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Performance of cooperative space time coding with spatially correlated fading and imperfect channel estimation Wan, Derrick Che-Yu 2008

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Performance of Cooperative Space Time Coding with Spatially Correlated Fading and Imperfect Channel Estimation by DERRICK CHE-YU WAN B.A.Sc., The University of British Columbia, 2005 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (Electrical and Computer Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) February 2008 © Derrick Che-Yu Wan, 2008 Abstract A performance evaluation of CSTC (Cooperative Space Time Coding) with spatially cor- related fading and imperfect channel estimation in Gaussian as well as impulsive noise is presented. Closed form expressions for the pairwise error probability conditioned on the estimated channel gains are derived by assuming the components of the received vector are independent given the estimated channel gains. An expurgated union bound using the limiting before averaging technique given the estimated channel gains is then obtained. Although this assumption is not strictly valid, simulation results show that the bound is accurate in estimating the diversity order as long the channel estimation is not very poor. It is found that CSTC with block fading channels can reduce the frame error rate (FER) relative to SUSTC (Single User Space Time Coding) with quasi-static fading channels. even when the channel gains for each user are strongly correlated and when the channel estimations are very poor. A decision metric for CSTC with spatially correlated fading, imperfect channel estima- tion, and impulsive mixture Gaussian noise is derived which yields lower FERs than the Gaussian noise decision metric. Simulation results show that the FER performance of CSTC with mixture Gaussian noise outperforms CSTC with Gaussian noise at low SNR. At high SNR, the FER performance of CSTC with Gaussian noise is better than the FER performance of CSTC with mixture Gaussian noise due to the heavy tail of the mixture Gaussian noise. Contents Abstract ^ Table of Contents^ iii List of Tables  ^iv List of Figures ^  vii List of Abbreviations and Symbols ^  viii Acknowledgements ^ 1 Introduction  ^1 1.1 Motivation and Goals  ^2 1.2 Outline of Thesis ^ 2 Background and Related Work ^  5 2.1 Relay Channel  ^5 2.2 Block Fading Channel ^  6 2.3 Impulsive Noise ^ 2.4 Cooperative Communication ^  8 2.4.1 Amplify and Forward (AF)  9 2.4.2 Detect and Forward (DF)^ l 2.5 Cooperative Space Time Coding (CSTC) ^  12 3 CSTC with Spatially Correlated Fading and Imperfect Channel Esti- mation ^  19 ii 3.1 System Model^ 10 3.2 Performance Analysis Based on Estimated Channel Gains ^ 2:3 3.3 Evaluation of Expurgated Union Bound with Limiting Before Averaging Technique ^  29 4 Comparison of Analytical and Simulated Results ^ 31 4.1 FER Comparison ^  31 4.1.1 Single User Space Time Coding (SUSTC) ^  32 4.1.2 Cooperative Space Time Coding (CSTC) 35 4.1.3 Comparison of SUSTC and CSTC^  39 4.2 Cooperation Gain Comparison ^  41 4.2.1 Perfect Channel Estimation (pe = 1) ^  41 4.2.2 Imperfect Channel Estimation (oe = 0.99 and 0.9) ^ 44 5 STC with Impulsive Noise ^  52 5.1 System Model with Mixture Gaussian Noise ^  52 5.2 Numerical Results^ 5'3 5.2.1 Perfect Channel Estimation (pe = 1) ^  51 5.2.2 Imperfect Channel Estimation (pe = 0.99)  56 6 Conclusion ^  58 6.1 Main Thesis Contributions ^  58 6.2 Topics for Further Studies  59 Appendices ^  60 A Dependencies among the components of the received vector ^ 60 B Extended Trellis ^  64 C SNR Definition  74 iii List of Tables 2.1 Convolutional Codes Suitable for CSTC with Two Transmit Antennas at Each User in Octal Notation. ^ 16 4.1 Cooperation Gains for pe = 1, Pr = 0.1, 0.25, and 0.5 ^ 41 4.2 Cooperation Gains for pe = 0.99, pr = 0.1, 0.25, and 0.5 47 4.3 Cooperation Gains for p e = 0.9, pr = 0.1, 0.25, and 0.5 ^ 49 iv List of Figures 1.1 Cooperative Diversity ^ 2.1 Relay Channel.  ^5 2.2 Comparison of Gaussian and Mixture Gaussian pdf's. All pdf's have a variance of 1 ^ 2.3 Amplify and Forward. ^ 9 2.4 Detect and Forward.  10 2.5 TDMA Implementation for Coded Cooperation ^11 2.6 Time Division Channel Allocations^  12 2.7 Transmission Scheme : Segment 1. 13 2.8 Transmission Scheme : Segment 2.^ 1:3 2.9 Block Diagram of the Model. 3.1 Modified Transmission Scheme : Segment 1. ^  21 3.2 Modified Transmission Scheme : Segment 2.  22 4.1 Comparison of Analytical and Simulation Results for Single User Space Time Coding with fixed p, at 1 and varying A, : BPSK modulation, (5, 7, 5, 7) convolutional code (upper bound: dashed lines; simulation: solid lines) ^  32 4.2 Comparison of Analytical and Simulation Results for Single User Space Time Coding with fixed p, at 0.99 and varying ps : BPSK modulation, (5, 7, 5, 7) convolutional code (upper bound: dashed lines; simulation: solid lines) ^  3-I 4.3 Comparison of Analytical and Simulation Results for Single User Space Time Coding with fixed p. at 0.9 and varying p s : BPSK modulation, (5. 7, 5, 7) convolutional code (upper bound: dashed lines: simulation: solid lines) ^  35 V 4.4 Comparison of Analytical and Simulation Results for Cooperative Space Time Coding with fixed pe at 1 and varying p s : BPSK modulation. (5, 7, 5. 7) convolutional code (upper bound: dashed lines; simulation: solid lines)   36 4.5 Comparison of Analytical and Simulation Results for Cooperative Space Time Coding with fixed pe at 0.99 and varying p, : BPSK modulation, (5, 7.5. 7) convolutional code (upper hound: dashed lines: simulation: solid lines)   37 4.6 Comparison of Analytical and Simulation Results for Cooperative Space Time Coding with fixed pe at 0.9 and varying ps BPSK modulation. (5, 7, 5, 7) convolutional code . 38 4.7 Comparison of the Simulation Results for Single User and Cooperative Space Time Coding at I), at 1 and different values of p, (Cooperation: dashed lines; Single User: solid lines)   39 4.8 Simulation Results for Cooperative Space Time Coding with A, = 0, p e = 1, and Pi'" =- 0.1, 0.25, 0.5  ^42 4.9 Simulation Results for Cooperative Space Time Coding with p s = 0.5, p, = 1, and 13;" = 0.1, 0.25, 0.5  ^12 4.10 Simulation Results for Cooperative Space Time Coding with p s = 0.8, p c = 1, and P'f " = 0.1, 0.25, 0.5^ 43 4.11 Simulation Results for Cooperative Space Time Coding with p = 1, p = 1, and Pl." = 0.1, 0.25, 0.5^ 13 4.12 Simulation Results for Cooperative Space Time Coding with p. = 0, p, = 0.99, and Pr =0.1,0.25 0.25  0 5^ 45 4.13 Simulation Results for Cooperative Space Time Coding with p. = 0.5. p e = 0.99, and P"' = 0.1, 0.25  0 5^   45 4.14 Simulation Results for Cooperative Space Time Coding with p„ = 0.8. p c = 0.99. and P = 0.1, 0.25  0 5^ 4.15 Simulation Results for Cooperative Space Time Coding with ps = 1, pc = 0.99, and P'1' = 0.1, 0.25, 0.5  ^46 4.16 Simulation Results for Cooperative Space Time Coding with p„ = 0, p„ = 0.9, and Pr " = 0.1, 0.25, 0.5  ^47 vi 4.17 Simulation Results for Cooperative Space Time Coding with p.. = 0.5. p„ = 0.0. and P'" = 0 1 0.25  0 5 ^-16 4.18 Simulation Results for Cooperative Space Time Coding with ps = 0.8. pc = 0.9, and Pf" =0.1,0.25  05 4.19 Simulation Results for Cooperative Space Time Coding with p s = 1, p„ = 0.9, and Pr = 0.1, 0.25, 0.5^ 49 5.1 SUSTC with pe = 1, different ps values, and mixture gaussian noise with r = 0.05 and ^ 500 (solid: Mixture Gaussian Decision Metric; dashed: Gaussian Decision Metric) ^ 5.1 5.2 SUSTC with pc = 1, different p, values, and mixture Gaussian noise with r = 0.05 and 4 = 500 (dashed: Gaussian Noise with GDM: solid: Mixture Gaussian Noise with MGDM) ^  55 5.3 SUSTC with p, = 0.99, different p, values, and mixture Gaussian noise with e = 0.05 and 4-^500 (solid: Mixture Gaussian Noise with MGDM; dashed: Gaussian NoiseQ! with GDM) ^  56 B.1 K = 3, g = [5 71 convolutioanl encoder^ (6 B.2 4-State Space Time Trellis Code.  (11 B.3 Extended Trellis for the 4-State Space Time Trellis Code^ 67 B.4 Extended Trellis after Expurgation for the 4 State Space Time Trellis Code. 72 vii List of Abbreviations and Symbols Acronyms AWG N^Additive white Gaussian noise AF Amplify and Forward BER^Bit Error Rate BPSK^Binary Phase Shift Keying CDMA^Code Division Multiple Access CRC^Cyclic Redundancy Check CSI Channel State Information CSTC^Cooperative Space Time Coding DF Detect and Forward FER^Frame Error Rate GDM^Gaussian Decision Metric MGDM^Mixture Gaussian Decision Metric MRRC^Maximum Ratio Receiving Combining MLE^Maximum Likelihood Estimation PEP Pairwise Error Probability Y. V.^ Random Variable RF Radio Frequency RCPC^Rate Compatible Punctured Convolutional Codes SNR^Signal to Noise Ratio ST Space Time STTC^Space Time Trellis Code viii STBC^Space Time Block Code STD Simple Transmit. Diversity SUSTC^Single User Space Time Coding UB Upper Bound VA^Viterbi Algorithm 3G 3rd Generation Operators and Notation Absolute value of a complex number Expectation Determinant Complex conjugate Matrix or vector transposition Matrix or vector hermitian transposition Real part of a complex number Imaginary part of a complex number ix Acknowledgments I would like to extend my sincere thanks to my supervisor, Dr. Cyril Leung. who has given me the chance to pursue my interest in wireless communication, and has always been there to provide me with invaluable guidance, and insights into my research. He has taught me the importance of fundamental knowledge in the field and how to solve a problem most efficiently, which will be a very important asset for pursuing my career. I would also like to thank Dr. Paul Lusina who has given me useful suggestions on my research. Finally. I would like to thank my mother and my sister who have always encouraged me to stay motivated and been supportive of my quest to pursue higher education. I would not have been able to complete this thesis without them. This work was partially supported by the Natural Sciences and Engineering Research Council Grant OGP0001731. Chapter 1 Introduction Two important goals for third generation (3G) cellular communication systems are to achieve high voice quality and provide high bit rate data services. However. due to the nature of the wireless propagation environment, time-varying multi-path fading causes the received signal strength to vary significantly, thereby making it difficult to achieve reliable communication [H. The concept of diversity, which is to provide the receiver with multiple versions of the information bearing signals that are subjected to fading. has been shown to be very effective in mitigating the effects of multi-path fading. Some well known forms of diversity are space, time, frequency, and polarization diversity [2]. Time and frequency diversity use different time slots and frequency bands to transmit signals. Multiple antennas with different polarizations for reception and transmission are used in polarization diversity. Space diversity uses different propagation paths. Antennas arc spatially separated so that the paths from transmit antennas to receive antennas undergo more or less independently fading. Space diversity, also known as antenna diversity, can be divided into two groups: reeeivt , and transmit diversity. Receive diversity makes use of multiple receive antennas that are well separated to ensure independent fading. Some known forms of receive diversity include selection, switched, equal gain, and maximal ratio combining diversity [21. Sim- ilarly, transmit diversity uses multiple transmit antennas which are spatially separated. Previous work on transmit, diversity can be broadly divided into two categories: syst ems 1 with feedback and systems without feedback. Generally, a system with feedback has a better performance. For example, with channel state information (CSI) at the transmit- ter, optimal transmit weights can be calculated to maximize the desired received signal power at the receiver and minimize the interference with other nearby receivers [3]. How- ever, extra signalling overhead is required for systems with feedback. A system without feedback generally has a poorer performance. The signalling overhead for such a system is lower and the the receiver is simpler [3]. Space-time (ST) coding over multiple transmit antennas is often used when CSI is unknown and can he divided into two forms: ST trellis coding (STTC) and ST block coding (STBC) [4]. For STTC. the space time encoder chooses the constellation points for each input symbol to simultaneously transmit from each antenna so that the coding and diversity gains are maximized. STTC often requires higher decoding complexity due to the complexity of the trellis struct ure. Maximum likelihood estimation (MLE) is often implemented using a viterbi algorit bin (VA) where the trellis path with the smallest accumulated distance is chosen [4]. An example of STBC is the Alamouti coding [5] which is attractive for its particularly simple decoding scheme. 1.1 Motivation and Goals Many wireless systems, such as cellular, ad-hoc, and sensor networks, have size, power and complexity constraints which limit the use of conventional transmit diversity meth- ods [2]. For example, in the uplink of cellular systems, the size of the mobile station is a constraint. More recently, a "cooperative diversity" technique in [6, 7] has been proposed which can achieve the same diversity order as conventional transmit diversity schemes. while alleviating the size, power, and complexity problems associated with conventional transmit diversity schemes. The idea in cooperative diversity is that each node in the network is assigned "partners . ' whenever possible. Each of the partners in the network transmits not only its OWII ill- 2 formation but also the information for its partners as well. Thus. it establishes a virtual antenna array and enables the nodes to achieve higher data rates and diversity order than what they could achieve on their own. Figure 1.1 shows an example Independent fading paths Figure 1.1: Cooperative Diversity where there are two mobile users, each with one antenna communicating with the des- tination. Diversity cannot be achieved by each user individually since there is only one transmit antenna at each user. However, by broadcasting its information to the other user and having the other user forward some version of the received information. along with its own data to the destination, transmit diversity can be achieved. The earlier proposed methods used for user cooperation often have the users repeat the information received from the other users [8, 9]. These repetition methods can be gener- ally divided into two forms: Amplify-and-Forward (AF) and Detect-and-Forward (DF). In AF, the users simply amplify and retransmit their received signals whereas in DF the users fully decode, re-encode, and re-transmit each other's information. Recent research work involves a combination of cooperative diversity with channel coding and space I inie coding. Instead of repeating the received information, the users attempt, to decode t h e partners' information and transmit added parity symbols according to sonic chosen coding scheme. Most of the studies use idealized assumptions such as independent fading and perfect channel estimation [6, 7, 8, 9i. In practice, insufficient antenna spacing can cause the 3 channel gains to be dependent and noise can cause the channel estimation to be imperfect at the receiver. Numerous studies have examined the bit error rate (BER) performance of conventional transmit diversity schemes in different channel conditions. For example. in [10, 11[, the performance degradations of the simple transmit diversity (STD) tech- nique with time-selective, spatially correlated fading, and imperfect channel estima Hi error were studied. Performance analyses of the space time codes with imperfect channel estimation error are provided in [12, 13[. In this thesis we analyze the performance degra- dation of one particular cooperative space time coding scheme due to spatially correlated fading and imperfect channel estimation. 1.2 Outline of Thesis In Chapter 2, a review of related work on cooperative diversity is presented. In Chapter 3, performance bounds on the proposed CSTC system with spatially correlated fading and imperfect channel estimation are derived. In Chapter 4, these bounds are compared to simulation results. In Chapter 5, the effect of impulsive noise is studied and t he gain obtained using a decision metric for non-Gaussian noise is examined. A summary of the main contributions and findings as well as some recommendations for future work arc provided in Chapter 5. 4 Chapter 2 Background and Related Work In this chapter, we present a summary of related published work on cooperative com- munication. In particular, the cooperative space time coding protocol model in [14] is reviewed as it is adopted in the thesis. 2.1 Relay Channel The basis of cooperative communication comes from the idea of the relay channel that was proposed in [15] and shown in Figure 2.1. Figure 2.1: Relay Channel. There is one sender, X, and one receiver, Y, with an intermediate node, R, which act s as a relay to help the communication from X to Y. First, X transmits its information to R and Y, then R transmits X's information to Y to help Y decode X's information inore successfully. This model can be viewed as consisting of two parts: a broadcast channel (X to R and Y) and a multiple access channel (R and X to Y). The work in [151 evaluated 5 certain non-faded relay channels, derived its lower and upper bounds on capacity, and concluded that the overall capacity is better than individual capacities achievable without relay in many cases. However, in [15] it was assumed that the relay can transmit and receive at the same time, which is not often practical. Later, the model was extended to multi-path fading [6, 8, 16], and additional constraints were added so that the source and relay transmit on orthogonal (in time or frequency) channels. 2.2 Block Fading Channel The block fading model is a useful approximation for a time varying fading channel when fading is not fast enough to be represented as a temporally i.i.d process, but it is also not slow enough to be well approximated by a quasi static model [17]. Information bit s arc coded/modulated into F blocks of length N symbols, resulting a codeword of length N F symbols. The NF symbol codeword is referred to as a frame and shown as C1,1 C1,2 C1,3 ••• C1,N C2,1 C2,2 C2,3 ••• C2,N c = (2.1) C F, 1 C F,2 C F,3 • • C F,N For a block fading channel, each of the F blocks is assumed to undergo different fading but the channel is time-invariant during each block. For example, the model used in t his thesis has a frame of two blocks when cooperation takes place with each block consisting of a codeword of length 130 symbols as follows Cl , ' C1,2 C1,3 ••• C1,130 C2,1 C2,2 C2,3 ••• C2.130_ C= (2. 2) 6 2.3 Impulsive Noise Impulsive noise has been used in characterizing man-made RF noise and low frequency atmospheric noise [18]. For such situations, the commonly used Gaussian Noise model is often not appropriate. Unlike Gaussian noise, the probability density funct (pdf) of impulsive noise has a heavier tail, causing a large deviation from the mean. SOW(' commonly used impulsive noise models, such as generalized Gaussian and Cauchy noise. Mixture Noise, Middleton Class A noise, and Laplace noise have been discussed in ]18]. One common feature that all of these non-Gaussian noise models share is that the tails of their noise density function decays at rates lower than the rate of delay of the Gaussian Noise model. In [19, 20], the performance bounds of systems with Class-A Middleton noise is analyzed. It is shown that the real part and the imaginary part of the complex Mixture Gaussian noise are statistically dependent. In this thesis, we focus on Gaussian Mixture Noise. Mixture noise is one kind of widely used impulsive noise model. Its pdf is pN (n) = (1 — e)77(n) + (n)^ (2.3) where € is the impulsive index, a constant value, 77(n) is a Gaussian representing ti n background noise, I(n) is some other density function with a heavier tail that represents the impulsive noise. When I(n) is also a Gaussian, p N (n) is a mixture Gaussian noise and the pdf is given by 1^n 2^ _] 1^[^77 2 PN (n) = (1 — e) ^ 20-2V27ro-2 exp [— 2a-7)2 ] + e  v27, a12 exp (2.4) where o-2 is the variance of the background noise and or is the variance of the impulsive noise. The total noise variance is (1 — c)a 2 + Eo-2 The ratio^and the impulsive index (/. are usually in the range [20,10000] and [0.01, 0.33] respectively [18]. Figure 2.2 compares the Gaussian Noise with Mixture Gaussian Noise. Different 4 and e values of the Mixture Gaussian Noise are provided. It is observed that a smaller / and 7 rt ► I Mixture Gpuy^ i ian : ^1 e = 0.05, a,/6n = 181^►^►► 1 t t^Mixture Ga2u4an : t t^1 = 0.05, a, /an = 25 ► i^te = 0.01, a l /an = 25 1^ t i i1 .^t 0.8 0.7 0.6 0.5 Mixture Ga.9ian : C 0 0.4 0.3 0.2 0.1 Gaussian -4^ -1^0 Data Figure 2.2: Comparison of Gaussian and Mixture Gaussian pdf's. All pdf's have a variaiicc of 1 4 -4 imply a heavier tail in the pdf. All pdf's have a variance of 1.a, 2.4 Cooperative Communication Cooperative communication is similar to the relay channel in that they both use the con- cept of relaying information from another node to achieve diversity. In the relay channel. the relay only helps the user to transmit information on a different path and does not carry its own information. However, in the cooperative communication model. each user not only transmits its own information; it also acts as a relay to its chosen "partner". As a result, the same diversity order as conventional transmit diversity schemes, such as A lani- outi and maximum ratio receive combining (MRRC) [14, 5] can be achieved. A general 8 concept of "User Cooperation" was first presented in [6, 7] and a general informal ion- theoretic model for two users was used to analyze the achievable rate region and ow agL probability. A CDMA-based implementation of the DF scheme was then proposed. It was observed in [6, 7] that under most scenarios, user cooperation allows an increase in system throughput and cell coverage, and a decreased sensitivity to channel variations. In general, cooperative communication methods can be divided into two general groups: amplify and forward and detect and forward. 2.4.1 Amplify and Forward (AF) Figure 2.3: Amplify and Forward. The AF method is shown in Figure 2.3. Each user first receives a noisy version of its partner's information. The users simply amplify and retransmit the noisy signal to the receiver. It is shown in [8] that although a noisy version of the partner's information is amplified, the receiver is still able to receive two independent versions of the desired information, thus allowing the receiver to better decode the information. The AF method was proposed in [21] and the BER performance was analyzed. It was shown that despite the noise propagation from the partner, AF outperforms non-cooperative transmission. Later in [9], the outage probability for AF in quasi-static Rayleigh fading was analyzed. It was found that AF signalling achieves a diversity order of two for two users. 9 2.4.2 Detect and Forward (DF) Figure 2.4: Detect and Forward. The DF method is shown in Figure 2.4. As in the AF method, each user receives a. noisy version of its partner's information. Instead of amplifying the noisy signal, the partner attempts to decode the information and retransmit an estimate of its partner's informa- tion to the receiver. In [8], it is shown that error propagation may arise if the partner decodes incorrectly and retransmits the erroneous information to the receiver. A hybrid DF method was proposed in which the partner only transmits the user's information when the signal to noise ratio (SNR) between the user and its partner is high. When the SNP between the user and its partner is low, the user reverts back to the non-cooperative transmission. The AF and DF methods discussed in [9} involve a user repeating the modulated symbols transmitted by its partner. From a channel coding point of view, this approach is not the most efficient. A new cooperative framework termed "coded cooperation" was introduced in [161 in which cooperative signaling is combined with channel coding. By having the users transmit on orthogonal channels enables the destination to separately detect each user. Figure 2.5 shows an implementation for a TDMA system used in 1221. 10 User 1: User 1 bits User 2 bits  Rx User 2^Inactive User 1 bits User 2 bits User 2:^Rx User 1^Inactive^User 2 bits^User 1 bits^Rx User 1^Inactive User 1^User 2^User 1 Transmission^Transmission^Transmission Figure 2.5: TDMA Implementation for Coded Cooperation. In [22), each user has a N-bit codeword to be transmitted. The two users cooperate by dividing their N-bit codewords into two time segments. In the first segment. each cser transmits a codeword with N 1 bits and attempts to decode its partner's transmission. Ti the user successfully decodes the partner's code word (through error detection such as CRC code), the user transmits N2 additional parity bits for its partner according to some overall coding scheme in the second segment where N i + N2 = N. Otherwise. the use! transmits the parity symbols for its own information. The users do not have knowledge of whether their own first segments were correctly decoded. Hence, the destination must know the decision made in the second segment by each user. One approach is for each user to send one additional bit in the second segment to indicate the decision made in the first segment. The other approach is for the destination to decode all possible scenario* thus increasing the complexity at the destination. It was indicated in [22) that the pro- posed model is flexible in that it can be implemented with either block or convolutional codes. Rate-compatible punctured convolutional code (RCPC) was implemented in [22 and the BER and FER performances were studied for both slow and fast fading. It was found that coded cooperation can achieve significant gains compared to a non-cooperative system while maintaining the same information rate, transmitting power, and bandwidth. In [23], "space time cooperation" was introduced which combined space time coding withh "coded cooperation." The main difference between space time cooperation and coded co- operation is that the users send both their own and their partner's parity bits in the secoiu I segment. However, some implementation issues such as the transmission in both channels in the second segments and coherent combining at the receiver were discussed. It was found that space time cooperation provides better performance than coded cooperation 11 in fast fading when the two user uplink channels have unequal average SNR. The uset with the better channel has to sacrifice its performance to help the user with t he worse channel in "coded cooperation" whereas in "space time cooperation", the performance for both users improves [23]. 2.5 Cooperative Space Time Coding (CSTC) The Cooperative Space Time Coding protocol proposed in [24, 14] is used throughout the thesis. Performance analysis of two users with multiple transmit and receive antennas was discussed in [14]. In this thesis, we consider two users, U 1 and U2, each with two transmit antennas and one destination node with one receive antenna. U2 U1 U1 Transmits Transmits for U1 U2 Transmits Transmits for U2 Cooperation U1Segment 1 U1 Segment 2 U2 Segment 1 U2 Segment 2  N/ 2^Et)^N/ 2 ^ (- 14^N/2^<--->^N/2 ^ (- I Transmits^Ul Transmits^U2 Transmits^U2 Transmits No 1.11 Segment 1 th Segment 2 U2 Segment 1 U2 Segment 2 Cooperation 1 4^N/2 <-14^N/2 k-1—>^N/2 E- —>^N/2 <— Figure 2.6: Time Division Channel Allocations. The cooperative model in [8, 22, 24, 14] uses time division channel allocations among the users as shown in Figure 2.6 for cooperation and no cooperation. In either case. there is a total of N time slots, which will term a frame, assigned to each user. The N time slot s are divided into two halves: segment 1 and segment 2. This leads to N/2 slots for U 1 t() broadcast its coded bits in the first segment and N/2 slots for possible cooperation in the second segment. Since the FER analysis for U 1 is symmetric to the FER analysis of U., we only analyze the performance of U1 for simplicity. 12 gil.1), independent gI2 Destination ) User 2 feedback Figure 2.7: Transmission Scheme : Segment 1. Case 1: U2 successfully decodes U l 's^Case 2: U2 fails to decode U l 's information information Destination g21 g22 J independent User 1 feedback User 2 independent g11 g12 ( Destination Figure 2.8: Transmission Scheme : Segment 2. Figures 2.7 and 2.8 explain the transmission in the two segments. The channel gains of antenna i (i = 1,2) for U 1 and U2 are denoted by and g2 . i and are assumed to be outcomes of independent zero-mean complex Gaussian r.v's with variance 0.5 per di- mension and are constant during the two segments for a given user. The path gains for the two users as well as those for different antennas of the same user are assumed to be independent as indicated in Figure 2.7 and 2.8. The first segment is used for U 1 to broad- cast its coded bits to the receiver and U2. In the second segment. U2 informed U t if the coded bits have been successfully received. If U2 successfully decodes U i 's infortilin ion. U2 transmits the remaining coded bits for U 1 in the second segment. If U2 fails to decode 13 SOURCE Convolutional Encoder, 221 V ANTENNA 2 V ANTENNA 1 110 BPSKModulator BPSK Modulator SOURCE Convolutional Encoder, 211 Convolutional Encoder, R22 BPSK Modulator Convolutional Encoder, 212 BPSK Modulator Ui 's information, U1 continues to transmit the remaining coded bits in the second t ime segment. This makes the scheme easier to be applied in most applications since Ihe new scheme is exactly the same as the single user space time coding scheme when there is no cooperation. When cooperation takes place, the destination observes a block fadne2, channel [25, 26] since the links from U 1 and U2 to the receiver are independent. When there is no cooperation, the destination observes a quasi-static fading channel. to [24) and [23], each user uses a cyclic redundancy check (CRC) code for error detection along with a space time code for error correction. A block diagram of the system is shown in Figure 2.9. Figure 2.9: Block Diagram of the Model. The information bits at each user, Ui = 1,2), are encoded by two convolutional encoders with generator polynomials fpk1' pk 2 } (k 1,2) where k denotes the segment index. The coded bits are then mapped to binary phase shift keying (BPSK) constellation. The out- put of the modulator from antenna i of segment k at time t is denoted by e tk ,. The coded bits from each user at time t are transmitted simultaneously from both antennas. The 14 received signal r t in the first segment at time t is given by + n t^(2 i = 1 where {t = 1, 2,^0 and L is the block length. When cooperation takes place. the received signal in the second segment is given by 92,ict2 i^nt (2.6) where {t^L + 1, L + 2, ..., 2L} and the noise over two segments, In t , t = 1. .... 2 LI . are outcomes of independent, zero-mean complex Gaussian r.v.'s with variance^per dimension. When there is no cooperation, the received signal in the second segment is rt = VE8 ^2^n t ( ) 7) z = 1 with It = L + 1, L + In [7], several suitable convolutional codes for cooperative space time coding that sat- isfy the algebraic design criteria in [25, 26] are obtained and shown in Table 2.1. The first column shows the constraint length K of the convolutional codes. The second col- umn lists the generator polynomials used for U 1 at the first time second segment. The third column lists the generator polynomials used for the coded bits that either U 1 or U2 transmits at the second time segment. Since each antenna transmits one encoded bit at a time using the corresponding rate 1 code for each user, the overall rate of the system is A bound on the frame error rate (FER) of the cooperative space time coding system was proposed in [24] as = (1 —)pfPF +^PY's pl3F p.iin pyS where Pm denotes the FER of the inter-user channel, P BF denotes the FER over the block fading channel when cooperation takes place, and PYs denotes the FER over the jocoop I 15 Table 2.1: Convolutional Codes Suitable for CSTC with Two Transmit. Antennas at Lich User in Octal Notation. Ui U2 K pi' P12 P21 P22 3 5,7 5,7 4 15,17 13,15 5 23,35 25,37 6 53,75 67,71 7 133,171 117,165 quasi-static Rayleigh fading channel from the user to the receiver when there is no coop- eration. Suppose the probability that the receiver decides erroneously in favor of a signal e^61,161,2•••61,m62,1e2,2•••62,Tri•-eL,IeL,2•••eL,rti ^ (2.9) assuming that C^el,le1,2•••Cl,TriC2,1C2,2•••62,rn,• • •CL,1CL,2•••CL ,n1 ^ (2.10) was transmitted was considered where m denotes the number of transmit antennas. \Vc can express the codeword difference matrix as 61,1 — c1,1 62,1 — C2,1 eL,1 — cL,1 61,2 — C1,2 62,2 — C2,2 eL,2^CL.2 B(c,e) = 61,3 — 61,3 e2,3 — C2.3 eL.3^CL.3 (2.11) el,m Cl,m e2,771. C2,m ••• • eL,m^CL,rn Suppose we have m transmit antennas at each user and one receive antenna and assume all channels, including the inter-user channel, have the same quality (Es , = Es..2 =^= E 16 using the pairwise error probability (PEP) expression (2) and (5) in (25], the union upper bound on the FER. (2.8) can be further expressed as pl3F pinDiaS < mifif 2E, < N̂ ) -(2m) E^1//eb4 0 e^c b=1 T71^ 77t • ( 4 No —^E(Fi ihi)m)(= 1/Ai)^(2.12) e^c i=1^c e^ c i-=-1 where ,u i =^and it 2 =^where 7i and Si are the nonzero eigenvalues of B(c, e)B(c, e) T for the fading block b 1 and b = 2 respectively. It is assumed that tlice codeword-difference matrices in both fading blocks, b = 1, 2 are of full rank. Similarly. A i denotes the i th nonzero eigenvalue of the product of the difference matrix and its con- jugate transpose for the quasi-static channel [25]. It is shown in [14] that for a good inter-user channel, P ir 2,-, 0, Pt"' ti Pr. Dc f oop ( tc  Es ^27-11 4N0 (2.13) where i denotes the most dominant term in the summation. This shows that when P'" is close to zero, cooperative space time coding achieves a diversity order of 2w with In antennas at each user. Similarly, for a poor inter-user channel[14] assumed hat^(P f^f^i ^ n 1), pCoop^p , pys (Esth /4No) m2 < Gin where Cin is a very small number due to the poor inter-user channel so that 0Coop f ploCoop f 1 (ES Cin 42V0 )^F (2.1-1) where F denotes the minimum product of the codes eigenvalues which dominates I he per- 17 formance at high SNR . . Hence, the maximum diversity order that the cooperative spar( time coding can achieve with poor inter-user channel is m [14]. 18 Chapter 3 CSTC with Spatially Correlated Fading and Imperfect Channel Estimation In this chapter, we obtain bounds on the performance of CSTC with convolutional encod- ing and BPSK modulation in spatially correlated Rayleigh fading with imperfect channel estimation. For clarity, we use uppercase letters to denote r.v.'s and the corresponding lowercase letters to denote their sample values. 3.1 System Model Since the coded bits for the two segments are transmitted at different time and the channel gains are independent from user to user as in Section 2.5, we analyze the FER perfor- mance for the first segment with quasi-static fading and extend the analysis to block fading channel for cooperation. If there is no cooperation in the second time segment. the analysis is still valid since a quasi-static channel is equivalent to a block fading channel with same channel gains for each block. Within user j, the channel gains and C 1 ,2 are assumed to be independent (j = 1,2) in Section 2.5. However, in reality. insufficient an- tenna spacing may cause spatial correlations between antennas. In practice. most cellular communication systems require an antenna spacing of 50A to 100A at the base station [27 19 to ensure independency between antennas. In this section, we discuss the performance of CSTC when the channel gains are spatially correlated and when the channel estimation is not assumed to be perfect at the receiver. Similar to the model used in Section 2.5, we use two transmit antennas at each user and one receive antenna at the destination. However, the two transmit antennas at each of the two users are close to each other and hence their channel gains are spatially correlated. The received signal at time t in segment 1 is given by rt = E, + n 1 (3.1) i=1 where the noise {n t , t = 1, 2, ..., L} are outcomes of independent, zero-mean complex Gaus- sian r.v. with variance iv? per dimension and L denotes the block length. As mentioned in the discussion of block fading channel in Section 2.2, a frame consists of two block s for CSTC model. The channel gains G 1 , 1 and G1,2 are zero-mean complex Gaussian r.v. with variances o-6 equal to 0.5 per dimension and are assumed to be constant wit bin the transmission of one block. The term ctk i corresponds to the coded bit to be transmitted at time t from antenna i in segment k (i, k = 1,2). The variance of a complex Gaussian r.v.. i.e., X is defined as the variance of either its real or imaginary component and is denoted by o-2x in this thesis. The real and imaginary parts are assumed to be independent. The channel gains are correlated for the two paths of each user but are independent from user to user. The correlation coefficient, ps of G 1 , 1 and G1,2 is Ps EHG 1,11 2 1EHG1,21 2 ) and the covariance matrix of G1,1 and G1,2 can be expressed as 2 aG^vs`-'G ps oG o CG E {G1 ,1^,21 (3.2) (3.3) 20 hi] n^Spatially Correlated h 12 U Destination In [14], the channel gains are assumed to be perfectly known at the receiver. In our model. the channel gains are to be estimated from the received signals. This can be done by using pilot symbol channel estimation [28] where a sample of the estimated channel gain from antenna 1 can be obtained as rt=.- E's c l 91,1+ nt (3.1)  Let .z1,1 -=^denotes the channel gain estimation error. Then Z 1 , 1 is a zero mean independent complex Gaussian r.v. with variance al and H1 , 1 is a zero mean complex Gaussian r.v. with variance „r2^ 2^ (3..5 ) Similarly, a 2H12^a61,2^The system model for our performance analysis is shown in Figures 3.1 and 3.2. V^V Figure 3.1: Modified Transmission Scheme : Segment 1. Figures 3.1 and 3.2 explain the transmission of the two segments. The transmission scheme is the same as that described in Section 2.5 except the actual channel gains are replaced by h 1 , i and h2,i where h 1 , i and h2, i denote the estimated channel gains from antenna i of U1 and 112 to the destination (i 1, 2). 21 Spatially Correlated h11 User 1 L7// 11 12 ( Destination • feedback • feedback Destination h21 h22 Spatially Correlated Uler2 Figure 3.2: Modified Transmission Scheme : Segment 2. Case 1: U2 successfully decodes U 1 's information Case 2: U2 fails to decode U i 's information The channel estimation error is assumed to be independent from antenna to ant elm?) independent of G 1 , i.e. = E[GLili,i] =^= 0 , (z,^= The correlation coefficient, pe , between G 1 i and H1, i is defined as E[G 1,,(G^+ Z1,i) 1, 2) crc 0-11 (3.0) (3.7) ( 3 . (3.9i =Pe E{IG 141 214 1101^^20-6)(2a2H) It follows that 0.2 G H =^2 P, and the variance of the channel estimator error can be expressed as „.2^(I^-1 ) 0.6 . Pe Also, we have = ERG + Z 1,1)(G 1,2 + Z1,2) * ] = 2 P 8 0'6 ^ 3.10! E[G1,1-B7,2) = Er1,1(C1,2 + Z 2)1 = 2P80-6^(3.11 E[G1,2Hij]^E[G1,2(G1,1 + Z1,1) * ] = 2 9Q -;.^(3.12) 22 From (3.8) and (3.10). the covariance matrix of H 1 .H12 is  ,,2]C7 2 Psu G (7 2[Ps 0-6 CH = 19 Ps G 2^• Psaa21s.Pe (3.13) 3.2 Performance Analysis Based on Estimated Chan- nel Gains In this section, we analyze the performance of the CSTC system by fixing the value of the inter-user FER, Pr, and deriving the cooperative FER , 13.1 '9°P based on the expression in [14] where pyoop^(1^pljn)^pp.? pi S^ (3.11) When the partner successfully decodes the user's information (Pr = 0). Pic.'""" reduces to the FER of the block fading channel. When the partner fails to decode the user's information (Pin = 1), Pic°°P reduces to the FER. of the quasi-static channel. To determine the maximum likelihood (ML) decoding metric for the received signals given H11 and H1,2, we first find the joint pdf G 1 , 1 and G1,2 given H1 , 1 and H 1 9. PG1,1 'G1,21 111,1,111,2^91,21111,1, h 1 , 2 ).^From (3.3) and (3.13), the covariance matrix of G i ,, ,G 1 , 2 , H1 , 1 , and H1,2 is 2ac. Psa6 a 2 2UG 2PsaG 2Psa6 P 8 01 GCGH = 2 G Psa6 2^2uG I Pe Psa6 (3.15) Psa6 (72 Psa6 ac/Pe and the joint pdf of G1,1, C1,2, H1,1, H1,2 can be written as 23 1^ 1PGI,I , G1.2. 1-1 11.111.2 (91,1. g1,2, h1,1, h1,2) = (27)2(det(CGH))1/2 exP(T ^1 2 XGH XG1 -1 ) (3. where XGH = [G1,1 G1,2 H1,1 H1,2] and CG-1H is the inverse of the covariance matrix CGH The pdf of H 1 , 1 and H1 , 2 is h1,2) = 2̂7r(det(CH))1/2 exp(--2XHTCH1XH) 1^1 (3.17) where XII = [H 1 , 1 111 , 2 ] and CH is given by (3.13). Using (3.16) and (3.17), we can write the joint pdf of G 1 , 1 ,G 1 , 2 given H 1 . 1 , H 1 , 2 as 91,211/1,1, h1,2) (91,1, 91,2, h1,1, h1,2) h1,2) 1 27a2D V 1 - exp( - m1,0 2^2Pd(91,1 -^- m1,2) + (91,2 - m1,2)2 ) (3.1)Pd2a 2D^- with mi,2 (h1,1(1 -^+ hi,2Ps(1 - aPe2 1 - 161 4 (h1,2(1 - ps2 142 ) + ht,1Ps(1 - Pe2 ))P 1 - P,t4 aC( 1^Pe2 )( 1^982 P,2 ) D 1 - 64 Pd = Ps( 1 Pe) 1 - Ps Pe (3.19) (3.20) (3.21) (3.22) As in [11], we can express 9 1 , i , 9 1 , 2 in terms of h 1 , 1 , h 1 , 2 as 91,1 = m1,1 + d1,1 = ahia + bh1,2^d1,1^ (3.23) 24 91.2 = /71 1,2^d1,2 = bh1,1^ah1,2+ di,2^(3.21) where pe2(1^ps2pe2) a ==  ^ (3.25) 1 — 1).10 ‘ b = P2eP 8( 1 — (3.26) 1 — and D 1 , 1 and D1,2 are zero mean correlated complex Gaussian with variance an correlation coefficient pd . The received vector for the first time segment can then be expressed as rt = V Esgucti,i + Es91,24,2 + nt -=.^1E87711,24,2 + VEsdi,ictl,i + VE.9d1,2ti,2 nt ^(3.27) With (3.19) to (3.22), R t given H1 , 1 and H1,2 can be written as [1] PRt /11 1 H1 '  2 (rt, h1,1, h1,2) PH1,1,H1,2 (h1,1, h 1,2) 1 ^ exp( 1 ir t — 1.11 2 .V2R-o-2 2 o-2Rtrf 11,1,111.2 where p, = ^.Es^rni,i ,--,i,i i=1 ,,.2^(1 — 4( 1 — ps2 /9e2 )^I 1^2 ^ No ' RcIFILI.111,2 = Es ^ ^+2Esci,ict,2pdaD + 2 •1 — P.i 4 (3.28) (3.29) (3.30) For the case where the channel gains with each user are independent, i.e. p,. = 0. (3.29) and (3.30) reduce to (3.31) i=1 a2Rt1H1, 1111,2 = E 5 (1^p e2 ) +^ (3.32) , which are the same results as shown in [311. 25 Suppose a codeword (3.33) is sent in segment 1 and the received vector is r^r2 r3•••T'L ^ (3.31) where L denotes the block length. Optimum decoding amounts to choosing a codeword 1 .0 1 „I.^I.^1^1 e^e1 ,1'1,2'2,1'„2,2 „ , (3.35) for which the likelihood P(r(e, H1,1 = h1,1, H1,2 = h1,2) is maximized. The components of the received vector, r given the estimated channel gains are assumed to be independent i n [29]. However, it is noted in [30] that this assumption is mathematically inaccurate. no explanation was given in [30], a discussion of this dependency is given in Appendix 13. We assume that the elements of the received vector, r, are independent, in order to ob- tain a closed form analytical bound which is then compared with our simulation result s. Taking the logarithm of the likelihood P(rie, H1,1 = h1,1, H1,2 = /11,2), we decide in favor of the codeword e which minimizes the quantity — log P(r t e tl i e t1 2 , H1,1 = hi,i, H1,2 = h1,2).^(3.36) t-i From (3.36), the decoder chooses e to minimize L^ 2 (^ irt — ^Es Ei=iet,i (i2 No + 4Esq,14,2Pda2D + 2E8 0 -11(pli- 4)p1-1 „„i ,i^+ 2 ,7rE ( 1 — PD( 1 — 19 ,2A) \+ 1 In 71-No + 47rEs e li 1— 1),'42^ ,i-t,2P-aD^.9 / (3.37) For the case where the channel estimation is not very poor (p,.^0.99). we can ISS1111)(' that 2Eset1,1q,2Pd0-2, ti o 26 with (3.44)d2 (c, e t=1 i= 1 Thus. by dropping the constant terms,(3.37) can be simplified as t=. A frame error occurs when E I rt — B/Es t==1  2 (3.3) (3.39) Expanding (3.39) and we get 2 X = r t V Es E^71 V Es ml,i i=1 =1 2 2 +^Es 2^ Es2 E^jet1,i1 2 > 0^(3.40) i= 1 i=1 where X given H1,1 and H1,2 is a real Gaussian with mean and variance as /Ix^—Es id^1,i^— el I 2,11 (3.41) Qx = 20- R2^,H,,2 EsL 2 I (3.42) i= 1 Hence, the probability of X I H1 , 1 , H1,2 greater than zero is Pr{X}Hi,i, Hx1 , z > 0} ( O 2 x) Es^\I \ 2 (c, e)^(^, ^d 3.43) 2N0 + 4E8 (1-P e. )(1-(''' ) ^ 1--p(4 / = 7 From (3.38) to (3.44), we can easily extend the quasi-static analysis to block fading \viler(' the probability of deciding in favor of e given the estimated channel gains for CSTC 27 with (3.48)d2 (c, e) = h i (ct — et,i)1 2, t=1 i=1 spatially correlated fading and imperfect channel estimation error over two segments is P(c ---> elf/j„ =^j = 1,2) Es = Q 2No + 4E, (1-4)(1-P ^d2(c. e) i-pi (3 45l with 2 d2 (C, e) = E^E b=1 t=1 i=1 b^ ht,')2e (3.46) and B refers to the number of blocks in the proposed model. The number of blocks for the block fading channel model used in the thesis is equivalent to the number of iimc segments used for coded bits transmission as in Figures 3.1 and 3.2. It is assumed t hat the derived closed form PEP conditioned on the estimated channel gains assuming the el- ements of the received vector are independent, is an Upper Bound to the actual simulatiolt. For the case with imperfect channel estimation (i.e. pe^1), independent channels (i.e. ps^0), and B = 1, (3.45) and (3.46) reduce to P(c—^-= hi, i = 1, 2) = Q E, d2(c ' e) '4 2N0 4E,(1 — (3.47) which is the same result as shown in [31] for space time trellis codes with imperfect cl ien- nel estimation. For the case with perfect channel estimation at the receiver (i.e. pe = 1), independent channels among each user (i.e. ps = 0), and B = 1, (3.45) and (3.46) reduce to Es P(c^= g i , i = 1, 2) = Q^nrod2 (c, e) (3.19) 28 with L^2 d2 (c, e) = gi (cf.i — ct.a^ (3.511) t=1 which is the same result as shown in [4] for perfect channel estimation and independent channel gains. 3.3 Evaluation of Expurgated Union Bound with Lim- iting Before Averaging Technique In the standard union bound technique, we get Pf f  r§jEc etc P(c^= h)pH (h)dh^(3.51) where H denotes the vector of estimated channel gains. p H (h) is the joint prohabilit■ density function of H obtained in (3.17), and IS1 denotes the total number of cociewords in the space time code, which is 2 L . As shown in [32], the union bound evaluated in a straight forward manner is quite loose for quasi-static, or block fading channels. Following [33], we will tighten the bound by performing expurgation of the standard union bound and use the limit before averaging technique proposed in [32] where we limit the conditional union upper bound on the error probability before averaging over the fading distribution. The expurgated union bound with limit before averaging technique is Pf < f min 1, — Isl P(c 01) c e^ c pH (h)dh  < I min 1, 1 \̂ , EQ !SI L""c etc Es 2N0 + 4E8^pf-4(1-1_)(lPP) d2(c. e) pH h Idh (3:52; 29 where P(c^01) corresponds to the expression derived in (3.45). The distance term. d2BF (c. 6) for PfBF is d2BF(c• e) = 1^Alb.1 21 " 6,1 + 2 ^b=1 ^ b=1 2 ^+ 2R{ (3.53) b=1 where B corresponds to the number of blocks (segments) and  1(c1^e lL1)1 2 t=1 1(et,2^e t b ,2)I 2 t= 1 (3.54) (3.55) (3.56)Bb t=1 The distance term, 4s (c, e) for PQs is d2^ ,■.-■-■^1 2 di +^12 A2QS^e)^=^2 1 ,, q,11^41,,,1,21 + 2R{ 2 (m1 ,1 1 . 2) 13 } (3.57) where L =^— t=1 L ==^1(ct,2^et,2)I 2 t=1 (3.58) (3.59) B =^t,1^et,1)(et,2^et,2) * •^ (3.60) t=1 Detail explanations about the extended trellis and the distance terms resulted from the expurgation technique are shown in Appendix B. 30 Chapter 4 Comparison of Analytical and Simulated Results For illustration purposes, we consider BPSK modulation and the constraint length 3. four-state cooperative space time trellis code in Table 2.1 with a frame size of 2(30 bits . Examples with a higher number of states were also simulated. However, only analyt ical results for a constraint length 3 code were obtained because the size of the expurgated transition matrix for codes with a higher number of states is too big. (See Appendix B for a discussion of the complexity) 4.1 FER Comparison The analytical Upper Bounds (UBs) and simulated FER curves are plotted as a function of the SNR for different (p5 pc ) values. By increasing p5 , the channel gains between the two transmit antennas with each user become more correlated whereas the estimates of the channel gains get worse with decreasing Pe . The SNR is defined as the ratio of the variance of the complex channel gains to the variance of the additive Gaussian noise, i.e.. a No  or ----L2E as in Appendix C. Analytical and simulated FER curves for two extreme cases: SUSTC (Pt = 1) and CSTC (Pt = 0) with spatially correlated fading and imperfect channel estimations are presented where Pr denotes the FER of the inter-user channel between users. 31 14 16 18 202^4^6^8^10^12 SNR (dB) ps - 0.0, p e = 1 p s - 0 5 pe 1 p s — 0 8, pe = 1 V ps — 1 , pe = 1 4.1.1 Single User Space Time Coding (SUSTC) The FER performance of the SUSTC with different (pe p4 values is discussed in this section. The simulation results are then compared with the corresponding UB curves. Figure 4.1: Comparison of Analytical and Simulation Results for Single User Space Time Coding withh fixed ps at 1 and varying p s : BPSK modulation, (5, 7, 5, 7) convolutional code (upper hound: dashed lines; simulation: solid lines) In [33], the FER performance of the 4-PSK, four-state space time trellis codes proposed in [4] with two transmit and one receive antennas over spatially correlated quasi-static fading channels with perfect channel estimation and a frame size of 130 symbols was ana- lyzed. It was observed in [33] that the UB captures the diversity and accurately predict s the performance of the space time trellis codes and is about 3.5 dB away front the re- sults obtained through simulation. As for the SUSTC model proposed in the thesis. full diversity order was achieved with perfect channel estimation (pe = 1) and independent 32 quasi-static Rayleigh fading channels (ps = 0) as in Figure 4.1. Diversity order is defined as the gradient of the performance curve as SNR approaches infinity. It is observed t hat the FER increases as ps increases from 0 to 1 with pe = 1. From the simulation curves at a target FER of 10 -i , there is about 0.5 dB degradation when p. increases from 0 to and about 1.8 dB degradation when p, increases from 0 to 0.8. The degradation grows to 6 dB as p, increases from 0 to 1. From the analytical UB curves, we observe that the expurgated UBs with limiting before averaging technique [32] are about 3.5 to 4 dB away from the corresponding simulation curves. Even though the bound is loose, it captures the FER degradations of the simulation results nicely as the degradations are about 0.17. 1.7 and 6.1 dB as ps increases from 0 to 0.5, 0.8 and 1 respectively. Figure 4.2 shows the FER performance of SUSTC with different p. values and = 0.99. It is observed that the FER increases as p, increases from 0 to 1 with pc = 0.99. For a target FER at 10 -1 , the simulation curve shows a 0.67 dB degradation when p s increases from 0 to 0.5 and a 2.2 dB degradation when ps increases from 0 to 0.8. The degradation grows to 6.3 dB as ps increases from 0 to 1 when the two channel gains are fully correlated within each user. The FER degradations for the UB curves are 0.73, 2.5, and 8.1 d13 as ps increases from 0 to 0.5, 0.8, and 1 respectively. If p e continues to decrease to 0.9 as shown in Figure 4.3, the target FER of 0.1 cannot be achieved. It is shown in Figure 4.3 that with pc = 0.9, as A, increases to 1, the simulated FER outperforms the cases with p. = 0, 0.5, and 0.8 at high SNR and a performance reversal occurs when p s increases from 0.8 to 1 in the UB curves. A possible explanation for this is as p. 1, the probability of both channels are good is higher. When both channels are good, the probability of decoding the frame more correctly is higher. From Figures 4.1, 4.2, and 4.3, the FER curves exhibit a floor. The floors represent the best performance that SUSTC can achieve given the corresponding pc value. "file analytical expression for the floor can be obtained from (3.47) and (3.52) with the distance term equal to (3.57). Generally, it is observed from Figures 4.1. 4.2, and 4.3 t hat the FER increases as ps increases from 0 to 1 and as pc decreases from 1 to 0. For p, 33 p s 0.0, pe — 0.99 p s — 0 5 pe = 0.99 0  p s — 0 8 pe = 0.99 p s — 1.0, pe = 0.99 cc w 1 0 0 10' 1 0 2 10 3 5^10 15 SNR (dB) 20^25^30 Figure 4.2: Comparison of Analytical and Simulation Results for Single User Space Time Coding with fixed pe at 0.99 and varying p : BPSK modulation, (5, 7, 5, 7) convolutional code (upper bound: dashed lines; simulation: solid lines) = 1, the FER degradations match well with the simulation results with increasing p,,. However, the degradations between the analytical and simulation results are different when pe decreases to 0.99 because the UB curves have higher floors which cause the U13 curves to reach the flattening part faster at the target FER. Similar to [11] where the BEH performance analysis of Alamouti coding with spatially correlated Rayleigh fading and channel estimation error was presented, it can be observed that spatial correlation causes the FER performance to degrade more as the channel estimation error increases. For a target FER of 10' and pe = 1, the degradations are about 0.5 and 1.8 as ps increases from 0 to 0.5 and 0.8 respectively. When pc = 0.99 , the degradations increase to 0.67 and 2.2 dB respectively. 34 1 0° ^ ps — 0.0, p e = 0.9 o^ p s — 0 5, pe = 0.9 ps — 0.8, p e 0.9 p s — 1.0, pe = 0.9 2^4^6^8^10^12^14^16^18^20 SNR (dB) Figure 4.3: Comparison of Analytical and Simulation Results for Single User Space Time Coding with fixed p. at 0.9 and varying p, : BPSK modulation, (5, 7, 5, 7) convolutional code (upper bound: dashed lines; simulation: solid lines) 4.1.2 Cooperative Space Time Coding (CSTC) The FER performance of the CSTC with different (pe ps ) values is discussed in t his sec- tion. The simulation results are then compared with the corresponding UB curves. In [34], the FER. performance of the 4-PSK, four-state space-time trellis code proposed in [4] with two transmit and two receive antennas over independent block fading chan- nels with a frame size of 520 symbols was analyzed. Full diversity order was able to be achieved with the given codes. As shown in Figure 4.4, CSTC with perfect channel est Una- tion (pe = 1) proposed in the thesis achieves full diversity order over the independent I )lock Rayleigh fading channels (p s = 0) and the FER performance degrades with increasing p„. 35 14 16 18 202^4^6^8^10^12 SNR (dB) 1O 2 - 1 0 -5 - p s — 0 0, pe = 1 p s — 0.5, pe = 1 ps — 0.8, pe = 1 p s — 1, pe = 1 0 Figure 4.4: Comparison of Analytical and Simulation Results for Cooperative Space Time Coding with fixed pe at 1 and varying p s : BPSK modulation, (5, 7, 5, 7) convolutional code (upper 1)01111(1: (lashed lines; simulation: solid lines) From the simulation curves at a target FER of 10 -2 , there is about 0.6 dB degradation when p, increases from 0 to 0.5 and about 2.0 dB degradation when p, increases from 0 to 0.8. The degradation grows to 7.3 dB as ps increases to 1. From the analytical UB curves. we observe that the expurgated UBs with the limiting before averaging technique 1:32 are about 3.5 to 4 dB away from the corresponding simulation curves. The degradations for the UB curves as p, increases match well with the simulation curves. As ps increases from 0 to 0.5, 0.8 and 1, the degradations for the UB curves are 0.6, 2.0, and 7.5 dB respectively. Figure 4.5 shows the FER performance of the simulation and UB curves with p, = 0.99 and different values of A,. As pe decreases from 1 to 0.99. with a target FEB at 36 ps — 0.0, p e = 0.99 p s — 0.5, pe = 0.99 p s — 0.8, p e = 0.99 p s = 1.0, pe = 0.99 8^10^12 SNR (dB) Figure 4.5: Comparison of Analytical and Simulation Results for Cooperative Space Time Coding with fixed pe at 0.99 and varying ps : BPSK modulation, (5, 7, 5, 7) convolutional code (upper bound: dashed lines; simulation: solid lines) there is about 0.68 dB degradation when p, increases from 0 to 0.5 and about 2.5 dB degradation when ps increases from 0 to 0.8. When p s increases from 0 to 1, the degrada- tion grows to 8 dB. The degradations for the UB curves are 0.9 and 3.3 dB as p, increase!, from 0 to 0.5 and 0.8 respectively. When ps increases to 1, the degradation can not be observed as the target FER cannot be obtained. If pe continues to decrease, ie. 0.9. as shown in Figure 4.6, the target FER of 0.01 cannot be achieved. Similar to the SUSTC with pe = 0.9, as ps increases to 1, the simulated FER of CSTC outperforms the ease where ps = 0, 0.5, and 0.8 at high SNR and a performance reversal occurs as A, increase from 0.8 to 1 in the analytical UB curves. Possible explanation for this because as p, 1. the probability of both channels within U 1 and U2 are good is higher. Thus, the prolm- 37 O p s = 0.0,p e =0.9 ^ p s = 0.5, p e = 0.9 • p s = 0.8, p e = 0.9 ^ p s = 1.0, p e = 0.9 bility of decoding the frame more correctly is higher. 2^4^6^8^10^12 ^ 14 ^ 16 ^ 18 ^ 20 SNR (dB) Figure 4.6: Comparison of Analytical and Simulation Results for Cooperative Space Time Coding wit!, fixed pe at 0.9 and varying p, : BPSK modulation, (5, 7, 5, 7) convolutional code A performance floor for CSTC can also be obtained at ps = 0 as shown in Figures 4.-1. 4.5, and 4.6. The floors represent the best performance that CSTC can achieve at a given pe value. The analytical UBs for the performance floor can be obtained from (3.47) and (3.52) with the distance term equal to (3.53). Generally, it is observed from Figures 4.5, and 4.6 that the FER. increases as ps increases from 0 to 1 and as p,, decreases front 1 to 0. Similar to SUSTC, it can be observed that spatial correlation causes the FEI1 performance to degrade more as the channel estimation error increases. For a target FEB of 10 -2 and pe = 1, the degradations are about 0.6 and 2.0 as p, increases front () to 0.5 and 0.8 respectively. When pe = 0.99, the degradations increase to 0.68 and 2.5 313. 38 Ips — 0.0, pe = 1 ps = 0.5, pe = 1 ps — 0.8, pe = 1 ps = 1, pe = 1 4.1.3 Comparison of SUSTC and CSTC The simulated FER curves for SUSTC and CSTC with spatially correlated fading and perfect channel estimation are presented in this section. 10° 10 1 1 0 2 10' 3 LU LL 10 5 10 6 2^4^6^8^10^12^14^16^18 ^ 20 SNR (dB) Figure 4.7: Comparison of the Simulation Results for Single User and Cooperative Space Time Coding at pe at 1 and different values of p s (Cooperation: dashed lines; Single User: solid lines) Figure 4.7 shows the FER. curves for SUSTC and CSTC as a function of SNR for different values of ps with p, = 1. It is observed that at high SNR, the diversity order for SUSTC and CSTC are approximated to be 2 and 4 respectively for p s equal to 0. 0.5, and 0.8. In particular, the diversity orders for SUSTC and CSTC obtained for p, = 0 confirm wit h the analysis shown at the end of Chapter 2 where the expected diversity order for SUSTC is equal to the number of transmit antennas (2 in the thesis) at each user whereas the expected diversity order for CSTC is equal to the sum of the number of transmit antenna, for both users (4 in the thesis) [14]. When ps increases to 1 (i.e. the two channels are 39 fully correlated), independent fading paths can no longer be assumed within each user. Thus, the diversity orders decrease to 1 and 2 for SUSTC and CSTC respectively. 40 4.2 Cooperation Gain Comparison An alternative way to analyze the simulated results to determine the performance improve- ment through cooperation compared to SUSTC is to use the cooperation gain defined in [35] as No—Coop^ pQS Gf = P^ =- fCoop^(1 — Pi")P7F Pfn P C/ S 1 (1 — Pin)8 f Pin^ (4.1) Nwhere P o—coopS^PYs and e f quasi-static FER for the users. lower FER. B F Pf is the ratio of cooperative blocking fading FEB When Gf > 1, the users benefit from cooperation with ;1 Most of the previous sections focus on the extreme cases where the users either perfect ly decode the partner's information (Pyi = 0) or fail to decode the partner's information (Pr = 1). In the following sections, the FER performance of SUSTC and CS —ff.' for different values of (p8 pe ) and inter-user channel qualities are presented. Cooperation gains are then computed from the simulation results in order to determine the benefits of CSTC over SUSTC under different channel conditions. 4.2.1 Perfect Channel Estimation (pe = 1) The FER curves and cooperation gains for SUSTC and CSTC with spatially correlated fading, perfect channel estimation, and different Pr values are presented in this section. 41 10' 1o 2 10 3 cc U. o^ Single User, Pr = 1 —a— P' n = 0.1f 0^ = 0.25 P lin = 0 ' 5 Cooperation, Pin = 0 10 SNR (dB) 1 o 6 i o 7 0 15 20 1 0 ° 10 -1 10 0 -3 u_ -e-- Single User, P;" = 1 n. , P i" = 0. 1 0^ P it" = 0.25 P in = 0 5 io ^Cooperation, P fin = 0 N \ \ 1 10^15^20SNR (dB) Figure 4.8: Simulation Results for Cooperative Space Time Coding with ps = 0, p,^1, and P" 0.1, 0.25, 0.5 Figure 4.9: Simulation Results for Cooperative Space Time Coding with p„,=-- 0.5, p 1, = 1, and 1 );" 0.1, 0.25, 0.5 42 ^o^ Single User, P fl° = 1 -^P tin = 0.1 = 0.25 p n = 0.5 - +. Cooperation, P lfn = 0 o^ Single User, Pi n = 1 = 0.1 • 0-• - P"=0.25 v P"=0.5 Cooperation, Wi n = 0 5 ^ 10 ^ 15 ^ 20 SNR (dB) Figure 4.10: Simulation Results for Cooperative Space Time Coding with p,^0.8, p„ =1rued Pi" -= 0,1, 0.25. 0.5 2^4^6^8^10^12 ^ 4^16^18^20 SNR (dB) Figure 4.11: Simulation Results for Cooperative Space Time Coding with p, = 1, p, = 1, end Pr =- 0.1, 0.25. 0.5 10° 10 0 cc U.1 10 L 2 10 -6 0 cc U- to' 43 Table 4.1: Cooperation Gains for pe = 1. Pi" = 0.1. 0.25. and 0.5 SNR^p , =^p.. = 0.5 p, = 0.8 n. P"' = 0.1f 8 4.9098 4.3508 3.2021 1.6220 12 7.9517 7.5144 6.3890 2.7062 16 9.6759 9.5377 9.1398 4.6209 20 9.9694 9.9575 9.9113 6.7045 P)" = 0.25 8 2.9727 2.7917 2.3424 1.4696 12 3.6837 3.6028 3.3659 1.6423 16 3.9558 3.9364 3.8783 2.8818 20 3.9959 3.9943 3.9881 3.4369 1) 'f" = 0.5 8 1.7934 1.7478 1.6183 1.2707 12 1.9444 1.9291 1.8818 1.3527 16 1.9926 1.9893 1.9793 1.7709 20 1.9993 1.9991 1.9980 1.8964 Figures 4.8 to 4.11 show the FER curves of the single user performance and the two-user cooperation systems with different ps values, perfect channel estimation, and differem inter-user channel qualities. From the FER curves, Table 4.1 is obtained. From Table 4.1. we observe that with perfect channel estimation (pe = 1), Gf decreases with increasing p. when Pr = 0.1, 0.25, and 0.5. For example, as p s increases from 0 to 0.8 at SNR. equals to 20 dB and Py = 0.1, G f decreases from 9.9694 to 9.9113. This is expected because as the channel gains get more correlated, the benefit from the diversity decreases. Even when ps increases to 1, where the two channel gains are fully correlated or when the inter-user channel quality gets very poor (Pr = 0.5), the users still benefit from cooperation as G 1 does not fall below 1. For example, as P7 increases from 0.1 to 0.5, G f decreases from 9.5377 to 1.9893 at p s = 0.5 and SNR equals to 16 dB. 4.2.2 Imperfect Channel Estimation (pe = 0.99 and 0.9) The FER. curves and cooperation gains for SUSTC and CSTC with spatially correlated fading, imperfect channel estimation. and different Pin values are presented in tins heel ion. 44 10' - -- Single User, Pr 1 o P"=0. 1 0^= 0.25 ^ Pr = 0.5 — Cooperation, P;" = 0 1 0° 10 ler U- 10' 4^6^8^10^12 SNR (dB) 4^16^18^20 O Single User, P1" = 1 -0- •••••^= 0.1 - -0■ - P 1° = 0.25 ^ - 0.5 Cooperation,^= 0 Figure 4.12: Simulation Results for Cooperative Space Time Coding with p. = 0, p,. = 0.99. and Pi". = 0.1, 0.25, 0.5 Figure 4.13: Simulation Results for Cooperative Space Time Coding with p s = 0.5. p„.= 0.99. and P;.' = 0.1, 0.25, 0.5 45 angle User, Pr = 1 o^- 0.1 0 ^_ 0.25 = 0.5 Cooperation. Pr = 0 N, 10° cc LL 10 2 O^ Sonia User, I.'," = 1 43—^= 0.1 -^= 0.25 = 0.5 i^ Cooperation, Pr = 0 8^10^12^14^16^18^20 SNR (dB) 1 0' 0 10 CC 10 2 LL W 10' 4^6^8^10^12 ^ 4^16^20 SNR (dB) Figure 4.14: Simulation Results for Cooperative Space Time Coding with p, = 0.8, p, = 0.99. tu b! 1) 1^' 0.1, 0.25, 0.5 Figure 4.15: Simulation Results for Cooperative Space Time Coding with p„ = 1. p,. = 0.99. and = 0.1, 0.25. 0.5 46 14 6 18 20 1 0 ° CCla 10 LL o^ Single User, Pr -{^= 0.1 ^ ..^= 0,25 v^ Pr - 0.5 Cooperation, P 2^4^6^8^10^12 SNR (dB) 1 0 0 Table 4.2: Cooperation Gains for pc = 0.99, Pyi = 0.1, 0.25, and 0.5 SNR ps = 0 ps = 0.5 p, = 0.8 p s = 1 P'" =0.1 8 4.4187 3.7508 2.7234 1.6328 12 7.2425 6.6980 5.2073 2.5997 16 8.8034 8.5972 7.7203 3.9387 20 9.3620 9.2076 8.6607 5.4077 P f "' = 0.25 8 2.8149 2.5717 2.1157 1.4770 12 3.5495 3.4355 3.0609 2.0525 16 3.8266 3.7937 3.6416 2.6438 20 3.9112 3.8885 3.8039 3.1175 P f  "' = 0.5 8 1.7539 1.6876 1.5422 1.2744 12 1.9188 1.8961 1.8144 1.5194 16 1.9702 1.9644 1.9365 1.7080 20 1.9850 1.9811 1.9662 1.8276 Figure 4.16: Simulation Results for Cooperative Space Time Coding with^= 0, p, = 0.9. and P;" 0.1, 0.25, 0.5 47 10° Single User, Pr = 1 - a-- P'," = 0.1 U ^- 0.25 Cooperation, Pi ° = 0 2 ^ 4 ^ 8^10^12^14^16^18^20 SNR (dB) Figure 4.17: Simulation Results for Cooperative Space Time Coding with p. = 0.5, p„ = 0.9. and Pr = 0.1, 0.25, 0.5 1 0° leo CCla 1 0 , LL —e— Single User, P i' n = 1 Pf ' = 0.1 = 0.25 ^ P:n - 0.5 Cooperation. P lf° = 0 4 ^ 8^10^12^14^16^18^20 SNR (dB) Figure 4.18: Simulation Results for Cooperative Space Time Coding with p, = 0.8. p„ = 0.9. and = 0.1, 0.25. 0.5 48 10° W 10 LL o^ Single User,^= 1 ^^ = 0.1 P'f" = 0.25 = 0.6 Cooperation, 1.;" = 0 2^4^6^8^10^12 SNR (dB) 1 0 0o 16^18^20 Figure 4.19: Simulation Results for Cooperative Space Time Coding with p, = 1. p, = 0.9, and^= 0.1, 0,25. 0.5 Table 4.3: Cooperation Gains for pe = 0.9, P = 0.1, 0.25. and 0.5 SNR p.,= 0 ,o., -,--. 0.5 p,, = 0.8 p s =1 = 0.1 8 2.3722 2.0732 1.6544 1.3719 12 3.1596 2.9190 2.2551 1.7470 16 3.6453 3.3647 2.8995 2,3077 20 3.7468 3.6360 3.2107 2.7396 Pf " = 0.25 8 1.9306 1.7587 1.4917 1.2919 12 2.3234 2.2117 1.8650 1.5536 16 2.5299 2.4135 2.2023 1.8947 20 2.5702 2.5262 2.3462 2.1238 P f " = 0.5 8 1.4735 1.4037 1.2816 1.1773 12 1.6122 1.5754 1.4476 1.3116 16 1.6755 1.6405 1.5722 1.4595 20 1.6871 1.6744 1.6195 1.5450 49 Figures 4.12 to 4.19 show the FER curves of the single user performance and the Iwo-user cooperation systems with different ps values, imperfect channel estimation (p c = 0.99 and 0.9), and different inter-user channel qualities (Pi" = 0.1, 0.25, and 0.5). From the FER curves, Tables 4.2 and 4.3 can be obtained. From Tables 4.2 and 4.3. we observe that even with imperfect channel estimation (pe = 0.9 and 0.99), the users still benefit from cooperation as G f is always greater than 1. It is also observed that Gf decreases with in- creasing ps and /11 . For example, as p, increases from 0.5 to 0.8 at SNR equals to 20 (113. = 0.1, and pc = 0.99, G f decreases from 9.2076 to 8.6607. This is expected because as the channels get more correlated, the benefit from the diversity decreases even when the channel estimation is not perfect. Similar to the case where the channel estimation is perfect as discussed in the previous section, the users still benefit from cooperation even when A, increases to 1 and when the inter-user channel quality becomes poor (P1= By comparing Tables 4.1, 4.2, and 4.3, we observe that • G1 decreases with increasing Pr. • G f decreases with increasing ps . • G f decreases with decreasing p c . The reason that causes G f to decrease with decreasing pc is not as clear as the reasons that cause G1 to decrease with increasing pr and ps because the statistics of the estimated errors are the same for both quasi-static and block fading channels. As p„ decreases from 1 to 0.99 with ps , pit.' and SNR equal to 0.5, 0.1, and 20dB, Gf decreases from 9.9575 to 9.2076. As pc decreases to 0.9, qf decreases to 3.6360. Recall from (4.1) that. O f = . As pc decreases from 1 to 0.99 and to 0.9, pr decreases faster than Pr, thus causing G1 to decrease with decreasing pc . From the Tables and discussions above, it can be concluded that even though the channel estimation is not perfect (pc = 0.99 and 0.9). the users still benefit from cooperation. It is assumed that the channel estimation correlation coefficient p c is fixed in the analysi discussed in this Chapter. However, the changes in SNR will affect the accuracy of 50 the channel estimation model. The influence of SNR to pe varies with different channel estimation models. For example, as in [11], we can define p, as pc =   for th( ^sik ^ channel estimation model described in this thesis. 51 Chapter 5 STC with Impulsive Noise In this chapter, the FER, performance of SUSTC with spatially correlated fading, channel estimation error, and impulsive noise is investigated. A decision metric is obtained in Section 5.1 and FER simulation results are presented in Section 5.2. 5.1 System Model with Mixture Gaussian Noise The system model is the same as that in Chapter 3 except the pdf's of the imaginary and real parts of the noise r.v. Nt are statistically dependent Gaussian mixture distributed as defined in (2.4). With the mean and variance of the received signal at time t given by (3.29) and (3.30). the pdf of Rt given the estimated channel gains is P(rt Ctk Hj,i^hj,z with ^ 1,2) = (1 ^1E) ^ exp V27ro-k 1^a rt ^E ^ exp ^27i- o-?, [ Irt — ^E, E,,=1 20-2, - V E, E i2= 1 etk, i 777 j ., 2a?.. (5.1) 2 =^( 1 — PD(1 c(7)( 1 —^ 2Es i it , c t̀ , 2pdal + ar2) (5.2) 52 and o- 2^E ( 1 — Pe2 )( 1 — Ps2 Pc ) 1 — 1°,14^ 2Esctk i c itc 2 pda 2Y = D, +^(5.3) where j denotes the user, c tic, i denotes the coded bit to be transmitted by antenna i in segment k at time t, and^(i,j = 1,2) is given in (3.19) and (3.20) for the respective iise! The expression in the numerator of the exponential for the first segment (t^1..... L) 2 -^Ec t ,^ (5.1) i=i When cooperation takes place, the expression in the numerator of the exponential for the second segment (t = L^1, ..., 2L) is ^ 2 ^ 1rt — V Es E (5.5) i=i When there is no cooperation, the expression in the numerator of the exponent ial for the second segment is 1, i 2 (.5.6) i= 1 with t^L^1,^2L. With the assumption that the components of the received vector given the estimated chan- nel gains are independent, the mixture Gaussian decision metric (MGDM) for SUSTC can be obtained by taking the logarithm of (5.1) to minimizes the quantity 2 L_EE log P(r t ictk i , ctk, , k=1 t=1 = h 1,1 , H1,2 = h1 ,2).^(5.7) Similarly, the MGDM for CSTC can be expressed as 2^L pt\rtIctk,,,^2,etk Hk_ EE log^= hk.i. Hk,2 = hk.2)• k=1 t=1 (5.8) 5.2 Numerical Results Simulations were performed with pe = 1 and 0.99 and highly impulsive noise (4^500. E = 0.05). The convolutional codes [5 7] and [5 7] were used for User 1 and User 2 respectively. 53 cc w o ps = 0, pe = 1 ^ p s = 0.5, p e = 1 p s = 0.8, p e = 1 pe = 1, pe = 1 -40^-30^-20^-10^0 ^ 10 ^ 20 ^ 30 SNR (dB) 5.2.1 Perfect Channel Estimation (A, = 1) \\Then p, = 1. (5.1) reduces to o ^P(rt IG•i,i = gi,i. , j = 1 . 2) = (1^E) v217ra2i exp^LI.' - N/Es 2 3- 212.=1 41'11).'-T1 1^in - VE, Ei2-1ct,igLi 2 ^ + C ^ exp ^2rol Figure 5.1: SUSTC with pe = 1, different p s values, and mixture gaussian noise with e = 0.05 alai = 500 (solid: Mixture Gaussian Decision Metric; dashed: Gaussian Decision Metric) The FER of SUSTC as a function of SNR for spatially correlated fading, perfect channel estimation (p, = 1), and mixture Gaussian noise is plotted in Figure 5.1 with c = 0.05 a 2and -L = 500. The solid line shows the performance when the proposed MGDM in (5.7) is used whereas the dashed line shows the performance with the Gaussian decision metric (GDM) in (3.38). It can be seen that the MGDM provides a better performance I han the GDM. This is expected since the MGDM in (5.7) is optimal when there is no channe] 2o1^ • (5.9) 54 -40 -30 -20 10-10^0 SNR (dB) 20 o ps = 0, pe = 1 ^ p s = 0.5, pe = 1 p s = 0.8, pe . 1 p s = 1, pe = 1 estimation error. With channel estimation error, (5.7) is no longer optimal because the components of the received vector are no longer independent. One disadvantage of the MGDM is that the impulsive noise parameters, i.e. E and z, are required at the receiver. Figure 5.2: SUSTC with p = 1, different ps values, and mixture Gaussian noise with F = 0.05 nod - 4 = 500 (dashed: Gaussian Noise with GDM; solid: Mixture Gaussian Noise with MCDM) The FER curves of SUSTC as a function of SNR with Gaussian noise and Mixture Gaus- sian noise are shown in Figure 5.2. It can be seen that the FER with Mixture Gaussian noise and MGDMof (5.7) (solid lines) is lower than with Gaussian noise and GDM of (3.38) at lower SNR values. As the SNR increases beyond 24 dB, the p, = 0 FER curve with Gaussian noise crosses that with mixture Gaussian noise. This is because of the heavier tail of the mixture Gaussian pdf as shown in Figure 2.2. The performance bounds for optimum reception under class-A impulsive noise was discussed in [20]. It is observed in [20] that as SNR, increases, the slopes of the curves with Gaussian and class-A impulsive noises are the same. From Figure 5.2, it can be observed that the asymptotic slopes of the curves are similar for ps = 0, 0.5, 0.8. It is expected that the asymptotic slopes of 55 the FER curves with Gaussian and Mixture Gaussian noises will be the same. The FEU curves at higher SNR values were not simulated due to the excessive times needed. 5.2.2 Imperfect Channel Estimation (pi = 0.99) Figure 5.3: SUSTC with pe = 0.99, different ps values, and mixture Gaussian noise with c = 0.05 and C6- = 500 (solid: Mixture Gaussian Noise with MGDM; dashed: Gaussian Noise with CDM) The FER of SUSTC as a function of SNR for spatially correlated fading, imperfect chart- nel estimation (p = 0.99), and Gaussian and mixture Gaussian (E. = 0.05 and 4 = AO; noises is plotted in Figure 5.3. Similar to the case with perfect channel estimation. it observed that SUSTC shows better performance with mixture Gaussian noise than withh Gaussian noise at low SNR. At high SNR, the FER with Gaussian noise is lower than that with mixture Gaussian noise. The performance of Cooperative Space Time Coding (CSTC) was not simulated in t his 56 chapter due to the time constraints. Since the diversity order of a system depends on the number of independent fading paths, it is expected that the performance of CSTC' be qualitatively similar to that of SUSTC except that the slopes of the CSTC curve ti will be steeper. 57 Chapter 6 Conclusion 6.1 Main Thesis Contributions A performance study of cooperative space time coding with spatially correlated fading and imperfect channel estimation in Gaussian as well as impulsive noise was presented. The main contributions of the thesis are listed below. • Closed form expressions for the pairwise error probability conditioned on the estimated channel gains with spatially correlated fading and imperfect channel estimation are de- rived for 1) single user space time coding (SUSTC) with quasi-static fading channels and 2) cooperative space time coding (CSTC) with block fading channels. An expurgated bound on the FER for both cases is then obtained using the limiting before average tech- nique in [33] and assuming the components of the received vector are independent given the estimated channel gains. • Simulation results for a constraint length 3 convolutional code were compared with the analytical bounds. The results show that even though the bound is not very tight. it is able to capture the performance degradation as p s increases perfectly when^= 1. As pe decreases, the performance degradation of the simulation results and the bounds arc different because of the channel estimation error. 58 • The cooperation gains show that the users always benefit from cooperation even the case where the channel gains within each user are strongly correlated (i), ti 1) and when the channel estimations are very poor (p, ti 0.9) because CSTC with block fading channels have a lower FER compared to SUSTC with quasi-static fading channels. • A detailed discussion on the dependency among the components of the received vecl oi for a system with space time trellis codes was presented. The discussion confirms with the observation in [301 that the components of the received vector are not independent and shows that the introduction of the channel estimation error is the reason for this dependency. • A decision metric for CSTC with spatially correlated fading, imperfect channel estima- tion, and mixture Gaussian noise is derived. Simulation results show that, by using the proposed mixture Gaussian decision metric for CSTC with mixture Gaussian noise. the FER performance of CSTC with mixture Gaussian noise is better than that of CSTC with Gaussian noise at low SNR. At high SNR, CSTC with Gaussian noise outperforms CSTC with mixture Gaussian noise because of the heavy tail of the mixture Gaussian noise. 6.2 Topics for Further Studies • Extensions of the thesis work to other modulation, channel fading, channel estiinat ion. or different impulsive noise models can be made. • More investigations can be carried out on CSTC with spatially correlated fading and poor channel estimation (p, < 0.9) to provide a more complete picture of the FER per- formance. 59 Appendix A Dependencies among the components of the received vector In [31], the performance of space-time codes with two transmit and one receive antennas and their design criteria in the presence of channel estimation errors was examined. In the performance analysis, the received signal at time t is denoted as 2 rt^Es^g i ct , i +^ (^) = 1 where the channel gains, g i and 92 , and noise component at time t are modeled as indepen- dent samples of a zero mean complex Gaussian r.v.'s with variance o-6 and 1 cL respectively. If a codeword C = C1 , 1C1 , 2 C2 , 1C2 , 2...CL , 1CL , 2 ^ (1.2) is sent, the corresponding received vector is r = r 1 r2 r3 ...rL^ (A.3) where L denotes the frame length. At the receiver, ML decoding is used. The codeword e = e1,1e1,2 62,1e2,2.-eLJeL,2 ^ (Al) is chosen for which Pr(rle,H = h) is maximized where H = [1-1 1 1/9] and h = [17 i h 2 ]. 60 It is assumed in [31] that, the components of the received vector are independent given the estimated channel gains. Then ML decoding simplifies to choosing the codewonl which minimizes E _ log Pr(rle, H = h) t=i We look at the covariance between the received signals at a two time instance to explain the dependencies among the components of the received vector. The covariance bet wee]] two complex r.v.'s X and Y is Cov{XY} = E{XY*} — E{X}E{Y*}^(.1.6) Let X = R 1 and Y = R2. We now compute E[R 1 R2*111] and E[R I IH]E[R2*IH] for Alamouti coding and the model in Chapter 3 for quasi-static fading to determine if the components of the received vectors are independent given the estimated channel gains. For Alamouti coding with quasi static fading, we have r l =^+ 9282 + ni ^ r2 = — 91s; + .C2si + 71 2^ (A.7) where s i and s 2 are the transmitted signals and the covariance of R 1 and R2 given the estimated channel gains, H 1 and H2, can be computed as E[R i g 1 11] =^+ ( G 2 IH )s 2 + Ni) (( — Gi.111)s2 + (G2 1H)si + -V2 r i = E[M,177124 — 77217712S 2 8 182^2 S^2 2 _]^co— 51827121^PdaDI Pdab'21 ) E[R11i] E[R2111] = E R Gi1l1) 3 1 + (G211-1 )82 + NER (—G1111 )s2 + (C2 H),!.5 1^:Y2/ J = E[m i m2 s 2, — rn i m2 4 + s 1 s2 m2 — s 1 8 2^(A.9) with m l , m2 , pd , and a 2D given in Chapter 3. With E{s i2 } = 1 for the case of BPSK modulation, we have ( E[R i R;IH] — E[R i lli]E[1?; H] = 0.^(Ali)) 61 It is shown in Chapter 3 that^= o-6,21H • With the assumption that the real and imaginary parts of the complex r.v.'s are independent, it can be shown that ErR I = E^R21 111. Thus, the covariance of R 1 and R2 given the estimated channels shows that the components of the received vector with Alamouti coding and quasi static fading are independent. For the single user space time model with quasi-static fading discussed in Chapter 3. we have rt = gict,1 + g2ct,2 + nt = mict,i + m2ct,2 + dict,i + d2 ct , 2 + n t^(A .11 ) where m i , m2 , d 1 , and d2 given in Chapter 3. (VEs is dropped from the equation to simplify the derivation as it does not affect the result.) The covariance of R 1 and R2 given H is E[R i^H] =^+ c i ,2(c211-1) + Ni)(c2,1( c111-1 ) + c2,2(G2 H) + :V2 )1 = E[cLic2,1m21 + c1,2c2,27772  + C1,1 c2,27ni rn2 + c1,2c2,11ni7n2] E[CiJC2,1^C1,2C2,2^C1,1C2,2Pd^C1,2C2 , 1pcd(7 2D ^ (A.12) E[R i IH]E[R; 1 1-1]^(Gi IH) + c1,2(G2IH) + Ni)]E[(c2,i(Gi 111) + c2,2(G2 H) + ,V 2 )* =^c2,1rn4 + ci,2c2,2rn + c1,1 c2,2mi m2 + c1.2c2,irni m21 E[R i^1H] — E[R.111-1]E[R;H] = E[c1,ic2,d0-2D + E[c ,2c2,2] 01) + E[cLic2,2]Pda 2D + E[c1,2c2,1]Pdai)^(A.1 1) For the single user space time model, we still have E[R 1 R2*IH] = E[R i*R2 114]. (7-(2-;1 ji G21H, and E[ciP = 1 (i,j = 1,2), but the covariance between R 1 and R2 are no longer zero. This shows that the components of the received vector are not independent given 62 the estimated channel gains. For the case with independent channel gains and perfect channel estimation as in the covariance of R 1 and R2 given G 1 and G2 can be easily shown to be zero. Following the above discussions, we conclude that the introduction of the channel estimat ion error conditioned on the estimated channel gains causes the components of the received \vet Hy to depend on each other. This confirms with the result in [30] but provides a more detail explanation on the dependencies among the components of the received vector. 63 Aj) , -= t=1 A 21 '), 2 — t=1 ^( ctb^etb,^12 1( Cb — e bt,2 t, 2 2 b * e t,2.) ) Appendix B Extended Trellis From (3.52), the distance term d2 (c, e) depends on the number of transmit antennas. Following a similar derivation as that in [34], the d2 (c, e) term for a block fading channel is d2BF(c, e) 2^L^2 = EE1E b=1 t=1 i=1 2 771b,t^)1 2 2 2 A1 b,1 ^-(1-6,1 ±Ei mb,21 2 A b2,2 ^6=1 ^ b=1 2 ^+ 214^mb,1 b,2)Bb}^ (13.1) b=1 where t=1 , L denotes the number of bits in each block, and b denotes the block (segment) index. As in Chapter 3, if the partner fails to decode the user's information, the user continues to transmit the remaining parts of the coded bits in the second time segment. Thus. 64 quasi-static channel can be viewed as a block fading channel except the channel gains are the same in both blocks (segments) and the d2 (c, e) term can be expressed as d2Qs(c, e) L = E — et,1) 2 t=1 i=1 = 217-n1,11 2 A1 +^ " 2 + 2R{2(rnm^)B} (13.5) where L Ai =^,i)12 t=1 L t,2 — et,2/1 2 t=1 B =^— et,i)(ct,2 — et,2) * t=i In order to find a bound for CSTC system, we need to know the weight enumerating function, or distance spectrum of the space time trellis code [36]. We made use of the technique in [33] where it keeps track of all the real variables Al, A2 , B as in (B.2) 10 (13..1) and (B.6) to (B.8) and claims that the weight enumerating function is the multiplicities of the these real variables over all codeword pairs. Since BPSK modulation is used. B is also a real variable. In this thesis, convolutional encoding with constraint length 4 and generator polynomial g = [5 7] is used for both users. The encoding diagram and the corresponding trellis are shown in Figures B.1 and B.2. The extended trellis for this code is shown in Figure B.3. To obtain the extended trel- lis, we start by looking at the 4 state trellis in Figure B.2 and computing the dist i.rice between the correct path starting from state i, S i and the erroneous path starting from state j. Si where i, j = 0, 1, 2, 3. Thus, the states of the extended trellis are of the form " S c) Sj e.) " where the correct path (c) and the erroneous path (e) are in state S, and S 65 Coded Bit for Transmit Antenna 1 -Input-0-1 Coded Bit for ► Transmit Antenna 2 Figure B.1: K = 3, g = [5 7] convolutioanl encoder. SO 0\̂ 0/00^________,c) SO N^ )/4-t- ----- S1 0<.----^1 PO^S1 -"MX, N S2^ S2 Input = 0 Input = 1 Example: 0 / 0 1 {^Coded bit from antenna 2 Coded bit from antenna 1 S3^--vort   S3 Figure B.2: 4-State Space Time Trellis Code. 66 S0 (` ) SO (e) 000 222 222 000 SO (c) S1 (e) 222 000 000 222 SO ( ` ) 52 (e) 0 020 200 200 020 SO (8) Se ) 200 020 020 200 S1 (8) SO (8) 0 222 000 000 222 S1 (8) S1 (8) 000 222 222 000 S1 (c) S2 (e) 200 020 020 200 S1 (8) S3 (8) 020 200 200 020 S2 (8) 50 (e) 020 200 200 020 S2 (c) S1 (8) 200 020 020 200 S2 (c) S2 (e) 000 22-2 22-2 000 S2 (c) 53 (8) 0 22-2 000 000 22.2 S3 (8) 50 (e) 0 200 020 020 200 S3 (8) Si (e) 020 200 200 020 S3 (8) S2 (e) 22-2 000 000 22-2 S3 (8) Se ) 000 22-2 22-2 000 • S0'`) SO (e) SO (c) S1 (e) O SO (c) S2 (e) • S0 (`) S3 (e) O 51 (c) SO (e) Si (c) S1 (e) 51 ( ` ) S2 (e) Si m S3 (e) S2 (c) SO (e) S2 (c) S1 (e) S2 (c) 52 (e) S2 (c) S3 (e) o S3 (e) SO (e) 0 S3 (c) S1 (e) o S3 (c) S2 (e) 53 (c) S3(e) Figure B.3: Extended Trellis for the 4-State Space Time Trellis Code. 67 of the original 4 state trellis in Figure B.2. The labels on the extended trellis take the form [X Y Z] and correspond to one state transition of the Euclidean distance bet ween the correct path and the erroneous path for the first transmit antenna, second transmit antenna, and the cross terms between the two antennas. The multiplicities of all slat( transitions for the entire correct and erroneous sequence are equivalent to (B.2) to (B.4). and (B.6) to (B.8) for block fading and quasi-static channel respectively. For example. let us assume the correct path and the erroneous path start at state zero, which is .. 3(Y . ',50 in Figure B.3 and the correct path goes to state 2 while the erroneous path stays in state zero, which is "S e) in the next time instance. Then, we have 11 (+1+1 as for BPSK modulation) and 00 (-1-1 as for BPSK modulation) as the outputs on the 4 state trellis for the correct and erroneous paths. Thus, the corresponding Euclidean distance for the two paths for the first transmit antenna, the second transmit antenna, and the cross t ern] for both transmit antennas can then be computed as 222. The 16 by 16 state transition matrix S for the extended trellis for the 4 state space time trellis code is then given by (B.9). - A (M2/3 ° 0 Ai,q,92 0 0 0 0 0 il?Ap 2 0 A?A2./3° 0 0 0 0 0 0 A?iqB° 0 il?A?B° 0 0 0 0 0 A?A?B° 0 A?A3B° 0 0 0 0 A?A23 2 0 A7A2B° 0 0 0 0 0 A?2,12B° 0 ATA2B2 0 0 0 0 0 0 A?A2/3° 0 ATA.B° 0 0 0 0 0 A?,,lp° 0 .2q4/3° 0 0 0 0 S1 to 8 — 0 0 0 0 Al,),22.9° 0 ,,,qA2Ba 0 0 0 0 0 zqAcp° 0 ,4?A23° 0 0 0 0 0 0 A?A?B° 0 .4,-A26 -2 0 0 0 0 0 ,,q./13B-2 0 A?A!;/3" 0 0 0 0 ANB° 0 ATAp() 0 0 0 0 0 AyAp° 0 A?A(.:;B° 0 0 0 0 0 0 ..-1/2B-2 0 .-/3" 0 0 0 0 0 A?A2B° 0 .-1;.-1/3^2 68 -^.,12.10 2 0 AC).42B° 0 0 0 0 0 ATAB' 0 .442,B2 0 0 0 0 0 0 .4.426" 0 ATA!,B° 0 0 0 0 0 ATAP° 0 AfAP° 0 0 0 0 ATAP° 0 ANB2 0 0 0 0 0 ANB 2 0 A(,)AP° 0 0 0 0 0 0 A(,),A.B° 0 ANB° 0 0 0 0 S9 to 16= 0 11?4B° 0 2,17A2B° 0 0 0 0 r^9 0 0 0 0 A?Ap° 0 A7A3B° 0 0 0 0 0 A?Ap° 0 44' 'B'' 0 0 0 0 0 0 .MA2B-2 0 AC)AP' 0 0 0 0 0 ATAP° 0 Af.43 B -2 0 0 0 0 A?AP° 0 2.-1,42B° 0 0 0 0 0 /q4B° 0 A? AB' 0 0 0 0 0 0 AMB° 0 Af AP -2 0 0 0 0 0 AYA2B-2 0 A?.4P(' where S 1 to 8 and S9 to 16 denote columns 1 to 8 and columns 9 to 16 of S. In each entry of the S, the exponent of A l denotes the Euclidean distance between the correct and erroneous sequences transmitted across the first transmit antenna. the exponent of A2 denotes the Euclidean distance between the correct and efrOtle011;, sequence transmitted across the second transmitted antennas. The exponent of B de- notes the value of the cross term for both transmit antennas. Thus, each entry in the matrix S takes the form Ai AP 3z and represents a one-step transition in the extended trellis from the state corresponding to the "row number," to the state corresponding to the "column number." If we were to compute the weight enumerating function directly from the extended trellis, the weight enumerating function will be the sum of the firs1 entries in the first row of the matrix S' where 1 denotes the length of each block in the block fading channel model and S 1 denotes the /th power of the matrix S. For evaluating the bound for the system, we make use of the "Limiting Before Averaging and Expurgation techniques in [33]. The idea of expurgating is to evaluate the bound by taking only the simple error events into account. Simple error events arc those error events where once the erroneous codeword re-merges with the correct codeword on the extended trellis, it is assumed that the two codewords stay at the same state until the 69 end. The expurgated state transition matrix is computed by introducing a new state. the 17th row of the extended matrix, to the extended state transition matrix, S. This new state represents the paths that have previously diverged through the trellis are re-mergile,1 for the first time and no longer diverge. Thus, only simple error events are considered. Once the transition reaches this new state, it is assumed both paths are at one of the four states in Figure B.2 and do not diverge again. Thus, a new column needs to be added to the transition matrix to store the distances of the transitions that have previous led the transition to one of the ,9 ( c) S(r, ,5"e)Se), SVS . (3e) states in Figure B.3. The 17 by 17 expurgated state transition matrix S exp for the 4 state space time trellis code can be computed as A? A (2) B° 0 ,,q Ap3 2 0 0 0 0 0 0 0 A? Ap° 0 0 0 0 0 0 APIP° 0 II? A? B° 0 0 0 0 0 ,,q Apr 0 A? Ap ° 0 0 0 0 0 0 AP1P° 0 0 0 0 0 A?4B° 0 AYAB2 0 0 0 0 0 0 AqA(2)B° 0 AcIAB° 0 0 0 0 0 A?AP° 0 A?AgB° 0 0 0 0 SeXpi to 8 0 0 0 0 AcAir 0 /14 A f B () 0 0 0 0 0 IINEP 0 A? .,,q Bu 0 0 0 0 0 0 AMP° 0 A;.4p3-2 0 0 0 0 0 0 0 ,4(i)A1: R" 0 0 0 0 A? iq B ° 0 Ay /A' B ( ' 0 0 0 0 0 AMP° 0 Ai/•°B° 0 0 0 0 0 0 0 0 AyA!,:/3" 0 0 0 0 0 A' AB' 0 .4;.1 3 - 2 0 0 0 0 0 0 0 0 70 ANB2 ANS ° o 0 A c,) AP ° AyAW 0 0 .4?.42B" A:AzB° 0 o iqA2B° 0 0 0 0 9^9A,A,B ° 0 .4(:A3B0 ?^9A,A.,B 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4?.41132^-^lf..1 .:; t3 2 Af A .1/3 2 - lf 4^[1 2 o 9^?,A,A,B ° 0 3^9A 4 A,B ° 0 0 ?^9A,A,B ° o 9^3.A,A,B ° 0 0 SexP9 to 17 0 0 0 0 A?A2B° 0 ATAW 0 0 0 0 A?A3B0 0 A?AP° 0 0 0 0 0 0 0 A,A23-1 0 (..)A.A,B ° 0 0 0 0 0 ANB° 0 0 4fA^B -2^1^-C,B -= 0 0 0 0 9^3.A,A,B ° 0 Af 4/30 0 0 0 0 0 ?^2,,1,4,B ° 0 ATAW 0 0 0 0 0 0 A?A3B° 0 0 A?A3B-2^A?A;B" 0 0 0 0 AjA23-1 0 ANB° 0 0 0 0 0 0 0 where Sexp i to 8 and S expg to 17 denote columns 1 to 8 and columns 9 to 17 of S„, p . The corresponding extended trellis with expurgation is shown in Figure B.4. Since the convolutional codes used in this thesis is terminated, we need to multiply the expurgated transition matrix by a termination vector to ensure the frame ends in t he xero state. The termination vector can be easily computed as 71 SO (c) SO (e) 0^ 0 50 (c) SO (e) SO (c) Si (e) 0^ 0 SO (c) Si (e) so ( c) se ) O so(c ) 52 (e) So (c) S3 (e) o^ So(c) S3 (e) S 1(c ) Se ) 0^ S11`) SO (e) S1 (c) Sim^ .0 Si m S1 (e) Si (C) S2(e)^ Si (c) S2 (e) Si (c) S3 (e) a^ Si (c) S3 (e) S2 (C) 50 (e) 0^ S2(c) SO (e) S2 (c) Si (e)^ S2(c) Si (e) S2 ( c ) 52 (e )^0 S2 ( c ) S2 ( ° ) s2 (c ) s3 (° )^5 2(c) 53 ( e ) S3 (c) So (e)^o S3 (c) SO (e) S3 (c) Si (e) d^ 0 S3 (c) Si (e) 53 ( c ) S2 1° )^53(c)S2(e) S3(c) 53(e)^ o 53(c) 53(e) SPECIAL STATED^ SPECIAL STATE Figure B.4: Extended Trellis after Expurgation for the 4 State Space Time^Code. 72 Sterm =  1 Az j2 n2 A 21 1-124 B 2 A4A2 B2 1^2 A?AP3 2 1 2 B2A4 "1' A 2 AB 21 2 A?A 42B 2 AA22B 21 1 iqA0 -2 AnIAB 2 1 1 Similar to the non-expurgated case, the weight enumerating function for the space I ime trellis code is obtained from the sum of the first entry in the vector S lexpSterrn • If different codes are used to encode the information bits for the two blocks (i.e. K = convolutional code with g = [15 17] for user 1 and g = [13 15] for user 2), then a different weight enumerating function has to be derived for each block. The example shown in t his Appendix is for the case where the users use the same code to encode the informal ion hit, in each block. The derivation is similar to that as in [33], but a different code is used. 73 Appendix C SNR Definition Recall from Chapter 2 that the received signal at time t of segment k from user 1 is r t = ^ JEsgi^± (C. 1) i=1 From this, the average signal power is EsE{IGLi1 2 }E{Ic itil 2 } +EsE{IGL21 2 }E{1421 2 } + 2E8E{G1,iq2}E{c i:Jc itc,2} = and the noise power is No Hence, the signal to noise ratio is 2 SNR = 2E a ^G NO Assuming Eb = 1, then E, REb = z since the overall rate, R, of the system is (C.2) (C.3) 74 Bibliography [1] J. G. Proakis, "Digital Communications", McGraw Hill, 4th edition, 2001. [2] T. S. Rappaport, "Wireless Communications Principles and Practice", Prentice Hall 2002. [3] A. Paulraj, R. Nabar and D. Gore, "Introduction to Space-Time Wireless Commu- nications", Cambridge University Press 2003. [4] V. Tarokh, N. Seshadri, and A. R. Calderbank, "Space-Time Codes for High Data Rate Wireless Communication: Performance Criterion and Code Contruction - , IEEE Transactions on Information Theory, vol. 44 , no. 2, pp 744-765, March 1998. [5] S. M. Alamouti, "A Simple Transmit Diversity Technique for Wireless Communi- cations", IEEE Journal on Selected Areas in Communications, vol. 16, no, 8. pp. 1451-1458, October 1998. [6] A. Sendonaris, E. Erkip, and B. Aazhang, "User cooperation diversity - Part I: System Description", IEEE Trans. Commun., vol. 51, pp. 1927-1938, November 2003. [7] A. Sendonaris, E. Erkip, and B. Aazhang, "User cooperation diversity - Part II: Implementation aspects and performance analysis", IEEE Trans. Commun., vol. 51. pp. 1939-1948, November 2003. [8] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, -Cooperative diversity in wireless networks: Efficient protocols and outage behavior", IEEE Transactions on InfOrnm- tion Theory, vol. 50. no. 12, pp. 3062-3080, December 2004. 75 [9] J. N. 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