CPM for RLS system

Institutionen för systemteknik

Department of Electrical Engineering

Examensarbete

RLS for CPM system

Examensarbete utfört i Datatransmission

av

Frans Bergquist

LiTH-ISY-EX--07/3871--SE 2007 TEKNISKA HÖGSKOLAN LINKÖPINGS UNIVERSITET

Department of Electrical Engineering Linköping University

S-581 83 Linköping, Sweden

Linköpings tekniska högskola Institutionen för systemteknik 581 83 Linköping

RLS for CPM system

Examensarbete utfört i datatransmission vid Linköpings tekniska högskola

av Frans Bergquist

LiTH-ISY-EX--07/3871--SE

Handledare: Johan Henriksson Examinator: Mikael Johansson

Presentationsdatum

2007-03-19

Publiceringsdatum (elektronisk version)

Institution och avdelning Institutionen för systemteknik Department of Electrical Engineering

URL för elektronisk version

http://www.ep.liu.se Publikationens titel RLS for CPM system Författare Frans Bergquist Sammanfattning

I detta examensarbete har ett fasmodulerat radiosystem simulerats, fokusering ligger på kanalutjämnare som är av typen recursive least square (RLS). RLS utjämnaren har testats med två olika gsm kanalmodeler, dels typical urban som simulerar radioförbindelser i stadsmiljö den andra modellen är rural area där sändare och mottagare kan se varandra. Tre olika resultat presenteras; med felrättande koder, utan felrättande koder och mängden icke korrekta datapaket.

Slutsatser dras om radiosystemets bandbredd när de olika kanalmodellerna används vid olika brusmängd. Även utjämnarens

förmåga att hantera inter-symbol interference och fading utvärderas också

Språk

Svenska

X Annat (ange nedan) Engelska Antal sidor 96 Typ av publikation Licentiatavhandling X Examensarbete C-uppsats D-uppsats Rapport

Annat (ange nedan)

ISBN (licentiatavhandling) ISRN

Serietitel (licentiatavhandling)

Abstract

The main goal of this thesis is to create a continuous phase modulated radio system with a recursive least square equalizer. The two tested channel models are typical urban and rural area. The result of the performance of this radio system is displayed in Matlab plots as the bit error rate. Three error rates are displayed; with error correction, without error correction and the rate of received incorrect message bursts.

Conclusions are also drawn of the performance of the radio system in kbit/sec of bandwidth when the dierent channel models are used. The performance is also divided into how the equalizer handles inter symbol interference or a fading channel without inter symbol interference.

Acknowledgements

I would like to thank Johan Henriksson my supervisor at Saab. He has spent countless hours teaching me the intricate workings of radio systems.

At the university Mikael Olofsson has pushed me to create the best master thesis I could and for this I am eternally grateful.

Micael Belin has been an exceptional opponent and he has tried hard to nd holes any and all inconsistencies in my work.

I am also deeply indebted to the best girlfriend in the world. She has used so much of her spare time helping and supporting me. It would not be possible for me to nish this work without her.

Large thanks goes to Åke Bergquist who always has wanted to share as much of his knowledge as possible.

Contents

I Background

1

1 Introduction 2 1.1 Overview . . . 2 1.2 Reading Instructions . . . 2 2 Problem Description 4 2.1 Task . . . 4 2.2 Method . . . 5 2.3 Initial Limitations . . . 6

II Radio System Theory

7

3 Radio System Introduction 8 3.1 CPM . . . 8

3.2 Data Transmission . . . 11

3.3 Channel . . . 14

3.4 Error Correcting Codes . . . 18

4 Equalizer Theory 23

4.1 Pilot Sequence . . . 23

4.2 Equalizer Example . . . 24

4.3 Zero Forcing Equalizer . . . 25

4.4 Forgetting Factor . . . 26

4.5 Minimum Mean Square Error Equalizer . . . 26

4.6 Theory for the RLS . . . 27

4.7 RLS . . . 27

5 Radio System Verication 30 5.1 Channel verication . . . 30 5.2 Performance . . . 30 5.3 Equalizer . . . 31 5.4 Modulation . . . 32

III Implementation

33

6 Implementation Overview 34 6.1 Work Flow . . . 34

6.2 Overview Design Goals . . . 35

6.3 Architecture . . . 36

7 Basic Radio System Construction 37 7.1 Requirement Specication . . . 37

7.2 Design Decisions . . . 37

8 Channel Equalizer Implementation 42 8.1 Requirement Specication . . . 42 8.2 Design Decision . . . 43 8.3 Problems . . . 44 8.4 Architecture . . . 45 8.5 Verication . . . 45 9 Fading Channel 49 9.1 Requirement Specication . . . 49 9.2 Design Decisions . . . 49 9.3 Problems . . . 50 9.4 Architecture . . . 50 9.5 Verication . . . 52

IV Conclusion

56

10 Discussion 57 10.1 Simplications in the Modeling . . . 57

10.2 Limitations in the Model . . . 58

11 Final Tests 61 11.1 Error Correction Performance . . . 61

11.2 AWGN . . . 61

11.3 Fading Channel without ISI . . . 65

11.4 Channel with ISI . . . 65

11.5 RA . . . 70

12 Conclusion 79 12.1 Fading Channel . . . 79 12.2 ISI . . . 79 12.3 RA . . . 80 12.4 TU . . . 80 12.5 Performance . . . 81 Bibliography 83 A Abbreviations 84

Part I

Chapter 1

Introduction

This chapter gives a short introduction to the thesis and also presents instructions on how to study the report if time is limited.

1.1 Overview

The main purpose of this thesis is to create a model of a continuous phase modulated radio system with a recursive least square equalizer. Focus will lie on the equalizer, which will require emphasis on the channel that the equalizer counteracts. The modeling program used is Matlab and its user friendly extension Simulink.

1.2 Reading Instructions

The thesis report is divided into four parts. The rst part covers the background of the project and provides information on its practical origins. Basic theory is reviewed in part II, where great importance is dedicated to describe the channel and equalizer parts. Part III is focused on the implementation of the system in Simulink. The last part provides the tests and Matlab plots essential for the evaluation of the system and potential weak points of the simulation are discussed in this part.

All readers are of course encouraged to study the entire report thoroughly to get a full understanding of the project and its conclusions. To considerably facilitate the

under-standing of the report some knowledge of signal theory is a useful prerequisite but not required.

However; if time is limited and the reader has already acquired knowledge of basic radio systems, chapter 3 will not necessitate excessive scrutiny. Chapter 4 does not present any new information to readers already familiar with advanced radio systems and should only require a quick glance from the experienced radio technician.

Students aspiring to write a thesis covering a similar subject can use this chapter to nd ideas and hints for useful literature. To other students; the information concerning the project and the Simulink implementation of the radio system located in part III might prove helpful.

For readers who are primarily interested in understanding the conclusions, chapter 2 and part IV are highly recommended.

Chapter 2

Problem Description

This chapter provides information on how the thesis project started and gives a short background to its origins. An introduction to the problem and the solution method suggested by the supervisor can also be found here.

2.1 Task

The general task was to examine the performance of a RLS equalizer in a CPM modulated radio system.

The radio system under scrutiny has the following specications. 1. Modulation

(a) Continuous phase modulation (b) 2 bits per symbol

(c) Rectangular pulse shape (d) 1/4 modulation index

(e) Max 400 MHz carrier wave frequency 2. Equalizer

(b) Fix the weight after the pilot sequence (c) Congurable variables

i. The ability to set the number of taps ii. The ability to set start correlation matrix iii. The ability to set start weights

iv. The ability to set forgetting factor 3. Error correction

(a) 1/3-rate convolution encoding (b) Viterbi decoding

(c) 4 message block interleaving 4. Frequency hopping

(a) Message burst i. 128 bits pilot ii. 2308 bits of data

iii. Transmitted symbol rate of 106 _{symbols per second} (b) Reset the equalizer after each burst

5. Channel

(a) Speed 3-200 km/h normally 70 km/h (b) Typical Urban

i. No Direct Wave

ii. 6-taps According to GSM (c) Rural Area

i. Direct wave

ii. 5-taps According to GSM

2.2 Method

Matlab is used as a foundation to model the equalizer. A Matlab program called Simulink has been purchased and utilized in the project, mainly because of Simulink's ability to create simulations with a graphical user interface.

Simulink

Simulink is a block-based simulation tool, traditionally used for system based design, con-trol and signal modeling. In these application areas extra toolboxes exists. The following toolboxes have been available for this project:

• Communications Blockset • Real-Time Workshop • Signal Processing Blockset • Simulink Extras

• Stateow

• Video and Image Processing Blockset

The Communication Blockset is the most frequently used toolbox; all modulation and channel models have originated from this block. The Communication Blockset also pro-vides Simulink with extra features, one of which is Frame based signal s.

2.3 Initial Limitations

A general limitation is that only blocks from Simulink will be used, except when the equalizer is considered - this system is based on another real radio system. The modeled radio system has some limitations due to the fact that some blocks employed by the real system do not exists in Simulink. This primarily concerns the error correcting codes which combined with a dierent demodulation gives the real system superior performance.

Part II

Chapter 3

Radio System Introduction

All blocks in gure 3.1 except the equalizer will be covered by this chapter. The trans-mitted data between the blocks is loosely dened in the gure and printed in grey. The only part of this diagram outside of the radio system's control is the channel. Al-most everything in the radio system is constructed to compensate for uncertainties in the channel.

Another feature of this radio model is the absence of a carrier wave. The carrier wave has been substituted by a complex value, which has an angle and an absolute value representing phase and amplitude. In a scatter plot the complex value is displayed with the real value as the x-axis and the imaginary as the y-axis.

3.1 CPM

CPM is an abbreviation for Continuous Phase Modulation which is a modulation tech-nique. This modulation technique uses the phase of the carrier wave to send information, whereas the amplitude of the carrier wave is xed.[7]

All information about a symbol is in the phase change. For every symbol sent the phase is changed as noted in Table 3.1.

Figure 3.1: Basic skeetch of a general Radio System.

Symbol Phase Change

00 +π 4 01 −π 4 10 +3π 4 11 −3π 4

−1 −0.5 0 0.5 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Quadrature In−Phase Scatter plot

Figure 3.2: Scatter Plot of the ideal signal when two samples per symbol are used. The plot shows all transmitted phases.

Errors will be introduced when the phase shift at the receiver is dierent from the trans-mitted phase shift. Therefore, the symbols are Gray coded; which means that only one bit changes between two adjacent symbols in the scatter plot. The Gray code will only intro-duce a single bit error when the dierence in phase change is small instead of introducing two errors as it otherwise could.

In Figure 3.2 there is a scatter plot of all the possible locations of the phase, although it is important to remember that for every symbol only the four locations are possible.

Phase change example

Let us assume that we would send 00 11 01. The sender will start at any value and the phase will then be change by the sender. To send the rst value the phase will change to add +π

4, this dierence has to be detected at the receiver to detect the transmitted bits. The transmitter will then move the phase −3π

4 for the next symbol and then − π 4. All of these changes will then have to be detected in the receiver for the correct bits to be transmitted.

Figure 3.3: This is how the phase will change when 00 11 01 is transmitted. The start value could be anywhere because it is only the change that is considered when data is sent.

Rectangular phase change

Phase shifts in CPM cannot be instant by denition and there are several ways to perform this shift. In Figure 3.4 the phase change is created by using a rectangular pulse shaping and in Figure 3.5 a raised cosine pulse is used. These two gures also illustrate the fact that from every point in the graph, the phase change could take four dierent ways depending on transmitted symbol. This radio system uses the rectangular pulse shaping. The main reason to have a continouse phase shift is to keep the bandwidth low. Two dierent plots of how the phase could be changed as illustrated in gure 3.4 and 3.5 [4]

3.2 Data Transmission

In this radio system all data is transmitted in bursts. Each burst consists of 2308 bits which have been convolution encoded and have had a pilot sequence inserted before the coded data. In gure 3.6 a sketch is displayed of how the messages are transmitted. The convolution encoding increases the size of the data with a factor of three in this particular system. The bursts are sent at dierent frequencies. Since the equalizer's task is to counteract the channel, the settings in the equalizer will be obsolete when the frequencies

0 0.5 1 1.5 2 2.5 3 −8 −6 −4 −2 0 2 4 6 8

Plot of raised cosine phase change

time

phase

0 0.5 1 1.5 2 2.5 3 −8 −6 −4 −2 0 2 4 6 8

Plot of raised cosine phase change

time

phase

Figure 3.5: Phase change when a raised cosine pulse shaping is used in a CPM radio system

change. The channel will start at dierent points in time at dierent frequencies. The properties of the channel will remain the same.

3.3 Channel

The channel is the ether between the sender and receiver. The channel introduces an uncertainty to the signal and it is in this sequence that bit errors occur in the radio system. This section explains how the channel changes the signal.

One addition in the channel is noise, originating from thermal noise (kT0). All resistances and semiconductors generate thermal noise which is modeled with an AWGN channel. AWGN is an abbreviation for Additive White Gaussian Noise. As the name implies the noise is white and gaussian which generates a noise with a constant power spectrum.[3] The Inter Symbol Interference (ISI) is interference from previous symbols transmitted by this radio system. The system is actually interfering with itself due to the short intervals between the transmitted symbols. Reections are reaching the receiver from one or more previous symbols at the same time as the current symbol is received.[7]

Fading Channels

A fading channel simulates reections and movements in the real world; this thesis will focus on two dierent scenarios thus two channel models. As can be seen in gure 3.7 the two dierent receivers are receiving several radio waves propagated using slightly dierent rays. The components are then combined when they reach the receiver, sometimes they have the same phase and the sum will have an increase in amplitude, other times the components could have an opposite phase and the resulting amplitude could be small. Sender and Receiver 1 has a line of sight which Sender and Receiver 2 lacks. This fact generates a dierent channel model between the two sets of receivers and the sender.

Multipath Rayleigh Fading Channel

A Multipath Rayleigh Fading Channel represents a reection in the channel model, which provokes a phase and amplitude change to the signal. A Multipath Rayleigh Fading Channel has the following properties:

Figure 3.7: Two dierent fading channel scenarios. The main dierence is that between receiver 1 and the transmitter there is a line of sight, which does not exist between receiver 2 and transmitter.

• An evenly distributed phase shift over 0 to 2π. • No change in the average power of the signal.

• A xed gain also connected to the channel to variations in average power of the scattered waves.

The Rayleigh fading channel is a model of the channel behavior in mobile environments where no direct wave exists. The model is created by several wave components with a dierence in traveling distance being combined. When the components vary in either fre-quency or path, the sum of the arrived components will cause variations to the amplitude. A single Multipath Rayleigh Fading Channel consists of many dierent wave components reaching the receiver from many dierent paths but they have originated from the same sender.

The Rayleigh distribution has very deep dips and low heights. This implies that it will only be below the mean value for short periods of time but during these instants it will be far below the mean, whereas the remaining time will generate a value only slightly higher than the mean.

0 1 2 3 4 x 105 10−2 100 0 1 2 3 4 5 x 105 −5 0 5 data 1 y mean

Figure 3.8: Amplitude and phase change in a Rayleigh distribution. The energy is pre-served and it is also possible to se the dips in the signal and the fast phase change in the those dips.

Variable Unit Eect Standard value

Delay Sec Decide which sample

is eected 0 − 5 · 10

6

Gain dB Decides the degree of

reduction of the sample due to long travel and reection

0-(-20)

Doppler shift Hz The relationship

between the traveling speed of the radio

multiplied by the carrier frequency to

the speed of light

0-100

Table 3.2: Standard parameters in a Rayleigh Multipath Channel and the standard value used in system.

Figure 3.9: Sketch of Channel model with 3 Rayleigh Multipath Channels. The delays create the inter symbol interference and the rayleigh channel will create the fading inter-ference. Gain is used to simulate loss of energy due to reections and travelling length.

Complete Channel Model

A complete fading channel model is created by using one or more fading channels. Each fading channel has the parameters in table 3.2 set. We will have two dierent delays. One that will create the inter symbol interference and one smaller that will create the phase shift, the smaller is created in the fading channel. These two delays could be related in reality, but since the size dierence they are not related in the channel model.

A simple way to see the fading channel model without the noise is a line with delays. These delays correspond to the model and are set by the user. Samples taken from this delay line has a Rayleigh distributed amplitude and phase change. The energy is also decreased due to reections and distance by using a xed gain from the channel model. Figure 3.10 illustrates the basic approach to a channel with both fading and AWGN. Notice that the AWGN must be placed after the fading channel, since the fading channel could be a lter and lters alter the constant power spectrum diagram of the AWGN.

Figure 3.10: Complete Channel with a fading channel model and an addition of white gaussian noise.

Rural Area

The main characteristic of the RA model is the line of sight component combined with a number of Rayleigh Multipath channel components with dierent delays and gains. The reections are very focused in the rst two samples. The samples are transmitted with a speed of 0.5 so Rayleigh multipath channel will aect two samples when they are placed between two dierent samples.

Typical Urban

The main characteristic of the TU model is that it has no line of sight; only several Rayleigh Multipath channel components with dierent delays and gains. Note that the main power is not in the rst tap. The energy is more spread out and not as many Rayleigh multipath channels aect every sample.

3.4 Error Correcting Codes

Because of the channel discussed in chapter 3.3, errors are introduced in radio systems. The amount of errors between a sender and a receiver is called bit error rate. To com-pensate for bit errors, error correcting codes are used. The message is initially coded and

Tap Delay Average power Type Sample aected i=current

1 0 0 none Si

2 0 -10.0 Rayleigh Si

3 0.2 -2.0 Rayleigh Si & Si−1

4 0.4 -10.0 Rayleigh Si & Si−1

5 0.6 -20.0 Rayleigh Si−1 & Si−2

Table 3.3: The following parameters are used to simulate a Rural Area model. This model is used to simulate a scenario where there is a line of sight between the sender and the receiver, combined with small amount of inter symbol interference. Most of the energy is placed in the direct line of sight.

Tap Delay Average power Type Sample

aected i=current

1 0 -3.0 Rayleigh Si

2 0.2 0.0 Rayleigh Si & Si−1

3 0.6 -2.0 Rayleigh S_i−1 &

S_i−2

4 1.6 -6.0 Rayleigh S_i−3 &

S_i−4

5 2.4 -8.0 Rayleigh S_i−4 &

S_i−5

6 5.0 -10.0 Rayleigh S_i−10

Table 3.4: The following parameters are used to simulate a Typical Urban model. The most important abilities that this channel simulates is a no direct wave combined with a lot of reections that create inter symbol interference from seven symbols. It is also notable that most of the energy is placed in the second reection.

00 01 10 11 00 01 10 11 00 01 10 11 00 01 10 11 One Zero

Figure 3.11: 1/2-rate State transmission. The gure shows how the state change when a one or zero are transmitted.

increased in size but the payload of information stays the same. After the transmission an attempt to recreate the original message is made in the receiver.

In this system the error correction is performed with a Convolution Encoding and Viterbi Decoding. The coding is done through the transmission of states and it is the transitions between those states that indicate which bit has been sent. Decoding the message is done by creating a Trellis tree and then analyzing the tree to nd the path with the least amount of errors. The optimal way through the tree shows the most likely message.

1/2-rate example

When a new bit of data is coded it begins in a start state. The data and the current state are used to calculate the next state, this new state comprises the coded data and the state that will be used to calculate the next coded data.[1]

In this example the transmitted data is 1 0 1.

In order to get the rst two bits of coded data we use the left tree in gure 3.11 since we want to code a 1. We start at state 00; the left tree species that the transition from state 00 generates the new state 00 and hence our coded data is also 00. The next bit to code is a 0 so we use the tree on the right; starting at state 00 which generates the next state 11, 11 is thus our new state and coded data. We use the same method to code the last bit. When the data 1 0 1 is coded the result is 00 11 01.[1]

Figure 3.12: 1/2-rate Trellis Tree. The tree used to decode the six bits back to original uncoded message.

Figure 3.13: 1/2-rate Trellis Tree decoding correct transmitted message

Let us follow the correct path in Figure 3.13.

An incorrect channel implies the risk of errors. We will send 00 11 01 but now the received message is 00 10 01 and all paths must be calculated to see which is most likely. In Figure 3.14 the numbers now display the amount of errors in a certain path.

From the gure we deduct that the correct way still has the least amount of errors and is the most likely path. However, if a larger number of errors are introduced to the coded message, the most likely path could prove to be incorrect which will introduce errors rather than removing them.

Figure 3.14: 1/2-rate Trellis Tree decoding altered transmitted message. The numbers now show the amount of errors detected.

3.5 Block Interleaving

Errors are likely to occur in short intervals because of the properties of fading channels. By rearranging the bits in several message bursts it is possible to distribute the errors more evenly.[6]

This radio system uses a four message four column interleaving. This means that every message is written into a separate column. By reading the rows the interleaver creates a number of messages equal to the one inserted in the matrix.

mx(y)

x = message number

y = bit number in the burst

m1(1) m2(1) m3(1) m4(1)

m1(2) m2(2) m3(2) m4(2)

m1(3) m2(3) m3(3) m4(3)

m1(4) m2(4) m3(4) m4(4)

... ... ... ...

m1(last) m2(last) m3(last) m4(last)

The rst message will start with the rst bits from the four messages and then the second bits and so on until a quarter of the rows have been read, this will be the rst message of the four interleaved. The second covers the messages in the second quarter of rows and so on.

Chapter 4

Equalizer Theory

In section 3.3 the signal altering abilities of a radio channel are discussed. This chapter is devoted to the method of counteracting the channel. In Figure 3.1 the channel equalizer is placed after the channel but before the demodulation. This means that it is analog and modeled with a complex value containing information on the angle and amplitude of the base band.

4.1 Pilot Sequence

All messages start with a pilot sequence. This sequence is used to initialize the equalizer, since it is known to both the sender and receiver. There are two dierent alternative procedures possible when the pilot sequence has ended.

1. Fix the equalizer

(a) When the equalizer is xed it is essential that the channel does not undergo rapid changes. The equalizer is updated when a new pilot sequence arrives. 2. Try to calculate the error with the acquired knowledge of the system and modulation

(a) Using the fact that the scatter plot of the original system is known, it is possible to assume that the ideal phase and amplitude are the closest point in the known scatter plot of the modulation.

Figure 4.1: Tapped Delay line

4.2 Equalizer Example

Figure 4.1 has a simple sketch of a channel model, the dierence between the taps is one sample. The taps will phase shift the signal and change the amplitude. This change is represented by a complex value Cn with the angle as the phase shift and the amplitude as the gain. The received signal is the following:

Ri = C1Si+ C2Si+1+ C3Si+2

Si is the transmitted sample at time i from the sender

In this case, how should the equalizer be congured? Let us assume that the equalizer looks like the gure 4.1. New values in the gain and phase shift in each tap are represented by complex values. We want to shift the phase back and erase as much as possible of the ISI. A fairly good guess is choosing C∗

nto invert the phase and leave as much of the energy as possible.

Let us dene the result of the equalizer at time i to eqai. A sample transmitted from the sender at time i is dened as Si, the change from tap k in the channel as Ck and the tap k in the equalizer to C∗

k. After som simple calculations we will get the following expression.

Si+2(C3C1∗) + Si+1(C2C1∗+ C3C2∗) + Si(C1C1∗+ C2C2∗+ C3C3∗) +

Si−1(C1C2∗+ C2C3∗) + Si−2(C1C3∗) = eqai+2

There is a native delay through the equalizer which is equal to the delay to the last tap; in this case two time units.

We want Si to be unaected by all complex values C and the other S. In the nal expression it is possible to see that every S except Si is multiplied with two complex values which do not have the same amplitude. If one of those amplitudes is low the signal will be low from that symbol.

After the equalizer we will have a fair chance of decoding the message correctly, if one reection is larger than the other we will almost certainly be left with the Si value. If the channel does not have a path with a higher gain then any of the other, then the added information from the other samples is dicult to remove.

From this example it is possible to specify some situations that are challenging to handle in the equalizer.

• Multiple reections with similar gain

• Multiple reections that cancel out the main reection

4.3 Zero Forcing Equalizer

A Zero Forcing Channel Equalizer sends an impulse in the pilot sequence and forces the lter to generate the pilot impulse as the end result.

A channel with a high attenuation at certain frequencies will create a channel equalizer with a high gain at these frequencies. This means that the noise from the AWGN part will be increased at these frequencies. In an environment where the channel is noisy this approach does not create a good solution[1].

Figure 4.2: Basic mean square error equalizer lter

4.4 Forgetting Factor

λis the weighting factor or forgetting factor. We have previously (in Section 3.3) discussed the fact that a fading channel changes over time.

0 < λ ≤ 1

A lower λ implies less importance to previous experience. A channel with a fast change should have a lower λ, otherwise the utilized values come from a time when the channel was dierent, signifying that the lter tries to cancel the wrong channel approximation. A higher λ will use more information to get an optimal solution.

4.5 Minimum Mean Square Error Equalizer

The mean square error equalizer method is based on minimizing the sum of the square dierence between the pilot and the received signal. This equalizer will not have a high gain at certain frequencies since this will increase the error from noise and thus increase the mean square error.[1]

We need to nd a number of complex values called weights to use in the equalizer. The basic lter that we use is illustrated in Figure 4.2.

ξ(n) =

n

X

i=1

λ|e(i)|2

e(i) is the dierence between the correct pilot symbol and the transmitted value. This is where λ is used to adapt to fast channels.

4.6 Theory for the RLS

A and B are M-by-M matrices and they are related with the following expression. A = B−1_{+ CD}−1_CH. D is a Positive-denite N-by-M matrix and C is a M-by-N matrix with the h annotation standing for hermitian. When this has been fulllled the A−1could be calculated with the following formula:[2]

A−1 _{= B − BC D + C}H_BC
−1 CHB

Correlation Matrix

The correlation matrix is a matrix with the entries corresponding to the delay. In the RLS case the correlation matrix has the dimension taps × taps. In this case the signal is assumed to be stationary.

The correlation has a relationship stating that r (−k) = r∗_(k), which creates a Herminian matrix. If all the values in the matrix were real the Hermitian matrix would be called symmetric. Below is an example of a Herminian matrix with the size M.[2]

     r (0) r (1) · · · r (M − 1) r∗_{(1) r (0) · · · r (M − 2)} ... ... ... _... r∗_{(M − 1) r}∗_{(M − 2) · · · r (0)}     

4.7 RLS

The RLS algorithm is an adaptive lter algorithm, this means that the lter weights are not xed but calculated. A normal lter has xed weights but lacks the ability to change

and remove changing disturbance. If it is possible to calculate the disturbance then the lter could adapt to simply remove the disturbance and not interfere with any other part of the signal. The RLS algorithm calculates the weights and has the ability to do this in realtime.

The RLS algorithm in its easiest form is rather simple ˆ

w (n) = R−1

x (an) rdx(n)

ˆ

w (n) is the weight vector for the lter, R−1

x (n) is the weightend autocorrelation matrix for x (n) and rdx(n) is the cross-correlation between pilot and the received symbol.

RLS Algorithm

Calculating ˆw (n) is uncomplicated but inverting the weightend autocorrelation matrix takes a lot of resources. This is where the Matrix Inversion Lemma could be used, this lemma creates a recursive algorithm; the RLS algorithm.

ˆ

w(0) =initial weights

P (0) = initial inverted Autocorrelation Matrix

π (n) = P (n − 1) u (n)

k (n) = π (n)

λ + uH_{(n) π (n)}

Algorithm 1 RLS computation[2] 1. Initial settings

(a) Set the initial weights.

(b) Set the start correlations matrix and then invert it. 2. Calculate k

(a) k is the gain vector

(b) k will alter the weight vector after each iteration 3. Update the weights

(a) This is done by taking the new k, multiplying it with the error and then adding it to the previous weights.

i. Small error ⇒ Small change ii. Large Error ⇒ Large change

4. Got to step 2 to update the algorithm when a new sample is received.

ˆ

w (n) = ˆw (n − 1) + k (n) ξ ∗ (n)

Chapter 5

Radio System Verication

In this chapter some of the fundamental parts of radio system verication are discussed.

5.1 Channel verication

This thesis focuses on the channel which contains statistical distributions. As with all statistical distributions they will be correct when time approaches innity, signifying that a longer simulation will probably give a more accurate result. This creates a problematic uncertainty in the result and in general, a longer simulation time generates a more accurate result. It is importent to rember this fact in simulations because the simulation could vary in a short timespan.

5.2 Performance

A good verication is the performance and it is the bit error rate (BER) at dierent amount of noise. One of the most important terminologies is the SNR - Signal to Noise Ratio.

Algorithm 2 SNR[5] Es N0 = Eb N0 + 10log (k) SN R =  Es N0   Tsym Tsamp 

Variable Value (standard value)

Es Signal Energy

Eb Bit Energy

N0 Noise Power Spectral Density k Information Bits per Symbol (2) Tsamp Sample Time (5 · 10−7)

Tsym Symbol Time (1 · 10−6)

SNR is the amount of signal compared to noise that aects the radio system. Higher noise or lower signal decrease the probability that the transmitted data is correct.

The concept of Bit Error Rate is calculating the ratio of incorrect transmitted data at dierent signal to noise ratios. The nal result of this test is a graph displaying the amount of error of a radio system at dierent values of Eb

N0.

Eb

N0 is very easy to use to compare

dierent radiosystems because it takes the number of bits per symbol in consideration. The BER graph is compared to the graph of MSK (Minimum Shift Key) for verication. The two graphs will be similar but not identical.

5.3 Equalizer

It is possible to calculate the dierence between the equalized signal and the pilot signal. At the beginning of every message burst the rst part is a pilot sequence, during the pilot sequence the equalizer is adapting to the channel. A characteristic of a working equalizer is the decrease of the error at the beginning of every message when the pilot is transmitted. After the pilot the dierence should be at a low level.

The equalizer should have weights which in some way correspond to the channel, in ac-cordance with the example in section 4.2. The simpliest example is when the equaliser is used with only a AWGN channel, the ideal conguration is that the equalizer does not

change anything in the signal. This means that the only the rst taps is used and no phase change is applied.

5.4 Modulation

The rst verication mode used is the spectral diagram. The spectral diagram of a CPM radio system is known and easy to compare. By studying the spectral diagram it is also possible to analyze the bandwidth of the radio system.

Part III

Chapter 6

Implementation Overview

This chapter gives an overview of the implementation strategy that has been used.

6.1 Work Flow

The implementation is planned to be in three phases. The main reason that these three phases was chosen was that plots and performance from the dierent parts were of interest. This implementation strategy also creates a system that is easier to verify. The work is also rather evenly spread out with a large part of the system being created in the rst part, a new channel equalizer is then done and when all of the channels are extensive evaluation is performed.

1. Basic radio system construction (a) Modulation and demodulation

(b) Message construction with pilot sequence (c) Error correcting codes

(d) Bit error calculation 2. Channel equalizer

(a) RA model (b) TU model

Chapters 7, 8 and 9 are devoted to the three seperate phases dealing with design, imple-mentation and verication.

6.2 Overview Design Goals

The system is divided into sub blocks illustrated in Figure 6.1. The main system consists of the following ve sub blocks:

1. Message

(a) Outputs the complete coded message with pilot sequence to the modulation block.

(b) Outputs the original data for error detection. 2. Modulation

(a) Modulates the complete data burst and sends it into the channel. 3. Channel

(a) The channel properties are applied to the radio signal. 4. Demodulation

(a) Equalizes the radio signal (b) Demodulates the data burst

(c) Performs the error correction

(d) Outputs the demodulated data for error detection (e) Outputs the error corrected signal for error detection 5. Error detection

(a) Detects all the errors between the signal from the message block and the one from the demodulation block

Figure 6.1: Architecture of the radio system.

6.3 Architecture

Figure 6.1 describes the architecture of the system.

The only block that is not implemented inte the rst phase is the equalizer. The equalizer block is the hardest block to use and create.

Chapter 7

Basic Radio System Construction

This chapter deals with the rst phase of the implementation which consists of creating the basic radio system without the equalizer and a simple AWGN channel.

7.1 Requirement Specication

The main steps are

• Error detection

• Error correcting codes • Pilot sequence

• Modulation

When these have been created a radiosystem completly without fading channel or the equalizer to counteract the fading channel has been created. This system has the ability to answer how the radio handle noise.

7.2 Design Decisions

Message

The model only communicates using complete data bursts between the blocks, see section 3.2 for more details on data bursts. This signies that all sub blocks in the model are going to receive complete data bursts. Because of the data driven nature of Simulink this method was preferred since it simplies the circumvention of incorrect messages. The model will still need to avoid data skew . Data skew is the phenomenon when an error in a block creates a shift, thus sending the last and rst part of dierent messages as a new message, which will later be erroneously used.

Large Scale Simulation

Simulink will be controlled using Matlab and Matlab scripts. A standard simulation will be run by the creation of a Matlab script. The script will set all variables and start a simulation. When it is completed the results will be saved and this procedure could be repeated in normal Matlab loops. This creates a versatile foundation to run long simulation without any human supervision.

7.3 Verication

BER

The objective of this test is to compare this model BER to the ideal BER of the MSK. The system will not prove as robust to noise as the MSK but the plots should ressemble each other.

−5 0 5 10 15 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 Eb/No error

Plot of BER in over an AWGN channel without equalizer

4−ary 1−rec CPM with Error Correction 4−ary 1−rec CPM

Ideal MSK

Figure 7.1: BER in phase one over an AWGN channel The x-axis is the Eb

N0 value the 4-ary 1-rec CPM is a very simular modulation technique

to the MSK so the curves should follow each other to the degree in the plot. It is also notable the the error correction does not work well when the amount of errors are high.

The interesting part is the dierence between the Ideal MSK and the CPM without error correction. The two curves resemble each other to the expected level. The last part of the error corrected curve does however require some explanation; when few errors are detected, a small change in the number of errors can generate signicant changes in the BER. This problem is solved with an increased simulation time, but as a general rule the result when the BER is on the brink of existence should be taken with a pinch of salt.

Scatter Plot

The specic characteristics of the scatter plot that are investigated are: • Phase changes according to gure 3.1 on page 9.

• Amplitude

• End values of the phase changes.

−1 0 1 −1 −0.5 0 0.5 1 Quadrature In−Phase Scatter plot −1 0 1 −1 −0.5 0 0.5 1 Quadrature In−Phase Scatter plot

Figure 7.2: Scatter plots in phase one.

The scatter plots of the channel should have 16 evenly distributed points. Eight points as the endpoints and the one point between every point because we have two samples per symbol. In the scatter plot on the left in gure 7.2 it is easy to see every point and that their respective locations are correct. The scatter plot on the right describes the situation after a small amount of noise has been applied in the AWGN channel.

Spectral Diagram

The spectral diagram of the CPM is well known and a convenient way of verication of the modulation[4]. In gure 7.3 the correct spectrum of a 4-ary CPM 1-rec radio system. There is information of the frequencies that the radiosystem uses in this graph. In this thesis no regard is taken to the fact that a radio system transmits on dierent frequencies.

−1 −0.5 0 0.5 1 10−7 10−6 10−5 10−4 10−3 10−2

Figure 7.3: Power spectrum estimation of a 4-ary 1-rec CPM system

x-axis is normalized Frequency (×π rad/sample) and the y-axis is power/frequency (dB/rad/sample)

The power spectrum also shows a correct CPM modulated power spectrum. The spectrum also shows how close in frequency it is possible to place two sender.

Chapter 8

Channel Equalizer

Implementation

8.1 Requirement Specication

In this step the equalizer should only work in a design where the channel consists of an Additive White Gaussian Noise. The AWGN channel does not have any signal altering abilities that the equalizer can aect. The AWGN is completly random in its nature and we cannot remove a completly ramdom property. When no fading channel exists the equalizer will only create problems. A small increase in the BER is therefore expected, because the ideal equalizer does not change anything in the signal and the equalizer will not only pass the recieved signal through. It will probably make very small alterations to it.

The steps to complete in this part are: • Create a RLS based equalizer

• Simulate frequency hopping by resetting the system every data burst. • Fixing the equalizer after the pilot, as described in section 4.1.

8.2 Design Decision

In this phase the major decision is how the equalizer should be constructed.

Dierent Solutions

A set of dierent solutions have been examined. They all have dierence pros and cons. 1. Matlab-code in a Simulink block.

(a) Easy to use the matrix and vector operations.

(b) Two dierent ways exists to insert Matlab code in Simulink block: level-1 and level-2. The extra commands that level-2 has are required for the construction of the equalizer. The main disadvantage with level-2 is htat it has a very questionable documentation.

2. C-code in a Simulink block. (a) Good documentation.

(b) Dicult to use the matrix and vector operations. 3. Modify the existing RLS equalizer block in Simulink.

(a) Ready block to use which has been veried.

(b) Everything is already created in Simulink; no part is created by using Matlab functions. Advanced Simulink functionalities-like looping blocks-complicate the system.

All three dierent solutions were investigated to nd the best solution. The best solution is the one in which it is easiest to create a fully functional equalizer. During the evaluation all three dierent solutions where examined. The rst solution to put matlab-code in a Simulink block was tried but the documentation turned the design work in to pure guessing almost immediatly. Both the matlab-code and the c-code required much more Simulink knowledge because of the endless conguration possibilities oered in those parts of Simulink. All of these congurations are much more intuitive when ready Simulink blocks are used as in the case of the ready RLS equalizer.

Almost all matlab functions are found as basic building blocks in Simulink. The RLS algorithm is created visually with blocks which was confusing in the beginning.

Modifying the Existing RLS Equalizer

As a foundation the standard RLS equalizer in the communications toolbox to Simulink was used. There are two main problems with this model which need to be solved. The existing RLS equalizer has a core based on algorithm 1. Surrounded by a number of control systems. It takes some time to understand how the system works and how the algorithm is implemented.

• The standard equalizer updates the weights after the pilot sequence has ended. The model does however require the weights to be locked as soon as the pilot sequence is over.

• Add frequency hopping ability to the system. If the system was sending on the same frequency all the time the equalizer settings could be reused as a starting point. The channel is dierent for dierent frequencies. In the freqeuncy hopping system the frequency is changing with every burst and the settings in the equalizer are obsolute when the frequency is changed.

8.3 Problems

Frame based

A few diculties were encountered; in particular upon the equalizer's adaption to fre-quency hopping. Since the equalizer worked continuously and processed dierent frames at the ending part of a data burst, the system ended up malfunctioning. This is due to the convolution implemention metod, which requires a number of samples from two dierent frames.

Simulation Performance

There is still a lot of room for performance improvement in the equalizer block. The current implementation method is rather straight forward and demands heavy calculation. This problem is solved with a powerful computer.

8.4 Architecture

Figure 8.1 is a sketch of the original Simulink creation. One iteration is performed for every input sample and the output samples are then assembled to create an equalized message burst to output.

Figure 8.1: Simulink RLS architecture. The system gets a complete message burst and the has to break it down to single symbols. The processed symbols are then combined to create a new complete meassage burst.

In order to lock the weights after the pilot sequence the error was forced to zero. Recalling the theory in section 4.7; new weights are calculated with the error size and a low error creates small weights changes, if there is no error the change to the weights is zero. This approach creates a small change to the original block and the iterative parts of the design remains intact. The system is easy to verify but becomes rather slow.

To reset the system after each message a couple of dierent changes was needed. Most of which were implemented with a reset signal.

8.5 Verication

The main focus in the verication is regarding equalizer as a white box. Only a small amount of energy is placed on the system as a whole, which could be the Achilles heel of this verication but the BER should suce.

Debugging

The equalizer that was used as a foundation had some bugs reported at the developer's homepage, none of which apply to this particular use.

An unusual debugging approach was used after a while - statistics. The main concept was to detect the error probability of every bit in a message. With this system anomalies could be found easily.

Error Convergence

The error should decrease constantly to a reasonable level during the pilot sequence.

0 2 4 6 8 0 50 100 150 200 0 0.5 1 1.5 2 message burst # Plot of error convolution

transmitted bit

error

Figure 8.2: Error between equalized and ideal signal when the channel delays the signal seven samples. At the beginning when the pilot sequence is used to congure the equalizer the error is high but more of the pilot the lesser the error.

Tap weight

If the channel only includes a delay the weights should adapt to the specic delay and have a high gain at that point.

0 20 40 60 0 10 20 30 0 0.2 0.4 0.6 0.8 message burst # Plot of fixed tap weights

tap

weight

Figure 8.3: Tap Weights when the channel delays the signal 7 samples. The equalizer should counteract the seven samples by shifting the signal seven samples forward in the message burst. It is also possible to see why the system has lost some of its noise resistance by looking at the rest of the taps. In the ideal case they would have been zero but noise creates an equalizer that is not as good as seven shifts forward.

BER

−5 0 5 10 15 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 Eb/No error

Plot of BER over an AWGN

CPM without RLS and Error Correction CPM without RLS without Error Correction Ideal MSK diff

CPM with RLS and Error Correction CPM with RLS without Error Correction

Figure 8.4: BER with RLS over an AWGN channel. As previously discussed the radio system with an equalizer is not as noise resistance. This is because the noise is aecting the conguration of the equalizer and this could congure it to less than ideal.

Chapter 9

Fading Channel

9.1 Requirement Specication

The two major design issues in this chapter are the two main channels: • TU channel

• RA channel

Both of these are created using existing Simulink blocks. The only channel blocks used are Rayleigh multipath channels and AWGN channels.

9.2 Design Decisions

This step in the design is not as extensive as the rst two but it is in this process that the whole design comes together.

• Due to errors in the channel blocks every delay is created in a separate channel block and then added together.

9.3 Problems

After some reseearch of the developer's documentation it was discovered that there were a number of bugs in the Simulink channel blocks. The Rician fading channel should not be used at all. That is why no Rician channel is used.

The Multipath channel showed problems when the numbers of parallel paths were in-creased and completely dierent results were obtained. The solution was to create every path individually, add them together and later adjust the signal power in the AWGN channel.

9.4 Architecture

Two screenshots of the architecture of the channel is found in gure 9.1 and 9.2.

The channel is created by setting up every reection seperatly and then adding them together before the AWGN channel. Then in the AWGN channel the increased energy is compensated to apply the correct amount of noise.

Figure 9.2: RA screen shot. The direct wave is seen as the signal with no blocks. Note that in each Multipath Fading block, the average gain, delay and speed can be adjusted. It is also possible to create mulitple reections in a single block. After some problems with this abilities this was not used.

9.5 Verication

Maximum Doppler Frequency

It is uncomplicated to check the Doppler frequency if the absolute value is saved and then ploted. The doppler frequency in the graph is calculated by counting the amounts of dips

in a single rayleigh channel. 0 0.2 0.4 0.6 0.8 1 10−2 10−1 100 101

Plot of Rayleigh distrubition

time[sec]

Amplitude

Figure 9.3: Rayleigh Distribution at 70 Km/h and 400MHz. Thickness of the graph is actually the inter symbol interference. The plot is the absolute value and every symbol are represented by a complex value, when one signal has most of the energy the interfering symbols will force the the absolute value to change very rapidly depending on what symbol is transmitted. The channel are also changíng very much slower then the transmitted symbols.

BER

The bit error rate is dicult to predict but it should be within reasonable levels. If almost no errors are detected or the BER is too similar to the one with only an AWGN channel the model is probably wrong. If the radiosystem is completly unusable the result is also probably wrong. The equalizer should also make a signicant improvement.

0 10 20 30 40 50 10−4 10−3 10−2 10−1 100 Eb/No error

Rural Area with error correction

tap 13 & lambda 0.9 tap 21 & lambda 0.9 tap 13 & lambda 0.99 tap 21 & lambda 0.99 tap 13 & lambda 0.995 tap 21 & lambda 0.995 Without RLS

Figure 9.4: BER when the Rural Area model is used.

The direct wave will make the system without the equalizer capable to receive symbols. When the noise is very high it will actually be better then the system with the equalizer, this is probably because the equalizer will not adapt to the channel very good. With a low lambda value the equalizer will not be as good. But the other congurations are very simular in performance.

0 10 20 30 40 50 10−4 10−3 10−2 10−1 100 Eb/No error

Typical Urban with error correction

tap 13 & lambda 0.9 tap 21 & lambda 0.9 tap 13 & lambda 0.99 tap 21 & lambda 0.99 tap 13 & lambda 0.995 tap 21 & lambda 0.995 Without RLS

Figure 9.5: BER when the Typical Urban model is used.

In this model the radiosystem is useless without the equalizer. The low forgetting factor will make the perfomance low and it is also notable that there is a rather large perfomance dierence between the two tap lengths. The typical urban has inter symbol interference and has a much longer delay than the rural area and this explains why the equalizer with more taps will give a higher performance.

Part IV

Chapter 10

Discussion

10.1 Simplications in the Modeling

The model is not a perfect version of the real world.

Frequency Hopping

This radio system has frequency hopping and mobility which creates a problem. The dierence in simulation and reality lies in how a channel changes when the carrier wave frequency is altered.

• At specic times the channels introduce a lot of errors

• Error correction is problematic when a high level of errors are introduced in a short interval

• A real life scenario presents dierent channels which are not entirely uncorrelated • Interleaving is used to spread out errors to help the error correction

The channel in the model changes continuously and does not present a major change when a frequency hop is performed.

1. Use the same channel and treat every dierent message burst separately in the radio parts, not exploiting the fact that the channel has not undergone any major change. 2. Use a completely new channel for each burst.

The location of the errors does however aect the error correction, which is why neither of these approaches is completely correct.

Limited Time Simulation

Because of the limited simulation time there will always be an uncertainty in the result, but this will probably decrease with a longer simulation. It is possible to see how a result stabilizes after a number of bits have passed through the channel. In gure 10.1 dierent results are received when dierent amounts of symbols have passed through a channel in a standard simulation. The value will change over time, but in general a longer simulation time renders a more accurate result.

In gure 10.1 the most correct result should be considered to be the one with most bits passed through the channel. The reader will notice that a good result is found as early as 3 · 106_{. In all the tests in this thesis the amount of bits passed through the channel is} 1.2 · 107 _{this should create a result which has a low error margin. But it is still impossible} to claim that the result is completly accurate.

10.2 Limitations in the Model

In this section the limitations of the model are discussed. All aws described here are not as native as in section 10.1 and it is possible to remove them over time using extensive resources.

Error Correction

The current model system only has the standard convolution error correction. In the real system a much more complex and powerful error correction will be used.

104 105 106 107 108 109 10−2

10−1 100

Bits transfered throught the channel

result

Plot of simulation time

tap 21 & conv forgetting 0.995

Figure 10.1: Result of a standard simulation at dierent simulation length

There is a very high degree of uncertainy when only a few bits have passed through the channel. A longer simulation will lead to a more accurate result. 1.2 · 107 _{bits are used} in all the simulations in this thesis. As can be seen in the graph the curve has stopped uctuating when this values is reached and correct values should be received.

RLS Block

The main disadvantage concerning the RLS block is the rather functional and fast im-plementation of the weight xing after the pilot sequence. It consumes calculation power without doing any calculation. It is also hard to use the block in any other model since some of the variables are set outside of the block.

Message Size

There are some bounds on the message size, mainly due to the fact that the modulation part needs messages with even sizes after the convolution and pilot sequence addition. The pilot also needs to be even to generate complete symbols in the modulation.

Parallelism

Simulink should be easy to convert to a parallel program to automatically take advantage of many computer cores. The straight forward solution is creating simulations that are divided into multiple small parts and running the small parts in dierent Matlab programs. As an example dierent Eb

Chapter 11

Final Tests

In this chapter the matlab plots of some of the results are presented.

11.1 Error Correction Performance

When a certain limit is crossed the error correction ceases to decrease the errors and starts to increase them. In chapter 10.1 it was stated that the real system will use a better error correction than the convolution correction in the modelled system. Figure 11.1 could be used to further understand the dierence between the real and modeled system.

11.2 AWGN

This test is performed only on an AWGN channel. The main conclusion that could be drawn from this is how the RLS handles noise.

X-axis BER on the channel Y-axis BER after decoding Table 11.1: Parameters for Figure 11.1

0 0.1 0.2 0.3 0.4 10−5 10−4 10−3 10−2 10−1 introduced errors error

Performance of the error correcting codes

Error correction

Errors without error correction

Figure 11.1: Error Correction Performance of the system used in this radio model. Notable things in this graph is the fact that when 18 % of the bits are wrong errors will be introduced instead of removed.

The Errors without error correction show the ratio of incorrect bits and the x-axes also display this ratio. Simulation parameters are found in table 11.1

Taps 21 Forgetting Factor 0.995

X-axis Eb

N0

Simulated Samples 1.2 · 107

Table 11.2: Parameters for Figure 11.2 and 11.3

−5 0 5 10 15 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 Eb/No error

Plot of BER in phase two

without RLS with Error Correction without RLS without Error Correction Calculated MSK nondiff

RLS with Error Correction RLS without Error Correction

Figure 11.2: AWGN channel with and without the equalizer and error correction. No Fading channel is used.

At 10−4 errors the 4-ary CPM 1-rec system without equalizer will have 1 dB lower noise resistance then the MSK and when the equalizer is added the resistance is further lowered 1.5 dB.

2 3 4 5 6 7 8 9 10 11 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 Eb/No error

Plot of BER in phase two

without RLS with Error Correction without RLS without Error Correction RLS with Error correct

RLS with Transmitted Calculated MSK nondiff

Figure 11.3: AWGN channel zoomed around 10−4 as the BER.

Speed 70km/h

Channel Rayleigh Multipath

RLS Taps 21

Lambda 0.995

X-axis Eb

N0

Simulated Samples 1.2 · 107

Table 11.3: Parameters for Figure 11.4 and 11.5

We lost approximately 1,5 dB at a BER of 10−4 with the error correction and 1 dB without the error correction when the 4-ary CPM 1-rec systems are compared with and without the equalizer. When the dierence between MSK and 4-ary CPM 1-rec systems is compared at 10−4 the decreased noise resistance are 1.5 dB and the two curves follow each other very well. This is expected because MSK has a similar modulation.

11.3 Fading Channel without ISI

This simulation is only performed with a single Rayleigh Multipath Channel. This channel model does not have any kind of ISI (Inter Symbol Interference), the only alteration to the signal is a phase and gain change. A doppler shift to simulate 70 km/h is also used. With this test it is possible to see how the system handles fading channels without inter symbol interference.

11.4 Channel with ISI

The following test is performed to get a measurement of how good the equalizer performs with the only addition of the previous sample and without a fading channel. The previous sample is given a gain. That gain is not in the dB scale but in the normal linear scale.

0 10 20 30 40 50 10−4 10−3 10−2 10−1 100 Eb/No error

One Rayleigh Channel with 21 Taps & Forgetting Factor 0.995

RLS with Error Correction RLS without Error Correction without RLS with Error Correction without RLS without Error Correction

Figure 11.4: BER on one Rayleigh multipath channel

The most notable dierence is the low performance of the error correction. This is be-cause the errors are very focused and there is more than 18 % errors when the erros are introduced. It is also notable that the errors are always introduced by the fading channel even when the noise is almost not existing, probably from the fact that the channel is changing.

0 10 20 30 40 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Eb/No error

Message Corruption One Rayleigh Channel with 21 Taps & Forgetting Factor 0.995

RLS with Error Correction RLS without Error Correction

Figure 11.5: Corrupted messages on one Rayleigh multipath channel

Even without the equalizer the number of correct message burst are high. With the equalizer the amount of correct bursts is dubbled and the only performance loss is the 128 bit of pilot sequence which is not needed when no equalizer is used.

The error is the amount of message bursts with a single incorrect bit. Simulation parameters are found in table 11.3

0 10 20 30 40 50 10−6 10−5 10−4 10−3 10−2 10−1 100 Eb/No error

Inter Symbol Interference with error correction

RLS gain 0 RLS gain 0.5 RlS gain 1 RLS gain 2 without RLS gain 0 without RLS gain 0.5 without RLS gain 1 without RLS gain 2

Figure 11.6: Test of how the radio system handles ISI with error correction

As expected the best situation is when no equalizer or ISI are present. The cases where no equalizer is used the system is working fairly well as long as the inter symbol interference are low. When the gain is increased to one and two the system breaks down and no correct bits are received. When the equalizer is used the ISI are removed and there is always less then one % of errors when EB

N0 > 15.

0 10 20 30 40 50 10−6 10−5 10−4 10−3 10−2 10−1 100 Eb/No error

Inter Symbol Interference without error correction

RLS gain 0 RLS gain 0.5 RlS gain 1 RLS gain 2 without RLS gain 0 without RLS gain 0.5 without RLS gain 1 without RLS gain 2

Figure 11.7: Test of how the radio system handles ISI without error correction It is expected that the system without the equalizer will have problems when the inter-ference are high. A notable result is that the equalizer does not work as well when it is interfered with a rather low amount of interfering symbols. The error correction works well when the graph is compared to the one with error corrections and this says that the errors are spread out.

Channel One Direct Wave One Delay with a Gain

RLS Taps 21

Lambda 0.995

X-axis Eb

N0

Simulated Samples 1.2 · 107

Table 11.4: Parameters for Figure 11.6, 11.7 and 11.8

Speed 70 km/h

Channel Rural Area

X-axis Eb

N0

Simulated Samples 1.2 · 107

Table 11.5: Parameters for Figure 11.9, 11.10 and 11.11

The same result should be obtained when the gain is 2 and 0.5. The main dierence comes from the fact that the system is timing in on the rst sample. The simulation shows that the equalizer is nessary when the inter symbol interference is high.

11.5 RA

This is a simulation that has the properties from table 3.3 on page 19.

11.6 TU

0 10 20 30 40 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Eb/No error

Inter Symbol Interference corrupted messages with error correction

RLS gain 0 RLS gain 0.5 RlS gain 1 RLS gain 2 without RLS gain 0 without RLS gain 0.5 without RLS gain 1 without RLS gain 2

Figure 11.8: ISI corrupted messages

There is a sharp line between where the system has almost no correct messages and almost every message is correct.

The error is the amount of message bursts with a single incorrect bit. Simulation parameters are found in table 11.4

Speed 70 km/h

Channel Typical Urban

X-axis Eb

N0

Simulated Samples 1.2 · 107

0 10 20 30 40 50 10−4 10−3 10−2 10−1 100 Eb/No error

Rural Area with error correction

tap 13 & lambda 0.9 tap 21 & lambda 0.9 tap 13 & lambda 0.99 tap 21 & lambda 0.99 tap 13 & lambda 0.995 tap 21 & lambda 0.995 Without RLS

Figure 11.9: Rural Area with Error Correction

The forgetting factor is the single most important parameter in the equalizer. Because of the fact that the symbols are not as spread out in the model the size of the equalizer is not as important.

0 10 20 30 40 50 10−2 10−1 100 Eb/No error

Rural Area without error correction

tap 13 & lambda 0.9 tap 21 & lambda 0.9 tap 13 & lambda 0.99 tap 21 & lambda 0.99 tap 13 & lambda 0.995 tap 21 & lambda 0.995 Without RLS

Figure 11.10: Rural Area without Error Correction

When gure 11.9 is compared to this one, the error correction works well. The forgetting factor should be high for the equalizer to work well.