Institutionen för systemteknik Department of Electrical Engineering


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1 Institutionen för systemteknik Department of Electrical Engineering Examensarbete RLS for CPM system Examensarbete utfört i Datatransmission av Frans Bergquist LiTHISYEX07/3871SE 2007 TEKNISKA HÖGSKOLAN LINKÖPINGS UNIVERSITET Department of Electrical Engineering Linköping University S Linköping, Sweden Linköpings tekniska högskola Institutionen för systemteknik Linköping
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3 RLS for CPM system Examensarbete utfört i datatransmission vid Linköpings tekniska högskola av Frans Bergquist LiTHISYEX07/3871SE Handledare: Johan Henriksson Examinator: Mikael Johansson Linköping
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5 Presentationsdatum Publiceringsdatum (elektronisk version) Institution och avdelning Institutionen för systemteknik Department of Electrical Engineering Språk Svenska X Annat (ange nedan) Engelska Antal sidor 96 Typ av publikation Licentiatavhandling X Examensarbete Cuppsats Duppsats Rapport Annat (ange nedan) ISBN (licentiatavhandling) ISRN Serietitel (licentiatavhandling) Serienummer/ISSN (licentiatavhandling) URL för elektronisk version 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 intersymbol interference och fading utvärderas också Nyckelord CPM, RLS, kanalutjämnare, radiomodellering, simulink
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7 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.
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9 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. At last but not least I need to thank my whole family.
10 Contents I Background 1 1 Introduction Overview Reading Instructions Problem Description Task Method Initial Limitations II Radio System Theory 7 3 Radio System Introduction CPM Data Transmission Channel Error Correcting Codes Block Interleaving
11 4 Equalizer Theory Pilot Sequence Equalizer Example Zero Forcing Equalizer Forgetting Factor Minimum Mean Square Error Equalizer Theory for the RLS RLS Radio System Verication Channel verication Performance Equalizer Modulation III Implementation 33 6 Implementation Overview Work Flow Overview Design Goals Architecture Basic Radio System Construction Requirement Specication Design Decisions Verication
12 8 Channel Equalizer Implementation Requirement Specication Design Decision Problems Architecture Verication Fading Channel Requirement Specication Design Decisions Problems Architecture Verication IV Conclusion Discussion Simplications in the Modeling Limitations in the Model Final Tests Error Correction Performance AWGN Fading Channel without ISI Channel with ISI RA TU
13 12 Conclusion Fading Channel ISI RA TU Performance Bibliography 83 A Abbreviations 84
14 Part I Background 1
15 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 2
16 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. 3
17 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 (a) RLS based weight calculation 4
18 (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) 1/3rate convolution encoding (b) Viterbi decoding (c) 4 message block interleaving 4. Frequency hopping (a) Message burst i. 128 bits pilot ii bits of data iii. Transmitted symbol rate of 10 6 symbols per second (b) Reset the equalizer after each burst 5. Channel (a) Speed km/h normally 70 km/h (b) Typical Urban i. No Direct Wave ii. 6taps According to GSM (c) Rural Area i. Direct wave ii. 5taps 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. 5
19 Simulink Simulink is a blockbased simulation tool, traditionally used for system based design, control and signal modeling. In these application areas extra toolboxes exists. The following toolboxes have been available for this project: Communications Blockset RealTime 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 provides 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. 6
20 Part II Radio System Theory 7
21 Chapter 3 Radio System Introduction All blocks in gure 3.1 except the equalizer will be covered by this chapter. The transmitted 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. Almost 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 xaxis and the imaginary as the yaxis. 3.1 CPM CPM is an abbreviation for Continuous Phase Modulation which is a modulation technique. 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
22 Figure 3.1: Basic skeetch of a general Radio System. Symbol Phase Change 00 + π 4 01 π π π 4 Table 3.1: The Phase Change to the base band signal in this radio system 9
23 Scatter plot Quadrature In Phase 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 transmitted 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 introduce 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 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 for the next symbol and then 4 π. All 4 of these changes will then have to be detected in the receiver for the correct bits to be transmitted. 10
24 Figure 3.3: This is how the phase will change when 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 11
25 8 Plot of raised cosine phase change phase time Figure 3.4: Phase change when a rectangular pulse shaping is used in a CPM radio system 12
26 8 Plot of raised cosine phase change phase time Figure 3.5: Phase change when a raised cosine pulse shaping is used in a CPM radio system Figure 3.6: Data Bursts 13
27 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 (kt 0 ). 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: 14
28 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 frequency 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. A Rayleigh Multipath Fading channel is only a part of the complete channel model. 15
29 data y mean x x 10 5 Figure 3.8: Amplitude and phase change in a Rayleigh distribution. The energy is preserved 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 Gain db Decides the degree of 0(20) reduction of the sample due to long travel and reection Doppler shift Hz The relationship between the traveling speed of the radio multiplied by the carrier frequency to the speed of light Table 3.2: Standard parameters in a Rayleigh Multipath Channel and the standard value used in system. 16
30 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 interference. 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. 17
31 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 compensate for bit errors, error correcting codes are used. The message is initially coded and 18
32 Tap Delay Average power Type Sample aected i=current none S i Rayleigh S i Rayleigh S i & S i Rayleigh S i & S i Rayleigh S i 1 & S i 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 Rayleigh S i Rayleigh S i & S i Rayleigh S i 1 & S i Rayleigh S i 3 & S i Rayleigh S i 4 & S i 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. 19
33 00 One Zero Figure 3.11: 1/2rate 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. 1/2rate 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 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 is coded the result is [1] The tree constructed when this message is decoded is illustrated in Figure
34 Figure 3.12: 1/2rate Trellis Tree. The tree used to decode the six bits back to original uncoded message. Figure 3.13: 1/2rate Trellis Tree decoding correct transmitted message Let us follow the correct path in Figure An incorrect channel implies the risk of errors. We will send but now the received message is 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. 21
35 Figure 3.14: 1/2rate 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. m x (y) x = message number y = bit number in the burst m 1 (1) m 2 (1) m 3 (1) m 4 (1) m 1 (2) m 2 (2) m 3 (2) m 4 (2) m 1 (3) m 2 (3) m 3 (3) m 4 (3) m 1 (4) m 2 (4) m 3 (4) m 4 (4).. m 1 (last) m 2 (last) m 3 (last) m 4 (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. 22
36 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. 23
37 4.2 Equalizer Example Figure 4.1: Tapped Delay line 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 C n with the angle as the phase shift and the amplitude as the gain. The received signal is the following: R i = C 1 S i + C 2 S i+1 + C 3 S i+2 S i 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. 24
38 Let us dene the result of the equalizer at time i to eqa i. A sample transmitted from the sender at time i is dened as S i, the change from tap k in the channel as C k and the tap k in the equalizer to C k. After som simple calculations we will get the following expression. S i+2 (C 3 C 1) + S i+1 (C 2 C 1 + C 3 C 2) + S i (C 1 C 1 + C 2 C 2 + C 3 C 3) + S i 1 (C 1 C 2 + C 2 C 3) + S i 2 (C 1 C 3) = eqa i+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 S i 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 S i 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 S i 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]. 25
39 4.4 Forgetting Factor Figure 4.2: Basic mean square error equalizer lter λ is 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. The value that needs to be minimized is: 26
40 n ξ(n) = λ e(i) 2 i=1 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 MbyM matrices and they are related with the following expression. A = B 1 + CD 1 C H. D is a Positivedenite NbyM matrix and C is a MbyN matrix with the h annotation standing for hermitian. When this has been fulllled the A 1 could be calculated with the following formula:[2] A 1 = B BC ( D + C H BC ) 1 C H B 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 27
41 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 ŵ (n) = R 1 x (an) r dx (n) ŵ (n) is the weight vector for the lter, Rx 1 (n) is the weightend autocorrelation matrix for x(n) and r dx (n) is the crosscorrelation between pilot and the received symbol. RLS Algorithm Calculating ŵ (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. ŵ(0) = initial weights P(0) = initial inverted Autocorrelation Matrix π (n) = P (n 1) u (n) k (n) = π (n) λ + u H (n) π (n) ξ (n) = d (n) ŵ (n 1) u (n) 28
42 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. ŵ (n) = ŵ (n 1) + k (n) ξ (n) P (n) = λ 1 P (n 1) λ 1 k (n)u H (n)p (n 1) 29
43 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. 30
44 Algorithm 2 SNR[5] E s N 0 = E b N log (k) SNR = ( ) E s N 0 ( ) Tsym T samp Variable Value (standard value) E s Signal Energy E b Bit Energy N 0 Noise Power Spectral Density k Information Bits per Symbol (2) T samp Sample Time ( ) T sym Symbol Time ( ) 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 E of error of a radio system at dierent values of b E N 0. b N 0 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 accordance 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 31
45 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. 32
46 Part III Implementation 33
47 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 3. Fading channel model 34
48 (a) RA model (b) TU model Chapters 7, 8 and 9 are devoted to the three seperate phases dealing with design, implementation 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 With this design as a foundation the construction is initiated. 35
49 6.3 Architecture Figure 6.1: Architecture of the radio system. 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. 36
50 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 The major design decisions in this phase will be analyzed and discussed in this section. 37
51 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. 38
52 10 0 Plot of BER in over an AWGN channel without equalizer error ary 1 rec Eb/No CPM with Error Correction 4 ary 1 rec CPM Ideal MSK Figure 7.1: BER in phase one over an AWGN channel The xaxis is the E b N 0 value the 4ary 1rec 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. 39
53 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 Scatter plot 1 Scatter plot Quadrature Quadrature In Phase In Phase 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 4ary CPM 1rec 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. 40
54 Figure 7.3: Power spectrum estimation of a 4ary 1rec CPM system xaxis is normalized Frequency ( π rad/sample) and the yaxis 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. 41
55 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
56 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. Matlabcode 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: level1 and level2. The extra commands that level2 has are required for the construction of the equalizer. The main disadvantage with level2 is htat it has a very questionable documentation. 2. Ccode 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 functionalitieslike looping blockscomplicate 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 matlabcode in a Simulink block was tried but the documentation turned the design work in to pure guessing almost immediatly. Both the matlabcode and the ccode 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. After the investigation the Simulink standard RLS block was chosen as a foundation. 43
57 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 frequency 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. 44
58 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. 45
59 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. Plot of error convolution error transmitted bit message burst # 2 0 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. 46
60 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. Plot of fixed tap weights weight tap message burst # 60 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. 47
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