User manual | SIMULATION OF WIRELESS COMMUNICATION SYSTEM USING OFDM PRINCIPLE

SIMULATION OF WIRELESS COMMUNICATION
SYSTEM USING OFDM PRINCIPLE
A THESIS SUBMITTED IN PARTIAL FULFILMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
Master of Technology
In
VLSI Design and Embedded System
By
D.Jagadeesh
Roll No: 20507003
Department of Electronics and Communication Engineering
National Institute of Technology
Rourkela
2005 - 2007
SIMULATION OF WIRELESS COMMUNICATION
SYSTEM USING OFDM PRINCIPLE
A THESIS SUBMITTED IN PARTIAL FULFILMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
Master of Technology
In
VLSI Design and Embedded System
By
D.Jagadeesh
Roll No: 20507003
Under the guidance of
Prof. G. Panda
Department of Electronics and Communication Engineering
National Institute of Technology
Rourkela
2005 - 2007
National Institute of Technology
Rourkela
CERTIFICATE
This is to certify that the thesis entitled, “Simulation of Wireless Communication System
Using OFDM Principle” submitted by Mr. Jagadeesh Darapureddy in partial fulfillment of
the requirements for the award of Master of Technology Degree in Electronics and
communication Engineering with specialization in “VLSI Design and Embedded System”
at the National Institute of Technology, Rourkela (Deemed University) is an authentic work
carried out by him under my supervision and guidance.
To the best of my knowledge, the matter embodied in the thesis has not been submitted to any
other University/ Institute for the award of any degree or diploma.
Date:
Prof. G. Panda
Dept. of Electronics & comm. Engineering
National Institute of Technology, Rourkela
Rourkela - 769008
ACKNOWLEDGEMENTS
This project is by far the most significant accomplishment in my life and it would be
impossible without people who supported me and believed in me.
I would like to extend my gratitude and my sincere thanks to my honorable, esteemed
supervisor Prof. G. Panda, Head, Department of Electronics and Communication
Engineering. He is not only a great lecturer with deep vision but also and most importantly a
kind person. I sincerely thank for his exemplary guidance and encouragement. His trust and
support inspired me in the most important moments of making right decisions and I am glad
to work with him.
I want to thank all my teachers Prof. G.S. Rath, Prof. K. K. Mahapatra, Prof. S.K.
Patra and Prof. S.K. Meher for providing a solid background for my studies and research
thereafter. They have been great sources of inspiration to me and I thank them from the
bottom of my heart.
I would like to thank all my friends and especially my classmates for all the
thoughtful and mind stimulating discussions we had, which prompted us to think beyond the
obvious. I’ve enjoyed their companionship so much during my stay at NIT, Rourkela.
I would like to thank all those who made my stay in Rourkela an unforgettable and
rewarding experience.
Last but not least I would like to thank my parents, who taught me the value of hard
work by their own example. They rendered me enormous support during the whole tenure of
my stay in NIT Rourkela.
D. Jagadeesh
Contents
Abstract
iv
List of figures and tables
vi
1. INTRODUCTION
1.1 Introduction
1
1.2 Motivation
1
1.3 Literature Survey
3
1.4 Contribution
4
1.5 Thesis Outline
4
2. MULTIPLE ACCESS TECHNIQUES
2.1 Introduction
5
2.2 Duplexing
5
2.3 Introduction to Multiple Access
6
2.3.1 FDMA (Frequency Division Multiple Access)
6
2.3.2 TDMA (Time Division Multiple Access)
7
2.3.3 CDMA (Code Division Multiple Access)
7
3. DIGITAL MODULATION TECHNIQUES
3.1 Basic CONCEPTS of Modulation
9
3.1.1 Three kinds of modulations
9
3.2 ASK
9
3.3 FSK
10
3.4 PSK
11
3.4.1 BPSK
12
3.4.2 QPSK
14
3.4.2.1 Gray Code
15
3.4.2.2 QPSK Modulation
15
3.4.3 MPSK
17
3.5 QAM
18
3.5.1 Rectangular QAM
19
i
4. DIFFERNET TRAMSFORMS
4.1 Discrete Fourier Transform (DFT)
21
4.1.1 Definition
21
4.1.2 Properties
22
4.1.3 Applications
23
4.2 Fast Fourier Transform (FFT)
24
4.3 Discrete Hartley Transform (DHT)
25
4.3.1 Definition
25
4.3.2 Properties
26
4.3.3 Advantages
27
4.4 Discrete Cosine Transform (DCT)
28
4.4.1 Definition:
29
4.4.2 Properties of DCT
29
4.4.3 Applications
29
5. ORTHOGONAL FREQUENCY DIVISION MULTIPLEXING
5.1 Evolution of OFDM
31
5.2 Introduction to OFDM
31
5.3 Importance of orthogonality
32
5.4 OFDM is a special case of FDM
34
5.5 An example of OFDM using 4 sub-carriers
35
5.5.1 Using inverse FFT to create the OFDM symbol
5.6 Fading
39
43
5.6.1 Defining fading
43
5.6.2 Use of cyclic prefix to mitigate delay spread:
46
5.7 Properties of OFDM
49
5.7.1 Spectrum and performance
49
5.7.2 OFDM is a special case of FDM:
50
5.7.3 Peak to average power ratio (PAPR)
51
5.7.4 Synchronization:
52
5.7.5 Coding
52
5.8 OFDM Applications
53
5.9 Parameters of real OFDM
54
ii
6. OFDM SIMULATION AND RESULTS
6.1 Working of the OFDM Model
55
6.2 OFDM System Simulation with BPSK
57
6.2.1 Simulation Results for OFDM with BPSK
6.3 OFDM system simulation with QPSK
58
59
6.3.1 Simulation Results for OFDM with QPSK
59
6.4 Comparison of OFDM systems with QPSK and BPSK
61
6.5 OFDM implementation using FFT
61
6.5.1 Simulation Results
62
6.6 OFDM implementation using DHT
63
6.6.1 Simulation Results
64
6.7 OFDM implementation using DCT
65
6.7.1 Simulation Results
65
6.8 Comparison of performance of OFDM systems
66
7. CONCLUSION
7.1 Summary
68
7.2 Scope of future work
69
REFERENCES
70
iii
ABSTRACT
FDMA, TDMA and CDMA are the well known multiplexing techniques used in
wireless communication systems. While working with the wireless systems using these
techniques various problems encountered are (1) multi-path fading (2) time dispersion which
lead to intersymbol interference (ISI) (3) lower bit rate capacity (4) requirement of larger
transmit power for high bit rate and (5) less spectral efficiency. In a typical terrestrial
broadcasting, the transmitted signal arrives at the receiver using various paths of different
lengths. Since multiple versions of the signal interfere with each other, it becomes difficult to
extract the original information. The use of orthogonal frequency division multiplexing
(OFDM) technique provides better solution for the above mentioned problems. OFDM
technique distributes the data over a large number of carriers that are spaced apart at precise
frequencies. This spacing provides the "orthogonality", which prevents the demodulator from
seeing frequencies other than their own. The benefits of OFDM are high spectral efficiency,
resiliency of RF interference, and lower multi-path distortion. OFDM is a powerful
modulation technique that is capable of high data rate and is able to eliminate ISI. The use of
FFT technique to implement modulation and demodulation functions makes it
computationally more efficient. The OFDM based wireless communication system design
includes the design of OFDM transmitter, and OFDM receiver. Using MATLAB, simulation
of OFDM was done with different modulation techniques using different transform
techniques. The digital modulation schemes such as BPSK and QPSK were selected to assess
the performance of the designed OFDM system.
Different transform techniques such as Discrete Fourier transform (DFT) and its
inverse (IDFT), discrete Hartley transform (DHT) and its inverse (IDHT), Discrete Cosine
Transform and its inverse (IDCT) are used to perform the modulation and demodulation
operations and compared the performance of the designed OFDM system.
For OFDM-based transceivers, the modulator needs to compute a long-length inverse
discrete Fourier transform (IDFT), and the demodulator needs to compute a long length DFT,
where the transform length is up to 512 or more. For such long-length DFT/IDFT
computations, a great number of complex multiplications are required and each of them
basically involves four real multiplications and two real additions. Clearly, the complexity of
an OFDM-based transceiver would be reduced if the corresponding modulator/demodulator
could be implemented using fast algorithms like Fast Fourier transform (FFT) algorithm.
iv
The complexity of an OFDM-based transceiver would also be reduced if the
corresponding modulator/demodulator could be implemented using purely real transforms
like discrete Hartley transform (DHT) and discrete cosine transform (DCT). Since the DHT
and IDHT definitions are identical, we can use the same hardware or program to implement
the modulator and demodulator of the OFDM system. Like the DFT, there have been a
number of fast algorithms and hardware architectures available for the DHT computation.
A discrete cosine transform (DCT) is a Fourier-related transform similar to the
discrete Fourier transform (DFT), but using only real numbers. The DCT is often used in
signal and image processing, especially for lossy data compression, because it has a strong
"energy compaction" property: most of the signal information tends to be concentrated in a
few low-frequency components of the DCT. The Discrete cosine transform (DCT) is also a
real valued transform, used to find where the power is concentrated in a signal. So it will
transmit those parts only where the power was concentrated.
From the simulation results, it is observed that the BPSK allows the BER to be
improved in a noisy channel at the cost of maximum data transmission capacity. Use of
QPSK allows higher transmission capacity, but at the cost of slight increase in the probability
of error. From the results, use of OFDM with QPSK is beneficial for short distance
transmission link, whereas for long distance transmission link OFDM with BPSK will be
preferable.
From the results it is also observed that the DHT-based OFDM achieves the same
transmission performance as the DFT-based OFDM from the BER vs SNR point of view, but
requires less computational complexity with reduced implementation cost, because of it is
purely based on the real valued DHT. And the DCT based OFDM has better BER
performance compared to the DFT, DHT based OFDM methods, because of its strong
"energy compaction" property and real valued nature. So, from the BER vs SNR,
implementation and performance points of view, DHT, DCT transform techniques are may be
alternatives to DFT for the designing of OFDM-based communication systems.
v
List of Figures
Fig 2.1 (a) Frequency Division Duplexing (b) Time Division Duplexing
6
Fig 2.2 Frequency division multiple access (FDMA)
7
Fig 2.3 Time division multiple access (TDMA).
7
Fig 2.4 Code division multiple access (CDMA).
8
Fig 3.1 Baseband information sequence – 0010110010
10
Fig 3.2 Binary ASK (OOK) signal
10
Fig 3.3 Binary FSK signal
10
Fig 3.4 Binary PSK Carrier
11
Fig 3.5 Constellation diagram for BPSK
12
Fig 3.6 Illustrating of the modulation of a message signal to yield a BPSK signal
13
Fig 3.7 Amplitude spectrum of BPSK
13
Fig 3.8 (a) Binary modulating signal, and (b) BPSK signal
14
Fig 3.9 Constellation diagram for QPSK with Gray coding.
14
Fig 3.10 QPSK Modulator
16
Fig 3.11 The Mathematic representation of the process of the QPSK modulator
16
Fig 3.12 QPSK modulation: (a) binary sequence and (b) QPSK signal
17
Fig 3.13 Constellation diagram for M-ary PSK
18
Fig 3.14 M-ary PSK modulator
18
Fig 3.15 Constellation diagram for rectangular 16-QAM.
20
Fig 4.1 DCT (bottom) compared to the DFT (middle) of an input signal (top)
30
Fig 5.1 (a) A Regular-FDM single carrier (b) Orthogonal-FDM
32
Fig 5.2 Multi-carrier FDM and Multi-code division multiplex
32
Fig 5.3 Example of OFDM spectrum for 5 orthogonal carriers
33
Fig. 5.4 The area under a sine and a cosine wave over one period
33
Fig 5.5 the area under a sine wave multiplied by its own harmonic
34
Fig. 5.6 Bandwidth utilization of FDM
34
Fig 5.7 Bandwidth utilization of OFDM
35
Fig. 5.8 A bit stream that will be modulated using a 4 carrier OFDM.
35
Fig. 5.9 General Block diagram of an OFDM system
36
Fig. 5.10 Sub-carrier 1 and the bits it is modulating
36
Fig. 5.11 Sub-carrier 2 and the bits that it is modulating
37
vi
Fig. 5.12 Sub-carrier 3 and 4 and the bits that they modulating
37
Fig 5.13 OFDM signal in time and frequency domain.
38
Fig 5.14 Functional diagram of an OFDM signal creation.
38
Fig 5.15 the generated OFDM signal
39
Fig 5.16 The two views of a signal
39
Fig 5.17 FFT and IFFT are a matched linear pair
41
Fig 5.18 The incoming block of bits can be seen as a four bin spectrum,
42
Fig 5.19 The OFDM link functions
43
Fig 5.20 Fading is particular problem when the link path is changing,
44
Fig .21 Reflected signals arrive at a delayed time period and interfere
with the main line of sight signal, if there is one.
44
Fig 5.22 (a) the signal and the channel frequency response.
(b) A fading channel frequency response.
(c) OFDM have many little sub-carriers.
45
Fig. 5.23 Delay spread
46
Fig. 5.24 (Composite interference) The PSK symbol and its delayed version
46
Fig 5.25 Move the symbol back so the arriving delayed signal peters
out in the gray region. No interference to the next symbol
46
Fig. 5.26 If we just extend the symbol, then the front of the symbol
which is important to us since it allows figuring out what the phase of this
symbol is, is now corrupted by the “splash”.
47
Fig. 5.27 If we move the symbol back and just put in convenient filler in this area, then not
only we have a continuous signal but one that can get corrupted and we don’t care since we
will just cut it out anyway before demodulating.
47
Fig. 5.28 Cyclic prefix is copy of back of the symbol we add to the front of the symbol. 48
Fig 5.29 Addition of Cyclic prefix to the OFDM signal further improves
its ability to deal with fading and interference.
48
Fig. 5.30 The whole process can be done only once to the OFDM signal,
rather than doing it to each and every sub-carrier.
49
Fig. 5.31 The spectrum of an OFDM signal (without addition of cyclic prefix)
is much more bandwidth efficient than QPSK.
49
Fig. 5.32 The spectrum of an OFDM signal with 1024 sub-carriers
50
Fig. 5.33 The spectrum of a QPSK signal
50
vii
Fig. 5.34 Spectrum of FDM
51
Fig. 5.35 Spectrum of OFDM
51
Fig.5.36 AN OFDM signal is very noise like. It looks just like a
composite multi-FDM signal.
52
Fig 6.1 OFDM model used for simulation
55
Fig 6.2 addition of guard band interval with symbol
56
Fig 6.3 BER versus SNR curve for OFDM with BPSK
59
Fig 6.4 BER versus SNR curve for OFDM with BPSK
60
Fig 6.5 Comparison of BER versus SNR of Two Modulation Schemes
61
Fig 6.6 BER versus SNR curve for FFT based OFDM
63
Fig 6.7 BER versus SNR curve for DHT based OFDM
65
Fig 6.8 BER versus SNR curve for DCT based OFDM
66
Fig 6.9 Comparison of Bit Error Rates of FFT, DHT, and DCT based OFDM systems
67
viii
List of Tables
Table 3.1 Four symbols mapping definitions for QPSK
15
Table 5.1 Serial to parallel conversion of data bits.
36
Table 6.1 BER results for OFDM model using BPSK modulation
58
Table 6.2 BER results for OFDM model using QPSK modulation
60
Table 6.3 BER results for FFT based OFDM
62
Table 6.4 BER results for DHT based OFDM
64
Table 6.5 BER results for DCT based OFDM
66
ix
Chapter 1
INTRODUCTION
1. INTRODUCTION
1.1 INTRODUCTION:
In a basic communication system, the data are modulated onto a single carrier
frequency. The available bandwidth is then totally occupied by each symbol. This kind of
system can lead to inter-symbol-interference (ISI) in case of frequency selective channel. The
basic idea of OFDM is to divide the available spectrum into several orthogonal subchannels
so that each narrowband subchannel experiences almost flat fading. Orthogonal frequency
division multiplexing (OFDM) is becoming the chosen modulation technique for wireless
communications. OFDM can provide large data rates with sufficient robustness to radio
channel impairments. Many research centers in the world have specialized teams working in
the optimization of OFDM systems. In an OFDM scheme, a large number of orthogonal,
overlapping, narrow band sub-carriers are transmitted in parallel. These carriers divide the
available transmission bandwidth. The separation of the sub-carriers is such that there is a
very compact spectral utilization. With OFDM, it is possible to have overlapping subchannels
in the frequency domain, thus increasing the transmission rate. The attraction of OFDM is
mainly because of its way of handling the multipath interference at the receiver. Multipath
phenomenon generates two effects (a) Frequency selective fading and (b) Intersymbol
interference (ISI).
The "flatness" perceived by a narrowband channel overcomes the frequency selective
fading. On the other hand, modulating symbols at a very low rate makes the symbols much
longer than channel impulse response and hence reduces the ISI. Use of s u i t a b l e error
correcting codes provides more robustness against frequency selective fading. The insertion
of an extra guard interval between consecutive OFDM symbols can reduce the effects of ISI
even more. The use of FFT technique to implement modulation and demodulation functions
makes it computationally more efficient. OFDM systems have gained an increased interest
during the last years. It is used in the European digital broadcast radio system, as well as in
wired environment such as asymmetric digital subscriber lines (ADSL). This technique is
used in digital subscriber lines (DSL) to provides high bit rate over a twisted-pair of wires.
1.2 MOTIVATION:
Multimedia is effectively an infrastructure technology with widely different origins in
computing, telecommunications, entertainment and publishing. New applications are
1
emerging, not just in the wired environment, but also in the mobile one. At present, only low
bit-rate data services are available to the mobile users. The radio environment is harsh, due to
the many reflected waves and other effects. Using adaptive equalization techniques at the
receiver could be the solution, but there are practical difficulties in operating this equalization
in real-time at several Mb/s with compact, low-cost hardware. A promising candidate that
eliminates a need for the complex equalizers is the Orthogonal Frequency Division
Multiplexing (OFDM), a multiple carrier modulation technique. OFDM is robust in adverse
channel conditions and allows a high level of spectral efficiency. It effectively mitigates
performance degradations due to multipath and is capable of combating deep fades in part of
the spectrum. The OFDM waveform can be easily modified to adjust to the delay spread of
the channel. OFDM can handle large delay spreads easier to due the independence of the
carriers and the flexibility of varying the cyclic prefix length. OFDM allows efficient
operation in both FDD and TDD mode as very short or no pre-ambles are needed. Multiple
access techniques which are quite developed for the single carrier modulations (e.g. TDMA,
FDMA) had made possible of sharing one communication medium by multiple number of
users. Multiple techniques schemes are used to allow many mobile users to share
simultaneously a finite amount of radio spectrum. The sharing is required to achieve high
capacity by simultaneously allocating the available bandwidth (or the available amount of
channels) to multiple users.
For the quality communications, this must be done without severe degradation in the
performance of the system. FDMA, TDMA and CDMA are the well known multiplexing
techniques used in wireless communication systems. While working with the wireless
systems using these techniques various problems encountered are (1) multi-path fading (2)
time dispersion which lead to intersymbol interference (ISI) (3) lower bit rate capacity (4)
requirement of larger transmit power for high bit rate and (5) less spectral efficiency.
Disadvantage of FDMA technique is its Bad Spectrum Usage. Disadvantages of TDMA
technique is Multipath Delay spread problem. In a typical terrestrial broadcasting, the
transmitted signal arrives at the receiver using various paths of different lengths. Since
multiple versions of the signal interfere with each other, it becomes difficult to extract the
original information. The use of orthogonal frequency division multiplexing (OFDM)
technique provides better solution for the above mentioned problems.
2
1.3 LITERATURE SURVEY:
The concept of using parallel data transmission by means of frequency division
multiplexing (FDM) was published in mid 60’s [23, 24]. Some early development with this
can be traced back to the 50s. A U.S. patent was filled and issued in January 1970. The idea
was to use parallel data streams and FDM with overlapping sub channels to avoid the use of
high-speed equalization and to combat impulsive noise, and multipath distortion as well as to
fully use the available bandwidth. The initial applications were in the military
communications. In the telecommunications field, the terms of discrete multi-tone (DMT),
multichannel modulation and multicarrier modulation (MCM) are widely used and sometimes
they are interchangeable with OFDM. In OFDM, each carrier is orthogonal to all other
carriers. However, this condition is not always maintained in MCM. OFDM is an optimal
version of multicarrier transmission schemes. Weinstein and Ebert [25] applied the discrete
Fourier transform (DFT) [12] to parallel data transmission system as part of the modulation
and demodulation process. In the 1980s, OFDM has been studied for high speed modems,
digital mobile communications [10] and high-density recording.
Various fast modems were developed for telephone networks. In 1990s, OFDM has
been exploited for wideband data communications over mobile radio FM channels [14],
wireless LAN [13] wireless multimedia communication, high-bit-rate digital subscriber lines
(HDSL) [16], asymmetric digital subscriber lines (ADSL) [20], very high speed digital
subscriber lines (VHDSL), digital audio broadcasting (DAB) [18] and HDTV terrestrial
broadcasting.
In a classical parallel data system, the total signal frequency band is divided into N
nonoverlapping frequency subchannels. Each subchannel is modulated with a separate
symbol and then the N subchannels are frequency-multiplexed. It seems good to avoid
spectral overlap of channels to eliminate interchannel interference. However, this leads to
inefficient use of the available spectrum. To cope with the inefficiency, the ideas proposed
from the mid-1960s were to use parallel data and FDM with overlapping subchannels, in
which, each carrying a signaling rate b is spaced b apart in frequency to avoid the use of
high-speed equalization and to combat impulsive noise and multipath distortion, as well as to
fully use the available bandwidth.
3
1.4 CONTRIBUTION:
Using MATLAB, simulation of OFDM was done with different modulation
techniques using different transform techniques. The digital modulation schemes such as
BPSK and QPSK were selected to assess the performance of the designed OFDM system by
finding their Bit Error rate for different values of SNR.
Different transform techniques such as Discrete Fourier transform (DFT) and its
inverse (IDFT), discrete Hartley transform (DHT) and its inverse (IDHT), Discrete Cosine
Transform and its inverse (IDCT) are used to perform the modulation and demodulation
operations in the implementation of OFDM and compared their performance by finding their
Bit Error rate for different values of SNR.
1.5 THESIS OUTLINE:
Following the introduction, the rest of the thesis is organized as follows. Chapter 2
gives a review on existing multiple access techniques used in wireless communication
systems. Chapter 3 describes different digital modulation techniques that are used in OFDM.
Chapter 4 discribes about different transform techniques that are available like FFT, DHT,
and DCT. Chapter 5 gives an over view of OFDM. It discribes OFDM basic principle, its
working model, properties, parameters, and applications. In Chapter 6 discribes OFDM
simulations and results. Then I made a conclusion to my work and the points to possible
directions for future work in Chapter 7.
4
Chapter 2
MULTIPLE ACCESS TECHNIQUES
2. MULTIPLE ACCESS TECHNIQUES
2.1 INTRODUCTION
Multiple access techniques are employed side by side in cellular systems. The need for
multiple access techniques arises from the necessity to share a limited resource of radio
spectrum amongst many users. Multiple access schemes are used to allow many mobile users
to share simultaneously a finite amount of radio spectrum. The sharing of spectrum is
required to achieve high capacity by simultaneously allocating the available bandwidth (or
the available amount of channels) to multiple users.
2.2 DUPLEXING
In wireless communications systems, it is often desirable to allow the subscriber to send
simultaneously while receiving information to the base station while receiving information
from the base station. This effect is called duplexing and a device called duplexer is used
inside each subscriber unit and base station. Duplexing may be done using frequency or time
domain techniques. Figure 2.1 illustrates FDD (Frequency Division Duplexing) and TDD
(Time Division Duplexing).
Frequency division duplexing (FDD) provides two distinct bands of frequencies for every
user. The forward band provides traffic form the base station to mobile, and the reverse
provides traffic from the mobile to base station.
Time division duplexing (TDD) uses time instead of frequency to provide both a forward and
reverse link. In TDD, multiple users share a single radio channel by taking turns in the time
domain. Individual users allowed to access the channel in assigned time slots, and each
duplex channel has both forward time slot and a reverse time slot to facilitate bidirectional
communication.
5
Figure 2.1: (a) FDD provides two simplex channels at the same time; (b) TDD provides two
simplex time slots at the same frequency.
2.3 INTRODUCTION TO MULTIPLE ACCESS:
Frequency division multiple access (FDMA), time division multiple access (TDMA), and
code division multiple access (CDMA) are the three major access techniques used to share
the available bandwidth in a wireless communication system
2.3.1 FDMA (Frequency Division Multiple Access): Frequency Division Multiple Access
was the initial multiple-access technique for cellular systems. In this technique a user is
assigned a pair of frequencies when placing or receiving a call. One frequency is used for
downlink (base station to mobile) and one pair for uplink (mobile to base). This is called
frequency division duplexing. That frequency pair is not used in the same cell or adjacent
cells during the call. During the period of the call, no other user can share the same channel.
If an FDMA channel is not in use, then it sits idle and cannot be used by other users to
increase or share capacity. It is essentially a wasted resource. Figure 2.2 illustrates a FDMA
system. Even though the user may not be talking, the spectrum cannot be reassigned as long
as a call is in place. Frequency Division Multiple Access (FDMA) is the most common
analog system. It is a technique whereby spectrum is divided up into frequencies and then
assigned to users. With FDMA, only one subscriber at any given time is assigned to a
channel. The channel therefore is closed to other conversations until the initial call is
finished, or until it is handed-off to a different channel. A “full-duplex” FDMA transmission
requires two channels, one for transmitting and the other for receiving. FDMA has been used
for first generation analog systems.
6
Figure2.2: Frequency division multiple access (FDMA).
2.3.2 TDMA (Time Division Multiple Access): Time Division Multiple Access (TDMA)
improves spectrum capacity by splitting each frequency into time slots. TDMA allows each
user to access the entire radio frequency channel for the short period of a call. Other users
share this same frequency channel at different time slots. The base station continually
switches from user to user on the channel. TDMA is the dominant technology for the second
generation mobile cellular networks. TDMA system divide the radio spectrum into time
slots, and in each slot only one user is allowed to transmit and receive. It can be seen from
Figure 2.3 that each user occupies a cyclically repeating time slot, so a channel may be
thought of as a particular time slot that reoccurs every frame, where N time slots comprise a
frame.
Figure 2.3: Time division multiple access (TDMA).
2.3.3 CDMA (Code Division Multiple Access): Code Division Multiple Access is based on
“spread” spectrum technology. Since it is suitable for encrypted transmissions, it has long
been used for military purposes. CDMA increases spectrum capacity by allowing all users to
occupy all channels at the same time. Transmissions are spread over the whole radio band,
and each voice or data call are assigned a unique code to differentiate from the other calls
carried over the same spectrum. CDMA allows for a “soft hand-off”, which means that
terminals can communicate with several base stations at the same time. In CDMA systems,
the narrowband message signal is multiplied by a very large bandwidth signal called the
7
spreading signal. The spreading signal is a pseudo-noise code sequence that has a chip rate
which is orders of magnitudes greater than the data rate of the message. All users in a CDMA
system, as seen from figure 2.4, use the same carrier frequency and may transmit
simultaneously. Each user has its own pseudorandom codeword which is approximately
orthogonal to all other code words. The receiver performs a time correlation operation to
detect only the specific desired codeword. All other code words appear as noise due to
decorrelation. For detection of the message signal, the receiver needs to know the codeword
used by the transmitter. Each user operates independently with no knowledge of the other
users.
Figure 2.4: Code division multiple access (CDMA).
8
Chapter 3
DIGITAL MODULATION TECHNIQUES
3. DIGITAL MODULATION TECHNIQUES
3.1 BASIC CONCEPTS OF MODULATION
3.1.1 Three kinds of modulations
Modulation is the process of facilitating the transfer of information over a medium.
Sound transmission in air has limited range for the amount of power your lungs can generate.
To extend the range your voice can reach, we need to transmit it through a medium other than
air, such as a phone line or radio. The process of converting information (voice in this case)
so that it can be successfully sent through a medium (wire or radio waves) is called
modulation.
There are three basic types of digital modulation techniques. These are
1. Amplitude-Shift Keying (ASK)
2. Frequency-Shift Keying (FSK)
3. Phase-Shift Keying (PSK)
All of these techniques vary a parameter of a sinusoid to represent the information
which we wish to send. A general carrier wave may be written:
C (t ) = A sin (2πft + φ )
A sinusoid has three different parameters than can be varied. These are its amplitude, phase
and frequency. Modulation is a process of mapping such that it takes your voice (as an
example of a signal) converts it into some aspect of a sine wave and then transmits the sine
wave, leaving the actual voice behind. The sine wave on the other side is remapped back to a
near copy of your sound.
The medium is the thing through which the sine wave travels. So wire is a medium
and so are air, water and space. The sine wave is called the carrier. The information to be
sent, which can be voice or data is called the information signal. Once the carrier is mapped
with the information to be sent, it is no longer a sine wave and we call it the signal.
3.2 ASK
In ASK, the amplitude of the carrier is changed in response to information and all else is kept
fixed. In Binary ASK Bit 1 is transmitted by a carrier of one particular amplitude. To transmit
0, we change the amplitude keeping the frequency constant. On-Off Keying (OOK) is a
special form of ASK, where one of the amplitudes is zero as shown in fig 3.1 and fig 3.2.
9
Figure 3.1 - Baseband information sequence – 0010110010
Binary ASK (t ) = s (t ) sin(2π ft )
Figure 3.2 - Binary ASK (OOK) signal
3.3 FSK
In FSK, we change the frequency in response to information, In Binary FSK one
particular frequency for a 1 and another frequency for a 0 is used as shown in fig 3.3 for the
same bit sequence as above. In the example below, frequency f1 for bit 1 is higher than f2
used for the 0 bit.
Binary FSK(t) =
sin(2π f 1t )
sin(2π f 2t )
for bit 1
for bit 0
Figure 3.3 - Binary FSK signal
10
3.4 PSK
In PSK, we change the phase of the sinusoidal carrier to indicate information. Phase
in this context is the starting angle at which the sinusoid starts. The transmitted signal is a
sinusoid of fixed amplitude. Phase shift keying is a modulation process whereby the input
signal, a binary PCM waveform, shifts the output waveform to one of a fixed number of
states. The general analytic expression for PSK is
Si (t ) = (2 E / T )Λ 1 2* cos [ω o + ϕ i (t )]
0 ≤ t ≤ T i=1,…,M Where the phase term
given by
ϕ i (t ) = 2π i M
ϕ i (t )
will have M discrete values, typically
i=1,.....,M E is the symbol energy, T is symbol time duration.
For Binary PSK It has one fixed phase usually 0° when the data is 1. To transmit 0,
we shift the phase of the sinusoid by 180°. Phase shift represents the change in the state of the
information in this case. ASK techniques are most susceptible to the effects of non-linear
devices which compress and distort signal amplitude. To avoid such distortion, the system
must be operated in the linear range, away from the point of maximum power where most of
the non-linear behavior occurs. The use of phase shift keying produces a constant amplitude
signal and was chosen for its simplicity and to reduce problems with amplitude fluctuations
due to fading.
Binary PSK(t) =
sin(2π ft)
sin(2π ft +π)
for bit 1
for bit 0
Figure 3.4 - Binary PSK Carrier (Note the 180°phase shifts at bit edges)
11
3.4.1 BPSK
In binary phase shift keying (BPSK) the transmitted signal is a sinusoid of fixed
amplitude. BPSK is the simplest form of PSK. It uses two phases which are separated by
180° and so can also be termed 2-PSK. It has one fixed phase when the data is at one level
and when the data is at another level the phase is different by 180°. It does not particularly
matter exactly where the constellation points are positioned, and in this figure they are shown
on the real axis, at 0° and 180°. This modulation is the most robust of all the PSK’s, since it
takes serious distortion to make the demodulator reach an incorrect decision. It is, however,
only able to modulate at 1 bit/symbol (as seen in the figure 3.4) and so is unsuitable for high
data-rate applications when bandwidth is limited.
Fig 3.5 Constellation diagram for BPSK.
The bit error rate (BER) of BPSK in AWGN can be calculated as:
⎛
P b = Q ⎜⎜
⎝
2Eb
No
⎞
⎟⎟
⎠
Where No 2 is noise power spectral density (W/Hz)
Where Tb is bit interval
Where
Eb = PsTb
is the energy contained in a bit duration
Where Ps is power of sinusoid of amplitude A
So, Ps =
1 2
A
2
In binary phase-shift keying (BPSK) the transmitted signal is a sinusoid of fixed amplitude.
If the sinusoid is of amplitude A it has a power Ps =
12
1 2
A so that A = 2 Ps .
2
Thus the transmitted signal is either
VBPSK (t ) = 2 Ps cos (ω ot )
VBPSK (t ) = 2 Ps cos (ω ot + π )
= − 2 Ps cos (ω ot )
In BPSK the data b(t) is a stream of binary digits with voltage levels which, as a matter of
convenience, we take to be at +1v and -1v. When b(t) = 1v we say it is at logic level 1 and
when b(t) = -1v we say it is at logic level 0. Hence VBPSK (t ) can be written as
VBPSK (t ) = b(t ) 2 Ps cos (ω ot )
So, the BPSK signal is generated as shown in the figure 3.6
0
1
0
1
1
VBPSK (t )
b(t)
A cos (ω ot )
Figure 3.6: Illustrating of the modulation of a message signal to yield a BPSK signal.
In practice, a BPSK signal is generated by applying the waveform cos (ω ot ) , as a carrier, to a
balanced modulator and applying the baseband signal b(t) as the modulating waveform. In
this sense BPSK can be thought of as an AM signal.
If
f c is the frequency of the sinusoid and T is the bit interval then the spectrum of the
resulting BPSK signal is shown in the figure 3.7
Figure 3.7: Amplitude spectrum of BPSK.
13
For binary sequence m(t) = 0 1 0 1 0 0 1, and if the sinusoid s(t) is of amplitude of A, then
the resulting BPSK signal will be as shown in the figure 3.8
Binary
sequence
Time
(a)
BPSK
signal
Time
(b)
Figure 3.8 (a) Binary modulating signal, and (b) BPSK signal.
3.4.2 QPSK
QPSK (4-ary PSK) involves changing the phase of the transmitted waveform. Each
finite phase change represents unique digital data. A phase-modulated waveform can be
generated by using the digital data to change the phase of a signal while its frequency and
amplitude stay constant. A QPSK modulated carrier undergoes four distinct changes in phase
that are represented as symbols and can take on the values of π/4, 3π/4, 5π/4, and 7π/4. Each
symbol represents two binary bits of data. The constellation diagram of a QPSK modulated
carrier is shown in Figure 3.9
S (t ) = x cos( wt ) + y sin( wt )
Figure 3.9: Constellation diagram for QPSK with Gray coding.
Each adjacent symbol only differs by one bit.
14
Note that
Iamplitude = ( symbol exp ression)* cos( phase)
Qamplitude = ( symbol exp ression)* sin( phase)
Table 3.1: Four symbols mapping definitions for QPSK
Symbol
S1
Bits
00
Expression
( 2E T )
Λ
π
Phase
π/4
I
2 (− 1 2)
Q
2 (− 1 2)
*
π
3π/4
−2Λ (− 1 2)
2Λ (− 1 2)
*
π
5π/4
−2Λ (− 1 2)
−2Λ (− 1 2)
*
π
7π/4
2Λ (− 1 2)
−2Λ (− 1 2)
*
1 2 cos( wt + )
4
S2
10
( 2E T )
Λ
1 2 cos( wt + 3 )
4
S3
11
( 2E T )
Λ
1 2 cos( wt + 5 )
4
S4
01
( 2E T )
Λ
1 2 cos( wt + 7 )
4
Λ
Λ
3.4.2.1 Gray Code
The system performance of a digital communication network can be enhanced by
incorporating a coding technique, within the system, known as Gray coding. The gray
encoder is used to map the data in such a way as to help reduce bit errors. A QPSK system
takes the input data bits, two at a time, and creates a symbol that represents on of four phase
states. The gray encoder therefore is used to map every two input data bits to one of four
unique symbol values so that the bit pairs that are used to generate the symbols are only one
bit different from each adjacent symbol. This technique proves to help with error
performance because if a symbol is received in error, it will contain only one error bit if it
was received in error to an adjacent symbol. This can be more easily observed by viewing the
QPSK constellation diagram shown in Fig 3.9 This QPSK constellation diagram shows
symbols, each represented by two data bits that were first gray encoded. One can see that
each adjacent symbol is represented by two data bits that vary by one bit. The performance of
digital communication networks can further be enhances by the use of error correcting codes.
3.4.2.2 QPSK Modulation
Figure 3.10 represents the process of a QPSK modulator. First, the input binary bit
stream is split into two bit streams which are the even and odd bit streams (quadrature and inphase streams) by the serial to parallel converter. Then, send alternating bits to I, Q channels:
even bits to Q channel, odd bits to I channel.
Second, using the method of NRZ, the even and odd bits are converted from a
unipolar sequence to a bipolar sequence (0 to -1). Next, multiply Q channel with a sine of fc
and multiply I channel with a sine but shifted by 90 degree which is –cosine. Notice that the
15
90 degrees block in the figure transmits the upper sine sequence to the lower -cosine
sequence.
Finally, combining or adding the upper (I) and lower (Q) parts and passing through a
harmonic or channel filter will get the QPSK modulated output.
Figure 3.10: QPSK Modulator
Figure 3.11: The Mathematic representation of the process of the QPSK modulator
For a binary sequence m(t) = 0 0 0 1 1 0 1 1, if the sinusoid s(t) is of amplitude of A, then
the resulting QPSK signal will be as shown in the figure 3.12. Phase of the sinusoid is shifted
by 90°, 180°, 270°, 360° for data 00, 01, 10, 11 respectively
16
Binary
Sequence
Time
QPSK
Signal
Time
(b)
Figure 3.12 QPSK modulation: (a) binary sequence and (b) QPSK signal.
3.4.3 MPSK
In BPSK we transmit each bit individually. Depending on whether b(t) is logic 0 or logic 1,
we transmit one or another of sinusoid for the bit time Tb, the sinusoids differing in phase by
2 π /2 = 180°. In QPSK we lump together two bits. Depending on which of the four two bits
words develops, we transmit one or another of four sinusoids of duration 2Tb, the sinusoids
differing in phase by amount 2 π /4 = 90°. The scheme can be extended. Let us lump together
N bits so that in this N-bit symbol, extending over the time NTb, there are 2 N = M possible
symbols. Now let us represent the symbols by sinusoids of duration NTb = Ts which differ
from on another by the phase of 2 π /M.
Thus in M-ary PSK the waveforms used to identify the symbols are
vm (t ) = 2 Ps cos(ω o + φ m ) (m = 0, 1, ……, M-1)
With the symbol phase angel given by
φ m = (2m + 1)
π
M
17
Figure 3.13: Constellation diagram for M-ary PSK
Figure 3.14 M-ary PSK modulator
3.5 QAM
In BPSK, QPSK, and M-ary PSK we transmit, in any symbol interval, one signal or
another which are distinguished from one another in phase but are all of the same amplitude.
In each of these individual systems the end points of the signal vectors in signal space falls on
the circumference of a circle. Now we have note that our ability to distinguished one signal
vector from another in the presence of noise will depend on the distance between the vector
end points. It is hence rather apparent that we shall be able to improve the noise immunity of
a system by allowing signal vectors to differ, not only in phase but also in amplitude. We call
this as amplitude and phase shift keying or Quadrature amplitude modulation (QAM).
18
ASK is also combined with PSK to create hybrid systems such as Quadrature
Amplitude Modulation (QAM) where both the amplitude and the phase are changed at the
same time. QAM is a modulation scheme which conveys data by changing (modulating) the
amplitude of two carrier waves. These two waves, usually sinusoids, are out of phase with
each other by 90° and are thus called quadrature carriers—hence the name of the scheme.
As for many digital modulation schemes, the constellation diagram is a useful
representation. In QAM, the constellation points are usually arranged in a square grid with
equal vertical and horizontal spacing, although other configurations are possible. Since in
digital telecommunications the data is usually binary, the number of points in the grid is
usually a power of 2 (2, 4, 8...). Since QAM is usually square, some of these are rare—the
most common forms are 16-QAM, 64-QAM, 128-QAM and 256-QAM. By moving to a
higher-order constellation, it is possible to transmit more bits per symbol. However, if the
mean energy of the constellation is to remain the same (by way of making a fair comparison),
the points must be closer together and are thus more susceptible to noise and other
corruption; this results in a higher bit error rate and so higher-order QAM can deliver more
data less reliably than lower-order QAM.
If data-rates beyond those offered by 8-PSK are required, it is more usual to move to
QAM since it achieves a greater distance between adjacent points in the I-Q plane by
distributing the points more evenly. The complicating factor is that the points are no longer
all the same amplitude and so the demodulator must now correctly detect both phase and
amplitude, rather than just phase.
64-QAM and 256-QAM are often used in digital cable television and cable modem
applications. In the US, 64-QAM and 256-QAM are the mandated modulation schemes for
digital cable. In the UK, 16-QAM and 64-QAM are currently used for digital terrestrial
television (Freeview and Top Up TV).
3.5.1 Rectangular QAM
Rectangular QAM constellations are, in general, sub-optimal in the sense that they do
not maximally space the constellation points for a given energy. However, they have the
considerable advantage that they may be easily transmitted as two pulse amplitude
modulation (PAM) signals on quadrature carriers, and can be easily demodulated. The nonsquare constellations achieve marginally better bit-error rate (BER) but are harder to
modulate and demodulate.
The first rectangular QAM constellation usually encountered is 16-QAM, its
constellation diagram is shown in fig. 3.15. A Gray coded bit-assignment is also given. The
19
reason that 16-QAM is usually the first is that a brief consideration reveals that 2-QAM and
4-QAM are in fact binary phase-shift keying (BPSK) and quadrature phase-shift keying
(QPSK), respectively. Also, the error-rate performance of 8-QAM is close to that of 16-QAM
(only about 0.5dB better), but its data rate is only three-quarters that of 16-QAM.
Figure 3.15: Constellation diagram for rectangular 16-QAM.
Expressions for the symbol error-rate of rectangular QAM are not hard to derive but yield
rather unpleasant expressions. For an even number of bits per symbol, k, exact expressions
are available. They are most easily expressed in a per carrier sense:
1 ⎞ ⎛
3 Es ⎞
⎛
Psc = 2 ⎜1 −
⎟ Q ⎜⎜ M − 1 No ⎟⎟
M ⎠ ⎝
⎝
⎠
So,
Ps = 1 − (1 − Psc )
2
The bit-error rate will depend on the exact assignment of bits to symbols, but for a Graycoded assignment with equal bits per carrier:
4⎛
1 ⎞ ⎛ 3k Eb ⎞
Pbc = ⎜1 −
⎟
⎟Q⎜
k⎝
M ⎠ ⎝⎜ M − 1 No ⎠⎟
So,
Pb = 1 − (1 − Pbc )
2
M = Number of symbols in modulation constellation
Eb = Energy-per-bit
Es = Energy-per-symbol = kEb with k bits per symbol
No = Noise power spectral density (W/Hz)
Pb = Probability of bit-error
Pbc = Probability of bit-error per carrier
Ps = Probability of symbol-error
Psc = Probability of symbol-error per carrier
20
Chapter 4
DIFFERENT TRANSFORMS
4. DIFFERNET TRAMSFORMS
4.1 DISCRETE FOURIER TRANSFORM (DFT)
In mathematics, the discrete Fourier transform (DFT), occasionally called the finite Fourier
transform, is a transform for Fourier analysis of finite-domain discrete-time signals. It is
widely employed in signal processing and related fields to analyze the frequencies contained
in a sampled signal, to solve partial differential equations, and to perform other operations
such as convolutions. The DFT can be computed efficiently in practice using a fast Fourier
transform (FFT) algorithm.
Since FFT algorithms are so commonly employed to compute the DFT, the two terms are
often used interchangeably in colloquial settings, although there is a clear distinction: "DFT"
refers to a mathematical transformation, regardless of how it is computed, while "FFT" refers
to any one of several efficient algorithms for the DFT. This distinction is further blurred,
however, by the synonym "finite Fourier transform" for the DFT.
4.1.1 Definition
The sequence of N complex numbers x0, ..., xN−1 is transformed into the sequence of N
complex numbers X0, ..., XN−1 by the DFT according to the formula:
N −1
X k = ∑ xn e
−
2π i
kn
N
n =0
k = 0,..., N − 1
Where e is the base of the natural logarithm, is the imaginary unit (i 2 = −1) , and π is pi. The
inverse discrete Fourier transform (IDFT) is given by
2π i
kn
1 N −1
xn = ∑ X k e N n = 0,..., N − 1
N k =0
Note that the normalization factor multiplying the DFT and IDFT (here 1 and 1/N) and the
signs of the exponents are merely conventions, and differ in some treatments. The only
requirements of these conventions are that the DFT and IDFT have opposite-sign exponents
and that the product of their normalization factors be 1/N.
21
A normalization of 1
N for both the DFT and IDFT makes the transforms unitary, which
has some theoretical advantages, but it is often more practical in numerical computation to
perform the scaling all at once as above (and a unit scaling can be convenient in other ways).
(The convention of a negative sign in the exponent is often convenient because it means that
Xk is the amplitude of a "positive frequency" 2πk / N. Equivalently, the DFT is often thought
of as a matched filter: when looking for a frequency of +1, one correlates the incoming signal
with a frequency of −1.)
4.1.2 Properties
Completeness
The discrete Fourier transform is an invertible, linear transformation f : C N → C N
with C denoting the set of complex numbers. In other words, for any N > 0, an N-dimensional
complex vector has a DFT and an IDFT which are in turn N-dimensional complex vectors.
Orthogonality
2π i
The vectors e N
kn
form an orthogonal basis over the set of N-dimensional complex vectors:
N −1
∑e
2π i
kn
N
n=0
⎛ − 2Nπ i k1n ⎞
⎜e
⎟ = N δ kk1
⎝
⎠
Where δ kk1 is the Kronecker delta. This orthogonality condition can be used to derive the
formula for the IDFT from the definition of the DFT.
Periodicity
If the expression that defines the DFT is evaluated for all integers k instead of just for
k = 0,....., N − 1 then the resulting infinite sequence is a periodic extension of the DFT,
periodic with period N.
The periodicity can be shown directly from the definition:
N −1
X k + N = ∑ xn e
−
2π i
(k + N )n
N
n=0
N −1
= ∑ xn e
n=0
−
2π i
kn
−2π in
N
e
N −1
= ∑ xn e
n=0
−
2π i
kn
N
= Xk
Where we have used the fact that e −2π i = 1 . In the same way it can be shown that the IDFT
formula leads to a periodic extension.
22
Circular convolution theorem and cross-correlation theorem
The cyclic or circular convolution x*y of the two vectors x = xk and y = yn is the vector x*y
with components
N −1
( x ∗ y ) n = ∑ xm yn − m n = 0,....., N − 1
m=0
Where we continue y cyclically so that
y− m = y N − m
m = 0,....., N − 1
The discrete Fourier transform turns cyclic convolutions into component-wise multiplication.
That is, if
zn = (x * y)n
Then
Z k = X k Yk
k = 0,......., N − 1
Where capital letters (X, Y, Z) represent the DFTs of sequences represented by small letters
(x, y, z). The direct evaluation of the convolution summation, above, would require O( N 2 )
operations, but the DFT (via an FFT) provides an O(NlogN) method to compute the same
thing.
4.1.3 Applications
The DFT has seen wide usage across a large number of fields; we only sketch a few examples
below (see also the references at the end). All applications of the DFT depend crucially on
the availability of a fast algorithm to compute discrete Fourier transforms and their inverses,
a Fast Fourier Transform.
Partial differential equations
Discrete Fourier transforms are often used to solve partial differential equations, where again
the DFT is used as an approximation for the Fourier series (which is recovered in the limit of
infinite N). The advantage of this approach is that it expands the signal in complex
exponentials e jnx which are eigen functions of differentiation: d dx e jnx = jn e jnx
Thus, in the Fourier representation, differentiation is simple—we just multiply by i n. A
linear differential equation with constant coefficients is transformed into an easily solvable
algebraic equation. One then uses the inverse DFT to transform the result back into the
ordinary spatial representation. Such an approach is called a spectral method.
23
Spectral analysis
When the DFT is used for spectral analysis, the { xn } sequence usually represents a
finite set of uniformly-spaced time-samples of some signal x(t ) where t represents time. The
conversion from continuous time to samples (discrete-time) changes the underlying Fourier
transform of x(t) into a discrete-time Fourier transform (DTFT), which generally entails a
type of distortion called aliasing. Choice of an appropriate sample-rate (see Nyquist
frequency) is the key to minimizing that distortion. Similarly, the conversion from a very
long (or infinite) sequence to a manageable size entails a type of distortion called leakage,
which is manifested as a loss of detail (aka resolution) in the DTFT. DFT is just a discrete
sampling of the DTFT, which is a function of a continuous frequency domain. The
inefficiency of performing multiplications and additions is more than offset by the inherent
efficiency of the FFT. As already noted, leakage imposes a limit on the inherent resolution of
the DTFT. So there is a practical limit to the benefit that can be obtained from a fine-grained
DFT.
Data compression
The field of digital signal processing relies heavily on operations in the frequency
domain (i.e. on the Fourier transform). For example, several lossy image and sound
compression methods employ the discrete Fourier transform: the signal is cut into short
segments, each is transformed, and then the Fourier coefficients of high frequencies, which
are assumed to be unnoticeable, are discarded. The decompressor computes the inverse
transform based on this reduced number of Fourier coefficients. (Compression applications
often use a specialized form of the DFT, the discrete cosine transform or sometimes the
modified discrete cosine transform).
Multiplication of large integers
The fastest known algorithms for the multiplication of large integers or polynomials
are based on the discrete Fourier transform: the sequences of digits or coefficients are
interpreted as vectors whose convolution needs to be computed; in order to do this, they are
first Fourier-transformed, then multiplied component-wise, then transformed back.
4.2 FAST FOURIER TRANSFORM (FFT)
A Fast Fourier Transform (FFT) is an efficient algorithm to compute the discrete
Fourier transform (DFT) and its inverse. The Discrete Fourier Transform (DFT) is used to
produce frequency analysis of discrete non-periodic signals. The FFT is a faster version of
the Discrete Fourier Transform (DFT).
24
The FFT utilizes some clever algorithms to do the same thing as the DTF, but in much
less time. FFTs are of great importance to a wide variety of applications, from digital signal
processing to solving partial differential equations to algorithms for quickly multiplying large
integers. Let x0, ...., xN-1 be complex numbers. The DFT is defined by the formula
N −1
X k = ∑ xn e
−
2π i
nk
N
k = 0,....,N − 1
n =0
Evaluating these sums directly would take O(N2) arithmetical operations An FFT is
an algorithm to compute the same result in only O(N log N) operations. In general, such
algorithms depend upon the factorization of N, but (contrary to popular misconception) there
are FFTs with O(N log N) complexity for all N, even for prime N.
Many FFT algorithms only depend on the fact that e
−
2π i
N
is a primitive root of unity.
Since the inverse DFT is the same as the DFT, but with the opposite sign in the exponent and
a 1/N factor, any FFT algorithm can easily be adapted for it as well.
4.3 DISCRETE HARTLEY TRANSFORM (DHT)
A discrete Hartley transform (DHT) is a Fourier-related transform of discrete, periodic data
similar to the discrete Fourier transform (DFT), with analogous applications in signal
processing and related fields. Its main distinction from the DFT is that it transforms real
inputs to real outputs, with no intrinsic involvement of complex numbers. Just as the DFT is
the discrete analogue of the continuous Fourier transform, the DHT is the discrete analogue
of the continuous Hartley transform, introduced by R. V. L. Hartley in 1942.
Because there are fast algorithms for the DHT analogous to the fast Fourier transform (FFT),
the DHT was originally proposed by R. N. Bracewell in 1983 as a more efficient
computational tool in the common case where the data are purely real. It was subsequently
argued, however, that specialized FFT algorithms for real inputs or outputs can ordinarily be
found with slightly fewer operations than any corresponding algorithm for the DHT.
4.3.1 Definition:
Formally, the discrete Hartley transform is a linear, invertible function H: Rn -> Rn
(Where R denotes the set of real numbers). The N real numbers x0, ...., xN-1 are transformed
into the N real numbers H0, ..., HN-1 according to the formula
N −1 ⎡ ⎛ 2π ⎞
⎛ 2π ⎞ ⎤
Hk = ∑ xn ⎢ cos ⎜
nk ⎟ + sin ⎜
nk ⎟ ⎥
n=0 ⎣ ⎝ N
⎠
⎝ N
⎠⎦
25
k = 0, ….., N-1
Where π is Pi.
The IDHT is given by
N −1
⎡ ⎛ 2π ⎞
⎛ 2π ⎞ ⎤
xn = (1 N )∑ H k ⎢ cos ⎜
nk ⎟ + sin ⎜
nk ⎟ ⎥
n = 0,...., N − 1
⎠
⎝ N
⎠⎦
k =0
⎣ ⎝ N
π
The combination cos( z ) + sin( z ) = 2 cos( z − ) is sometimes denoted cas ( z ) , and should be
4
contrasted with the e − iz = cos( z ) − i sin( z ) that appears in the DFT definition (where i is the
imaginary unit). Like for the DFT, the overall scale factor in front of the transform and the
sign of the sine term are a matter of convention, and differ in some treatments, but do not
affect the essential properties.
4.3.2 Properties:
The transform can be interpreted as the multiplication of the vector (x0, ...., xN-1) by an
N-by-N matrix; therefore, the discrete Hartley transform is a linear operator. The matrix is
invertible; the inverse transformation, which allows one to recover the xn from the Hk, is
simply the DHT of Hk multiplied by 1/N. That is, the DHT is its own inverse (involutary), up
to an overall scale factor.
The DHT can be used to compute the DFT, and vice versa. For real inputs xn, the DFT
output Xk has a real part (Hk + HN-k)/2 and an imaginary part (HN-k - Hk)/2. Conversely, the
DHT is equivalent to computing the DFT of xn multiplied by 1+i, then taking the real part of
the result.
As with the DFT, a cyclic convolution z = x*y of two vectors x = (xn) and y = (yn) to
produce a vector z = (zn), all of length N, becomes a simple operation after the DHT. In
particular, suppose that the vectors X, Y, and Z denote the DHT of x, y, and z respectively.
Then the elements of Z are given by:
Zk = ⎡⎣ Xk (Yk + YN − k ) + XN − k (Yk − YN − k ) ⎤⎦ 2
ZN − k = ⎡⎣ XN − k (Yk + YN − k ) − Xk (Yk − YN − k ) ⎤⎦ 2
Where we take all of the vectors to be periodic in N (XN = X0, etcetera). Thus, just as the DFT
transforms a convolution into a point wise multiplication of complex numbers (pairs of real
and imaginary parts), the DHT transforms a convolution into a simple combination of pairs
of real frequency components. The inverse DHT then yields the desired vector z. In this way,
a fast algorithm for the DHT (see below) yields a fast algorithm for convolution. (Note that
this is slightly more expensive than the corresponding procedure for the DFT, not including
the costs of the transforms below, because the pair wise operation above requires 8 real-
26
arithmetic operations compared to the 6 of a complex multiplication. This count doesn't
include the division by 2, which can be absorbed e.g. into the 1/N normalization of the
inverse DHT.)
Fast algorithms:
Just as for the DFT, evaluating the DHT definition directly would require O(N2)
arithmetical operations. There are fast algorithms similar to the FFT, however, that compute
the same result in only O(N log N) operations. Nearly every FFT algorithm, from CooleyTukey to Prime-Factor to Winograd, has a direct analogue for the discrete Hartley transform.
In particular, the DHT analogue of the Cooley-Tukey algorithm is commonly known as the
fast Hartley transform (FHT) algorithm, and was first described by Bracewell in 1984. As
mentioned above, DHT algorithms are typically slightly less efficient (in terms of the number
of floating-point operations) than the corresponding DFT algorithm (FFT) specialized for real
inputs (or outputs). On present-day computers, performance is determined more by cache and
CPU pipeline considerations than by strict operation counts, and a slight difference in
arithmetic cost is unlikely to be significant. Since FHT and real-input FFT algorithms have
similar computational structures, neither appears to have a substantial a priori speed
advantage. As a practical matter, highly optimized real-input FFT libraries are available from
many sources (e.g. from CPU vendors such as Intel), whereas highly optimized DHT libraries
are less common.
On the other hand, the redundant computations in FFTs due to real inputs are much
more difficult to eliminate for large prime N, despite the existence of O(N log N) complexdata algorithms for such cases, because the redundancies are hidden behind intricate
permutations and/or phase rotations in those algorithms. In contrast, a standard prime-size
FFT algorithm, Rader's algorithm, can be directly applied to the DHT of real data for roughly
a factor of two less computation than that of the equivalent complex FFT. This DHT
approach currently appears to be the only known way to obtain such factor-of-two savings for
large prime-size FFTs of real data.
4.3.3 Advantages
The Discrete Hartley Transform is closely related to the Discrete Fourier Transform,
but unlike the DFT, the DHT has the advantage of producing real numbers. Furthermore, it is
(quasi-)symmetrical. The Hartley transform produces real output for a real input, and is its
own inverse. It therefore can have computational advantages over the discrete Fourier
transform.
27
4.4 DISCRETE COSINE TRANSFORM (DCT)
A discrete cosine transform (DCT) is a Fourier-related transform similar to the
discrete Fourier transform (DFT), but using only real numbers. DCTs are equivalent to DFTs
of roughly twice the length, operating on real data with even symmetry (since the Fourier
transform of a real and even function is real and even), where in some variants the input
and/or output data are shifted by half a sample.
Like other transforms, the Discrete Cosine Transform (DCT) attempts to decorrelate
the image data. After decorrelation each transform coefficient can be encoded independently
without losing compression efficiency. Like any Fourier-related transform, discrete cosine
transforms (DCTs) express a function or a signal in terms of a sum of sinusoids with different
frequencies and amplitudes. Like the discrete Fourier transform (DFT), a DCT operates on a
function at a finite number of discrete data points. The obvious distinction between a DCT
and a DFT is that the former uses only cosine functions, while the latter uses both cosines and
sines (in the form of complex exponentials). However, this visible difference is merely a
consequence of a deeper distinction: a DCT implies different boundary conditions than the
DFT or other related transforms.
The Fourier-related transforms that operate on a function over a finite domain, such as
the DFT or DCT or a Fourier series, can be thought of as implicitly defining an extension of
that function outside the domain. That is, once you write a function f(x) as a sum of sinusoids,
you can evaluate that sum at any x, even for x where the original f(x) was not specified. The
DFT, like the Fourier series, implies a periodic extension of the original function. A DCT,
like a cosine transform, implies an even extension of the original function.
However, because DCTs operate on finite, discrete sequences, two issues arise that do not for
the continuous cosine transform. First, one has to specify whether the function is even or odd
at both the left and right boundaries of the domain (i.e. the min-n and max-n boundaries in
the definitions below, respectively). Second, one has to specify around what point the
function is even or odd. In particular, consider a sequence abcd of four equally spaced data
points, and say that we specify an even left boundary. There are two sensible possibilities:
either the data is even about the sample a, in which case the even extension is dcbabcd, or the
data is even about the point halfway between a and the previous point, in which case the even
extension is dcbaabcd (a is repeated).
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4.4.1 Definition:
Let x (n) be a sequence of length N
Its DCT is given by
N −1
X (k ) = ∑ x(n) cos(2πnk / N )
n =0
k = 0 , 1, 2 , ...., N − 1
And the IDCT is given by
x(n) =
1
N
N −1
∑ X ( k ) cos(2π nk / N )
n = 0 , 1, 2 , . . . . , N − 1
k =0
4.4.2 Properties of DCT
Decorrelation
The principle advantage of image transformation is the removal of redundancy between
neighboring pixels. This leads to uncorrelated transform coefficients which can be encoded
independently. The amplitude of the autocorrelation after the DCT operation is very small at
all lags. Hence, it can be inferred that DCT exhibits excellent decorrelation properties.
Energy Compaction
Efficacy of a transformation scheme can be directly gauged by its ability to pack input data
into as few coefficients as possible. This allows the quantizer to discard coefficients with
relatively small amplitudes without introducing visual distortion in the reconstructed image.
DCT exhibits excellent energy compaction for highly correlated images.
Orthogonality
DCT basis functions are orthogonal. Thus, the inverse transformation matrix of A is equal to
its transpose i.e. A−1 = AT . Therefore, and in addition to its decorrelation characteristics, this
property renders some reduction in the pre-computation complexity.
4.4.3 Applications
The DCT, is often used in signal and image processing, especially for lossy data
compression, because it has a strong "energy compaction" property: most of the signal
information tends to be concentrated in a few low-frequency components of the DCT, see
figure 4.1, approaching the Karhunen-Loève transform (which is optimal in the decorrelation
sense) for signals based on certain limits of Markov processes.
29
For example, a DCT is used in JPEG image compression, MJPEG, MPEG, and DV
video compression. There, the two-dimensional DCT of N × N blocks are computed and the
results are quantized and entropy coded. In this case, N is typically 8 and the DCT-II formula
is applied to each row and column of the block. The result is an 8 × 8 transform coefficient
array in which the (0,0) element is the DC (zero-frequency) component and entries with
increasing vertical and horizontal index values represent higher vertical and horizontal spatial
frequencies. A related transform, the modified discrete cosine transform, or MDCT (based on
the DCT-IV), is used in AAC, Vorbis, and MP3 audio compression.
DCTs are also widely employed in solving partial differential equations by spectral
methods, where the different variants of the DCT correspond to slightly different even/odd
boundary conditions at the two ends of the array.
DCTs are also closely related to Chebyshev polynomials, and fast DCT algorithms are
used in Chebyshev approximation of arbitrary functions by series of Chebyshev polynomials,
for example in Clenshaw-Curtis quadrature.
Fig 4.1: DCT (bottom) compared to the DFT (middle) of an input signal (top)
30
Chapter 5
ORTHOGONAL FREQUENCY DIVISION
MULTIPLEXING
5. ORTHOGONAL FREQUENCY DIVISION MULTIPLEXING
5.1 EVOLUTION OF OFDM
Frequency Division Multiplexing (FDM)
Frequency Division Multiplexing (FDM) has been used for a long time to carry more than
one signal over a telephone line. FDM divides the channel bandwidth into subchannels and
transmits multiple relatively low rate signals by carrying each signal on a separate carrier
frequency. To ensure that the signal of one subchannel did not overlap with the signal from
an adjacent one, some guard-band was left between the different subchannels. Obviously, this
guard-band led to inefficiencies.
Orthogonal Frequency Division Multiplexing (OFDM)
In order to solve the bandwidth efficiency problem, orthogonal frequency division
multiplexing was proposed, where the different carriers are orthogonal to each other. With
OFDM, it is possible to have overlapping subchannels in the frequency domain, thus
increasing the transmission rate. This carrier spacing provides optimal spectral efficiency.
Today, OFDM has grown to be the most popular communication system in high-speed
communications. OFDM is becoming the chosen modulation technique for wireless
communications. OFDM can provide large data rates with sufficient robustness to radio
channel impairments.
5.2 INTRODUCTION TO OFDM
Orthogonal Frequency Division Multiplexing (OFDM)
Modulation - a mapping of the information on changes in the carrier phase, frequency or
amplitude or combination.
Multiplexing - method of sharing a bandwidth with other independent data channels.
OFDM is a combination of modulation and multiplexing. Multiplexing generally refers to
independent signals, those produced by different sources. In OFDM the question of
multiplexing is applied to independent signals but these independent signals are a sub-set of
the one main signal. In OFDM the signal itself is first split into independent channels,
modulated by data and then re-multiplexed to create the OFDM carrier.
OFDM is a special case of Frequency Division Multiplex (FDM). In an OFDM scheme, a
large number of orthogonal, overlapping, narrow band sub-carriers are transmitted in parallel.
These carriers divide the available transmission bandwidth. The separation of the sub-carriers
is such that there is a very compact spectral utilization.
31
As an analogy, a FDM channel is like water flow out of a faucet, a whole bunch of
water coming all in one stream; In contrast the OFDM signal is like a shower from which
same amount of water will come as a lot of small streams. In a faucet all water comes in one
big stream and cannot be sub-divided. OFDM shower is made up of a lot of little streams.
Fig 5.1 (a) A Regular-FDM single carrier (b) Orthogonal-FDM
The advantage one over the other is that if I put my thumb over the faucet hole, I can
stop the water flow but I cannot do the same for the shower. So although both do the same
thing, they respond differently to interference. Both methods carry the exact same amount of
data. But in case of any interfere to some of these small streams, only some part of data in the
OFDM method will suffer.
These small streams when seen as signals are called the sub-carriers in an OFDM
system and they must be orthogonal for this idea to work. The independent sub-channels can
be multiplexed by frequency division multiplexing (FDM), called multi-carrier transmission
or it can be based on a code division multiplex (CDM), in this case it is called multi-code
transmission.
Fig 5.2 Multi-carrier FDM and Multi-code division multiplex
5.3 IMPORTANCE OF ORTHOGONALITY
The main concept in OFDM is orthogonality of the sub-carriers. The "orthogonal"
part of the OFDM name indicates that there is a precise mathematical relationship between
the frequencies of the carriers in the system. It is possible to arrange the carriers in an OFDM
Signal so that the sidebands of the individual carriers overlap and the signals can still be
received without adjacent carriers interference. In order to do this the carriers must be
mathematically orthogonal. The Carriers are linearly independent ( i . e . orthogonal) if the
32
carrier spacing is a multiple of 1/T s . Where, T s is the symbol duration. The orthogonality
among the carriers can be maintained if the OFDM signal is defined by using Fourier
transform procedures. The OFDM system transmits a large number of narrowband carriers,
which are closely spaced. Note that at the central frequency of the each sub channel there is
no crosstalk from other sub channels.
Fig 5.3Example of OFDM spectrum for 5 orthogonal carriers
Since the carriers are all sine/cosine wave, we know that area under one period of a sine or a
cosine wave is zero. This is easily shown.
Fig. 5.4 - The area under a sine and a cosine wave over one period is always zero.
If a sine wave of frequency m multiplied by a sinusoid (sine or cosine) of a frequency n,
f (t ) = sin mwt × sin nwt
Where both m and n are integers, since these two components are each a sinusoid, the integral
is equal to zero over one period. The integral or area under this product is given by
=∫
2π
0
2π 1
1
cos( m − n) wt − ∫
cos( m + n) wt
0 2
2
=0-0
33
So when a sinusoid of frequency n multiplied by a sinusoid of frequency m/n, the area
under the product is zero. In general for all integers n and m, sin mx, cos mx, cos nx, sin nx
are all orthogonal to each other. These frequencies are called harmonics.
f (t ) = sin wt *sin nwt
Sine wave multiplied
by another of a
different harmonic
Fig 5.5 the area under a sine wave multiplied by its own harmonic is always zero
The orthogonality allows simultaneous transmission on a lot of sub-carriers in a tight
frequency space without interference from each other. In essence this is similar to CDMA,
where codes are used to make data sequences independent (also orthogonal) which allows
many independent users to transmit in same space successfully.
5.4 OFDM is a special case of FDM
Frequency Division Multiplexing FDM is a special case of FDM. If I have a bandwidth that
goes from frequency say a to b, I can subdivide this into a frequency of equal spaces. In
frequency space the modulated carriers would look like this.
a
Fig. 5.6 – bandwidth utilization of FDM
b
The frequencies a and b can be anything, integer or non-integer since no relationship
is implied between a and b. same is true of the carrier center frequencies which are based on
frequencies that do not have any special relationship to each other. If frequency C1 and Cn
were such that for any n, an integer, the following holds.
cn = n × c1
So that
C2= 2C1
C3= 3C1
C4= 4C1
All three of these frequencies are harmonic to c1.
34
Fig 5.7 bandwidth utilization of OFDM
In this case, since these carriers are orthogonal to each other, when added together,
they do not interfere with each other. In FDM, since we do not generally have frequencies
that follow the above relationship, we get interference from neighbor carriers. To provide
adjacent channel interference protection, signals are moved further apart. Each carrier may be
placed apart allowing for a 10% guard band. The frequencies would not be orthogonal but in
FDM we don’t care about this. It’s the guard band that helps keep interference under control.
5.5 An example of OFDM using 4 sub-carriers
In OFDM we have N carriers, N can be anywhere from 16 to 1024 in present
technology and depends on the environment in which the system will be used. Let’s examine
the following bit sequence we wish to transmit and show the development of the ODFM
signal using 4 sub-carriers. The signal has a symbol rate of 1 and sampling frequency is 1
sample per symbol, so each transition is a bit.
Fig. 5.8 – A bit stream that will be modulated using a 4 carrier OFDM.
First few bits are 1, 1, -1, -1, 1, 1, 1, -1, 1, -1, -1, -1, -1, 1, -1, -1, -1, 1,…
Let’s now write these bits in rows of fours see Table. 5.1, since we are using only four sub
carriers. We have effectively done a serial to parallel conversion.
Each column represents the bits that will be carried by one sub-carrier. Let’s start with the
first carrier, c1. From the Nyquist sampling Theorem, we know that smallest frequency that
can convey information has to be twice the information rate. In this case, the information rate
per carrier will be 1/4 or 1 symbol per second total for all 4 carriers. So the smallest
frequency that can carry a bit rate of 1/4 is 1/2 Hz. But we picked 1 Hz for convenience. Had
I picked 1/2 Hz as my starting frequency, then my harmonics would have been 1, 3/2 and 2
Hz. I could have chosen 7/8 Hz to start with and in which the harmonics would be 7/4, 7/2,
21/2 Hz.
35
Table 5.1 – Serial to parallel conversion of data bits.
C1
1
C2
1
C3
-1
C4
-1
1
1
1
-1
1
-1
-1
-1
-1
1
-1
-1
-1
1
1
-1
-1
-1
1
1
Fig. 5.9 General Block diagram of an OFDM system
We pick BPSK as our modulation scheme for this example. (For QPSK, just imagine the
same thing going on in the Q channel, and then double the bit rate while keeping the symbol
rate the same.) Note that I can pick any other modulation method, QPSK, 8PSK 32-QAM or
whatever.
Carrier 1 - We need to transmit 1, 1, 1 -1, -1, -1 which are superimposed on the BPSK carrier
of frequency 1 Hz. First three bits are 1 and last three -1.
Fig. 5.10 – Sub-carrier 1 and the bits it is modulating (the first column of Table 5.1)
36
Carrier 2 - The next carrier is of frequency 2 Hz. It is the next orthogonal/harmonic to
frequency of the first carrier of 1 Hz. Now take the bits in the second column, marked c2, 1,
1, -1, 1, 1, -1 and modulate this carrier with these bits as shown in Fig. 5.11.
Fig. 5.11 Sub-carrier 2 and the bits that it is modulating (the 2nd column of Table 5.1)
Carrier 3 – Carrier 3 frequency is equal to 3 Hz and fourth carrier has a frequency of 4 Hz.
The third carrier is modulated with -1, 1, 1, -1, -1, 1 and the fourth with -1, -1, -1, -1, -1, -1, 1
from Table 5.1.
Fig. 5.12 – Sub-carrier 3 and 4 and the bits that they modulating (the 3rd and 4th
columns of Table 5.1)
Now we have modulated all the bits using four independent carriers of orthogonal
frequencies 1 to 4 Hz. We have taken the bit stream, distributed the bits, one bit at a time to
the four sub-carriers as shown in figure below.
37
Fig. 5.13 – OFDM signal in time and frequency domain.
Now add all four of these modulated carriers to create the OFDM signal, often produced by a
block called the IFFT.
Fig. 5.14 – Functional diagram of an OFDM signal creation. The outlined part is often
called an IFFT block.
38
Fig. 5.15 the generated OFDM signal, Note how much it varies compared to the
Underlying constant amplitude sub-carriers.
In short-hand, we can write the process above as
N
c(t ) = ∑ mn (t )sin(2π nt )
n =1
It is basically an equation of an Inverse FFT.
5.5.1 Using Inverse FFT to create the OFDM symbol
The above equation is essentially an inverse FFT.
(a) Time domain view (b) Frequency domain view
Fig. 5.16 - The two views of a signal
39
Forward FFT takes a random signal, multiplies it successively by complex
exponentials over the range of frequencies, sums each product and plots the results as a
coefficient of that frequency. The coefficients are called a spectrum and represent “how
much” of that frequency is present in the input signal. The results of the FFT in common
understanding is a frequency domain signal.
We can write FFT in sinusoids as
N −1
⎛ 2π kn ⎞ N −1
⎛ 2π kn ⎞
x(k ) = ∑ x (n) sin ⎜
⎟ + j ∑ x (n) cos ⎜
⎟
⎝ N ⎠ n =0
⎝ N ⎠
n =0
Here x(n) are the coeffcients of the sines and cosines of frequency 2πk/N, where k is the
index of the frequencies over the N frequencies, and n is the time index. x(k) is the value of
the spectrum for the kth frequeny and x(n) is the value of the signal at time n. In Fig. 5.16(b),
the x(k =1) = 1.0 is one such value.
The inverse FFT takes this spectrum and converts the whole thing back to time domain signal
by again successively multiplying it by a range of sinusoids.
The equation for IFFT is
N −1
⎛ 2π kn ⎞ N −1
⎛ 2π kn ⎞
X ( n) = ∑ x( k ) sin ⎜
⎟ + j ∑ x( n) cos ⎜
⎟
⎝ N ⎠ n=0
⎝ N ⎠
n =0
The difference between the last two equations is the type of coefficients the sinusoids are
taking, and the minus sign, and that’s all. The coefficients by convention are defined as time
domain samples x(k) for the FFT and X(n) frequency bin values for the IFFT. The two
processes are a linear pair. Using both in sequence will give the original result back.
Column 1 of Table 5.1 the signal bits, can be considered the amplitudes of a certain
range of sinusoids. So we can use the IFFT to produce a time domain signal. Here we pretend
that the input bits are not time domain representations but are frequency amplitudes which if
you are thinking clearly, will see that that is what they are. In this way, we can take these bits
and by using the IFFT, we can create an output signal which is actually a time-domain
OFDM signal.
The IFFT is a mathematical concept and does not really care what goes in and what
goes out. As long as what goes in is amplitudes of some sinusoids, the IFFT will crunch these
numbers to produce a correct time domain result. Both FFT and IFFT will produce identical
results on the same input. But most of us are not used to thinking of FFT/IFFT this way. We
insist that only spectrums go inside the IFFT.
40
(a) a time domain signal comes out as a spectrum out of a FFT and IFFT.
They both do the same thing.
(b) A frequency domain signal comes out as a time domain signal out of a IFFT.
c) The pair return back the original input.
(d) The pair return back the input no matter what it is.
(e) the pair is commutable so they can be reversed and they will still return the original input.
Fig 5.17 – FFT and IFFT are a matched linear pair.
41
Keeping with that mindset, each row of Table 5.1 can be considered a spectrum as
plotted below. These rows aren’t actually spectrums, but that does not matter. Each row
spectrum has only 4 frequencies which are 1, 2, 3 and 4 Hz. Each of these spectrums can be
converted to produce a time-domain signal which is exactly what an IFFT does. Only in this
case, the input is really a time domain signal disguising as a spectrum.
Fig. 5.18 – The incoming block of bits can be seen as a four bin spectrum, The IFFT
converts this “spectrum” to a time domain OFDM signal for one symbol, which actually
has four bits in it.
The key components of an OFDM system are the inverse FFT at the transmitter and
FFT at the receiver. IFFT quickly computes the time-domain signal instead of having to do it
one carrier at time and then adding. An N-point FFT only requires Nlog(N) multiplications,
which is much more computationally efficient than an equivalent system with equalizer in
time domain. Calling this functionality IFFT may be more satisfying because we are
producing a time domain signal, but it is also very confusing. Because FFT and IFFT are
linear processes and completely reversible, it should be called a FFT instead of a IFFT. The
results are the same whether you do FFT or IFFT. In literature you will see it listed as IFFT
everywhere. The functional block diagram of how the signal is modulated/demodulated is
shown below in fig. 5.19.
42
Fig 5.19 – The OFDM link functions
5.6 FADING
The attraction of OFDM is mainly because of its way of handling the multipath
interference at the receiver. Multipath phenomenon generates two effects (a) Frequency
selective fading and (b) Intersymbol interference (ISI). The "flatness" perceived by a narrowband channel overcomes the frequency selective fading. On the other hand, modulating
symbols at a very low rate makes the symbols much longer than channel impulse response
and hence reduces the ISI. Use of s u i t a b l e error correcting codes provides more
robustness against frequency selective fading. The insertion of an extra guard interval
between consecutive OFDM symbols can reduce the effects of ISI even more.
5.6.1 Defining fading
If the path from the transmitter to the receiver either has reflections or obstructions,
we can get fading effects. In this case, the signal reaches the receiver from many different
routes, each a copy of the original. Each of these rays has a slightly different delay and
slightly different gain. The time delays result in phase shifts which added to main signal
component (assuming there is one.) causes the signal to be degraded. See fig.5.20.
Fading is big problem for signals. The signal is lost and demodulation must have a way
of dealing with it. In fading, the reflected signals that are delayed add to the main signal and
cause either gains in the signal strength or deep fades. And by deep fades, we mean that the
signal is nearly wiped out. The signal level is so small that the receiver can not decide what
was there.
The maximum time delay that occurs is called the delay spread of the signal in that
environment. This delay spread can be short so that it is less than symbol time or larger. Both
cases, cause different types of degradations to the signal. The delay spread of a signal
changes as the environment is changing.
Fig. 5.22 shows the spectrum of the signal, the dark line shows the response we wish the
channel to have. It is large enough that it allows the signal to go through without bending or
distortion. A fading response of the channels is something like shown in fig.8b, we note that at some
frequencies in the band, the channel does not allow any information to go through, so called deep
fades frequencies. This form of channel frequency response is called frequency selective fading.
43
Because it does not occur uniformly across the band. It occurs at selected frequencies. If the
environment is changing, then this response is also changing. In general when the delay Spread is
less than one symbol, we get what is called flat fading. When delay spread is much larger than one
symbol that it is called frequency-selective fading.
Rayleigh fading is a term used when there is no direct component and all signals reaching the
receiver are reflected. This type of environment is called Rayleigh fading.
K −1
hc (t ) = ∑ α kδ (t − τ k )
k =0
α k = Complex path gain
τ 0 = Normalized path delay relative to LOS
Δk = τ k - τ 0 difference in path time
Fig. 5.20 –Fading is particular problem when the link path is changing, such as for a
moving car or inside a building or in a populated urban area with tall building.
If we draw the interferences as impulses, they look like this
Fig. 5.21– Reflected signals arrive at a delayed time period and interfere with the main
line of sight signal, if there is one. In pure Raleigh fading, we have no main signal, all
components are reflected.
44
Fig. 5.22 – (a) the signal we want to send and the channel frequency response are well
matched. (b) A fading channel has frequencies that do not allow anything to pass. Data
is lost sporadically. (c) With OFDM, where we have many little sub-carriers, only a
small sub-set of the data is lost due to fading.
An OFDM signal offers an advantage in a channel that has a frequency selective fading response. As
we can see, because of the fading response of the channel, only two sub-carriers are affected, all the
others arc perfectly OK. Instead Of the whole symbol being knocked out, we lose just a small
subset of the (l/N) bits. With proper coding, this can be recovered.
The BER performance of an OFDM signal in a fading channel is much better than the
performance of QPSK/FDM which is a single carrier wideband signal. The BER of the OFDM
signal is same as the BER of 8PSK signal in Gaussian channel. But in channels that are fading, the
OFDM offers far better BER than a wide band signal of exactly the same modulation. The
advantage here is coming from the diversity of the multi-carrier such that the fading applies only to
a small subset. In FDM carriers, often the signal is shaped with a Root Raised Cosine shape to
reduce its bandwidth, in OFDM since the spacing of the carriers is optimal, there is a natural
bandwidth advantage and use of RRC does not buy us as much.
45
5.6.2 Use of cyclic prefix to mitigate delay spread:
You are driving in rain, and the car in front splashes a bunch of water on you. What do you
do? You move further back, you put a little distance between you and the front car, far
enough so that the splash won’t reach you. If we equate the reach of splash to delay spread of
a splashed signal then we have a better picture of the phenomena and how to avoid it.
Fig. 5.23 – Delay spread is like the undesired splash you might get from the car
ahead of you. In fading, the front symbol similarly throws a splash backwards which we
wish to avoid.
Increase distance from car in front to avoid splash. The reach of splash is same as the delay
spread of a signal. Fig. 5.23 shows the symbol and its splash. In composite, these splashes
become noise and affect the beginning of the next symbol as shown in fig. 5.24
Fig. 5.24 (Composite interference) The PSK symbol and its delayed version
To mitigate this noise at the front of the symbol, we will move our symbol further away from
the region of delay spread as shown below. A little bit of blank space has been added between
symbols to catch the delay spread.
Fig 5.25 – Move the symbol back so the arriving delayed signal peters out in the gray
region. No interference to the next symbol
46
But we can not have blank spaces in signals. This is won’t work for the hardware
which likes to crank out signals continuously. So it’s clear it need to have something there.
So, just let the symbol run longer as a first choice.
Fig. 5.26 – If we just extend the symbol, then the front of the symbol which is important
to us since it allows figuring out what the phase of this symbol is, is now corrupted by
the “splash”.
We extend the symbol into the empty space, so the actual symbol is more than one
cycle. But now the start of the symbol is still in the danger zone, and this start is the most
important thing about our symbol since the slicer needs it in order to make a decision about
the bit. We do not want the start of the symbol to fall in this region, so lets just slide the
symbol backwards, so that the start of the original symbol lands at the outside of this zone.
And then fill this area with something.
Fig. 5.27 – If we move the symbol back and just put in convenient filler in this area, then
not only we have a continuous signal but one that can get corrupted and we don’t care
since we will just cut it out anyway before demodulating.
1. We want the start of the symbol to be out of the delay spread zone so it is not corrupted.
2. We start the signal at the new boundary such that the actual symbol edge falls out side this zone.
So, Slide the symbol to start at the edge of the delay spread time and then fill the guard space with a
copy of what turns out to be tail end of the symbol. We will be extending the symbol so it is
1.25 times as long, to do this, copy the back of the symbol and glue it in the front. In reality, the
symbol source is continuous, so all we are doing is adjusting the starting phase and making the
symbol period longer.
47
Fig. 5.28-Cyclic prefix is copy of back of the symbol we add to the front of the symbol.
This procedure is called adding a cyclic prefix. Since OFDM, has a lot of carriers, we would do
this to each and every carrier. But that's only in theory. Since the OFDM signal is a linear
combination, the whole process can be done only once to the OFDM signal, rather than doing it
to each and every sub-carrier. We can add cyclic prefix just once to the composite OFDM signal.
The prefix is any where from 10% to 25% of the symbol time. We add the prefix after doing
the IFFT just once to the composite signal. After the signal has arrived at the receiver, first
remove this prefix, to get back the perfectly periodic signal so it can be FFT'd to get back the
symbols on each carrier.
However, the addition of cyclic prefix which mitigates the effects of link fading and inter
symbol interference, increases the bandwidth.
Fig 5.29 – Addition of Cyclic prefix to the OFDM signal further improves its ability to
deal with fading and interference.
Here is an OFDM signal with period equal to 32 samples. We want to add a 25% cyclic shift
to this signal.
1. First we cut pieces that are 32 samples long.
2. Then we take the last .25 (32) = 8 samples, copy and append them to the front as shown.
48
Fig. 5.30 – The whole process can be done only once to the OFDM signal, rather than
doing it to each and every sub-carrier.
5.7 Properties of OFDM
5.7.1 Spectrum and performance
Fig. 5.31 – The spectrum of an OFDM signal (without addition of cyclic prefix) is much
more bandwidth efficient than QPSK.
49
Unshaped QPSK signal produces a spectrum such that its bandwidth is equal to (1+ α)Rs. In
OFDM, the adjacent carriers can overlap in the manner shown here. The addition of two
carriers, now allows transmitting 3Rs over a bandwidth of -2Rs to 2Rs or total of 4Ts. This
gives a bandwidth efficiency of 4/3 Hz per symbol for 3 carriers and 6/5 for 5 carriers.
As more and more carriers are added, the bandwidth approaches,
N +1
Bits per Hz.
N
So the larger the number of carriers, the better.
Here is a spectrum of an OFDM signal. Note that the out of band signal is down by 50 dB
without any pulse shaping.
Fig. 5.32 – the spectrum of an OFDM signal with 1024 sub-carriers
Compare this to the spectrum of a QPSK signal, note how much lower the sidebands are for
OFDM and how much less is the variance.
Fig. 5.33 – the spectrum of a QPSK signal
50
5.7.2 OFDM is a special case of FDM:
Frequency division multiplexing:
Each signal will occupy separate frequency band. To provide adjacent channel interference
protection, signals are moved further apart. So here the spectrum is not used completely.
Fig. 5.34 Spectrum of FDM
OFDM frequency dividing:
Large number of orthogonal, overlapping, narrow band sub carriers is transmitted. Since
these carriers are orthogonal to each other, when added together, they do not interfere with
each other. In FDM, we do not generally have frequencies that follow the above relationship.
Here spectrum is used almost twice. So there is a earn in spectral efficiency
Fig. 5.35 Spectrum of OFDM
5.7.3 Peak to average power ratio (PAPR)
If a signal is a sum of N signals each of max amplitude equal to 1 v, then it is conceivable
that we could get a max amplitude of N that is all N signals add at a moment at their max
points. The PAPR is defined as
x (t )
R=
Pavg
2
For an OFDM signal that has 128 carriers, each with normalized power of 1 w, then the max
PAPR can be as large as log (128) or 21 dB. This is at the instant when all 128 carriers
combine at their maximum point, unlikely but possible. The RMS PAPR will be around half
this number or 10-12 dB. The large amplitude variation increases in-band noise and increases
the BER when the signal has to go through amplifier non-linearities.
51
Fig.5.36 AN OFDM signal is very noise like. It looks just like a composite multi-FDM
signal.
The OFDM time-domain signal has a relatively large peak-to-average ratio. This
tends to reduce the power efficiency of the RF amplifier. Non-linear amplification destroys
the orthogonality of the OFDM signal and introduced out-of-band radiation. Several ideas are
used to mitigate it.
1.Clipping
We can just clip the signal at a desired power level. This reduces the PAPR but introduces other
distortions and ICI.
2. Selective Mapping
Multiply the data signal by a set of codes, do the IFFT on each and then pick the one with the least
PAPR. This is essentially doing the process many times using a CDMA like code.
3. Partial IFFT
Divide the signal in clusters, do IFFT on each and then combine these. So that if we
subdivide 128 carrier in to a group of four 32 carriers, each, the max PAPR of each will be 12 dB
instead of 21 for the full. Combine these four sequences to create the transmit signal.
5.7.4 Synchronization:
The other problem is that tight synchronization is needed. OFDM is sensitive to frequency,
clock and phase offset. Often pilot tones are served in the sub carrier space. These are used to
lock on phase and to equalize the channel.
5.7.5 Coding:
The sub-carriers are typically coded with Convolutional coding prior to going through IFFT.
The coded version of OFDM is called COFDM or Coded OFDM.
52
5.8 OFDM APPLICATIONS:
OFDM is digital transmission technique developed into a popular scheme for
wideband digital communication systems. It is well suited for wideband, high data rate
transmissions. The main advantage is that less equalization is necessary. The OFDM use has
increased greatly in the last 10 years. Nowadays, OFDM is mainly used for one to many
(broadcast) communications like radio or television broadcasting. Examples are digital
broadcasting systems such as DAB and DVB.
It is now proposed for Digital audio broadcasting such as in Eureka 147 standard and
Digital Radio Mondiale (DRM). Digital Audio Broadcasting (DAB) is an international,
standardized digital broadcasting system developed by the European EUREKA-147 Project.
OFDM is used for modem/ADSL application where it coexists with phone line. For ADSL use, the
channel, the phone line, is filtered to provide a high SNR. OFDM here is called Discrete Multi Tone
(DMT.) HDSL: High bit rate Digital Subscriber Line is another implementation for
symmetric speeds (uplink rate = downlink rate).
HiperLAN2 is the all new high performance radio technology, specifically suited for
operating in LAN environments. HiperLAN2 is a technology being developed within the
European Telecommunications Standardisation Institute (ETSI). OFDM is the modulation
used in the physical layer of HiperLAN2.
OFDM is also in use in wireless internet modem and this usage is called 802.1la.
Examples of applications are:
ADSL and VDSL broadband access via telephone network copper wires.
IEEE 802.11a and 802.11g Wireless LANs.
Terrestrial digital TV systems DVB-T, DVB-H, T-DMB and ISDB-T.
IEEE 802.16 or WiMax Wireless MAN.
IEEE 802.20 or Mobile Broadband Wireless Access (MBWA).
Flash-OFDM cellular system.
Some Ultra wideband (UWB) systems.
Power line communication (PLC).
Digital audio, video broadcasting.
Wireless ATM transmission system.
Proposed scheme for UMTS air interface for bit rates higher than 384 kbps
53
5.9 PARAMETERS OF REAL OFDM
OFDM is used in wireless internet modem and this usage is called 802.11a. Let’s take a look
at some Parameters of this application of OFDM. The summary of these are given below.
Data rates
6 Mbps to 48 Mbps
Modulation
BPSK, QPSK, 16 QAM and 64 QAM
Coding
Convolutional concatenated with Reed Solomon
FFT size
64 with 52 sub-carriers uses, 48 for data and 4 for pilots.
Sub carrier frequency spacing
20 MHz divided by 64 carriers or .3125 MHz
FFT period
Also called symbol period, 3.2 μsec = 1/Δf
Guard duration
One quarter of symbol time, 0.8 μsec
Symbol time
4 μsec
54
Chapter 6
OFDM SIMULATION AND RESULTS
6. OFDM SIMULATION AND RESULTS
6.1 WORKING OF THE OFDM MODEL:
Random data
generator
Serial to
parallel
converter
Digital
modulation
BPSK / QPSK
IFFT
Parallel to
serial converter
Guard
interval
insertion
Additive white
Gaussian noise
channel
Serial
data out
Parallel to
serial converter
Digital
demodulation
BPSK / QPSK
FFT
Serial to
parallel
converter
Guard
interval
removal
Fig 6.1 OFDM model used for simulation
Figure 6.1 shows the basic block diagram of OFDM transmitter and receiver used for
simulation. OFDM is generated by choosing the spectrum required, based on the input data,
and modulation scheme used. Each carrier to be produced is assigned data to be transmitted.
The required amplitude and phase of the carrier is then calculated based on the modulation
scheme (typically BPSK, QPSK, or QAM). For example, if we have to transmit incoming 8
bit digital data, we have to choose 8 different carrier signals, which are orthogonal to each
other. Each carrier is assigned to a different bit and its amplitude and phase are chosen
according to modulation scheme used. The required spectrum is then converted back to its
time domain signal using an Inverse Fourier Transform.
In most applications, an Inverse Fast Fourier Transform (IFFT) is used [6, 7]. They are
already in time domain, but here we pretend that the input bits are not time domain
representations but are frequency amplitudes which if you are thinking clearly, will see that
that is what they are. In this way, we can take these bits and by using the IFFT, we can create
an output signal which is actually a time-domain OFDM signal. The IFFT is a mathematical
concept and does not really care what goes in and what goes out. As long as what goes in is
amplitudes of some sinusoids, the IFFT will crunch these numbers to produce a correct time
55
domain result. Both FFT and IFFT will produce identical results on the same input. We insist
that only spectrums go inside the IFFT. IFFT quickly computes the time-domain signal
instead of having to do it one carrier at time and then adding. Calling this functionality IFFT
may be more satisfying because we are producing a time domain signal, but it is also very
confusing. Because FFT and IFFT are linear processes and completely reversible, it should be
called a FFT instead of a IFFT. The results are the same whether you do FFT or IFFT. In
literature you will see it listed as IFFT everywhere. This block can also be a FFT as long as
on the receive side, you do the reverse. The IFFT performs the transformation very
efficiently, and provides a simple way of ensuring the carrier signals proceed are orthogonal.
The reverse process guarantees that the carriers generated are orthogonal.
Consider the model shown in Fig 6.1. The random data generator generates the data
system. This input serial data stream is formatted into the word size required for transmission.
For example, 1 bit/word for BPSK & 2 bits/word for QPSK and then shifted into a parallel
format. The data is then transmitted in parallel by assigning each data word to one carrier in
the transmission. The data to be transmitted on each carrier is then mapped into a Phase Shift
Keying (PSK) format. The data on each symbol is mapped to a phase angle based on the
modulation method. For example, in QPSK the phase angles used are 0°, 90°, 180°, and 270°.
The use of phase shift keying produces a constant amplitude signal and was chosen for its
simplicity and to reduce problems with amplitude fluctuations due to fading.
After the required spectrum is worked out, an Inverse Fourier Transform is used to
find the corresponding time domain waveform. The guard period is then added to the start of
each symbol as shown in the fig 6.2.
Guard period
Symbol period
Data bits
Half Guard Period time contains
Zeros amplitude transmission
Cyclic extension of symbol
Fig 6.2 addition of guard band interval with symbol
56
One of the most important properties of OFDM transmissions is its high level of robustness
against multipath delay spread. This is a result of the long symbol period used, which
minimizes the inter-symbol interference. The level of multipath robustness can be further
increased by the addition of a guard period between transmitted symbols. The guard period
allows time for multipath signals from the previous symbol to die away before the information
from the current symbol is gathered. The guard period used, can be made up of two sections.
Half of the guard period time is a zero amplitude transmission called Zero padding and the
other half of the guard period is a cyclic extension of the symbol to be transmitted. After the
guard has been added, the symbols are then converted back to a serial time waveform. This is
then the base band signal for the OFDM transmission. A channel model is then applied to the
transmitted signal. The model allows for the signal to noise r a t i o . I t is set by adding a
known amount of white noise to the transmitted signal. The channel output is given to the
receiver. The receiver basically does the reverse operation to the transmitter. The guard
period is removed from the received signal. The FFT of each symbol is then taken to find the
original transmitted spectrum. The phase angle of each transmission carrier is then evaluated
and converted back to the data word by demodulating the received phase. The data words are
then combined, which gives the same word size as that of original data.
6.2 OFDM SYSTEM SIMULATION WITH BPSK:
The random binary generator block generates random binary numbers that are applied
to BPSK modulator. This modulates using the binary phase shift keying method. The output
is a baseband representation of the modulated signal. The input must be a discrete-time
binary-valued signal. If the input bit is 0 or 1, then the modulated symbol is exp(θ) or -exp(θ)
respectively, where θ is the Phase offset parameter. This frequency domain data is then
applied to IFFT block as shown in the Fig 6.1
The IFFT block computes the Inverse Fast Fourier Transform (IFFT) of length-M
input, where M must be a power of two. While working with other input sizes, the Zero Pad
blocks can be used to pad or to truncate the length so that it w i l l be of size M. The output is
always frame-based, and each output frame contains the M-point Inverse Discrete Fourier
Transform (IDFT) of the corresponding input. Thus IFFT converts the frequency domain data
in to time domain signal and at the same time maintains the orthogonality among the carriers.
The AWGN channel block adds white Gaussian noise to a real or complex input signal.
When the input signal is real, this block adds real Gaussian noise and produces a real output
signal. When the input signal is complex, this block adds complex Gaussian noise and
produces a complex output signal.
57
At the receiver basically the reverse operation to the transmitter will be done. After the
removal of guard band the FFT block computes the Fast Fourier Transform (FFT) of lengthM input, where M must be a power of two. To work with other input sizes, use the zero pad
block to pad or truncate the length-M dimension to a power-of-two length. The data is
converted back into frequency domain so that it can be processed by the BPSK Demodulator
block. This block demodulates a signal that was modulated using the binary phase shift
keying method. The input is a baseband representation of the modulated signal. The input
must be a discrete-time complex signal. The block maps the point exp(θ) and - exp(θ) to 0
and 1, respectively, where θ is the Phase offset parameters.
The bit error rate calculated by comparing the input data from a transmitter with input
data from a receiver. It calculates the bit error rate by dividing the total number of unequal
pairs of data elements by the total number of input data elements from source.
6.2.1 Simulation Results for OFDM with BPSK:
No. of bits transmitted = 12000
No. carriers used = 6
Bits per each carrier = 2000
TABLE 6.1: BER results for OFDM model using BPSK modulation
BER(using BPSK)
SNR(dB)
0
1
2
3
4
5
6
7
8
9
0.0757
0.0564
0.0388
0.0215
0.0118
0.0057
0.0022
8.3x 10-4
2.5x 10-4
0
These are categorized as tabular results, and graphical results. Signal to Noise ratio
(SNR) also called as Eb /No, where Eb is bit energy and No is noise energy. SNR values in dB
are adjusted every time by adding noise in the AWGN channel. For particular SNR value
system is simulated and corresponding probability of error (Bit Error Rate, BER) is
calculated. These results are noted in Table 6.1. Figure 6.3 shows the nature of the BER versus
SNR curve. As we go on increasing the SNR value, bit error rate reduces.
58
BER Vs SNR using BPSK
-1
10
-2
BER
10
-3
10
-4
10
0
1
2
3
4
SNR(in dB)
5
6
7
8
Fig 6.3 BER versus SNR curve for OFDM with BPSK
6.3 OFDM SYSTEM SIMULATION WITH QPSK:
Simulation of OFDM with QPSK modulation technique is similar to the simulation of
OFDM with BPSK modulation. The difference is that the BPSK modulator/demodulators are
replaced by the QPSK modulator/demodulators. The QPSK modulator modulates using the
quaternary phase shift keying method. The output is a baseband representation of the
modulated signal. The QPSK demodulator demodulates a signal that was modulated using the
quaternary phase shift keying method.
6.3.1 Simulation Results for OFDM with QPSK:
The entire simulation process to be carried out is similar to that of OFDM with BPSK
scheme. For particular SNR value system is simulated and corresponding probability of error
(Bit Error Rate, BER) is calculated. These results are noted in Table 6.2. Figure 6.4 shows
the nature of the BER versus SNR curve. As we go on increasing the SNR value, bit error
rate reduces.
No. of bits transmitted = 12000
No. carriers used = 6
Bits per each carrier = 2000
59
TABLE 6.2: BER results for OFDM model using QPSK modulation
SNR(dB)
0
1
2
3
4
5
6
7
8
9
10
11
BER(using QPSK)
0.2088
0.1802
0.1475
0.1125
0.0807
0.0554
0.0313
0.0172
0.0090
0.0035
0.0015
5.83x 10-4
BER Vs SNR using QPSK
0
10
-1
BER
10
-2
10
-3
10
-4
10
0
2
4
6
SNR(in dB)
8
10
Fig 6.4 BER versus SNR curve for OFDM with BPSK
60
12
6.4 COMPARISON OF OFDM SYSTEMS WITH QPSK AND BPSK
Figure 6.5 shows comparison of the BER versus SNR curves obtained in the OFDM systems
with BPSK and QPSK modulation schemes. It is clearly observed that the curve in latter
system is always above the curve in the earlier system
BER Vs SNR
0
10
ofdm using BPSK
ofdm using QPSK
-1
BER
10
-2
10
-3
10
-4
10
0
2
4
6
SNR(in dB)
8
10
12
Fig 6.5 Comparison of BER versus SNR of Two Modulation Schemes
6.5 OFDM IMPLEMENTATION USING FFT
In the OFDM system implementation Fast Fourier Transform (FFT) and its inverse
(IFFT) were used to perform the modulation and demodulation operations. BPSK is used as
digital modulation technique. The Fourier transform allows us to relate events in time domain
to events in frequency domain. OFDM system Use discrete Fourier transform (DFT) to
modulate and demodulate parallel data. The fast Fourier transform (FFT) is merely a rapid
mathematical method for computer applications of DFT. In most applications, an Inverse Fast
Fourier Transform (IFFT) is used. Both transmitter and receiver can be implemented using
efficient FFT techniques that reduce the number of operations from N2 in DFT, down to
NlogN. It is the availability of this technique, and the technology that allows it to be
implemented on integrated circuits at a reasonable price, that has permitted OFDM to be
developed as far as it has. The orthogonality of subchannels in OFDM can be maintained and
individual subchannels can be completely separated by the FFT at the receiver when there are
61
no intersymbol interference (ISI) and intercarrier interference (ICI) introduced by
transmission channel distortion.
IFFT quickly computes the time-domain signal instead of having to do it one carrier at time
and then adding. The IFFT performs the transformation very efficiently, and provides a
simple way of ensuring the carrier signals proceed are orthogonal. The reverse process
guarantees that the carriers generated are orthogonal. FFT based OFDM is computationally
efficient due to the use of FFT techniques to implement modulation and demodulation
functions.
6.5.1 Simulation Results
Results are noted in Table 6.3. Figure 6.6 shows the nature of the BER versus SNR
curve. As we go on increasing the SNR value, bit error rate reduces.
No. of bits transmitted = 12000
No. of carriers used = 6
Bits per each carrier = 2000
Spacing between the each carrier = 6 KHz
Carrier frequencies are 6 KHz, 12 KHz, 18 KHz, 24 KHz, 30 KHz, and 36 KHz
Modulation technique used is BPSK
TABLE 6.3: BER results for FFT based OFDM
SNR(dB)
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
BER(using FFT)
0.09042
0.07660
0.06800
0.05730
0.05042
0.04420
0.03683
0.03090
0.02375
0.02040
0.01308
0.01190
6.75x 10-3
5.20x 10-3
2.92x 10-3
6.70x 10-4
4.17x 10-4
62
BER Vs SNR using FFT
-1
10
-2
BER
10
-3
10
-4
10
0
2
4
6
8
SNR(in dB)
10
12
14
16
Fig 6.6 BER versus SNR curve for FFT based OFDM
6.6 OFDM IMPLEMENTATION USING DHT
In the OFDM system implementation real-valued discrete Hartley transform (DHT) and its
inverse (IDHT) were used to perform the modulation and demodulation operations. BPSK is
used as digital modulation technique. Since the DHT and IDHT definitions are identical, we
can use the same hardware or program to implement the modulator and demodulator.
For the current OFDM-based transceivers, the modulator needs to compute a long-length
inverse discrete Fourier transform (IDFT), and the demodulator needs to compute a long
length DFT, where the transform length is up to 512 or more. For such long-length
DFT/IDFT computations, a great number of complex multiplications are required and each of
them basically involves four real multiplications and two real additions. Clearly, the
complexity of a OFDM-based transceiver would be reduced if the corresponding
modulator/demodulator could be implemented using purely real transforms while fast
algorithms similar to the fast Fourier transform (FFT) algorithm can still be applied.
The DHT involves only real-valued arithmetic and has an identical inverse. Like the DFT,
there have been a number of fast algorithms and hardware architectures available for the
DHT computation. From the results it is seen that the DHT-based OFDM method achieves
63
the same BER performance as the DFT-based OFDM method, but requires less
computational complexity.
6.6.1 Simulation Results
Results are noted in Table 6.4. Figure 6.7 shows the nature of the BER versus SNR
curve. As we go on increasing the SNR value, bit error rate reduces.
No. of bits transmitted = 12000
No. of carriers used = 6
Bits per each carrier = 2000
Spacing between the each carrier = 6 KHz
Carrier frequencies are 6 KHz, 12 KHz, 18 KHz, 24 KHz, 30 KHz, and 36 KHz
Modulation technique used is BPSK
TABLE 6.4: BER results for DHT based OFDM
SNR(dB)
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
BER(using DHT)
0.1072
0.0882
0.0775
0.0649
0.0523
0.0447
0.0378
0.0340
0.0246
0.0189
0.0137
0.0106
0.0080
0.0042
2.6x10-3
5.0x 10-4
3.3x 10-4
64
BER Vs SNR using DHT
0
10
-1
BER
10
-2
10
-3
10
-4
10
0
2
4
6
8
SNR(in dB)
10
12
14
16
Fig 6.7 BER versus SNR curve for DHT based OFDM
6.7 OFDM IMPLEMENTATION USING DCT
In the OFDM system implementation real-valued Discrete Cosine Transform (DHT) and its
inverse (IDCT) were used to perform the modulation and demodulation operations. BPSK is
used as digital modulation technique. The DCT uses only real arithmetic, as opposed to the
complex-valued DFT. This reduces the signal-processing complexity/power consumption.
For the current OFDM-based transceivers, the modulator needs to compute a long-length
inverse discrete Fourier transform (IDFT), and the demodulator needs to compute a long
length DFT, where the transform length is up to 512 or more. For such long-length
DFT/IDFT computations, a great number of complex multiplications are required and each of
them basically involves four real multiplications and two real additions. The DCT basis is
well known to have excellent spectral compaction and energy concentration properties.
6.7.1 Simulation Results
Results are noted in Table 6.5. Figure 6.8 shows the nature of the BER versus SNR
curve. As we go on increasing the SNR value, bit error rate reduces.
65
No. of bits transmitted = 12000
No. of carriers used = 6
Bits per each carrier = 2000
Spacing between the each carrier = 6 KHz
Carrier frequencies are 6 KHz, 12 KHz, 18 KHz, 24 KHz, 30 KHz, and 36 KHz
Modulation technique used is BPSK
TABLE 6.5: BER results for DCT based OFDM
SNR(dB) BER(using DCT)
0
0.0237
1
0.0210
2
0.0162
3
0.0109
4
0.0083
5
0.0050
6
0.0028
7
0.0014
8
4.167x 10
9
1.667x 10
10
0
BER Vs SNR using DCT
-1
10
-2
BER
10
-3
10
-4
10
0
1
2
3
4
5
SNR(in dB)
6
7
8
Fig 6.8 BER versus SNR curve for DCT based OFDM
66
9
6.8 COMPARISON OF PERFORMANCE OF OFDM SYSTEMS
Figure 6.9 shows comparison of the BER versus SNR curves obtained in the FFT,
DHT, and DCT based OFDM systems with BPSK as a modulation technique. It is observed
that for all the systems BER is getting reduced as SNR increases. The BER performance of
FFT and DHT is almost same. It is also observed that the DCT based BER curve is always
below the curves of other two systems.
BER Vs SNR
0
10
ofdm using FFT
ofdm using DCT
ofdm using DHT
-1
BER
10
-2
10
-3
10
-4
10
0
2
4
6
8
SNR(in dB)
10
12
14
16
Fig 6.9 Comparison of Bit Error Rates of FFT, DHT, and DCT based OFDM systems
67
Chapter 7
CONCLUSION
7. CONCLUSION
This chapter gives a summary of the work presented in this thesis. An outline for the
future work based on this is also given.
7.1 SUMMARY
The OFDM makes efficient use of available spectrum by allowing overlapping among
the carriers. It basically converts the high data rate stream in to several parallel lower data
rate streams and thereby eliminating the frequency selective fading. It has been seen that the
OFDM is a powerful modulation technique that is capable of high data rate and is able to
eliminate ISI. It is computationally efficient due to the use of FFT techniques to implement
modulation and demodulation functions.
Using MATLAB software, the performance of OFDM system was tested for two
digital modulation techniques namely BPSK and QPSK. The performance of OFDM system was
also tested for three different transform techniques namely FFT, DHT, and DCT.
From the simulation results, it is observed that the BPSK allows the BER to be
improved in a noisy channel at the cost of maximum data transmission capacity. Use of
QPSK allows higher transmission capacity, but at the cost of slight increase in the probability
of error. This is because of the fact that QPSK uses two bits per symbol. Hence QPSK is easily
affected by the noise. Therefore OFDM with QPSK requires larger transmit power. From the
results, use of OFDM with QPSK is beneficial for short distance transmission link, whereas
for long distance transmission link OFDM with BPSK will be preferable.
From the results it is also observed that the DHT-based OFDM achieves the same
transmission performance as the DFT-based OFDM from the BER vs SNR point of view, but
requires less computational complexity with reduced implementation cost, because of it is
purely based on the real valued DHT. And the DCT based OFDM has better BER
performance compared to the DFT, DHT based OFDM methods, because of its strong
"energy compaction" property and real valued nature. So, from the BER vs SNR results we
conclude that implementation and performance points of view, DHT, DCT transform
techniques are may be alternatives to DFT for the designing of OFDM-based communication
systems.
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7.2 SCOPE OF FUTURE WORK
The following are the some of the interesting extensions of the present work:
1) An interesting topic for future research is to perform more extensive performance
comparisons between FFT based OFDM, DHT based OFDM, and DCT based OFDM
systems under additional real-world channel impairments, such as multipath fading,
time dispersion whish leads to inter symbol interference (ISI).
2) The main problems with OFDM signal is very sensitive to carrier frequency offset,
and its high Peak to Average Power Ratio (PAPR). So, these three transform based
OFDM systems can be tested for these problems.
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