# Orthogonal

**Frequency
Division Multiplexing**

# Contents

## 1. Introduction.
2

2.
OFDM Basic
Principles
3

3.
Other OFDM
Systems10.

4.
Benefits of OFDM and Performance
Criteria12

5.
Digital Audio Broadcasting An
Application of OFDM .
17

6.
Other Developments
.20

7.
References .
21

#### Introduction

Orthogonal
Frequency Division Multiplexing (OFDM) is a multicarrier transmission technique
used in applications catering to both Wired and Wireless Communications.
However, in the wired case, the usage of the term Discrete Multi-Tone is more
appropriate. The OFDM technique divides the frequency spectrum available into
many closely spaced carriers, which are individually modulated by low-rate data
streams. In this sense, OFDM is similar to FDMA (The bandwidth is divided into
many channels, so that, in a multi-user environment, each channel is allocated
to a user). However, the difference lies in the fact that the carriers chosen
in OFDM are much more closely spaced than in FDMA (1kHz in OFDM as opposed to
about 30kHz in FDMA), thereby increasing its spectral usage efficiency. The
orthogonality between the carriers is what facilitates the close spacing of
carriers.

The
orthogonality principle essentially implies that each carrier has a null at the
center frequency of each of the other carriers in the system while also
maintaining an integer number of cycles over a symbol period.

The motivation
for using OFDM techniques over TDMA techniques is twofold. First, TDMA limits
the total number of users that can be sent efficiently over a channel. In
addition, since the symbol rate of each channel is high, problems with
multipath delay spread invariably occur. In stark contrast, each carrier in an
OFDM signal has a very narrow bandwidth (i.e. 1 kHz); thus the resulting symbol
rate is low. This results in the signal having a high degree of tolerance to
multipath delay spread, as the delay spread must be very log to cause
significant inter-symbol interference (e.g. > 500usec).

#### OFDM Basic Principles

__Orthogonality__

To generate OFDM signals successfully the relationship between all carriers
must be carefully controlled in order to maintain orthogonality. Shown below is
the frequency spectrum depicting the various carriers/channels (used
interchangeably). Rectangular windowing of transmitted pulses results in a
sinc-shaped frequency response for each channel. As can be seen, whenever any
particular carrier frequency attains peak amplitude, the remaining carriers
have a null point.

Fig. Frequency spectrum showing N channels for an
OFDM system with N carriers over a bandwidth W

##### OFDM Generation

The spectrum
required is first chosen based on the input data and the modulation scheme used
(typically Differential BPSK, QPSK or QAM). Data to be transmitted is assigned
to each carrier that is to be produced. Amplitudes and phases of the carriers
are calculated based on the chosen scheme of modulation. The required spectrum
is then converted back to its time domain signal by employing Inverse Fourier
Transform algorithms like the Inverse Fast Fourier Transform (Cooley-Tukey
Algorithm)

The next step is
that of adding a guard period to the symbol to be transmitted. This ensures
robustness against multipath delay spread. This step can be achieved by having
a long symbol period, which minimizes intersymbol interference. The level of
robustness can be further increased by the addition of a guard period between
successive symbols. The most popular and effective method of doing this, is the
addition of a **cyclic prefix. **A
cyclic prefix is a copy of the last part of the OFDM symbol, which is prepended
to the transmitted symbol. This makes the transmitted signal periodic and does
not affect the orthogonality of the carriers. Further, this also plays a
decisive role in avoiding inter-symbol and inter-carrier interference.

Fig.
The Cyclic Prefix is a copy of the last part of the OFDM signal

A cyclic prefix
does however introduce a loss in the signal-to-noise ratio, but this effect is
usually negligible as compared to its effect on mitigating interference.

A schematic
diagram is shown next and a mathematical model of a base band OFDM system is
now developed.

##### Continuous-Time Model

Fig.
Base band OFDM system Model

Since the first
OFDM systems did not use digital modulation and demodulation schemes, the
continuous-time OFDM model shown above can be considered as the ideal OFDM
system. To build the mathematical model, we start with the waveforms used in
the transmitter and proceed all the way to the receiver.

###### Transmitter

We assume an
OFDM system with N carriers, a bandwidth of W Hz and a symbol length of T
seconds, of which T_{cp }seconds is the length of the cyclic prefix.
The transmitter uses the following waveforms:

_{}

=
0 otherwise
.Eqn. 1

where *T = (N/W) + T*_{cp}_{ .}

A note must also
be made of the fact that *f*_{k}(t)
= *f*_{k}(t + N/W)
when *t* is within the cyclic prefix.
Since *f*_{k}(t) is a
rectangular pulse modulated on the carrier frequency *kW/N*, the common interpretation of OFDM is that it uses N carriers,
each carrying a low bit-rate. The waveforms *f*_{k}(t)
are used in the modulation and the transmitted base band signal for OFDM symbol
number *l* is

_{}
..Eqn.
2

where *x*_{0,l}, x_{1,l}
,x_{N-1,l}
are complex numbers obtained from a set of signal constellation points.
When an infinite sequence of OFDM symbols is transmitted, the output from the
transmitter is a juxtaposition of individual OFDM symbols:

_{}
Eqn.3

###### The Physical Channel

An important
assumption is that the effect of the impulse response of the physical channel
(which may or may not be time invariant), is restricted to the time period

t I [0,T_{cp}], i.e. to the length of the cyclic prefix. The
received signal then becomes :

.Eqn.
4

where *n(t)* is additive, white and complex
Gaussian noise.

We now move on
to the receiver.

###### Receiver

A filter bank, matched
to the last part [T_{cp},T] of the transmitter waveforms *Φ*_{k}(t), i.e. ,

_{} Eqn. 5

= 0 otherwise

This operation
effectively removes the cyclic prefix in the receiver stage of the system. All
the ISI is contained in the Cyclic Prefix and does not manifest itself in the sampled output obtained at the
receiver filterbank. We can now remove the time index, *l*, when calculating the sampled output at the *k*th matched filter.

_{} Eqn.6

Considering the
channel to be fixed over the OFDM symbol interval and denoting it by g(τ),
Eqn.6, after simplification gives us the following result:

_{} Eqn.7

where *G(f)* is the Fourier transform of *g(τ)* and *n*^{}_{k}
is additive white Gaussian noise.

We move on next
to the Discrete Time Model for the OFDM system

##### Discrete-Time Model

The modulation
and demodulation (with *Φ*_{k}(t)
&* ψ*_{k}(t)) in the
continuous-time model are replaced by the Inverse Discrete Fourier Transform
and the Discrete Fourier transform respectively while the channel is a
Discrete-Time convolution. The Cyclic Prefix operates in the same way in this
system and calculations are essentially performed in the same fashion. As in
all other cases, the integrals are changed to summations when in the
Discrete-Time domain. An end-to-end discrete-time model is shown below:

Fig.
Discrete-Time OFDM System

The employment
of a cyclic prefix longer in duration than the channel, transforms the linear
convolution into a cyclic convolution, when seen from the receiver end of the
system. Denoting the cyclic convolution by
* , we can depict the whole OFDM
system by the following equation :

y_{l } = DFT(IDFT(x_{l}) * g_{l} + n_{l})
= DFT(IDFT(x_{l}) * g_{l}) + n^{}_{l}
.Eqn.8

where *y*_{l} contains the N received
data points, *x*_{l} the N
transmitted constellation points, *g*,
the channel impulse response (padded with zeroes to obtain a length, N) and *n*_{l}, the channel noise. Since
the channel noise is assumed to be white and Gaussian, the term, *n*_{l}=DFT(n_{l}) represents
uncorrelated Gaussian noise. Using the result that the DFT of two cyclically
convolved signals is equivalent to the product of their individual DFTs, we
obtain

y_{l }= x_{l}
. DFT(g_{l}) + n_{l } =
x_{l . }h_{l }+ n_{l}
Eqn.9

where the symbol
. denotes element-by-element multiplication.

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##### Other OFDM Systems

While describing
OFDM systems in the previous sections certain assumptions were made. These have
been listed below:

§
A Cyclic Prefix is used

§
The impulse response of the
channel is shorter than the Cyclic Prefix

§
Since fading effects are slow
enough, the channel is considered time-invariant over the symbol interval

§
Rectangular Windowing of the
transmitted pulses

§
Transmitter and Receiver are in
perfect synchronism

§
Channel noise is additive, white,
and complex Gaussian

Also worth
mentioning is the fact that the transmitted energy increases with an increase
in the duration of the Cyclic Prefix, while the expressions for received and
sampled signals stay the same. The transmitted energy per carrier is given by

_{} ..Eqn.10

and the SNR loss because of the discarded
Cyclic Prefix in the receiver becomes

SNR_{loss }=
-10 log_{10}(1- γ) ..Eqn.11

where γ = T_{cp}/T is the relative
length of the Cyclic Prefix. Therefore, a longer Cyclic Prefix would mean a
higher SNR loss. Typically, the relative length of the Cyclic Prefix is small
and the ICI- and ISI- free transmission motivates the usage of OFDM , the SNR
loss being less than 1 dB for γ<0.2.

Depending on the
channel characteristics and desired complexity of the synchronization circuitry
in the receiver we could design certain other OFDM systems.

__Case 1 :__ If the Channel response
is bad and increasing the length of the Cyclic Prefix makes the SNR loss a
substantial quantity, we can resort to adding a guard time interval between
symbols. The **Guard Period** is
characterised by a zero transmission, i.e., transmitting silence. This scheme
also has another advantage. It might help simplifying synchronization
circuitry. Simple envelope detection might be enough because of the presence of
the guard period

__Case 2 :__ The power spectrum of
the OFDM system decays as *f*^{-2}
since we were using a rectangular window for the transmitted pulses. In certain
cases, this may not be good enough and methods have been proposed to shape the
spectrum. Shown below is the spectrum where a **raised cosine** pulse is used. In this case the roll off region also
acts as a guard space. If the flat part is the OFDM symbol, including the
cyclic prefix, both ICI and ISI are avoided. The spectrum with this kind of
pulse shaping is shown further below, where it is compared with a rectangular
pulse. It is easily seen that this kind of spectrum falls much more quickly and
reduces the interference to adjacent bands.

Other types of pulse shaping such as overlapped and
well-localized pulses have also been investigated.

Amplitude

time

Fig. Puldse Shaping using Raised
Cosine Fig.
Normalised Spectrum with Rectangular

Grey indicates the part including
CP and signal
-pulse(solid) and raised-cosine(dashed)

#### Benefits of OFDM and Performance Criteria

The four main criteria for evaluating the performance of
the OFDM system are tolerance to multipath delay spread, peak power clipping,
channel noise and time synchronization errors. The performance of different
OFDM systems under varied channel conditions, keeping in mind the above
criteria is now discussed.

Multipath delay Spread Immunity

In a MATLAB simulation of a
practical OFDM system modeled by Eric Lawrey [2], the following assumptions
were made:

Carrier
Modulation Used: DBPSK, DQPSK or D16PSK

FFT
Size : 2048

Number
of Carriers Used : 800

Guard
Time: 512 samples (25%)

Guard
Period Type: Half Zero, Half a cyclic extension of the symbol

__Note:__ A point to note here would be that most OFDM systems
in effect are COFDM (C=Coded), meaning, Forward Error Correction is applied to
the OFDM signal. Typically, an 800 carrier system would allow a maximum of 100
users to operate. Each user is allocated 8 carriers so that even if some
carriers are lost due to frequency selective fading, the rest will allow the
lost data to be recvered using the error correction scheme.

It was found that a delay spread of 256
samples (corresponding to approximately 80milliseconds) occurred only if it was
assumed that a reflection that traveled 24km extra path length suffered only a
3dB attenuation. As can be seen from the BER v/s Multipath Delay Spread curve,
there is little or no delay associated with reflections that reached the
receiver within the lifetime of the guard period. The Delay Spread increases
rapidly only after the guard period has ended (due to ISI). However, if a
signal is attenuated by more than the noise tolerance of the OFDM signal, no
significant effect will occur on the BER.

The maximum BER occurs when the delay spread is longer
than the symbol interval itself. Such a case would definitely increase the
Inter Symbol Interference.

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__Peak Power Clipping:__

The OFDM signal showed high degrees of tolerance (BER is
not affected adversely) even if it was heavily clipped. The clipping
distortions mostly arise from the Power Amplifier transmitting the signal. The
signal can be clipped by as high as 9dB without a significant effect on the
BER. This could be used to our advantage, meaning, the OFDM signal could be
clipped by up to 6dB so that the Peak-to-RMS ratio can be reduced, thus alowing
an increased transmitted power.

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__Gaussian Noise Tolerance of OFDM:__

Since the transmitted signal is similar to standard FDM,
it is found that the SNR performance is similar to standard single carrier
digital transmission. The BER is found to be adversely affected, if the SNR
drops below 6dB. The SNR tolerance is mostly dependant on the kind of
modulation used (i.e. QPSK, BPSK, 16PSK etc.) as shown in the plot.

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__Time-Synchronization
Errors__

The Synchronization factor in an OFDM system is the most
critical one. When the receiver is initially turned on, it is not in
synchronization with the transmitter. For this reason, data transmission in an
OFDM system might need data to be sent in frames. At the beginning of each
frame a null symbol is transmitted, so that the receiver can detect incoming
data using simple envelope detection techniques. However, the noise in the
signal might interfere with the envelope detection process. In general, it has
been found that the receiver synchronizes itself with the transmitter in a time
interval less than or equal to the guard interval.

#### Digital Audio Broadcasting An application of OFDM

fig.
Shows a block diagram representation of the DAB system

DAB is a means
of providing current AM & FM listeners with a new service that offers :

·
Improved sound quality (CD
comparable)

·
Increased service availability,
especially reception in moving vehicles

·
Flexible coverage scenarios

·
High Spectrum Efficiency

As can be seen in the block diagram, the source
(Music etc.) is encoded using the MPEG Layer-II Audio standards and is
broadcast using COFDM. The Eureka-147 DAB signal consists of multiple carriers
assembled using COFDM within a 1.536 MHz channel bandwidth. Four possible modes
of operation define the channel coding configuration, specifying the total number
of carriers, the carrier spacing and also the guard interval duration. Each
channel provides a raw data rate of 2304 Kbps; after error protection, a useful
data rate, anywhere between 600 to 1800 Kbps is available to the service
provider, depending on the user-specified multiplex configuration. This useful
data rate can be divided into a large number of possible configurations of
audio and data programs.

The more interesting issue(in the context of this paper) is the
implementation of the COFDM system. Though there are several processing stages required to
generate and receive an OFDM signal, the most processing intensive stage are
the ones in which the Fast Fourier Transform or its Inverse transform are
implemented.

The complexity of performing an FFT is dependent on
the size of the FFT. However, it can be seen that because the symbol period
increases with a larger FFT, the extra processing required is minimal. For e.g.
the 2048-point FFT requires only 1.1 times the
time required for processing a 1024-point FFT. To get an estimate of the
processing power required to implement a practical phone system, lets consider
an example.

Given Data:

Total bandwidth : 1.25 Mhz

User capacity : 64 Users

Carriers per User: 13

Modulation Used : QPSK

FFT Size : 2048

No.of complex calculations
required for a 2048-pt. FFT : 33792

Guard Period : 512 samples

From the above data we can compute

No. of Active carriers : 832

Data Rate of each user : 39 Kbps

Useful Symbol time : 666 msecs

Total Symbol time : 833 msecs

The maximum time that can be taken in performing the
calculations is once every symbol, thus once every 833 msecs. Now, if we assume,
the processor requires 2 instructions to perform a single complex calculation,
and there is an overhead of 30% for scheduling of tasks and other processing.
The minimum processing power of the DSP being used, must then be MIPS = (33792 x 2 x 1.3 x 10^{-6})/(833
x 10^{-6})=105

Since a transmitter requires a minimum of 105 MIPS,
a transceiver would definitely require a minimum of 210 MIPS. DSPs that are lesser priced do not have such processing
capabilities (e.g. ADSP-2185L is 40MIPS).

The DAB system uses the Texas Instruments DSP
TMS-320C62x family of processors. These are 16/32-bit fixed point processors
valued between $25 to $180. The processing capabilities of these processors are
around 1200 MIPS.

^{}

#### Other Developments

The history of
OFDM dates back to the mid 60s when Chang published his paper on the synthesis
of bandlimited signals for multichannel transmission. Shortly after Chang
presented his paper, Saltzberg performed an analysis of the performance, where
he concluded that the strategy should concentrate more on reducing cross talk
between adjacent channels than on perfecting the channels. A major contribution
was made in 1971, by Weinstein and Ebert. They used the DFT to perform baseband
modulation and demodulation. Thereafter Peled and Ruiz introduced the concept
of the Cyclic Prefix in 1980.

Currently, work
is on in the following fields among others :

·
Algorithms to reduce Peak to
Average Power Ratio in multicarrier communication systems

·
Coding strategies for OFDM with
antenna diversity for high Bit-Rate Mobile data Applications

·
Multiuser Subcarrier Allocation
for OFDM using Adaptive Modulation

·
Fading and carrier frequency
Offset Robustness for different Pulse Shaping Filters in OFDM

#### References

1.
Ove Edfors, Magnus Sandell, Jan-Jaap van
de Beek, Daniel Landstrom, Frank Sjoberg *An
Introduction to orthogonal frequency division multiplexing* University of Lulea, September,1996

2.
Eric Lawrey *The suitability of OFDM as a modulation technique for wireless
communication, with a CDMA comparison* James Cook University, October,1997

3.
Knud Knudsen, Bob Heise, Mohsen
Hosseinian , Michael Fattouche *A 26 Mbps
OFDM Transceiver* Wi-LAN Inc.,Canada

4.
Adrian Bohdanowicz, Chris van den Bos,
Maarten Ditzel, Wouter A. Serdijn, Gerard J.M. Janssen, Ed. F.A. Deprettere *Wireless Link using OFDM Modulation:
Performance prediction, Modeling and Implementation*

5.
Michael Speth *OFDM Receivers for Broadband Transmission*

6.
Radio Broadcast Technologies Research
division of the Communications Research Centre (CRC), Ottawa, Canada
* Digital Radio Broadcasting* www.drb.crc.doc.ca

7.
Texas Instruments *ETS 300 401 DSP DAB modulator* www.ti.com