CHAPTER 1

1.1 INTRODUCTION

ORTHOGONAL frequency division multiplexing (OFDM) has been attracting

substantial attention due to its excellent performance under severe channel

condition. The rapidly growing application of OFDM includes WiMAX,

DVB/DAB and 4G wireless systems.

1.2 OVERVIEW

Initial proposals for OFDM were made in the 60s and the 70s. It has

taken more than a quarter of a century for this technology to move from the

research domain to the industry. The concept of OFDM is quite simple but

the practicality of implementing it has many complexities. So, it is a fully

software project.OFDM depends on Orthogonality principle. Orthogonality

means, it allows the sub carriers, which are orthogonal to each other,

meaning that cross talk between co-channels is eliminated and inter-carrier

guard bands are not required. This greatly simplifies the design of both the

transmitter and receiver, unlike conventional FDM; a separate filter for each

sub channel is not required.

Orthogonal Frequency Division Multiplexing (OFDM) is a digital multi

carrier modulation scheme, which uses a large number of closely spaced

orthogonal sub-carriers.

A single stream of data is split into parallel streams each of which is coded

and modulated on to a subcarrier, a term commonly used in OFDM systems.

Each sub-carrier is modulated with a conventional modulation scheme (such

as quadrature amplitude modulation) at a low symbol rate, maintaining data

1

rates similar to conventional single carrier modulation schemes in the same

bandwidth. Thus the high bit rates seen before on a single carrier is reduced

to lower bit rates on the subcarrier.

In practice, OFDM signals are generated and detected using the Fast

Fourier Transform algorithm. OFDM has developed into a popular scheme for

wideband digital communication, wireless as well as copper wires. Actually;

FDM systems have been common for many decades. However, in FDM, the

carriers are all independent of each other. There is a guard period in

between them and no overlap whatsoever. This works well because in FDM

system each carrier carries data meant for a different user or application. FM

radio is an FDM system. FDM systems are not ideal for what we want for

wideband systems. Using FDM would waste too much bandwidth. This is

where OFDM makes sense. In OFDM, subcarriers overlap. They are

orthogonal because the peak of one subcarrier occurs when other

subcarriers are at zero. This is achieved by realizing all the subcarriers

together using Inverse Fast Fourier Transform (IFFT). The demodulator at the

receiver parallel channels from an FFT block. Note that each subcarrier can

still be modulated independently.

2

CHAPTER 2

2.1 BACKGROUND:

Most first generations systems were introduced in the mid 1980’s, and

can be Characterized by the use of analog transmission techniques and the

use of simple multiple access techniques such as Frequency Division Multiple

Access (FDMA). First generation telecommunications systems such as

Advanced Mobile Phone Service (AMPS) only provided voice communications.

They also suffered from a low user capacity, and security problems due to

the simple radio interface used. Second generation systems were introduced

in the early 1990’s, and all use digital technology. This provided an increase

in the user capacity of around three times. This was achieved by

compressing the voice waveforms before transmission.

Third generation systems are an extension on the complexity of

second-generation systems and are expected to be introduced after the year

2000. The system capacity is expected to be increased to over ten times

original first generation systems. This is going to be achieved by using

complex multiple access techniques such as Code Division Multiple Access

(CDMA), or an extension of TDMA, and by improving flexibility of services

available. The telecommunications industry faces the problem of providing

3

telephone services to rural areas, where the customer base is small, but the

cost of installing a wired phone network is very high. One method of

reducing the high infrastructure cost of a wired system is to use a fixed

wireless radio network. The problem with this is that for rural and urban

areas, large cell sizes are required to get sufficient coverage.

Fig.2.1 shows the evolution of current services and networks to the

aim of combining them into a unified third generation network. Many

currently separate systems and services such as radio paging, cordless

telephony, satellite phones and private radio systems for companies etc, will

be combined so that all these services will be provided by third generation

telecommunications systems.

Fig: 2.1 Evolution of current networks to the next generation of

wireless networks.

Currently Global System for Mobile telecommunications (GSM)

technology is being applied to fixed wireless phone systems in rural areas.

However, GSM uses time division multiple access (TDMA), which has a high

symbol rate leading to problems with multipath causing inter-symbol

interference. Several techniques are under consideration for the next

4

generation of digital phone systems, with the aim of improving cell capacity,

multipath immunity, and flexibility. These include CDMA and OFDM. Both

these techniques could be applied to providing a fixed wireless system for

rural areas. However, each technique as different properties, making it more

suited for specific applications.

OFDM is currently being used in several new radio broadcast systems

including the proposal for high definition digital television (HDTV) and digital

audio broadcasting (DAB). However, little research has been done into the

use of OFDM as a transmission method for mobile telecommunications

systems. In CDMA, all users transmit in the same broad frequency band

using specialized codes as a basis of channelization. Both the base station

and the mobile station know these codes, which are used to modulate the

data sent. OFDM/COFDM allows many users to transmit in an allocated band,

by subdividing the available bandwidth into many narrow bandwidth carriers. Each

user is allocated several carriers in which to transmit their data.

The transmission is generated in such a way that the carriers used are

orthogonal to one another, thus allowing them to be packed together much

closer than standard frequency division multiplexing (FDM). This leads to

OFDM/COFDM providing a high spectral efficiency.

Orthogonal Frequency Division Multiplexing is a scheme used in the

area of high-data-rate mobile wireless communications such as cellular

phones, satellite communications and digital audio broadcasting. This

technique is mainly utilized to combat inter-symbol interference.

2.2 MULTIPLE ACCESS TECHNIQUES:

5

Multiple access schemes are used to allow many simultaneous users to

use the same fixed bandwidth radio spectrum. In any radio system, the

bandwidth, which is allocated to it, is always limited. For mobile phone

systems the total bandwidth is typically 50 MHz, which is split in half to

provide the forward and reverse links of the system

.

Sharing of the spectrum is required in order increase the user capacity

of any wireless network. FDMA, TDMA and CDMA are the three major

methods of sharing the available bandwidth to multiple users in wireless

system. There are many extensions, and hybrid techniques for these

methods, such as OFDM, and hybrid TDMA and FDMA systems. However, an

understanding of the three major methods is required for understanding of

any extensions to these methods.

2.3 FREQUENCY DIVISION MULTIPLE ACCESS (FDMA):

In Frequency Division Multiple Access (FDMA), the available bandwidth

is subdivided into a number of narrower band channels. Each user is

allocated a unique frequency band in which to transmit and receive on.

During a call, no other user can use the same frequency band.

Each user is allocated a forward link channel (from the base station to

the mobile phone) and a reverse channel (back to the base station), each

being a single way link. The transmitted signal on each of the channels is

continuous allowing analog transmissions. The bandwidths of FDMA channels

are generally low (30 kHz) as each channel only supports one user. FDMA is

used as the primary breakup of large allocated frequency bands and is used

as part of most multi-channel systems.

6

Fig 2.2 FDMA showing that the each narrow Fig 2.3 FDMA spectrum where the available

band channel allocated to a single user B.W is subdivided into narrowband channels

2.3 TIME DIVISION MULTIPLE ACCESS(TDMA):

Time Division Multiple Access (TDMA) divides the available spectrum

into multiple time slots, by giving each user a time slot in which they can

transmit or receive. Fig. 1.4 shows how the time slots are provided to users

in a round robin fashion, with each user being allotted one time slot per

frame. TDMA systems transmit data in a buffer and burst method, thus the

transmission of each channel is non-continuous.

Fig 2.4 TDMA scheme, where each user is allocated a small time slot

7

The input data to be transmitted is buffered over the previous frame

and burst transmitted at a higher rate during the time slot for the channel.

TDMA can not send analog signals directly due to the buffering required, thus

are only used for transmitting digital data. TDMA can suffer from multipath

effects, as the transmission rate is generally very high. This leads the

multipath signals causing inter-symbol interference. TDMA is normally used

in conjunction with FDMA to subdivide the total available bandwidth into

several channels. This is done to reduce the number of users per channel

allowing a lower data rate to be used. This helps reduce the effect of delay

spread on the transmission. Fig.2.5 shows the use of TDMA with FDMA. Each

channel based on FDMA, is further subdivided using TDMA, so that several

users can transmit of the one channel. This type of transmission technique is

used by most digital second generation mobile phone systems. For GSM, the

total allocated bandwidth of 25MHz is divided into 125, 200 kHz channels

using FDMA. These channels are then subdivided further by using TDMA so

that each 200 kHz channel allows 8-16 users.

Fig.2.5 TDMA/FDMA hybrid, showing that the bandwidth is split into

frequency channels and time slots.

2.4 CODE DIVISION MULTIPLE ACCESS(CDMA):

8

Code Division Multiple Access (CDMA) is a spread spectrum technique

that uses neither frequency channels nor time slots. In CDMA, the narrow

band message (typically digitized voice data) is multiplied by a large

bandwidth signal, which is a pseudo random noise code (PN code). All users

in a CDMA system use the same frequency band and transmit

simultaneously. The transmitted signal is recovered by correlating the

received signal with the PN code used by the transmitter. Fig. 2.6 shows the

general use of the spectrum using CDMA.

Some of the properties that have made CDMA useful are: Signal hiding

and non-interference with existing systems, Anti-jam and interference

rejection, Information security, Accurate Ranging, Multiple User Access,

Multipath tolerance.

Fig. 2.6 Code Division Multiple Access (CDMA)

Fig.2.7 shows the process of a CDMA transmission. The data to be

transmitted (a) is spread before transmission by modulating the data using a

PN code. This broadens the spectrum as shown in (b). In this example the

process gain is 125 as the spread spectrum bandwidth is 125 times greater

the data bandwidth. Part (c) shows the received signal. This consists of the

required signal, plus background noise, and any interference from other

CDMA users or radio sources.

9

The received signal is recovered by multiplying the signal by the

original spreading code. This process causes the wanted received signal to

be dispread back to the original transmitted data. However, all other signals,

which are uncorrelated to the PN spreading code used, become more spread.

The wanted signal in (d) is then filtered removing the wide spread

interference and noise signals.

Fig.2.7 Basic CDMA Generation.

CDMA Generation:

CDMA is achieved by modulating the data signal by a pseudo random

noise sequence (PN code), which has a chip rate higher then the bit rate of

the data. The PN code sequence is a sequence of ones and zeros (called

chips), which alternate in a random fashion. The data is modulated by

modular-2 adding the data with the PN code sequence. This can also be done

by multiplying the signals, provided the data and PN code is represented by

1 and -1 instead of 1 and 0. Fig. 2.8 shows a basic CDMA transmitter.

10

Fig. 2.8 Simple direct sequence modulator

The PN code used to spread the data can be of two main types. A short

PN code (Typically 10-128 chips in length), can be used to modulate each

data bit. The short PN code is then repeated for every data bit allowing for

quick and simple synchronization of the receiver. Fig.2.9 shows the

generation of a CDMA signal using a 10-chip length short code. Alternatively

a long PN code can be used. Long codes are generally thousands to millions

of chips in length, thus are only repeated infrequently. Because of this they

are useful for added security as they are more difficult to decode.

Fig.2.9 Direct sequence signals

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CHAPTER-3CHAPTER-3

3.1 OFDM INTRODUCTION:

The OFDM technology was first conceived in the 1960s and 1970s

during research into minimizing ISI, due to multipath. The expression digital

communications in its basic form is the mapping of digital information into a

waveform called a carrier signal, which is a transmitted electromagnetic

pulse or wave at a steady base frequency of alternation on which information

can be imposed by increasing signal strength, varying the base frequency,

varying the wave phase, or other means. In this instance, orthogonality is an

implication of a definite and fixed relationship between all carriers in the

collection. Multiplexing is the process of sending multiple signals or streams

of information on a carrier at the same time in the form of a single, complex

signal and then recovering the separate signals at the receiving end.

12

Modulation is the addition of information to an electronic or optical

signal carrier. Modulation can be applied to direct current (mainly by turning

it on and off), to alternating current, and to optical signals. One can think of

blanket waving as a form of modulation used in smoke signal transmission

(the carrier being a steady stream of smoke). In telecommunications in

general, a channel is a separate path through which signals can flow. In

optical fiber transmission using dense wavelength-division multiplexing, a

channel is a separate wavelength of light within a combined, multiplexed

light stream. This project focuses on the telecommunications definition of a

channel.

3.2 OFDM PRINCIPLES:

OFDM is a special form of Multi Carrier Modulation (MCM) with densely

spaced sub carriers with overlapping spectra, thus allowing for multiple-

access. MCM) is the principle of transmitting data by dividing the stream into

several bit streams, each of which has a much lower bit rate, and by using

these sub-streams to modulate several carriers. This technique is being

investigated as the next generation transmission scheme for mobile wireless

communications networks.

3.3 FOURIER TRANSFORM :

Back in the 1960s, the application of OFDM was not very practical. This

was because at that point, several banks of oscillators were needed to

generate the carrier frequencies necessary for sub-channel transmission.

Since this proved to be difficult to accomplish during that time period, the

scheme was deemed as not feasible.

13

However, the advent of the Fourier Transform eliminated the initial

complexity of the OFDM scheme where the harmonically related frequencies

generated by Fourier and Inverse Fourier transforms are used to implement

OFDM systems. The Fourier transform is used in linear systems analysis,

antenna studies, etc., The Fourier transform, in essence, decomposes or

separates a waveform or function into sinusoids of different frequencies

which sum to the original waveform. It identifies or distinguishes the

different frequency sinusoids and their respective amplitudes.

The Fourier transform of f(x) is defined as:

F (ω )=∫−∞

∞

f ( x )⋅¿ e− jωx dx ¿

and its inverse is denoted by:

f ( x )= 12 π ∫

−∞

∞

F (ω )⋅e jωx dω

However, the digital age forced a change upon the traditional form of the

Fourier transform to encompass the discrete values that exist is all digital

systems. The modified series was called the Discrete Fourier Transform

(DFT). The DFT of a discrete-time system, x(n) is defined as:

Χ ( k )=∑n=0

N −1

x (n)⋅e− j

2 πN

kn

1 k N

and its associated inverse is denoted by:

14

(1)

(2)

(3)

x (n )= 1N

∑n=0

N −1

Χ ( k )⋅ej2 πN

kn

1 n N

However, in OFDM, another form of the DFT is used, called the Fast Fourier

Transform (FFT), which is a DFT algorithm developed in 1965. This “new”

transform reduced the number of computations from something on the order

of

N2 to

N2⋅log2 N .

3.4 ORTHOGONALITY:

In geometry, orthogonal means, "involving right angles" (from Greek

ortho, meaning right, and gon meaning angled). The term has been

extended to general use, meaning the characteristic of being independent

(relative to something else). It also can mean: non-redundant, non-

overlapping, or irrelevant. Orthogonality is defined for both real and complex

valued functions. The functions m(t) and n(t) are said to be orthogonal with

respect to each other over the interval a < t < b if they satisfy the condition:

∫a

b

ϕm( t )ϕm

¿

( t )dt=0 ,Where n m

3.5 OFDM CARRIERS:

15

(4)

(5)

(6)

As for mentioned, OFDM is a special form of MCM and the OFDM time

domain waveforms are chosen such that mutual orthogonality is ensured

even though sub-carrier spectra may over-lap. With respect to OFDM, it can

be stated that orthogonality is an implication of a definite and fixed

relationship between all carriers in the collection. It means that each carrier is

positioned such that it occurs at the zero energy frequency point of all other

carriers. The sinc function, illustrated in Fig.3.1 exhibits this property and it is used

as a carrier in an OFDM system.

fu is the sub-carrier spacing

Fig 3.1.OFDM sub carriers in the frequency domain

3.6 OFDM:

Orthogonal Frequency Division Multiplexing (OFDM) is a multicarrier

transmission technique, which divides the available spectrum into many

carriers, each one being modulated by a low rate data stream. OFDM is

similar to FDMA in that the multiple user access is achieved by subdividing

the available bandwidth into multiple channels that are then allocated to

16

users. However, OFDM uses the spectrum much more efficiently by spacing

the channels much closer together. This is achieved by making all the

carriers orthogonal to one another, preventing interference between the

closely spaced carriers.

Coded Orthogonal Frequency Division Multiplexing (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. For

this discussion the terms OFDM and COFDM are used interchangeably, as the

main focus of this thesis is on OFDM, but it is assumed that any practical

system will use forward error correction, thus would be COFDM.

In FDMA each user is typically allocated a single channel, which is used

to transmit all the user information. The bandwidth of each channel is

typically 10 kHz-30 kHz for voice communications. However, the minimum

required bandwidth for speech is only 3 kHz. The allocated bandwidth is

made wider then the minimum amount required preventing channels from

interfering with one another. This extra bandwidth is to allow for signals from

neighboring channels to be filtered out, and to allow for any drift in the

center frequency of the transmitter or receiver. In a typical system up to

50% of the total spectrum is wasted due to the extra spacing between

channels.

This problem becomes worse as the channel bandwidth becomes

narrower, and the frequency band increases. Most digital phone systems use

17

vocoders to compress the digitized speech. This allows for an increased

system capacity due to a reduction in the bandwidth required for each user.

Current vocoders require a data rate somewhere between 4- 13kbps, with

depending on the quality of the sound and the type used. Thus each user

only requires a minimum bandwidth of somewhere between 2-7 kHz, using

QPSK modulation. However, simple FDMA does not handle such narrow

bandwidths very efficiently. TDMA partly overcomes this problem by using

wider bandwidth channels, which are used by several users. Multiple users

access the same channel by transmitting in their data in time slots. Thus,

many low data rate users can be combined together to transmit in a single

channel, which has a bandwidth sufficient so that the spectrum can be used

efficiently.

There are however, two main problems with TDMA. There is an

overhead associated with the change over between users due to time

slotting on the channel. A change over time must be allocated to allow for

any tolerance in the start time of each user, due to propagation delay

variations and synchronization errors. This limits the number of users that

can be sent efficiently in each channel. In addition, the symbol rate of each

channel is high (as the channel handles the information from multiple users)

resulting in problems with multipath delay spread.

OFDM overcomes most of the problems with both FDMA and TDMA.

OFDM splits the available bandwidth into many narrow band channels

(typically 100-8000). The carriers for each channel are made orthogonal to one

another, allowing them to be spaced very close together, with no overhead as in the

FDMA example. Because of this there is no great need for users to be time multiplex

as in TDMA, thus there is no overhead associated with switching between users.

18

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, allowing

then to be spaced as close as theoretically possible. This overcomes the

problem of overhead carrier spacing required in FDMA.Each carrier in an

OFDM signal has a very narrow bandwidth (i.e. 1 kHz), thus the resulting

symbol rate is low. This results in the signal having a high tolerance to

multipath delay spread, as the delay spread must be very long to cause

significant ISI (e.g > 500usec).

3.7 OFDM GENERATION:

To generate OFDM successfully the relationship between all the

carriers must be carefully controlled to maintain the orthogonality of the

carriers. For this reason, 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 modulation scheme

(typically differential BPSK, QPSK, or QAM).

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. The IFFT performs the transformation very

efficiently, and provides a simple way of ensuring the carrier signals

produced are orthogonal.

19

The Fast Fourier Transform (FFT) transforms a cyclic time domain

signal into its equivalent frequency spectrum. This is done by finding the

equivalent waveform, generated by a sum of orthogonal sinusoidal

components. The amplitude and phase of the sinusoidal components

represent the frequency spectrum of the time domain signal.

. The IFFT performs the reverse process, transforming a spectrum

(amplitude and phase of each component) into a time domain signal. An IFFT

converts a number of complex data points, of length, which is a power of 2,

into the time domain signal of the same number of points. Each data point in

frequency spectrum used for an FFT or IFFT is called a bin. The orthogonal

carriers required for the OFDM signal can be easily generated by setting the

amplitude and phase of each bin, then performing the IFFT. Since each bin of

an IFFT corresponds to the amplitude and phase of a set of orthogonal

sinusoids, the reverse process guarantees that the carriers generated are

orthogonal.

Fig. 3.2 OFDM Block Diagram

20

Fig.3.2 shows the setup for a basic OFDM transmitter and receiver.

The signal generated is a base band, thus the signal is filtered, then stepped

up in frequency before transmitting the signal. OFDM time domain

waveforms are chosen such that mutual orthogonality is ensured even

though sub-carrier spectra may overlap. Typically QAM or Differential

Quadrature Phase Shift Keying (DQPSK) modulation schemes are applied to

the individual sub carriers. To prevent ISI, the individual blocks are separated

by guard intervals wherein the blocks are periodically extended.

3.8 MODULATION TECHNIQUES:

Quadrature Amplitude Modulation(QAM):

This modulation scheme is also called quadrature carrier multiplexing.

Infact, this modulation scheme enables to DSB-SC modulated signals to

occupy the same transmission BW at the receiver o/p. it is, therefore, known

as a bandwidth-conservation scheme. The QAM Tx consists of two separate

balanced modulators, which are supplied, with two carrier waves of the same

freq but differing in phase by 90. The o/p of the two balanced modulators

are added in the adder and transmitted.

Fig. 3.3 QAM System

21

The transmitted signal is thus given by

S (t) = X1 (t) A cos (2Fc t) + X2 (t) A sin (2Fc t)

Hence, the multiplexed signal consists of the in-phase component ‘A

X1 (t)’ and the quadrature phase component ‘–A X2 (t)’.

Balanced Modulator:

A DSB-SC signal is basically the product of the modulating or base

band signal and the carrier signal. Unfortunately, a single electronic device

cannot generate a DSB-SC signal. A circuit is needed to achieve the

generation of a DSB-SC signal is called product modulator i.e., Balanced

Modulator.

We know that a non-linear resistance or a non-linear device may be

used to produce AM i.e., one carrier and two sidebands. However, a DSB-SC

signal contains only 2 sidebands. Thus, if 2 non-linear devices such as

diodes, transistors etc., are connected in balanced mode so as to suppress

the carriers of each other, then only sidebands are left, i.e., a DSB-SC signal

is generated. Therefore, a balanced modulator may be defined as a circuit in

which two non-linear devices are connected in a balanced mode to produce a

DSB-SC signal.

Quadrature Phase Shift Keying(QPSK):Quadrature Phase Shift Keying(QPSK):

In communication systems, we have two main resources. These are:

1. Transmission Power

2. Channel bandwidth

If two or more bits are combined in some symbols, then the signaling

rate will be reduced. Thus, the frequency of the carrier needed is also

22

reduced. This reduces the transmission channel B.W. Hence, because of

grouping of bits in symbols; the transmission channel B.W can be reduced. In

QPSK two successive bits in the data sequence are grouped together. This

reduces the bits rate or signaling rate and thus reduces the B.W of the

channel. In case of BPSK, we know that when sym. Changes the level, the

phase of the carrier is changed by 180. Because, there were only two sym’s

in BPSK, the phase shift occurs in 2 levels only. However, in QPSK, 2

successive bits are combined. Infact, this combination of two bits forms 4

distinct sym’s. When the sym is changed to next sym, then the phase of the

carrier is changed by 45 degrees.

S.No I/p successive bits symbol phase shift in

carrier

I=1 1(1v) 0(-1v) S1 /4

I=2 0(-1v) 0(-1v) S2 3/4

I=3 0(-1v) 1(1v) S3 5/4

I=4 1(1v) 1(1v) S4 7/4

Generation of QPSK:

Here the i/p binary seq. is first converted into a bipolar NRZ type of

signal. This signal is denoted by b (t). It represents binary ‘1’ by ‘+1V’ and

binary ‘0’ by ‘-1V’. The demultiplexer divides b (t) into 2 separate bit streams

of the odd numbered and even numbered bits. Here Be (t) represents even

numbered sequence and Bo (t) represents odd numbered sequence. The

symbol duration of both of these odd numbered sequences is 2Tb. Hence,

each symbol consists of 2 bits.

23

Fig.3.4 Generation of QPSK

It may be observed that the first even bit occurs after the first odd bit.

Hence, even numbered bit sequence Be (t) starts with the delay of one bit

period due to first odd bit. Thus, first symbol of Be (t) is delayed by one bit

period due to first odd bit. Thus, first symbol of Be (t) is delayed by on bit

period ‘Tb’ with respect to first symbol of Bo (t). This delay of Tb is known as

offset. This shows that the change in the levels of Be (t) and Bo (t) can’t

occur at the same time due to offset or staggering. The bit stream Be (t)

modulates carrier cosine carrier and B0(t) modulates sinusoidal carrier.

These modulators are the balanced modulators. The 2 carriers are Ps.cos

(2Fc.t) and Ps.sin (2Fc.t) have been shown in fig. Their carriers are

known as quadrature carriers. Due to the offset, the phase shift in QPSK

signal is /2.

3.9 FFT & IFFT:

In practice, OFDM systems are implemented using a combination of

FFT and IFFT blocks that are mathematically equivalent versions of the DFT

and IDFT, respectively, but more efficient to implement.

An OFDM system treats the source symbols (e.g., the QPSK or QAM

symbols that would be present in a single carrier system) at the Tx as though

they are in the freq-domain. These sym’s are used as the i/p’s to an IFFT

block that brings the sig into the time domain. The IFFT takes in N sym’s at a

24

time where N is the num of sub carriers in the system. Each of these N i/p

sym’s has a symbol period of T secs. Recall that the basis functions for an

IFFT are N orthogonal sinusoids. These sinusoids each have a different freq

and the lowest freq is DC. Each i/p symbol acts like a complex weight for the

corresponding sinusoidal basis fun. Since the i/p sym’s are complex, the

value of the sym determines both the amplitude and phase of the sinusoid

for that sub carrier.

The IFFT o/p is the summation of all N sinusoids. Thus, the IFFT block

provides a simple way to modulate data onto N orthogonal sub carriers. The

block of N o/p samples from the IFFT make up a single OFDM sym. The length

of the OFDM symbol is NT where T is the IFFT i/p symbol period mentioned

above.

Fig.3.5 FFT & IFFT diagram

After some additional processing, the time-domain sig that results from

the IFFT is transmitted across the channel. At the Rx, an FFT block is used

to process the received signal and bring it into the freq domain. Ideally,

the FFT o/p will be the original sym’s that were sent to the IFFT at the Tx.

When plotted in the complex plane, the FFT o/p samples will form a

constellation, such as 16-QAM. However, there is no notion of a

constellation for the time-domain sig. When plotted on the complex plane,

the time-domain sig forms a scatter plot with no regular shape. Thus, any

25

Rx processing that uses the concept of a constellation (such as symbol

slicing) must occur in the frequency- domain.

3.10 GUARD PERIOD:

One of the most important properties of OFDM transmissions is the

robustness against multipath delay spread. This is achieved by having a long

symbol period, which minimizes the ISI. The level of robustness, can in fact is

increased even more by the addition of a guard period b/w transmitted sym’s. The

guard period allows time for multipath sig’s from the pervious symbol to die away

before the information from the current symbol is gathered.

The most effective guard period to use is a cyclic extension of the

symbol. If a mirror in time, of the end of the symbol waveform is put at the

start of the symbol as the guard period, this effectively extends the length of

the symbol, while maintaining the orthogonally of the waveform. Using this

cyclic extended symbol the samples required for performing the FFT (to

decode the sym), can be taken anywhere over the length of the sym. This

provides multipath immunity as well as sym time synchronization tolerance.

As long as the multipath delay echos stay within the guard period

duration, there is strictly no limitation regarding the signal level of the echos:

they may even exceed the signal level of the shorter path! The signal energy

from all paths just adds at the input to the receiver, and since the FFT is

energy conservative, the whole available power feeds the decoder.

If the delay spread is longer than the guard interval then they begins

to cause ISI. However, provided the echoes are sufficiently small they do not

26

cause significant problems. This is true most of the time as multipath echo’s

delayed longer than the guard period will have been reflected of very distant

objects. Other variations of guard periods are possible. One possible

variation is to have half the guard period a cyclic extension of the symbol, as

above, and the other half a zero amplitude signal. This will result in a signal

as shown in Fig.3.6.

Using this method the symbols can be easily identified. This possibly

allows for symbol timing to be recovered from the signal, simply by applying

envelop detection. The disadvantage of using this guard period method is

that the zero period does not give any multipath tolerance, thus the effective

active guard period is halved in length. It is interesting to note that this

guard period method has not been mentioned in any of the research papers

read, and it is still not clear whether symbol timing needs to be recovered

using this method.

Fig.3.6 Section of an OFDM signal showing 5 symbols, using a guard period which is half a cyclic extension of the symbol, and half a zero amplitude signal.

27

CHAPTER-4CHAPTER-4

4.1 PROPAGATION OF CHANNEL CHARACTERISTICS:

In an ideal radio channel, the received signal would consist of only a

single directpath signal, which would be a perfect reconstruction of the

transmitted signal. However in a real channel, the signal is modified during

transmission in the channel.

It is known that the performance of any wireless system’s performance

is affected by the medium of propagation, namely the characteristics of the

28

channel. In telecommunications in general, a channel is a separate path

through which signals can flow. In the ideal situation, a direct line of sight

between the transmitter and receiver is desired. But alas, it is not a perfect

world; hence it is imperative to understand what goes on in the channel so

that the original signal can be reconstructed with the least number of errors.

The received signal consists of a combination of attenuated, reflected,

refracted, and diffracted replicas of the transmitted signal. On top of all this,

the channel adds noise to the signal and can cause a shift in the carrier

frequency if the transmitter, or receiver is moving (Doppler effect).

Understanding of these effects on the signal is important because the

performance of a radio system is dependent on the radio channel

characteristics.

4.2 ATTENUATION:

Attenuation is the “drop in the signal power when transmitting from

one point to another. It can be caused by the transmission path length,

obstructions in the signal path, and multipath effects”. Fig.4.1 shows some

of the radio propagation effects that cause attenuation. Any objects, which

obstruct the line of sight signal from the transmitter to the receiver, can

cause attenuation.

29

Fig.4.1.Some channel characteristics

Shadowing of the signal can occur whenever there is an obstruction

between the transmitter and receiver. It is generally caused by buildings and

hills, and is the most important environmental attenuation factor. Shadowing

is most severe in heavily built up areas, due to the shadowing from buildings.

However, hills can cause a large problem due to the large shadow they

produce.

Radio signals diffract off the boundaries of obstructions, thus

preventing total shadowing of the signals behind hills and buildings.

However, the amount of diffraction is dependent on the radio frequency

used, with low frequencies diffracting more then high frequency signals.

Thus high frequency signals, especially, Ultra High Frequencies (UHF), and

microwave signals require line of sight for adequate signal strength. To over

come the problem of shadowing, transmitters are usually elevated as high as

possible to minimize the number of obstructions. Typical amounts of

variation in attenuation due to shadowing are shown in Table 3.1.

30

Table.4.1 Typical attenuation in a radio channel.

Shadowed areas tend to be large, resulting in the rate of change of the

signal power being slow. For this reason, it is termed slow-fading, or

lognormal shadowing.

4.3 MULTIPATH EFFECTS:4.3 MULTIPATH EFFECTS:

(a)(a) Rayleigh fading:Rayleigh fading:

In a radio link, the RF signal from the transmitter may be reflected

from objects such as hills, buildings, or vehicles. This gives rise to multiple

transmission paths at the receiver. Fig.4.2 show some of the possible ways

in which multipath signals can occur.

Fig.4.2 Multipath Signals

31

The relative phase of multiple reflected sig’s can cause constructive or

destructive interference at the Rx. This is experienced over very short

distances (typically at half wavelength distances), thus is given the term fast

fading. These variations can vary from 10-30dB over a short distance.

Fig.4.3 Typical Rayleigh fading while the mobile unit is moving.

The Rayleigh distribution is commonly used to describe the statistical

time varying nature of the received signal power. It describes the probability

of the signal level. Being received due to fading. Table.4.2 shows the

probability of the signal level for the Rayleigh distribution.

Table 4.2 Cumulative distributions for Rayleigh distribution

32

(b) Frequency Selective Fading:

In any radio transmission, the channel spectral response is not flat. It

has dips or fades in the response due to reflections causing cancellation of

certain frequencies at the receiver. Reflections off near-by objects (e.g.

ground, buildings, trees, etc) can lead to multipath signals of similar signal

power as the direct signal. This can result in deep nulls in the received signal

power due to destructive interference. For narrow bandwidth transmissions if

the null in the frequency response occurs at the transmission frequency then

the entire signal can be lost. This can be partly overcome in two ways.

By transmitting a wide bandwidth signal or spread spectrum as CDMA,

any dips in the spectrum only result in a small loss of signal power, rather

than a complete loss. Another method is to split the transmission up into

many small bandwidth carriers, as is done in a COFDM/OFDM transmission.

The original signal is spread over a wide bandwidth thus; any nulls in the

spectrum are unlikely to occur at all of the carrier frequencies. This will result

in only some of the carriers being lost, rather then the entire signal. The

information in the lost carriers can be recovered provided enough forward

error corrections are sent.

4.4 DELAY SPREAD:

33

The received radio signal from a transmitter consists of typically a

direct signal, plus reflections of object such as buildings, mountings, and

other structures. The reflected signals arrive at a later time than the direct

signal because of the extra path length, giving rise to a slightly different

arrival time of the transmitted pulse, thus spreading the received energy.

Delay spread is the “time spread between the arrival of the first and last

multipath signal seen by the receiver”.

In a digital system, the delay spread can lead to inter-symbol

interference. This is due to the delayed multipath signal overlapping with the

following symbols. This can cause significant errors in high bit rate systems,

especially when using time division multiplexing (TDMA). Fig.4.4 shows the

effect of inter-symbol interference due to delay spread on the received

signal. As the transmitted bit rate is increased the amount of inter-symbol

interference also increases. The effect starts to become very significant

when the delay spread is greater then ~50% of the bit time.

34

Fig.4.4 Multi delay spread

shows the typical delay spread that can occur in various environments. The

maximum delay spread in an outdoor environment is approximately 20usec,

thus significant intersymbol interference can occur at bit rates as low as

25kbps.

Inter-symbol interference can be minimized in several ways. One

method is to reduce the symbol rate by reducing the data rate for each

channel (i.e. split the bandwidth into more channels using frequency division

multiplexing). Another is to use a coding scheme which is tolerant to inter-

symbol interference such as CDMA.

4.5 DOPPLER SHIFT:

When a wave source and a receiver are moving relative to one another

the frequency of the received signal will not be the same as the source.

When they are moving toward each other the frequency of the received

signal is higher then the source, and when they are approaching each other

the frequency decreases. This is called the

Doppler Effect. An example of this is the change of pitch in a car’s horn as

it approaches then passes by. This effect becomes important when

developing mobile radio systems. The amount the frequency changes due to

the Doppler effect depends on the relative motion between the source and

35

receiver and on the speed of propagation of the wave. The Doppler shift in

frequency can be written:

Where f is the change in frequency of the source seen at the receiver, fo is

the frequency of the source, v is the speed difference between the source

and transmitter, and c is the speed of light.

For example: Let fo = 1GHz, and v = 60km/hr (16.7m/s) then the Doppler

shift will

be:

This shift of 55Hz in the carrier will generally not effect the

transmission. However,

Doppler shift can cause significant problems if the transmission technique is

sensitive to carrier frequency offsets (for example COFDM) or the relative

speed is higher (for example in low earth orbiting satellites).

4.6 INTER SYMBOL INTERFERENCE:

As communication systems evolve, the need for high symbol rates

becomes more apparent. However, current multiple access with high symbol

rates encounter several multi path problems, which leads to ISI. An echo is a

copy of the original signal delayed in time. ISI takes place when echoes on

36

different-length propagation paths result in overlapping received symbols.

Problems can occur when one OFDM symbol overlaps with the next one.

There is no correlation between two consecutive OFDM symbols and

therefore interference from one symbol with the other will result in a

disturbed signal.

In addition, the symbol rate of communications systems is practically

limited by the channel’s bandwidth. For the higher symbol rates, the effects

of ISI must be dealt with seriously. Several channel equalization techniques

can be used to suppress the ISIs caused by the channel. However, to do this,

the CIR – channel impulse response, must be estimated.

Recently, OFDM has been used to transmit data over a multi-path

channel. Instead of trying to cancel the effects of the channel’s ISIs, a set of

sub-carriers can be used to transmit information symbols in parallel sub-

channels over the channel, where the system’s output will be the sum of all

the parallel channel’s throughputs.

This is the basis of how OFDM works. By transmitting in parallel over a

set of sub-carriers, the data rate per sub-channel is only a fraction of the

data rate of a conventional single carrier system having the same output.

Hence, a system can be designed to support high data rates while deferring

the need for channel equalizations.

In addition, once the incoming signal is split into the respective

transmission sub-carriers, a guard interval is added between each symbol.

Each symbol consists of useful symbol duration, Ts and a guard interval, t,

37

in which, part of the time, a signal of Ts is cyclically repeated. This is shown

in Fig.4.5.

Fig.4.5 Combating ISI using a guard interval

As long as the multi path propagation delays do not exceed the

duration of the interval, no inter-symbol interference occurs and no channel

equalization is required.

4.7 CHANNELS:

The transmission signal models of the electromagnetic wave which

travels form transmitter to receiver. Along the way the wave encounters a

wide range of different environments. Channel models represent the attempt

to model these different environments. Their aim is to introduce well defined

disturbances to the transmission signal. In this lecture we discuss channel

models which are typical for DAB transmission. We consider the effects of

noise, movement, and signal reflection. The general strategy is to have a

38

pictorial representation of the channel environment before we introduce the

mathematical model.

Overview Diagram

The following figure shows again the block diagram of communication

system. Such a system consists of ‘Sender’, ‘Channel’ and ‘Receiver’. In this

lecture we focus on the channel aspect of the communication system. In the

block diagram, s(t) is the transmission signal and ˆs(t) is the received

transmission signal.

(a) Frequency offset channel

The frequency offset channel introduces a static frequency offset. One

possible cause for such a frequency offset is a slow drifting time base,

normally a crystal oscillator, in either transmitter or receiver. The frequency

offset channel tests the frequency correction circuit in the receiver. The

following figure shows the block diagram of the Frequency shift channel.

The mathematical model follows as:

.

39

(b) AWGN channel

For the Additional White Gaussian Noise (AWGN) channel the received

signal is equal to the transmitted signal with some portion of white Gaussian

white noise added. This channel is particularly important for discrete models

operating on a restricted number space, because this allows one to optimise

the circuits in terms of their noise performance. The block diagram of the

AWGN channel is given in the next figure.

s(t) = s(t) + n(t)

where n(t) is a sample function of a Gaussian random process. This

represents white Gaussian noise.

(c) Multi path channel

The multipath channel is the last of the static channels. It reflects the

fact that electromagnetic waves can travel over various paths from the

transmission antenna to the receiver antenna. The receiver antenna sums up

all the different signals. Therefore, the mathematical model of the multipath

environment creates the received transmission signal by summing up scaled

and delayed versions of the original transmission signal. This superposition

of signals causes ISI.

40

The following figure shows a multipath environment.

The block diagram, shown in the next figure, details a DSP model for the multipath

environment.

The mathematical model follows as:

(d) Fading channels

Fading channels represent a mathematical model for wireless data

exchange in a physical environment which changes over time. These

changes arise for two reasons:

41

1. The environment is changing even though the transmitter and

receiver are fixed; examples are changes in the ionosphere, movement

of foliage and movement of reflectors and scatterers.

2. Transmitter and receiver are mobile even though the environment

might be static.

3. The next figure shows a multipath fading environment. The fading is

modeled by the fact that the environment is changing.

The block diagram, shown in the next figure, details a DSP model for the multipath

environment

Mathematically the DSP model can be formulated as follows:

42

DSP model and mathematical description are close to the

underlying physical phenomena. This makes them unsuitable for practical

channel models. To establish practical channel models we employ statistical

methods to abstract and generalize the fading channel models. In the

following two subsections we discuss Rayleigh and Rician fading channels.

Both represent statistical channel modes, the difference between them is

that the Rayleigh model does not assume a direct or prominent path and the

Ricien model assumes a direct path. The last channel model extends the

ideas of Rayleigh and Rician fading channels with mobility aspects. The

resulting mobile fading channels model the degrading effects in the

frequency domain of wireless multipath channels.

(e) Rayleigh fading:

Rayleigh fading is caused by multipath reception. The mobile antenna

receives a large number, say N, reflected and scattered waves. Because of

wave cancellation effects, the instantaneous received power seen by a

moving antenna becomes a random variable, dependent on the location of

the antenna.

To simplify the derivation of the fading models an un-

modulated carrier of the form s(t) = Acos(2pifct) as transmission signal is

used. Based on the block diagram the complex envelope of the received

signal is:

43

where ai (t) is the gain factor and Ti (t) is the delay for a specific path i at a

specific time t.

where rRa (t) is a sample function of a Rayleigh distributed random process:

and the is uniformly distributed in the interval [0, 2pi).

The general form of this channel model is:

again, and are amplitude and phase from a particular

measurement of a rayleigh distributed random process. This channel is

called rayleigh fading channel.

(f) Rician fading

The model behind Rician fading is similar to that for Rayleigh fading,

except that in Rician fading a strong dominant component is present. This

dominant component can for instance be the line-of-sight wave. Refined

Rician models also consider

1. that the dominant wave can be a phasor sum of two or more dominant

signals, e.g. the line-of- sight, plus a ground reflection. This combined

signal is then mostly treated as a deterministic (fully predictable)

process

44

2. that the dominant wave can also be subject to shadow attenuation.

This is a popular

Assumption in the modeling of satellite channels. Besides the dominant

component, the mobile antenna receives a large number of reflected and

Scattered waves.

A Rician fading channel indicates that there is a prominent or direct

path over which the electromagnetic wave can travel. Compared to the

Rayleigh channel model, Equation 1, the Rician fading channel model has an

additional Acos(2pifct) component to reflect the prominent path:

Above Equation can be written as:

Where rRi (t) is a sample function of a random process with a Rician

distributed probability density function (pdf):

Where I0 is the zero order modified Bessel functions of the first kind given

by:

and the distribution of is:

45

Where is the error function defined as:

The ratio , referred as the K-factor, relates the power in un faded

and faded components. Values of K >> 1 indicate less severe fading,

whereas K << 1 indicates severe fading.

The general form of the Rician channel model is:

Where rRi (t) and are amplitude and phase of a particular

measurement of a rician distributed random process.

CHAPTER 5

5.1 PAPR INTRODUCTION:

However, OFDM is not without drawbacks. One critical problem is its

high peak-to-average power ratio (PAPR). High PAPR increases the

complexity of analog-to-digital (A/D) and digital-to-analog (D/A) converters,

and lowers the efficiency of power amplifiers. Over the past decade various

PAPR reduction techniques have been proposed, such as block coding,

selective mapping (SLM) and tone reservation, just to name a few . Among

46

all these techniques the simplest solution is to clip the transmitted signal

when its amplitude exceeds a desired threshold. Clipping is a highly

nonlinear process, however. It produces significant out-of-band interference

(OBI). A good remedy for the OBI is the so-called companding. The technique

‘soft’ compresses, rather than ‘hard’ clips, the signal peak and causes far

less OBI. The method was first proposed in, which employed the classical 𝜇-

law transform and showed to be rather effective. Since then many different

companding transforms with better performances have been

Published. This paper proposes and evaluates a new companding algorithm.

The algorithm uses the special airy function and is able to offer an improved

bit error rate (BER) and minimized OBI while reducing PAPR effectively. The

paper is organized as follows. In the next section the PAPR problem in OFDM

is briefly reviewed.

5.2 PAPR IN OFDM

• OFDM is a powerful modulation technique being used in many new and

emerging broadband communication systems.

– Advantages:

• Robustness against frequency selective fading and time

dispersion.

• Transmission rates close to capacity can be achieved.

• Low computational complexity implementation (FFT).

– Drawbacks:

• Sensitivity to frequency offset.

• Sensitivity to nonlinear amplification.

• Compensation techniques for nonlinear effects

47

– Linearization (digital predistortion).

– Peak-to-average power ratio (PAPR) reduction.

– Post-processing.

• PAPR-reduction techniques:

– Varying PAPR-reduction capabilities, power, bandwidth and

complexity requirements.

– The performance of a system employing these techniques has

not been fully analyzed

– PAPR is a very well known measure of the envelope fluctuations

of a MC signal

– Used as figure of merit.

– The problem of reducing the envelope fluctuations has turned to

reducing PAPR.

– In this paper we ...

– present a quantitative study of PAPR and NL distortion

– simulate an OFDM-system employing some of these techniques

Motivation: evaluate the performance improvement capabilities of PAPR-

reducing methods.

5.3 ORTHOGONAL FREQUENCY DIVISION MULTIPLEXING

48

• An OFDM signal can be expressed as

If the OFDM signal is sampled at , the complex samples can be

described as

Peak-to-average power ratio

• Let be the m-th OFDM symbol, then its PAPR is defined as

•

49

The CCDF of the PAPR of a non-oversampled OFDM signal is

• CCDF of PAPR increases with the number of subcarriers in the OFDM

system.

– It is widely believed that the more subcarriers are used in a

OFDM system, the worse the distortion caused by the

nonlinearity will be.

– In-band and out-of-band distortion

• If N is large enough, the OFDM signal can be approximated as a

complex Gaussian distributed random variable. Thus its envelope is

Rayleigh distributed

where the variance of the real and imaginary parts of the signal is

• Buss gang theorem

50

NL

1 2

1

2

x x

x tR

x t

1 2 1 2 1 2

1

2

wherex y x y x x

x tR R R

y t

1 2x t x t xy xxR R

An interesting result is that the output of a NL with Gaussian input (OFDM)

can be written as:

Considerations on PAPR reduction

• In order to improve the system performance, PAPR should predict the

amount of distortion introduced by the nonlinearity

– PAPR increases with the number of subcarriers in the OFDM

signal.

– The distortion term and the uniform attenuation and rotation of

the constellation only depend on the back-off.

The effect of a nonlinearity to an OFDM signal is not clearly related to

its PAPR

• The effective energy per bit at the input of the nonlinearity is

• where Eo is the average energy of the signal at the input of the

nonlinearity, K is the

• number of bits per symbol and ηp is the power efficiency.

• There will only be a a BER performance improvement when the

effect of reducing the in-band distortion becomes noticeable and

more important than the loss of power efficiency.

• This is not taken into account in the majority of the PAPR reducing

methods.

51

Let (0),(1), ⋅ ⋅ ⋅,𝑋(𝑁 −1) represent the data sequence to be transmitted in an

OFDM symbol with 𝑁 subcarriers. The baseband representation of the OFDM

symbol is given by:

where 𝑇 is the duration of the OFDM symbol. According to the central limit

theorem, when 𝑁 is large, both the real and imaginary parts of 𝑥(𝑡) become

Gaussian distributed, each with zero mean and a variance of E[∣𝑥(𝑡)∣2]/2, and

the amplitude of the OFDM symbol follows a Rayleigh distribution.

Consequently it is possible that the maximum amplitude of OFDM signal may

well exceed its average amplitude. Practical hardware (e.g. A/D and D/A

converters, power amplifiers) has finite dynamic range; therefore the peak

amplitude of OFDM signal must be limited. PAPR is mathematically defined

as:

It is easy to see from above that PAPR reduction may be achieved by

decreasing the numerator max[∣𝑥(𝑡)∣2], increasing the denominator (1/T) ⋅ ∫ 𝑇 0 ∣𝑥(𝑡)∣2 𝑑𝑡, or both.

The effectiveness of a PAPR reduction technique is measured by the

complementary cumulative distribution function

(CCDF), which is the probability that PAPR exceeds some threshold, i.e.:

CCDF = Probability (PAPR > 𝑝0), where 𝑝0 is the threshold.

52

CHAPTER 6

6.1 SIMULATION RESULTS:

NEW COMPANDING ALGORITHM

OBI is the spectral leakage into alien channels. Quantification of the

OBI caused by companding requires the knowledge of the power spectral

density (PSD) of the companded signal. Unfortunately analytical expression

of the PSD is in general mathematically intractable, because of the nonlinear

companding transform involved. Here we take an alternative approach to

estimate the OBI. Let (𝑥) be a nonlinear companding function, and (𝑡) =

sin(𝜔𝑡) be the input to the compander. The companded signal (𝑡) is: (𝑡) =

[(𝑡)] = 𝑓 [sin(𝜔𝑡)] . Since (𝑡) is a periodic function with the same period as

(𝑡), (𝑡) can then be expanded into the following Fourier series:

53

where the coefficients 𝑐(𝑘) is calculated as:

Notice that the input x in this case is a pure sinusoidal signal, any (𝑘) ∕= 0 for ∣𝑘∣ > 1 is the OBI produced by the nonlinear companding process. Therefore,

to minimize the OBI, (𝑘) must approach to zero fast enough as 𝑘 increases. It

has been shown that (𝑘) ⋅ 𝑘−(𝑚+1) tends to zero if 𝑦(𝑡) and its derivative up

to the 𝑚-th order are continuous [8], or in other words, 𝑐(𝑘) converges at the

rate of 𝑘−(𝑚+1). Given an arbitrary number n, the 𝑛-th order derivative of 𝑦(𝑡), 𝑑𝑛𝑦/𝑑𝑡𝑛, is a function of 𝑑𝑖𝑓(𝑥)/𝑑𝑥𝑖, (𝑖 = 1, 2, ⋅ ⋅ ⋅ , 𝑛), as well as

sin(𝜔𝑡) and cos(𝜔𝑡), i.e.:

sin(𝜔𝑡) and and cos(𝜔𝑡) are continuous functions, 𝑑𝑛𝑦/𝑑𝑡𝑛 is continuous if

and only if 𝑑𝑖𝑓(𝑥)/𝑑𝑥𝑖 (𝑖 = 1, 2, ⋅ ⋅ ⋅ , 𝑛) are continuous. Based on this

observation we can conclude:

Companding introduces minimum amount of OBI if the companding function

(𝑥) is infinitely differentiable. The functions that meet the above condition

are the smooth functions. We now propose a new companding algorithm

using a smooth function, namely the airy special function. The companding

function is as follows:

where airy(⋅) is the airy function of the first kind. 𝛼 is the parameter that

controls the degree of companding (and ultimately PAPR). 𝛽 is the factor

54

adjusting the average output power of the compander to the same level as

the average input power:

where 𝐸[⋅] denotes the expectation. The decompanding function is the

inverse of (𝑥):

where the superscript-1 represents the inverse operation. Notice that the

input to the decompander is a quantized signal with finite set of values. We

can therefore numerically pre-compute 𝑓−1(𝑥) and use table look-up to

perform the decompanding in practice. Next we examine the BER

performance of the algorithm. Let (𝑡) denote the output signal of the

compander, (𝑡) the white Gaussian noise. The received signal can be

expressed as:

The decompanded signal ˜(𝑡) simply is:

Notice that the signal-to-noise ratio (SNR) in a typical additive white

Gaussian noise (AWGN)

channel is much greater.

55

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

0

0.2Companding for proposed alogorithm

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

0

0.2DE Companding for proposed alogorithm

alp=5

alp=7.5

alp=12.5

1 2 3 4 5 6 7 8-20

0

20Companding for Exponential copanding

d=5

d=7.5d=12.5

Fig.6.1 Companding and decompanding profile

The simulated PSD of the companded signals is illustrated in Fig.6.2.

The proposed algorithm produces OBI almost 3dB lower than the exponential

algorithm, 10dB lower than the 𝜇-law. The result is in line with our

expectation. The 𝜇-law function has a singularity in its second order

derivative at x = 0 and therefore is expected to have the strongest OBI.

56

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-400

-350

-300

-250

-200

-150

-100

-50

0

Normalized Frequency ( rad/sample)

Mag

nitu

de (

dB)

Magnitude Response (dB)

OriginalProposedExponentialmu law

Fig.6.2 Power spectral density of original and companded signals

Using the first order Taylor series expansion,

From the given Equation shows that if (𝑡) falls into the range of the

decompanding function 𝑓−1(𝑢) where 𝑑𝑓−1(𝑢)/𝑑𝑢∣ 𝑢=𝑦(𝑡) < 1, the noise 𝑤(𝑡) is suppressed, and if 𝑦(𝑡) is out of the range, 𝑑𝑓−1(𝑢)/𝑑𝑢∣ 𝑢=𝑦(𝑡) > 1

and the noise is enhanced. Therefore, if the parameter 𝛼 in (8) is properly

chosen such that more (𝑡) is within the noise-suppression range of 𝑓−1(𝑢), it

is possible to achieve better overall BER performance. It is worth to mention

though that BER and PAPR affect each other adversely and therefore there is

a tradeoff to make.

The OFDM system used in the simulation consists of 64 QPSK-

modulated data points. The size of the FFT/IFFT is 256, meaning a 4.

oversampling. Given the compander input power of 3dBm, the parameter 𝛼 57

in the companding function is chosen to be 30. Consequently about 19.6

percent of (𝑡) is within the noise-suppression range of the decompanding

function. Two other popular companding algorithms, namely the 𝜇-law

companding [3] and the exponential companding [5], are also included in the

simulation for the purpose of performance comparison.

Fig.6.3.depicts the CCDF of the three companding schemes. The new

algorithm is roughly 1.5dB inferior to the exponential, but surpasses the 𝜇-

law by 2dB.

0 2 4 6 8 10 12 1410

-2

10-1

100

Exponential

Proposed

No companding

Fig.6.3.Complementary cumulative distribution function of original and

companded signals (compander input power = 3dBm, 𝛼 = 30).

The BER vs. SNR is plotted in Fig.6.4. Our algorithm outperforms the

other two. To reach a BER of 10−3, for example, the required SNR are 8.9dB,

10.4dB and 11.7dB respectively for the proposed, the exponential and the 𝜇-

law companding schemes, implying a 1.5dB and 2.8dB improvement with the

new algorithm. The amount of improvement increases as SNR becomes

58

higher. One more observation from the simulation is: unlike the exponential

companding whose performance is found almost unchanged under different

degrees of companding, the new algorithm is flexible in adjusting its

specifications simply by changing the value of 𝛼 in the companding function.

1 2 3 4 5 6 7 8 9 10 1110

-5

10-4

10-3

10-2

10-1

100

Performance analysis

-----EbNo

----

BE

R

No companding

Proposed

Exponential

Fig.6.4.Bit error rate vs. SNR for original and companded signals in AWGNchannel (compander input power = 3dBm, 𝛼 = 30)

59

CONCLUSION

In this project,a new companding algorithm was proposed. Both theoretical

analysis and computer simulation show that the algorithm offers improved

performance compared to exponential companding and decompanding in

terms of BER and OBI while reducing PAPR effectively as well as shown the

simulated results on PSD of original and companded signals.

60

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