subcarriers in
the signal. The amount of this phase rotation is proportional to the
subcarrier frequency. Provided the time offset is held constant from
symbol to symbol, the phase rotation due to a time offset can be removed
out as part of the channel equalization. In multipath environments ISI
reduces the effective length of the guard period leading to a
corresponding reduction in the allowable time offset error.

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 Time offset errors greater than the guard period result in a rapid loss in
performance, as the section of the symbol that the FFT is applied to will
contain some of the neighboring symbol, leading to ISI. So, the symbol
timing offset may vary over an interval equal to the guard time without
causing ICI or ISI, as depicted in Fig. 4.17, ICI and ISI occur only when
the FFT interval extends over a symbol boundary. To minimize
sensitivity to timing errors, the system should be designed such that
timing error is small compared with the guard interval.
4.8. Peak-to-Average Power Ratio (PAPR)
OFDM signals have a higher PAPR than single-carrier signals. The
reason is that in the time domain, a multi-carrier signal is the sum of
many narrowband signals. At some time instances, this sum is large and
at other times is small, which means that the peak value of the signal is
substantially larger than the average value. When N signals are added
with the same phase, they produce a peak power that is N times the
average power. Although the PAPR is moderately high for OFDM, high
magnitude peaks occur relatively rarely and most of the transmitted
power is concentrated in signals of low amplitude, as shown in Fig. 4.18.
This high PAPR is one of the most important implementation challenges
that face OFDM, especially in broadcasting applications.
OFDM is significantly more sensitive to the nonlinear distortions
caused by the HPA than the corresponding single carrier systems.
Nonlinearity of any amplifier causes signal distortion and inter-
modulation products resulting in unwanted out of band power and higher
BER. One way to avoid nonlinear distortion is to operate the amplifier in
its linear region. Unfortunately such solution is not power efficient and
thus not suitable for battery operated wireless communication
applications. The power efficiency of an HPA can be increased by
reducing the PAPR of the transmitted signal. For example, the efficiency

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of a class A amplifier is halved when the input PAPR is doubled or the
operating point (average power) is halved.
To alleviate the nonlinear effects, numerous approaches have been
pursued. The first plan of attack is to reduce PAPR at the transmitter.
The second is to linearize the HPA characteristic using one of the
linearization techniques. Another set of techniques focuses on OFDM
signal reconstruction at the receiver in spite of the introduced
nonlinearities. A further approach is to attempt to transform the OFDM
signal prior to the HPA, and applying the inverse transform at the
receiver prior to demodulation. This approach includes Constant-
Envelope OFDM (CEOFDM), which uses a phase modulator as the
transformer.

Fig. 4.17. Example of an OFDM signal with three subcarriers, showing

the earliest and latest possible symbol timing instants that do not cause
ISI or ICI.

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Fig. 4.18. High PAPR of OFDM transmitted signal.

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