study of ofdm performance over awgn channels...

STUDY OF OFDM PERFORMANCE OVER AWGN CHANNELS

Ender Bolat

Undergraduate Project Report submitted in partial fulfillment of

the requirements for the degree of Bachelor of Science (B.Sc.)

in

Electrical and Electronic Engineering Department Eastern Mediterranean University

July 2003

Approval of the Electrical and Electronic Engineering Department ______________________________ Assoc. Prof. Dr. Derviş Z. Deniz Chairman This is to certify that we have read this thesis and that in our opinion it is fully adequate, in cope and quality, as an Undergraduate Project. _________________________________ ______________________________

Supervisor . Co-Supervisor

Members of the examining committee Name Signature 1. 2. 3. Date:

I

ABSTRACT

STUDY OF OFDM PERFORMANCE OVER AWGN CHANNELS

by

Ender Bolat 999293

Electrical and Electronic Engineering Department

Eastern Mediterranean University

Supervisor: Asst. Prof. Dr. Erhan A. Ince

Keywords: wireless communications, terrestrial digital video broadcasting, OFDM, AWGN, SNR, symbol error rate The next generation wireless communications systems need to be of a higher standard in order to provide the customers with the multitude of high quality services they demand. In recent years, Orthogonal Frequency Division Multiplexing (OFDM) has been successfully used in terrestrial digital video broadcasting and showed it is a strong candidate for the modulation technique of future wireless systems. This project is concerned with how well OFDM performs when transmitted over an Additive White Gaussian Noise (AWGN) channel only. In order to investigate this, a simulation model was created and implemented using MATLAB. The OFDM signal was transmitted over the AWGN channel for various signal-to-noise ratio (SNR) values. To evaluate the performance, for each SNR level, the received signal was demodulated and the received data was compared to the original information. The result of the simulation is shown in a plot of the symbol error rate versus SNR, which provides information about the system’s performance. The plot shows that OFDM performance is good over this type of channel.

II

ACKNOWLEDGEMENTS First of all I’m grateful to Allah for giving me strength and wisdom throughout all my life and especially to finish this project. I thank my family for their love, their moral and financial support they had given me. This helped me a lot. I thank my project supervisor Assist. Prof. Dr. Erhan A. Ince for the help he has given me in completing this project. I hope we will see each other again and maybe work together in the future. Last, but not least, I thank all my friends, both here and back home, who have been there for me when I needed them. They are NOT ordered according to their importance to me; it’s just the order they came to my mind; so here are some of them: Umut Beyazitli, Ilyas Haciomeroglu , Imran Javaid, Abdallah. S. Abdallah, Abdisalam Houssein, Issa Housein Djama, Chingiz Abdurrahmanov, Tarlan Bilalov, the romanian group in Cyprus ( Osman Suliman, Behruz Saganai, Deniz Serif, Olgun Memedula, Leila Septar, Enise Sali), my friends back home ( Anca Bertea-my girlfriend, Alexandru Mamo, Costin Niculescu, Iustin Ocnarescu, Dima Lascu, Dinu Caragheorghe, Adrian Mergiani, Flaviu Goia), my cousins (Elif and Ervin Bolat, Aylin Medina Bagas, Elis Bekir, Timur Regep, Merghin Bectemir, Belgin Bectemir, Kemal and Leila Azis, Asan Kaia) etc.

III

TABLE OF CONTENTS ABSTRACT I ACKNOWLEDGEMENTS II TABLE OF CONTENTS III LIST OF FIGURES IV LIST OF TABLES V CHAPTER 1 Introduction 1 CHAPTER 2 Theory of OFDM 2 2.1 General considerations 2 2.2 Drawbacks of OFDM 3 2.3 Principles of OFDM 3 CHAPTER 3 OFDM Transmission 5 3.1 Terrestrial digital video broadcasting 5 3.2 FFT Implementation 8 CHAPTER 4 OFDM Reception 17 CHAPTER 5 Conclusion 24 APPENDIX The MATLAB code used 25 REFERENCES 30

IV

LIST OF FIGURES

1) Figure 2.1: Basic OFDM system

2) Figure 3.1: Terrestrial Digital Video Broadcasting

3) Figure 3.2: OFDM symbol generation block diagram

4) Figure 3.3: Time response of signal carriers

5) Figure 3.4: Frequency response of signal carriers

6) Figure 3.5: Impulse response of g (t)

7) Figure 3.6: Time response of signal u at (C)

8) Figure 3.7: Frequency response of signal u at (C)

9) Figure 3.8: D/A filter response

10) Figure 3.9: Time response of signal uoft at (D)

11) Figure 3.10: Frequency response of signal uoft at (D)

12) Figure 3.11: Time response of signal s(t) at (E)

13) Figure 3.12: Frequency response of signal s(t) at (E)

14) Figure 4.1: An OFDM receiver

15) Figure 4.2: Original 4-QAM constellation

16) Figure 4.3: Received 4-QAM constellation for SNR=2dB



19) Figure 4.6: Eye pattern for the received constellation in an ideal channel

20) Figure 4.7: Eye pattern for the received constellation for SNR=2dB



23) Figure 4.10: Simulated and theoretical symbol error rate

V

LIST OF TABLES Table 3.1: Parameters of the 2k DVB-T

1

Chapter1

Introduction

High capacity and variable bit rate information transmission with high bandwidth

efficiency are just some of the requirements that the modern transceivers have to meet in

order for a variety of new high quality services to be delivered to the customers. Because

in the wireless environment signals are usually impaired by fading and multipath delay

spread phenomenon, traditional single carrier mobile communication systems do not

perform well. In such channels, extreme fading of the signal amplitude occurs and Inter

Symbol Interference (ISI) due to the frequency selectivity of the channel appears at the

receiver side. This leads to a high probability of errors and the system’s overall

performance becomes very poor. Techniques like channel coding and adaptive

equalization have been widely used as a solution to these problems. However, due to the

inherent delay in the coding and equalization process and high cost of the hardware, it is

quite difficult to use these techniques in systems operating at high bit rates, for example,

up to several M bps. An alternative solution is to use a multi carrier system. Orthogonal

Frequency Division Multiplexing (OFDM) is an example of it and it is used in several

applications such as asymmetric digital subscriber lines (ADSL), a system that makes

high bit-rates possible over twisted-pair copper wires. It has recently been standardized

and recommended for digital audio broadcasting (DAB) in Europe and it is already used

for terrestrial digital video broadcasting (DVB-T). The IEEE 802.11a standard for

wireless local area networks (WLAN) is also based on OFDM. The purpose of this

project is to investigate how OFDM performs in an Additive White Gaussian Noise

(AWGN) channel only. In this channel only one path between the transmitter and the

receiver exists and only a constant attenuation and noise is considered. Therefore no

multipath effect is taken into account. This is a basic investigation and it is intended as a

basis of understanding OFDM better in order for future studies of this technique in

multipath channels.

2

Chapter 2

Theory of OFDM

2.1 General considerations

OFDM is a technique for transmitting data in parallel by using a large number of

modulated sub-carriers. These sub-carriers (or sub-channels) divide the available

bandwidth and are sufficiently separated in frequency (frequency spacing) so that they

are orthogonal. The orthogonality of the carriers means that each carrier has an integer

number of cycles over a symbol period. Due to this, the spectrum of each carrier has a

null at the center frequency of each of the other carriers in the system. This results in no

interference between the carriers, although their spectra overlap. The separation between

carriers is theoretically minimal so there would be a very compact spectral utilization.

OFDM systems are attractive for the way they handle ISI, which is usually introduced by

frequency selective multipath fading in a wireless environment. Each sub-carrier is

modulated at a very low symbol rate, making the symbols much longer than the channel

impulse response. In this way, ISI is diminished. Moreover, if a guard interval between

consecutive OFDM symbols is inserted, the effects of ISI can completely vanish. This

guard interval must be longer than the multipath delay. Although each sub-carrier

operates at a low data rate, a total high data rate can be achieved by using a large number

of sub-carriers. ISI has very small or no effect on the OFDM systems hence an equalizer

is not needed at the receiver side.

In the OFDM system, Inverse Fast Fourier Transform/Fast Fourier Transform

(IFFT /FFT) algorithms are used in the modulation and demodulation of the signal. The

length of the IFFT/FFT vector determines the resistance of the system to errors caused by

the multipath channel. The time span of this vector is chosen so that it is much larger than

the maximum delay time of echoes in the received multipath signal.

OFDM is generated by firstly choosing the spectrum required, based on the input

data, and modulation scheme used. Each carrier to be produced is assigned some data to

transmit. The required amplitude and phase of the carrier is then calculated based on the

3

modulation scheme (typically differential BPSK, QPSK, or QAM). Then, the IFFT

converts this spectrum into a time domain signal.

The FFT transforms a cyclic time domain signal into its equivalent frequency

spectrum. Finding the equivalent waveform, generated by a sum of orthogonal sinusoidal

components, does this. The amplitude and phase of the sinusoidal components represent

the frequency spectrum of the time domain signal.

2.2 Drawbacks of OFDM

There are two main drawbacks:

The large dynamic range of the signal, also known as the peak-to-average-power ratio

(PAPR). Solutions to deal with this problem have been (and still are) developed and

one of the most used ones is clipping.

Sensitivity to frequency errors.

Most research centers throughout the world are mainly focusing their work on these two

topics in their attempt to optimize OFDM.

2.3 Principles of OFDM

The main features of a practical OFDM system are as follows:

Some processing is done on the source data, such as coding for correcting errors,

interleaving and mapping of bits onto symbols. An example of mapping used is

QAM.

The symbols are modulated onto orthogonal sub-carriers. This is done by using

IFFT

Orthogonality is maintained during channel transmission. This is achieved by

adding a cyclic prefix to the OFDM frame to be sent. The cyclic prefix consists of

the L last samples of the frame, which are copied and placed in the beginning of

the frame. It must be longer than the channel impulse response.

4

Synchronization: the introduced cyclic prefix can be used to detect the start of

each frame. This is done by using the fact that the L first and last samples are the

same and therefore correlated. This works under the assumption that one OFDM

frame can be considered to be stationary.

Demodulation of the received signal by using FFT

Channel equalization: the channel can be estimated either by using a training

sequence or sending known so-called pilot symbols at predefined sub-carriers.

Decoding and de-interleaving

A block diagram showing a simplified configuration for an OFDM transmitter and

receiver is given in Figure 2.1.

(a) Transmitter

(b) Receiver

Figure 2.1: Basic OFDM system

The OFDM signal generated by the system in Figure 2.1 is at baseband; in order to

generate a radio frequency (RF) signal at the desired transmit frequency filtering and

mixing is required. OFDM allows for a high spectral efficiency as the carrier power and

modulation scheme can be individually controlled for each carrier. However in broadcast

systems these are fixed due to the one-way communication.

Modulation (QPSK, QAM

etc.)

IFFT

D/A

Data in

Baseband OFDM signal

Modulation (QPSK, QAM etc.)

FFT

A/D

Data out

Baseband OFDM signal

5

Chapter 3

OFDM Transmission

3.1 Terrestrial digital video broadcasting (DVB-T)

A simplified block diagram of the European DVB-T standard is shown in the

figure below. A digital signal processor (DSP) performs most of the processes described

in this diagram.

Figure 3.1: Terrestrial Digital Video Broadcasting

Terrestrial Digital Video Broadcasting (DVB-T) standard has been developed in

Europe and has been implemented as a working system since March 1997.It uses Coded

Orthogonal Frequency Division Multiplexing (COFDM) as modulation scheme [2].

COFDM is the same as OFDM except that forward error correction is applied to the

signal before transmission. This is to overcome errors in the transmission due to lost

carriers from frequency selective fading, channel noise and other propagation effects. The

main focus of this project is on OFDM, but in real-life applications any practical system

will use forward error correction, thus would be COFDM.

MPEG-2 Source coding

and multiplexing

Splitter

MUX Adaptation, Outer Coder and Interleaver, Inner Coder

MUX Adaptation, Outer Coder and Interleaver, Inner Coder

Inner Interleaver,Mapper,Frame adaptation

Inner Interleaver,Mapper,Frame adaptation

Pilot & TPS

OFDM

Guard interval

D/A

Front End

6

The terrestrial network operator can choose one of the two modes of operation [4]:

2k mode: suitable for single transmitter operations and small single frequency

networks (SFN) with limited transmitter distances. It employs 1705 carriers.

8k mode: suitable for both single transmitter operations and small and large single

frequency networks (SFN). It employs 6817 carriers.

Existing DVB-T modes produce a transport capacity of 5 to 15 Mbps (1-3 TV programs)

suitable for mobile receivers.

The expression for one OFDM symbol starting at t = ts is given in [1] as follows:

Tttttts

Ttttts

ss

ss

Ni

Ni

tstT

ifcj

Nsi

s

sd

+>∧<=

+≤≤

= ∑

=

−=

−+

−

+

,0)(

,Re)(2

2

)))(5.0(2(

2/ exp π

3.1

where di are complex modulation symbols, Ns is the number of sub-carriers, T the symbol

duration, and fc the carrier frequency. A particular version of 3.1 is given in the DVB-T

standard as the emitted signal. The expression is

( )

Ψ⋅= ∑∑ ∑

∞

= = =0

67

0,,,,

2max

min

Re)(m l

k

kkklmklm

tfj tcets cπ

3.2

Where:

++≤≤+=Ψ

••••−−∆− •••

else

TmltTmle ssTmTlt

Tkj

klm

ssu

,0

)168()68(,)68('2

,,

π

3.3

7

Where:

k denotes the carrier number;

l denotes the OFDM symbol number;

m denotes the transmission frame number;

K is the number of transmitted carriers;

TS is the symbol duration;

TU is the inverse of the carrier spacing;

∆ is the duration of the guard interval;

fc is the central frequency of the radio frequency (RF) signal;

k` is the carrier index relative to the center frequency, k` = k-(Kmax + Kmin)/2;

cm, o, k complex symbol for carrier k of the data symbol no.1 in frame number m;

cm, 1,k complex symbol for carrier k of the data symbol no.2 in frame number m;

cm, 67,k complex symbol for carrier k of the data symbol no.68 in frame number m;

This project is based on the 2k mode of the DVB-T standard, intended for mobile

reception of digital TV. In this mode, the transmitted OFDM signal is organized in

frames, each having duration TF. Each frame consists of 68 OFDM symbols. Four frames

make one super-frame. Each symbol is constituted by a set of K=1705 carriers (actually

sub carriers) and transmitted with a duration of Ts, composed of a useful part with a

duration TU and a guard interval with a duration ∆. In addition to the data, the DVB-T

signal contains reference information (scattered pilot cells, continual pilot carriers, TPS

carriers), defined by the standard, which can be used by the receiver for e.g.

synchronization and channel estimation. Since this project deals only with AWGN

channel there is no need for those and all sub carriers are used for data modulation.

I will provide a description of the steps involved in the generation and reception

of an OFDM signal, more precisely the signal used in the 2k mode of the DVB-T

standard. The generation of the OFDM signal will concentrate only on the blocks labeled

OFDM, D/A, and Front End in the figure 3.1.

8

The numerical values for the OFDM parameters in the 2k mode are given in the table

below:

Parameter 2kmode

Elementary period T 7/64 µs Number of carriers K 1705 Value of carrier number Kmin 0 Value of carrier number Kmax 1704 Duration TU 224 µs Spacing between carriers Kmin and Kmax (K-1)/ TU

7.61 MHz

Carrier spacing 1/ TU 4464 Hz Allowed guard interval ∆/ TU 1/4 1/8 1/16 1/32 Duration of symbol part TU 2048xT

224 µs Duration of guard interval ∆ 512xT

56 µs 256xT 28 µs

128xT 14 µs

64xT 7 µs

Symbol duration Ts=∆+ TU 2560xT 280 µs

2304xT 252 µs

2176xT 238 µs

2112xT231 µs

Table 3.1: Parameters of the 2k mode DVB-T

As mentioned before, OFDM is implemented using IFFT/FFT algorithms. Then

subsequent up-conversion gives the real signal s(t) centered on the RF transmit carrier

frequency fc.

3.2 FFT Implementation

A practical implementation became a reality in the 1990’s due to the availability of

digital signal processors (DSP) that made the FFT affordable [1]. The OFDM spectrum is

centered on fc. This means that sub-carrier 1 is located (7.61/2) MHz to the left of the

carrier and sub-carrier 1705 is located (7.61/2) MHz to the right of the carrier. A simple

way to achieve centering is to use a 2N-IFFT [1] and T/2 as the elementary period. As

you can see from the table, the OFDM symbol duration TU is specified considering a

2048-IFFT (N=2048); thus we will use a 4096-IFFT.Next, a suitable simulation period

9

must be selected. T is defined as the elementary period for a baseband signal; however,

since the simulation is of a passband signal, a relationship between T and 1/Rs, a time-

period that considers at least twice the carrier frequency, must be found. For simplicity,

an integer relation was chosen, namely Rs=40/T.This gives a carrier frequency of around

90 MHz, which is in the range of a VHF channel five, a common TV channel in any city.

The block diagram below shows the generation of one OFDM symbol:

Figure 3.2: OFDM symbol generation block diagram

In the Fig 3.2, the name of the variable used in the MATLAB code is under each

encircled letter. Next, I will describe each of the steps specified in the figure above. The

total number of sub-carriers in this system is 1705.However, the size of IFFT/FFT vector

is 4096.Therefore, we add 4096-1705=2391 zeros to the signal info at (A) to achieve

over-sampling and to center the spectrum. In Figure 3.3 and Figure 3.4, you can observe

the result of this operation and that the signal carriers at (B) has a time period of T/2.

1705

4QAM Symbols

4096 IFFT

g (t)

T/2

fp=1/T LPF

A B C D E

fc

info carriers u uoft

s(t)

10

0 0.2 0.4 0.6 0.8 1 1.2

x 10-6

-40

-20

0

20

40

60carriers inphase

Time(sec)

Am

plitu

de

0 0.2 0.4 0.6 0.8 1 1.2

x 10-6

-100

-50

0

50

100

150carriers quadrature

Time(sec)

Am

plitu

de

Figure 3.3: Time response of signal carriers

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 107

0

0.5

1

1.5carriers FFT

Frequency(Hz)

Am

plitu

de

0 2 4 6 8 10 12 14 16 18

x 106

-90

-80

-70

-60

-50

-40

-30

Frequency(Hz)

Pow

er S

pect

ral D

ensi

ty (d

B/H

z)

carriers Welch PSD estimate

Figure 3.4: Frequency response of signal carriers

The signal carriers are a discrete-time baseband signal. The next step is to produce a

continuous-time signal. In order to achieve this, a transmit filter g (t) is applied to the

complex signal carriers. The impulse response of this filter is shown next:

11

0 1 2 3 4 5 6 7

x 10-8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Pulse g(t)

Time(sec)

Am

plitu

de

Figure 3.5: Impulse response of g (t)

The output of the filter is shown in the following figures, both in time-domain and

frequency-domain.

0 0.2 0.4 0.6 0.8 1 1.2

x 10-6

-40

-20

0

20

40

60u inphase

Time(sec)

Am

plitu

de

0 0.2 0.4 0.6 0.8 1 1.2

x 10-6

-100

-50

0

50

100

150u quadrature

Time(sec)

Am

plitu

de

Figure 3.6: Time response of signal u at (C)

12

0 0.5 1 1.5 2 2.5 3 3.5 4

x 108

0

10

20

30

40

50

Am

plitu

de

Frequency(Hz)

0 0.5 1 1.5 2 2.5 3 3.5

x 108

-120

-100

-80

-60

-40

-20

Frequency (Hz)

Pow

er S

pect

ral D

ensi

ty (d

B/H

z)

Welch PSD Estimate

Figure 3.7: Frequency response of signal u at (C)

The frequency response of Figure 3.7 is periodic, since it is of a discrete-time system.

The bandwidth of the spectrum shown in this figure is given by Rs. The period of the

signal U is 2/T, thus the transition bandwidth for the reconstruction or digital-to-analog

(D/A) filter is (2/T=18.286)-7.61=10.675 MHz. If a 2048-IFFT (N-IFFT) was used, the

transition bandwidth would have been only (1/T=9.143)-7.61=1.533 MHz, which

requires a very sharp roll-off, hence high complexity, in the D/A filter to avoid aliasing.

The digital-to-analog (D/A) filter chosen is a Butterworth filter of order 13 and cut-off

frequency close to 1/T.The filter’s response is shown below:

13

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 108

-700

-600

-500

-400

-300

-200

-100

0

100D/A filter response

Frequency(Hz)

Am

plitu

de(d

B)

Figure 3.8: D/A filter response

The output of the filter can be seen in Figure 3.9 and Figure 3.10.

2 4 6 8 10 12 14

x 10-7

-60

-40

-20

0

20

40

60

Am

plitu

de

Time(sec)

2 4 6 8 10 12 14

x 10-7

-100

-50

0

50

100

150

Am

plitu

de

Time(sec)

Figure 3.9: Time response of signal uoft at (D)

14

0 0.5 1 1.5 2 2.5 3 3.5 4

x 108

0

10

20

30

40

50uoft FFT

Frequency(Hz)

Am

plitu

de

0 0.5 1 1.5 2 2.5 3 3.5

x 108

-120

-100

-80

-60

-40

-20

Frequency(Hz)

Pow

er S

pect

ral D

ensi

ty (d

B/H

z)uoft Welch PSD estimate

Figure 3.10: Frequency response of signal uoft at (D)

The delay produced by the filtering operation is of approximately 2x10-7, as it is

obvious when comparing Figure.3.6 and Figure.3.9. Disregarding this, the filtering

performs as expected since we now have only the baseband spectrum. Recall that carriers

1 to 852 are located to the left of 0 Hz and carriers 853 to 1705 are to the right. This

signal is, as mentioned previously, a baseband signal. The next step is to convert it to a

passband signal using quadrature multiplex double-sideband amplitude modulation. In

this type of modulation, an in-phase signal mI (t) and a quadrature signal mQ (t) are

modulated using the formula

( ) ( ) ( ) ( )fctttftts mm QcI ππ 22)( sincos +=

3.4

The in-phase signal corresponds to the real part of the complex modulation symbols,

whereas the quadrature signal corresponds to the imaginary part of the same complex

modulation symbols. For this project, these are 4QAM symbols. Using the formula

above, the signal out of the transmitter s (t) becomes:

15

( ) ( ) ( ) ( )tfttftts cQ

cI uoftuoft ππ 2sin2cos)( +=

3.5

The time and frequency response of the complete OFDM signal s (t) is shown next:

2 4 6 8 10 12 14

x 10-7

-150

-100

-50

0

50

100

150s(t)

Time(sec)

Am

plitu

de

Figure 3.11: Time response of signal s (t) at (E)

16

0 0.5 1 1.5 2 2.5 3 3.5 4

x 108

0

5

10

15

20

25s(t) FFT

Frequency(Hz)A

mpl

itude

0 2 4 6 8 10 12 14 16 18

x 107

-120

-100

-80

-60

-40

-20

Frequency(Hz)

Pow

er S

pect

ral D

ensi

ty (d

B/H

z)

s(t) Welch PSD estimate

Figure 3.12: Frequency response of signal s (t) at (E)

The next step is to transmit the signal through an AWGN channel, receive it and check

the errors. The simulation is based on multiple signal-to-noise-ratio (SNR); meaning that

the signal is received for various SNR values and error check is performed.

17

Chapter 4

OFDM Reception

The design of an OFDM receiver is open since there are only transmission

standards. Most of the research and innovation is done in the receiver. For example, the

frequency sensitivity drawback is mainly a transmission channel prediction problem,

something that is done at the receiver. In this report, I will present only a basic receiver

structure that follows the inverse of the transmission process. The block diagram is

presented in Figure 4.1.

Figure 4.1: An OFDM receiver

OFDM is very sensitive to timing and frequency offsets. The delay produced by the

reconstruction and demodulation filters is about td = 64/Rs for my program. This delay was

taken care of when I did the simulation. As you can see from the block diagram in the

Figure 4.1, the reception process is straightforward: the received OFDM signal is first low-

pass filtered to get the corresponding baseband signal and sampled. The output of the FFT

modulation block is the received constellation. This one passes through a 4QAM slicer,

which assigns the received symbols into the four possible constellation points. The error,

which is a symbol error, is calculated by comparing the original constellation with the one

that is outputted by the 4QAM slicer. As in the case of the transmitter, I indicated the

Fp=2fc

Fs=2/T

4096 FFT

4QAMSlicer

F G H I J

fc

r(t)=s(t)+n

rtilde r_info r_data info_h a_hat

18

names of the variables used in the simulation and the output processes in the reception. The

original constellation is shown in Figure 4.2 whereas the received constellation is shown in

Figure 4.3, Figure 4.4 and Figure 4.5 for corresponding SNR values of 2 dB, 6 dB and 12

dB.

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2Original constellation

Figure 4.2: Original 4-QAM constellation

19

-3 -2 -1 0 1 2 3

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

infoh Received Constellation

Figure 4.3: Received 4-QAM constellation for SNR=2dB

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-1.5

-1

-0.5

0

0.5

1

1.5



20

-1.5 -1 -0.5 0 0.5 1 1.5

-1.5

-1

-0.5

0

0.5

1



It is clear that as the SNR is increased the received constellation gets less affected

by the noise, hence there will be less errors. However, for low values of SNR we have ISI

introduced by the noise at the receiver side. This is presented in Figure 4.6, Figure 4.7,

Figure 4.8 and Figure 4.9.

21

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4-1

-0.5

0

0.5

1

Time

Am

plitu

de

Eye Diagram for In-Phase Signal

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4-1

-0.5

0

0.5

1

Time

Am

plitu

de

Eye Diagram for Quadrature Signal

Figure 4.6: Eye pattern for the received constellation in an ideal channel

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4-3

-2

-1

0

1

2

3

Time

Am

plitu

de


-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4-3

-2

-1

0

1

2

3

Time

Am

plitu

de


Figure 4.7: Eye pattern for the received constellation for SNR=2dB

22

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4-3

-2

-1

0

1

2

Time

Am

plitu

de


-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4-3

-2

-1

0

1

2

Time

Am

plitu

de



-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4-1.5

-1

-0.5

0

0.5

1

1.5

Time

Am

plitu

de


-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4-1.5

-1

-0.5

0

0.5

1

1.5

Time

Am

plitu

de



23

The theoretical probability of symbol error for rectangular QAM constellation is

given in [3] as follows:

( )211 PP MM −−=

4.1

where

( )

⋅−

⋅⋅

−=

NEP M

QM

avM

013112

4.2

Here Eav is the average energy per bit; M = 2k represents the number of levels and

k is the number of bits per symbol. Equations 4.1 and 4.2 are for the case of k even. For k

odd, there is no exact result. However, the symbol-error probability is upper bounded as

( )

⋅−≤

NEP M

kQ av

M01

34

4.3

The result of the simulation is given in Figure 4.10. The theoretical curve was

generated using 4.3, without the scaling factor (i.e. only using the Q-function without the 4

in front), although for my project k was even (i.e. k = 2 for 4QAM). This was suggested in

[3].

24

0 1 2 3 4 5 6 7 810-5

10-4

10-3

10-2

10-1

100

SNR/bit in dB

Sym

bol E

rror R

ate

Simulated error rateTheoretical probability of error

Figure 4.10: Simulated and theoretical symbol error rate

25

Chapter 5

Conclusion

The simulation done in MATLAB worked well. The Additive White Gaussian

Noise (AWGN) corrupted the transmitted signal and this resulted in a different received

4QAM constellation than the original constellation. For small SNR values the calculated

error rate was quite large and ISI was produced due the relative high power of noise. As

SNR was increased the error rate was decreasing, as expected. In fact, for a SNR value

greater than 8 dB, the error was zero. This is a quite different than expected and it is due to

the fact that the program is simulating only 68 OFDM symbols (i.e. one frame), sent one by

one. If the number of transmitted OFDM symbols is increased, than a more accurate error

rate can be obtained, but this necessitates a high processing power PC and time. Letting this

aside, the system’s performance was good since the simulated error rate for small SNR

values was a little bit above the theoretical probability curve. The difference between the

two curves is less than 0.5 dB. As the SNR is increased we observe that the simulated

symbol error rate intersects and then drops below the theoretical error curve. There are

more aspects of OFDM that need to be researched since this simulation was only a basic

one. As an example, there are a lot of improvements that can be brought to the program,

such as the addition of guard interval, coding the original information, simulation over a

multipath channel etc.

26

APPENDIX

MATLAB code used for simulation

clear all;

clc;

close all;

%*********************The 2k DVB-T parameters************************************

Tu=224e-6; %useful OFDM symbol period

T=Tu/2048; %baseband elementary period

G=0; %choice of 1/4, 1/8, 1/15 and 1/32

delta=G*Tu; %guard band duration

Ts=Tu+delta; %total OFDM symbol period

Kmax=1705; %number of subcarriers

Kmin=0;

FS=4096; %IFFT/FFT length

q=10; %carrier period to elementary period ratio

fc=q*1/T; %carrier frequency

Rs=4*fc; %simulation period

t=0:1/Rs:Tu;

tt=0:T/2:Tu;

%*******************************************************************************

repeat=68; % one OFDM frame( 68 OFDM symbols) is sent, symbol by symbol

SNR_dB = 0:2:16 ; %Signal-to-noise ratio in dB

error = zeros(1,length(SNR_dB));

27

%OFDM TRANSMISSION

%DATA GENERATOR

for z=1:repeat

for w=1:length(SNR_dB)

sM=2;

[x,y]=meshgrid((-sM+1):2:(sM-1),(-sM+1):2:(sM-1));

alphabet=x(:) + 1i*y(:);

N=Kmax+1;

rand('state',0);

a=-1+2*round(rand(N,1)).'+i*(-1+2*round(rand(N,1)).');

A=length(a);

info=zeros(FS,1);

info(1:(A/2))= [ a(1:(A/2)).'];

info((FS-((A/2)-1)):FS)= [ a(((A/2)+1):A).'];

carriers=FS.*ifft(info,FS);

%UPCONVERTER

L=length(carriers);

chips=[carriers.';zeros((2*q)-1,L)];

p=1/Rs:1/Rs:T/2;

g=ones(length(p),1);

dummy=conv(g,chips(:));

u=[dummy;zeros(46,1)];

[b,aa]=butter(13,1/20);

uoft=filter(b,aa,u);

delay=64; % Reconstruction filter delay

28

s_tilde=(uoft(delay+(1:length(t))).').*exp(1i*2*pi*fc*t);

s=real(s_tilde);

%***********************************************************

% Here based on the power of the received signal plus the

% desired SNR we generate and add the AWGN noise to create

% the corrupt signal

noisedst = awgn(s,SNR_dB(w),'measured');

%***********************************************************

%OFDM RECEPTION

%DOWNCONVERTER

r_tilde=exp(-1i*2*pi*fc*t).*noisedst; % (F)

%CARRIER SUPPRESSION

[B,AA]=butter(3,1/2);

r_info=2*filter(B,AA,r_tilde); %Baseband signal continous-time (G)

%SAMPLING

r_data=real(r_info(1:(2*q):length(t)))+1i*imag(r_info(1:(2*q):length(t)));

%Baseband signal discrete-time (H)

29

%FFT

info_2N=(1/FS).*fft(r_data,FS); % (I)

info_h=[info_2N(1:A/2) info_2N((FS-((A/2)-1)):FS)];

%SLICING

for k=1:N,

a_hat(k)=alphabet((info_h(k)-alphabet)==min(info_h(k)-alphabet)); % (J)

end;

figure(1);

plot(info_h((1:A)),'.k');

title('info_h Received Constellation');

axis square;

axis equal;

grid on;

figure(2);

plot(a_hat((1:A)),'or');

title('a_hat 4-QAM');

axis square;

axis equal;

grid on;

axis([-1.5 1.5 -1.5 1.5]);

error(w)= error(w) + length(find(a_hat~=a));

end

end

30

figure(3);

semilogy(SNR_dB,error/(repeat*N),'b<-');

grid on;

ylabel('Symbol Error Rate');

xlabel('SNR/bit in dB')

save result error SNR_dB N repeat

% error,SNR ,N and repeat variables are saved for possible future use

% and to avoid re-runing the simulation

31

REFERENCES

[1] OFDM Simulation using MATLAB. Retrieved May 9, 2003, from http://users.ece.gatech.edu/~mai/tutorial/OFDM/Tutorial_web.pdf

[2] Broadcast papers. Retrieved May 9, 2003, from

http://www.broadcastpapers.com/tvtran/ HarrisDVBTDeliverMobRec01.htm

http://www.broadcastpapers.com/tvtran/ITISMagicsOfDTV10.htm

[3] Proakis, John G. and Salehi, Masoud, Contemporary Communications Systems using MATLAB, CA: Brooks/Cole, 2000.

[4] DVB-T standard. Retrieved May 20, 2003, from

http://www.kjmbc.co.kr/old/beta/ofdm/ofdm.html

study of ofdm performance over awgn channels...

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