what Digital Communications

Digital Communications
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Published Date:25-10-2017
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1 Introduction to Digital Communications This first chapter introduces some of the basics of digital communications that are needed in subsequent chapters. Section 1.1 starts with a brief introduction of fundamental multiple access techniques and the general structure of communication systems. Some of the most important parts of the systems are discussed in more detail later in the chapter. Section 1.2 addresses the mobile radio channel with its statistical properties and the way of modeling it. Some analysis concerning signal detection and the theoretical performance of linear schemes for different transmission channels are presented in Sections 1.3 and 1.4. Finally, Section 1.5 explains the principle of diversity and delivers basic results for outage and ergodic error probabilities. 1.1 Basic System Model 1.1.1 Introduction Vector channels, or multiple-input multiple-output (MIMO) channels, represent a very gen- eral description for a wide range of applications. They incorporate SISO (Single-Input Single-Output), MISO (Multiple-Input Single-Output) and SIMO (Single-Input Multiple- Output) channels as special cases. Often, MIMO channels are only associated with multiple antenna systems. However, they are not restricted to this case but can be used in a much broader context, for example, for any kind of multiuser communication. Therefore, the aim of this work is to study MIMO systems for two specific examples, namely Code Division Multiple Access (CDMA) and multiple antenna systems. Besides a unified description using vector notations, detection algorithms are derived that make the similarity of both systems obvious. Figure 1.1 illustrates the considered scenario in a very abstract form. Generally, a com- mon channel that may represent a single cell in a cellular network is accessed by N inputs I and N outputs. In this context, the term channel is not limited to the physical transmission O ¨ 2 INTRODUCTION TO DIGITAL COMMUNICATIONS 1 1 2 2 channel N I N O Figure 1.1 Principle of multiple access to a common channel medium, that is, the radio channel, but has a more general meaning and also incorporates parts of a digital communication system. The boundary between the transmitter and the receiver on the one side, and the channel on the other side is not strict and depends on the considered scenario. Detailed information about specific descriptions can be found in subsequent chapters. Principally, single-user and multiuser communications are distinguished. In the single- user case, the multiple inputs and outputs of a vector channel may correspond to different transmit and receive antennas, carrier frequencies, or time slots. Due to the fact that the data stems from a single user, intelligent signaling at the transmitter can be performed, for example, for efficient receiver implementations. In Chapter 6, multiple antenna systems will be used in several ways, depending on the channel characteristics and the system requirements. Multiple antennas can be employed for increasing the system’s diversity degree (Alamouti 1998; Naguib et al. 1998; Seshadri and Winters 1994; Tarokh et al. 1999a) and, therefore, enhance the link performance. The link reliability can also be improved by beamforming, which enlarges the signal to noise ratio (SNR). Alternatively, several data streams can be multiplexed over spatially separated channels in order to multiply the data rate without increasing bandwidth (Foschini 1996; Foschini and Gans 1998; Golden et al. 1998). On the contrary, the N inputs and N outputs may correspond to N independent I O u 1 user signals in the multiuser case. With reference to conventional mobile radio commu- nications, a central base station coordinates the transmissions of uplink and downlink, that is, from mobile subscribers to the base station and vice versa, respectively. In the downlink, a synchronous transmission can be easily established because all the signals originate from the same transmitter. Moreover, knowledge about interactions between dif- ferent users and their propagation conditions can be exploited at the base station. This allows sophisticated signaling in order to avoid mutual interference and to distribute the total transmit power onto different users efficiently. In many cases, intelligent signal- ing at the base station coincides with low complexity receivers, being very important for mobile terminals with limited battery power. However, optimal signaling strategies 1 In multiuser MIMO scenarios, transmitters and receivers are equipped with multiple antennas. These systems are a part of current research and possess an extremely high degree of freedom. They will be briefly introduced in Chapter 6.INTRODUCTION TO DIGITAL COMMUNICATIONS 3 for the downlink is only known for a few special cases and is still subject to intensive research. Establishing synchronous transmissions in the uplink requires high efforts because mobile subscribers transmit more or less independently. They only communicate with the base station and not among themselves. In most practical cases, they have no information about their influence on other users so that mutual interference cannot be avoided prior to transmission. Hence, sophisticated detection algorithms have to be deployed at the base station to compensate for this drawback. 1.1.2 Multiple Access Techniques Looking at the transmission of multiple data streams sharing a common medium, their separation is managed by multiplexing techniques in single-user scenarios or multiple access techniques in multiuser communications. In order to ensure reliable communica- tion, many systems try to avoid interference by choosing orthogonal access schemes so that no multiple access interference (MAI) disturbs the transmission. However, in most cases, orthogonality cannot be maintained due to the influence of the mobile radio chan- nel. The next subsections introduce the most important multiplexing and multiple access strategies. Time Division Multiplexing (TDM) and Multiple Access (TDMA) This relatively common multiple access technique divides the time axis into different time slots, each of length T according to Figure 1.2. Each data packet or burst is assigned slot to a certain slot, whereby a user can also occupy several slots. A defined number N slot of slots build a frame that is periodically repeated. Hence, each user has periodical access to the shared medium. Due to the influence of the transmission channel (cf. Section 1.2) and the restrictions of practical filter design, guard intervals of length T have to be inserted between successive slots in order to avoid interference between them. Within these intervals, no information is transmitted so that they represent redundancy and reduce the spectral efficiency of the communication system. Frequency Division Multiplexing (FDM) and Multiple Access (FDMA) Alternatively, the frequency axis can be divided into N subbands each of width B as illus- f trated in Figure 1.3. The data streams are now distributed on different frequency bands, f 1 2 3 N 1 slot T T t slot Figure 1.2 Principle of time division multiple access www.allitebooks.com4 INTRODUCTION TO DIGITAL COMMUNICATIONS Figure 1.3 Principle of frequency division multiple access rather than on different time slots. However, in mobile environments, the signals’ band- widths are spread by the Doppler effect, so that neighboring subbands interfere. Thus, gaps of an appropriate width f combating this effect at the expense of a reduced spectral efficiency are required for Frequency division multiple access (FDMA). Code Division Multiplexing (CDM) and Multiple Access (CDMA) In contrast to both the preceding schemes, CDMA allows simultaneous access on the chan- nel in the same frequency range. The basic principle is to spectrally spread the data streams with specific sequences called spreading codes (Spread Spectrum technique). The signals can be distinguished by assigning them individual spreading codes. This opens a third dimension, as can be seen in Figure 1.4. One intuitive choice would lead to orthogonal codes, ensuring a parallel transmission of different user signals. However, the transmission channel generally destroys the orthogonality and multiuser interference (MUI) becomes a limiting factor concerning spectral efficiency (cf. Chapters 4 and 5). Space Division Multiplexing (SDM) and Multiple Access (SDMA) The fourth access scheme exploits the resource space (Figure 1.4 and Figure 1.5). Spatially separated data streams can simultaneously access the channel in the same frequency band, Figure 1.4 Principle of code and space division multiple accessINTRODUCTION TO DIGITAL COMMUNICATIONS 5 4 3 2 1 N u Figure 1.5 Principle of space division multiple access provided that the locations of transmit and receive antennas are appropriately chosen. In mobile environments, this requirement is sometimes difficult to fulfill, because users are changing their position during the connection. Therefore, quasi-static scenarios or combina- tions with the aforementioned access techniques are often considered. Mutual interference is also likely to occur in Space division multiple access (SDMA) systems because the trans- mitter and the receiver have no perfect channel knowledge of what would be necessary to totally avoid interference. As expected, all the mentioned access schemes can be combined. The well-known Global System for Mobile (GSM) Communications and Digital Cellular System (DCS)- 1800 standards both combine Time division multiple access (TDMA) and FDMA. In UMTS (Universal Mobile Telecommunications System) or IMT-2000 (International Mobile Com- munications) systems, CDMA is used in connection with TDMA and FDMA (Dahlman et al. 1998; Ojanpera and Prasad 1998b; Toskala et al. 1998). While TDMA, FDMA, and ¨ CDMA have already been used for a fairly long time, SDMA is rather recent in compari- son. This development is a result of the demand to use licenses that are assigned to certain frequency bands as efficiently as possible. Hence, all the resources have to be exploited for reaching this goal. 1.1.3 Principle Structure of SISO Systems Since the behavior and the properties of a MIMO system vastly depend on the characteristics of the underlying SISO systems between each pair of transmit and receive antennas, this subsection describes their principle structure. Figure 1.6 shows a simplified time-discrete block diagram of a SISO communication link. The time-discreteness is expressed by square brackets and the time indices i, ,and k indicate different symbol rates of the corresponding signals. Neglecting a lot of the fundamental components of practical systems like source coding, analog-to-digital conversion and so on, the transmitter consists of three blocks that are of special interest here: a forward error correction (FEC) encoder, an interleaver  and a signal mapper. Due to our focus on digital communications, the inputs and the outputs of an FEC encoder and an interleaver are binary, while the output of the signal mapper depends on the type of modulation and can take on symbols out of an M-ary m alphabet X. Conventionally, M= 2 is a power of two. The receiver is comprised of the corresponding counterparts of the above-mentioned blocks in reverse order. The first6 INTRODUCTION TO DIGITAL COMMUNICATIONS di b xk signal FEC  mapper encoder time-discrete channel ˆ r yk di SP/ FEC −1  demapper decoder Figure 1.6 Principle structure of digital communication systems block performs some kind of signal processing (SP) – depending on the channel – and the −1 demapping. It is followed by a de-interleaver  and an FEC decoder delivering estimates ˆ di of the transmitted information bits. All the blocks mentioned thus far will be subsequently described in more detail. How- ever, some remarks on the time-discrete channel depicted in Figure 1.6 are necessary at this point. In order to simplify the description and to concentrate on the main focus of this work, all time-continuous components of the modulator and the demodulator are declared as parts of a time-discrete channel model generally described in the equivalent baseband (cf. Section 1.2). Therefore, the only parts of the modulator and the demodulator that appear sep- arately in Figure 1.6 are the signal mapper and the demapper. These assumptions coincide with a citation of Massey (1984): ‘the purpose of the modulation system is to create a good discrete channel from the modulator input to the demodulator output, and the purpose of the coding system is to transmit the information bits reliably through this discrete channel at the highest practicable rate.’ However, it is not always easy to strictly separate both devices (e.g. for coded modulation (Biglieri et al. 1991; Ungerboeck 1982)). Interleaving Interleaving plays an important role in many digital communication systems for manifold reasons. In the context of mobile radio communications, fading channels often lead to bursty errors, that is, several successive symbols may be corrupted by deep fades. This has a crucial impact on the decoding performance, for example, of convolutional codes because of its sensitivity to bursty errors (compare the decoding of convolutional codes in Chapter 3). In order to overcome this difficulty, interleaving is applied. At the transmitter, an interleaver simply permutes the data stream b in a specified manner, so that the sym- bols are transmitted in a different order. Consequently, a de-interleaver has to be employed at the receiver, in order to reorder the symbols back into the original succession. More- over, we will see in Section 3 that interleaving is also employed in concatenated coding schemes.INTRODUCTION TO DIGITAL COMMUNICATIONS 7 b write ˜ b b0 b3 b6 b9 read b1 b4 b7 b10 b2 b5 b8 b11 Figure 1.7 Structure of block interleaver of length L = 12 π Block interleaver There are several types of interleaving. The simplest one is termed block interleaver,which divides a sequence into blocks of length L . The symbols b within each block are then π permuted by writing them column-wise into an array consisting of L rows and L row col columns, and reading them row by row. An example with L = 3and L = 4 is shown row col in Figure 1.7. The input sequence b0,b1, ... b11 develops into the following due to interleaving b0,b3,b6,b9,b1,b4,b7 b10,b2,b5,b8,b11. It is recognized that there is a spacing between the originally successive symbols of L = 4. I This gap is called interleaving depth. The optimum number of rows and columns and, there- fore, the interleaving depth depends on several factors that are discussed in subsequent chapters. Convolutional interleaving For the sake of completeness, convolutional interleaving should be mentioned here. It provides the same interleaving depth as block interleaving, but with lower delays and lesser memory. However, since this interleaver is not addressed later in the chapter, further details are not discussed and instead the reader is referred to (Viterbi and Omura 1979). Random interleaving The application of block interleaving in concatenated coding schemes generally leads to a weak performance. Due to the regular structure of the interleaver it may ensue that the temporal distance between pairs of symbols does not change by interleaving, resulting in poor distance properties of the entire code (cf. Section 3.6). Therefore, random or pseudo- random interleavers are often applied in this context. Pseudo-random interleavers can be generated by calculating row and column indices with modulo arithmetic. For concatenated coding schemes, interleavers are optimized with respect to the constituent codes. Interleaving delay A tight restriction to the total size of interleavers may occur for delay sensitive applications such as full duplex speech transmission. Here, delays of only around 10 ms are tolerable.8 INTRODUCTION TO DIGITAL COMMUNICATIONS Since the interleaver has to first be completely written before it can be read out, its size L directly determines the delay t= L · T . π π s 1.2 Characteristics of Mobile Radio Channels 1.2.1 Equivalent Baseband Representation Wireless channels for mobile radio communications are challenging media that require care- ful system design for reliable transmission. As SISO channels, they represent an important building block of vector channels. Therefore, this section describes their time-discrete, equivalent baseband representation in more detail. Using a representation in the equiva- lent baseband is beneficial for simulation purposes, because the carrier whose frequency is generally much higher than the signal bandwidth need not to be explicitly considered. Figure 1.8 depicts the entire channel model that comprises all time-continuous analog com- ponents, including those from the transmitter and the receiver. The whole structure describes a time-discrete model, whose input xk is a sequence of generally complex-valued symbols of duration T according to some finite symbol alphabet X. The output sequence typically s yk has the same rate 1/T and its symbols are distributed within the complex plane C. s The input xk is first transformed by the transmit filter g (t) of bandwidth B into a T time-continuous, band limited signal  x(t)= T · xk· g (t− kT ) (1.1) s T s k called complex envelope. Denoting the symbols of the alphabet X by X and assuming that µ  ∞ 2 −1 the energy of the transmit filter impulse response is defined to g (t) dt= T ,a T s −∞ √ analog part of jω t 0 2e transmitter xk x(t) x (t) BP g (t) Re· T h (t, τ) BP 1 −jω t 0 √ e 2 n (t) BP + y(t) y (t) y (t) BP yk BP g k g (t) W R jH· analog part of receiver Figure 1.8 Structure of the time-discrete, equivalent baseband representation of a mobile radio channelINTRODUCTION TO DIGITAL COMMUNICATIONS 9  (f ) H H BP BP  (f )  (f ) XX X X a) b) BP BP c)  (f ) N N BP BP 2 σ 2 2 X σ /2 σ /2 X X N /2 0 2B 2B 2B 2B B −f +f f −f +f f −f +f f 0 0 0 0 0 0 Figure 1.9 Power spectral densities for a) complex envelope, b) bandpass signal and c) transmission channel for a rectangular shape of G (jω) T single symbol T · xk· g (t− kT ) has the average energy s T s  ∞ 2 2 2 2 E = T · EX · g (t) dt= T · EX (1.2) s µ T s µ s −∞ 2 2 2 resulting in an average power of σ = E /T = EX . For zero-mean and indepen- s s µ X dent identically distributed (i.i.d.) symbols xk, the average spectral density of x(t) is (Kammeyer 2004; Kammeyer and Kuhn ¨ 2001; Proakis 2001) 2 2 2  (jω)= T ·G (jω) · EX = E ·G (jω) . (1.3) XX s T µ s T Obviously, it largely depends on the spectral shape of the transmit filter g (t), and not on T the kind of modulation scheme. Proceeding toward transmission, the real-valued bandpass signal √ √     jω t   0 x (t)= 2· Re x(t)· e = 2· x (t) cos(ω t)− x (t) sin(ω t) (1.4) BP 0 0 is obtained by shifting x(t) into the bandpass region with the carrier frequency ω = 2πf 0 0 √ and taking the real part. The factor 2 in (1.4) keeps the signal power and the symbol energy constant during modulation. The average spectral density of x (t) has the form BP  E s 2 2  (jω)= · G (jω− jω ) +G (jω+ jω ) . (1.5) X X T 0 T 0 BP BP 2 Besides the shift to ±ω , it differs from  (jω) by the factor 1/2 due to the total 0 XX transmit power constraint. Figure 1.9 sketches the spectral densities for a rectangular shape of G (jω) with B= 1/(2T ). T s Now, x (t) is transmitted over the mobile radio channel, which is generally represented BP by its time-variant impulse response h (t, τ) and an additive noise term n (t) with BP BP spectral density N /2 0 y (t)= h (t, τ)∗ x (t)+ n (t). (1.6) BP BP BP BP The convolution in (1.6) is defined by  ∞ h (t, τ)∗ x (t)= h (t, τ)x (t−τ)dτ. (1.7) BP BP BP BP 0 2 −1 The impulse response g (t) itself has the dimension s because its spectrum G (jω)=Fg (t) has no T T T dimension.10 INTRODUCTION TO DIGITAL COMMUNICATIONS a) b) c)  (f ) YY + +  (f )  (f ) Y Y Y Y BP BP BP BP  (f ) NN  (f )  + + (f ) N N BP BP N N 2N 0 BP BP N 0 N /2 0 2B −f +f f −f +f f −f B +f f 0 0 0 0 0 0 Figure 1.10 Power spectral densities for a) received bandpass signal, b) analytical signal and c) received complex envelope In order to express y in the equivalent baseband, negative frequencies are eliminated with BP the Hilbert transformation (Kammeyer 2004; Proakis 2001)  ∞ a(τ)   Ha(t)= dτ FHa(t)=−j sgn(ω)A(jω). (1.8) t− τ −∞ Therefore, adding jHy (t) to the received signal y (t) yields the complex analytical BP BP signal 2Y (jω) forω 0 BP + +   y (t)= y (t)+ jHy (t) Y (jω)= (1.9) BP BP BP BP 0else whose spectrum vanishes for negative frequencies. However, for positive frequencies, the spectrum is doubled or, equivalently, the spectral power density is quadrupled (Figure 1.10). + AccordingtoFigure 1.8, y (t) is shifted back into the baseband and weighted by a factor BP √ 1/ 2 in order to keep the average power constant. With reference to the background noise, this leads to a spectral density of N . 0 As is shown in the Appendix A.1, the output of the receive filter g (t) in Figure 1.8 R has the form  ˜ y(t)= g (t)∗ h(t, τ)∗ x(t)+ n(t)= xk· h(t, kT )+ n(t) (1.10) R s k + −jω t 0 where h(t, τ)= 1/2· h (t, τ)e denotes the equivalent baseband representations of BP √ + −jω t 0 channel impulse response and n(t)= g (t)∗ (n (t)e / 2) the filtered background R ˜ noise. The filter h(t, kT ) is comprised of a transmit and receive filter as well as the s channel impulse response, and represents the response of a time-discrete channel on an 3 impulse transmitted at time instant kT . s The optimum receive filter g (t) that maximizes the SNR at its sampled output has R to be matched to the concatenation of channel impulse response and the transmit filter ∗ (Forney 1972; Kammeyer 2004), that is, g (t)= f (−t) with f(t)= g (t)∗ h(t, τ) holds. R T In order to avoid interference between successive symbols, the transmit and receive filters are generally chosen such that their convolution fulfills the first Nyquist criterion (Nyquist 1928; Proakis 2001). This criterion also ensures that the filtered and sampled noise remains 3 ˜ Note that the second parameter of h(t,kT ) does not represent delay τ, but the transmission time kT . s sINTRODUCTION TO DIGITAL COMMUNICATIONS 11 white, and a symbol-wise detection is still optimum. However, even if g (t)∗ g (t) fulfills T R the first Nyquist criterion, the channel impulse response h(t, τ) between them destroys this property and the background noise n(t) is colored. Therefore, a prewhitening filter g k working at the sampling rate 1/T and decorrelating the noise samples n(t) is W s t=kT s 4 required. Finally, the time-discrete equivalent baseband channel delivers a complex-valued signal yk= g k∗ y(t) by sampling y(t) at rate 1/T and filtering it with g k. W t=kT s W s Throughout this work, g (t) is assumed to be a perfect lowpass filter of bandwidth T B= 1/(2T ). With g (t) matched to h(t, τ)∗ g (t) and a perfect prewhitening filter, the s R T received signal ykhas theform L t  yk= hk,κ· xk− κ+ nk (1.11) κ=0 where L denotes the total filter length of the time-discrete channel model hk,κ working t at rate 1/T and nkistermed Additive White Gaussian Noise (AWGN). It is described s in more detail in the next subsection, followed by a description of the frequency-selective fading channel. 1.2.2 Additive White Gaussian Noise Every data transmission is disturbed by noise stemming from thermal noise, noise of elec- tronic devices, and other sources. Due to the superposition of many different statistically independent processes at the receive antenna, the noise n (t) is generally modeled as BP white and Gaussian distributed. The attribute white describes the flat spectral density that corresponds with uncorrelated successive samples in the time domain. For Gaussian dis- tributed samples, this is equivalent with statistical independence. A model reflecting this behavior is the AWGN channel. As mentioned in the last section, its two-sided spectral power density N /2 results in infinite power due to the infinite bandwidth. Therefore, this 0 model only gains practical relevance with a bandwidth limitation, for example, by filtering with g (t). R In this subsection, the channel is assumed to be frequency-nonselective and time invari- ant so that h(t, τ)= δ(τ) holds and the transmit and receive filters are perfect lowpass filters (cf. Figure 1.9 and 1.11b). They fulfill the first Nyquist condition (Nyquist 1928), that is, 5 their spectra are symmetric with respect to the Nyquist frequency f = 1/(2T ). Therefore, N s the sampled equivalent baseband noise nk= n(t) remains white (cf. Figure 1.11) t=kT s (Kammeyer 2004) and has a spectral density of N (cf. (1.9) and Figure 1.10). N is equally 0 0   distributed onto the real part n k and the imaginary part n k, each with a density of N /2. They are independent of each other resulting in the joint density 0   p (n)= p (n )· p (n ) N N N 2 2 2 n n n − − − 2 2 1 1 1 2 2σ 2σ σ   N N N = e · e = e (1.12) 2 2 2 πσ 2πσ 2πσ N   N N 4 In practice, the receive filter g (t) is only matched to g (t) due to lower implementation costs and imperfect R T knowledge of the channel impulse response. 5 For perfect lowpass filters, B= f = 1/(2T ) holds, that is, 2BT symbols can be transmitted per channel N s s usage.12 INTRODUCTION TO DIGITAL COMMUNICATIONS  (f ) NN b) a) N 0 nk B xk yk f +f −f 0 0 Figure 1.11 a) Model and b) spectral density of AWGN channel in the equivalent baseband representation of the complex baseband noise. The power of n(t) and, therefore, nkis N 0 2 2 2 σ = 2B· N = = σ + σ . (1.13) 0   N N N T s ∗ The perfect lowpass filter g (t)= g (−t) with bandwidth B is matched to g (t) R T T (Kammeyer 2004; Proakis 2001) and maximizes the SNR at its output. With the signal 2 power σ = E /T we obtain the signal to noise ratio s s X 2 2 σ σ E /T E X s s s BP X SNR= = = = . (1.14) 2 2 N /T N σ σ 0 s 0 N N BP as a characteristic measure of the AWGN channel in the baseband, as well as in the bandpass regime. 1.2.3 Frequency-Selective Time-Variant Fading For mobile radio systems, the propagation of radio waves is disturbed by scattering, reflections, and shadowing. Generally, many replicas of the same signal arrive at the receive antenna with different delays, attenuations, and phases. Moreover, the channel is time-variant due to the movements of the transmitter or the receiver. A channel with N propagation paths can be described by its equivalent baseband impulse response N−1  h(t, τ)= h(t, ν)· δ(τ− τ ), (1.15) ν ν=0 where t denotes the observation time and h(t, ν) the complex-valued weighting coefficient corresponding to the ν-th path with delay τ . ν Statistical Characterization Due to the stochastic nature of mobile radio channels, they are generally classified by their statistical properties. The autocorrelation function ∗ φ (t, τ)= Eh (t, τ)h(t+ t, τ) (1.16) HH of h(t, τ) with respect to t is an appropriate measure for this classification. The faster the channel changes, the faster φ (t, τ) vanishes in the direction of t. This relationship HHINTRODUCTION TO DIGITAL COMMUNICATIONS 13 can also be expressed in the frequency domain. The Fourier transformation of φ (t, τ) HH with respect to t yields the scattering function  (f ,τ)=Fφ (t, τ). (1.17) HH d HH The Doppler frequency f originates from the relative motions between the transmitter and d the receiver. Integrating over τ leads to the Doppler power spectrum ∞   (f )=  (f ,τ)dτ, (1.18) HH d HH d 0 describing the power distribution with respect to f . The range over which  (f ) is d HH d almost nonzero is called Doppler bandwidth B . It represents a measure for the time variance d of the channel and its reciprocal 1 t = (1.19) c B d denotes the coherence time. For t  T , the channel is slowly fading, for t  T , it changes c s c s remarkably during the symbol duration T . In the latter case, it is called time-selective and s time diversity (cf. Section 1.4) can be gained when channel coding is applied. Integrating  (f ,τ) versus f instead of τ delivers the power delay profile HH d d f dmax   (τ)=  (f ,τ)df (1.20) HH HH d d −f dmax that describes the power distribution with respect to τ. The coherence bandwidth defined by 1 B = (1.21) c τ max represents the bandwidth over which the channel is nearly constant. For frequency-selective channels, B B holds, that is, the signal bandwidth B is much larger than the coherence c bandwidth and the channel behaves differently in different parts of the signal’s spectrum. In this case, the maximum delay τ is larger than T so that successive symbols overlap, max s resulting in linear channel distortions called intersymbol interference (ISI). If the coefficients hk,κ in the time domain are statistically independent, frequency diversity is obtained (cf. Section 1.4). For B B , the channel is frequency-nonselective, that is, its spectral density c is constant within the considered bandwidth (flat fading). Examples for different power delay profiles can be found in Appendix A.2. Modeling Mobile Radio Channels Typically, frequency-selective channels are modeled with time-discrete finite impulse response (FIR) filters following the wide sense stationary uncorrelated scattering (WSSUS) approach (Hoher ¨ 1992; Schulze 1989). According to (1.11), the signal is passed through a tapped-delay-line and weighted at each tap with complex channel coefficients hk,κas shown in Figure 1.12. www.allitebooks.com14 INTRODUCTION TO DIGITAL COMMUNICATIONS xk T T T s s s hk, 0 hk, 1 hk, 2 hk,L − 1 t nk yk + Figure 1.12 Tapped-delay-line model of frequency-selective channel with L taps t Although the coefficients are comprised of transmit and receive filters, as well as the channel impulse response h(t, τ) and the prewhitening filter g k, (as stated in W Section 1.2.1), they are assumed to be statistically independent (uncorrelated scattering). The length L = τ /T of the filter depends on the ratio of maximum channel delay t max s τ and symbol duration T . Thus, the delay axis is divided into equidistant intervals and max s for example, the n propagation paths falling into the κ-th interval compose the coefficient κ n −1 κ  j2πf kT +jϕ d,i s i hk,κ= e (1.22) i=0 with ϕ as the initial phase of the i-th component. The power distribution among the taps i according to the power delay profiles described in Appendix A.2 (Tables A.1 and A.3) can be modeled with the distribution of the delays κ. The more delays that fall into a certain interval, the higher the power associated with this interval. Alternatively, a constant number of n propagation paths for each tap can be assumed. In this case, n−1  j2πf kT+jϕ s d,i i hk,κ= ρ · e (1.23) κ i=0 holds and the power distribution is taken into account by adjusting the parameters ρ .The κ Doppler frequencies f in (1.22) and (1.23) depend on the relative velocity v between the d,i transmitter and the receiver, the speed of light c and the carrier frequency f 0 0 v f = · f · cosα. (1.24) d 0 c 0 In (1.24), α represents the angle between the direction of arrival of the examined propagation path and the receiver’s movement. Therefore, its distribution also determines that of f d leading to Table A.2. Maximum and minimum Doppler frequencies occur for α=0and α= 180, respectively, and determine the Doppler bandwidth B = 2f . The classical d dmax Jakes distribution depicted in Figure 1.13 A √ f≤ f d dmax 2 1−(f /f ) d dmax  (f )= (1.25) HH d 0else,INTRODUCTION TO DIGITAL COMMUNICATIONS 15 2 1.5 1 0.5 0 −1.5 −1 −0.5 0 0.5 1 1.5 f /f → d dmax Figure 1.13 Distribution of Doppler frequencies for isotropic radiations (Jakes spectrum) is obtained for isotropic radiations without line-of-sight (LoS) connection. For referred directions of arrival, Gaussian distributions with appropriate means and variances are often assumed (cf. Table A.2 for τ 0.5 µs). Unless otherwise stated, nondissipative channels 2 assume meaning, so that E hk,κ= 1 holds. H κ In the following part, the focus is on the statistics of a single channel coefficient and, therefore, drop the indices k and κ. For a large number of propagation paths per tap, real and imaginary parts ofH are statistically independent and Gaussian distributed stochastic √ 2 2 processes and the whole magnitudeH= H +H is Rayleigh distributed 2 2 2 2ξ/σ · exp(−ξ /σ)ξ≥ 0 H H p (ξ)= (1.26) H 0else 2 2 with mean E h= πσ /2. In (1.26), σ denotes the average power ofH. The instan- H H H taneous power which is chi-squared distributed with two degrees of freedom 2 2 1/σ · exp(−ξ/σ)ξ≥ 0 H H p (ξ)= (1.27) 2 H 0else while the phase is uniformly distributed in −π, π. If a LoS connection exists between the transmitter and the receiver, the total power P of the channel coefficient h is shared among a constant LoS and a Rayleigh fading 2 component with a variance of σ . The power ratio between both parts is called Rice factor H 2 2 2 2 K= σ /σ . Hence, the LoS component has a power of σ = Kσ and the channel LoS H LoS H coefficient becomes 2 h= σ · K+ α (1.28) H 2 with total power P = (1+ K)σ . The fading process α consists of real and imaginary parts H 2 that are statistically independent zero-mean Gaussian processes each with variance σ /2. H p (f /f )→ f d dmax d16 INTRODUCTION TO DIGITAL COMMUNICATIONS 2 b) P = 1 a) σ = 1 H 1 2 K= 0 K= 0 K= 1 K= 1 0.8 K= 3 K= 6 1.5 K= 5 K= 11 0.6 1 0.4 0.5 0.2 0 0 0 1 2 3 4 0 1 2 3 ξ→ ξ→ Figure 1.14 Rice distributions for different Rice factors K As shown in (Proakis 2001), the magnitude ofH is Ricean distributed   2 2 2 2 2ξ/σ · exp − ξ /σ − K · I 2ξ K/σ ξ≥ 0 0 H H H p (ξ)= (1.29) H 0else . In (1.29), I (·) denotes the zeroth-order modified Bessel function of first kind (Benedetto 0 and Biglieri 1999). With reference to the squared magnitude, we obtain the density   2 2 2 1/σ · exp − ξ/σ − K · I 2 ξK/σ ξ≥ 0 0 H H H p 2(ξ)= (1.30) H 0else . The phase is no longer uniformly distributed. 2 Figure 1.14a shows some Rice distributions for a constant fading variance σ =1and H varying Rice factor. For K= 0, the direct component vanishes and pure Rayleigh fading is 2 obtained. In Figure 1.14b, the total average power is fixed to P =1and σ = P/(K+ 1) H is adjusted with respect to K. For a growing Rice factor, the probability density function becomes more narrow and reduces to a Dirac impulse for K→∞. This extreme case corresponds to the AWGN channel without any fading. The reason for especially discussing the above channels is that they somehow repre- sent extreme propagation conditions. The AWGN channel represents the best case because noise contributions can never be avoided perfectly. The frequency-nonselective Rayleigh fading channel describes the worst-case scenario. Finally, Rice fading can be interpreted as a combination of both, where the Rice factor K adjusts the ratio between AWGN and fading parts. 1.2.4 Systems with Multiple Inputs and Outputs So far, this section has only described systems with a single input and a single output. Now, the scenario is extended to MIMO systems that have already been introduced in p (ξ)→ H p (ξ)→ HINTRODUCTION TO DIGITAL COMMUNICATIONS 17 h k,κ 1,1 n k 1 h k,κ 2,1 y k x k 1 1 n k 2 x k y k 2 2 h k,κ 2,N I n k N O h k,κ x k N ,N y k O I N N I O Figure 1.15 General structure of frequency-selective MIMO channel Subsection 1.1.1. However, at this point we are restricted to a general description. Specific communication systems are treated in Chapters 4 to 6. According to Figure 1.1, the MIMO system consists of N inputs and N outputs. Based I O on (1.11), the output of a frequency-selective SISO channel can be described by L−1 t  yk= hk,κ· xk− κ+ nk. κ=0 This relationship now has to be extended for MIMO systems. As a consequence, N I signals x k, 1≤ µ≤ N , form the input of our system at each time instant k and we µ I obtain N output signals y k, 1≤ ν≤ N . Each pair (µ, ν) of inputs and outputs is O ν O connected by a channel impulse response h k,κ as depicted in Figure 1.15. Therefore, ν,µ the ν-th output at time instant k can be expressed as N L−1 I t   y k= h k,κ· x k− κ+ n k (1.31) ν ν,µ µ ν µ=1 κ=0 where L denotes the largest number of taps among all the contributing channels. Exploiting t vector notations by comprising all the output signals y k into a column vector ykand ν all the input signals x k into a column vector xk, (1.31) becomes µ L−1 t  yk= Hk,κ· xk− κ+ nk. (1.32) κ=0 In (1.32), the channel matrix has the form   h k,κ ··· h k,κ 1,1 1,N I  . .  . . . . Hk,κ= . (1.33)   . . . h k,κ ··· h k,κ N ,1 N ,N O O I Finally, we can combine the L channel matrices Hk,κ to obtain a single matrix Hk= t T T T Hk, 0···Hk,L − 1. With the new input vector x k= xk ···xk− L − 1 t L t t we obtain yk= Hk· x k+ nk. (1.34) L t18 INTRODUCTION TO DIGITAL COMMUNICATIONS 1.3 Signal Detection 1.3.1 Optimal Decision Criteria This section briefly introduces some basic principles of signal detection. Specific algorithms for special systems are described in the corresponding chapters. Assuming a frame-wise transmission, that is, a sequence x consisting of L discrete, independent, identically dis- x tributed (i.i.d.) symbols xk is transmitted over a SISO channel as discussed in the last section. Moreover, we are restricted to an uncoded transmission, while the detection of coded sequences is subject to Chapter 3. The received sequence is denoted by y and com- prises L symbols yk. y Sequence Detection For frequency-selective channels, y suffers from ISI and has to be equalized at the receiver. The optimum decision rule for general channels with respect to the frame error probability P looks for the sequence x˜ that maximizes the a posteriori probability PrX = x˜ y, that f is, the probability that x˜ was transmitted under the constraint that y was received. Applying Bayes’ rule ˜ PrX = x PrX = x˜Y= y= p (y)· , (1.35) Yx˜ p (y) Y we obtain the maximum a posteriori (MAP) sequence detector   xˆ= argmax Prx˜ y= argmax p (y)· Prx˜ (1.36) Yx˜ L L x x x˜∈X x˜∈X L 6 x where X denotes the set of sequences with length L and symbols xk∈ X. x It illustrates that the sequence MAP detector takes into account the channel influence by p (y) as well as a priori probabilities Prx˜ of possible sequences. It has to be emphasized Yx˜ that p (y) is a probability density function since y is distributed continuously. On the Yx˜ contrary, Prx˜ y represents a probability because x˜ serves as a hypothesis taken from a L x finite alphabet X and y represents a fixed constraint. If either Prx˜ is not known, a priori to the receiver or all sequences are uniformly distributed resulting in a constant Prx˜, we obtain the maximum likelihood (ML) sequence detector xˆ= argmax p (y). (1.37) Yx˜ L x x˜∈X Under these assumptions, it represents the optimal detector minimizing P . Since the sym- f bols xkin x˜ are elements of a discrete set X (cf. Section 1.4), the detectors in (1.36) and (1.37) solve a combinatorial problem that cannot be fixed by gradient methods. An L x exhaustive search within the set of all possible sequences x˜∈ X requires a computational effort that grows exponentially withX and L and is prohibitive for most practical cases. x An efficient algorithm for an equivalent problem – the decoding of convolutional codes (cf. Section 3.4) – was found by Viterbi in 1967 (Viterbi 1967). Forney showed in (Forney 1972) that the Viterbi algorithm is optimal for detecting sequences in the presence of ISI. 6 For notational simplicity, p (y) is simplified to p (y) and equivalently PrX= x˜ to Prx˜.The term YX=˜ x Y˜ x p (y) can be neglected because it does not depend on x˜. YINTRODUCTION TO DIGITAL COMMUNICATIONS 19 Orthogonal Frequency Division Multiplexing (OFDM) and CDMA systems offer different solutions for sequence detection in ISI environments. They are described in Chapter 4. Symbol-by-Symbol Detection While the Viterbi algorithm minimizes the error probability when detecting sequences, the optimal symbol-by-symbol MAP detector  xˆk= argmax PrX k= X y= argmax PrX = x˜ y µ X ∈X X ∈X µ µ L x x˜∈X x˜k=X µ  = argmax p (y)· Prx˜ (1.38) Yx˜ X ∈X µ L x x˜∈X x˜k=X µ minimizes the symbol error probability P . Obviously, the difference compared to (1.36) is s the fact that all sequences x˜ with x˜k= X contribute to the decision, and not only to the µ most probable one. Both approaches need not deliver the same decisions as the following example demonstrates. Consider a sequence x= x0,x1 of length L = 2 with binary x symbols xk∈X ,X . The conditional probabilities Prx˜ y= Pr˜ x0, x˜1 y are 0 1 exemplarily summarized in Table 1.1. While the MAP sequence detector delivers the sequence xˆ= X ,X with the highest 0 1 a posteriori probability Prx˜ y= 0.27, the symbol-by-symbol detector decides in favor to  PrX 0= X y= PrX 0= X ,X 1= X y µ µ ν X ∈X ν (and an equivalent expression for x1) resulting in the decisions xˆ0=ˆ x1= X . How- 0 ever, the difference between both approaches is only visible at low SNRs and vanishes at low error rates. Again, for unknown a priori probability or uniformly distributed sequences, the corre- sponding symbol-by-symbol ML detector is obtained by  xˆk= argmaxp (y)= argmax p (y). (1.39) YX k=X Yx˜ µ X ∈X X ∈X µ µ L x x˜∈X x˜k=X µ Table 1.1 Illustration of sequence and symbol-by-symbol MAP detection Pr˜ x0, x˜1 y˜ x1= X x˜1= X Pr˜ x0 y 0 1 x˜0= X 0.26 0.27 0.53 0 x˜0= X 0.25 0.22 0.47 1 Pr˜ x1 y 0.51 0.4920 INTRODUCTION TO DIGITAL COMMUNICATIONS Memoryless channels For memoryless channels like AWGN and flat fading channels and i.i.d. symbols xk,  the a posteriori probability Prx˜ y can be factorized into Pr˜ xk yk. Hence, the k detector no longer needs to consider the whole sequence, but can instead decide symbol by symbol. In this case, the time index k can be dropped and (1.38) becomes xˆ= argmax PrX = X y. (1.40) µ X ∈X µ Equivalently, the ML detector in (1.39) reduces to xˆ= argmaxp (y). (1.41) YX µ X ∈X µ 1.3.2 Error Probability for AWGN Channel This section shall describe the general way by which to determine the probabilities of decision errors. The derivations are restricted to memoryless channels but can be extended to channels with memory or trellis-coded systems. In these cases, vectors instead of symbols have to be considered. For a simple AWGN channel, y= x+ n holds and the probability density function p (y) in (1.41) has the form YX µ 2 2 1 −y−X /σ µ N p (y)= · e (1.42) YX µ 2 πσ N (cf. (1.12)). With (1.42), a geometrical interpretation of the ML detector in (1.41) shows 2 that the symbol X out of X that minimizes the squared Euclidean distancey− X is µ µ determined. Let us now define the decision region   2 2 D = yy− X y− X ∀ X = X (1.43) µ µ ν ν µ for symbol X comprising all symbols y∈C whose Euclidean distance to X is smaller µ µ than to any other symbol X = X . The complementary set is denoted by D . Assuming ν µ µ that X was transmitted, a detection error occurs fory/∈D or equivalently y∈D .The µ µ µ  complementary set can be expressed by the union D = D of the sets µ µ,ν ν=µ   2 2 D = yy− X y− X (1.44) µ,ν µ ν containing all symbols y whose Euclidean distance to a specific X is smaller than to ν X . This does not mean that X has the smallest distance of all symbols to y∈ D . µ ν µ,ν The symbol error probability can now be approximated by the well-known union bound (Proakis 2001)        P (X )= Pr y∈D = Pr y∈ D s µ µ µ,ν   ν= µ    ≤ Pr y∈ D . (1.45) µ,ν ν=µ

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