Section 6.12 Summary and Discussion
`
`437
`
`referred to as a muttiple-input, multiple-output (MIMO) wireless communication
`system, which includes receive diversity and transmit diversity as special cases of space
`diversity. The novel feature of the MIMO systemis that, in a rich Rayleigh scattering
`environment, it can provide a high spectral efficiency, which may be explainedasfol-
`lows: The signals radiated simultaneously by the transmit antennasarrive at the input
`of each receive antenna in an uncorrelated mannerdueto the rich scattering mecha-
`nism of the channel. The net result is the potential for a spectacular increase in the
`spectral efficiency of the wireless link. Most
`importantly,
`the spectral efficiency
`increases roughly linearly with the numberof transmit or receive antennas, whichever
`is less. This result assumes that the receiver has knowledge of channel state informa-
`tion. The spectralefficiency of the MIMOsystem can be further enhanced byincluding
`a feedback channel from the transmitter to the receiver, whereby the channelstate is
`also made available to the transmitter, and with it, the transmitter is enabled to exer-
`cise control over the transmitted signal.
`Increasing spectral efficiency in the face of multipath fading is one important
`motivation for using MIMOtransmission schemes. Another important motivation is
`the developmentof space-time codes, whose aim is the joint coding of multiple transmit
`antennas so as to provide protection against channel fading, noise, and interference. In
`this context, of particular interest is a class of block codes referred to as orthogonal and
`generalized complex orthogonal space-time block codes. In this class of codes, the
`Alamouti code, characterized by a two-by-two transmission matrix, is the only full-rate
`complex orthogonal space-time block code. The Alamouti codesatisfies the condition
`for complex orthogonality or unitarity in both the spatial and temporal sense. In con-
`trast, the generalized complex orthogonal space-time codes can accommodate more
`than two transmit antennas; they are therefore capable of providing a larger coding
`gain than the Alamouti code for a prescribed bit error rate and total transmission rate
`at the expense of a reduced code rate and increased computational complexity. How-
`ever, unlike the Alamouti code, the generalized complex orthogonal space-time codes
`satisfy the condition for complex orthogonality only in the temporal sense. Accord-
`ingly, the complex orthogonal space-time codes,including the Alamouti code and gen-
`eralized forms, permit the use of linear receivers.
`The complex orthogonal property of the Alamouti codeis exploited in the devel-
`opmentof a differential space-time block coding scheme, which eliminates the need for
`channel estimation and thereby simplifies the receiver design. This simplification is,
`however, attained at the expense of degradation in receiver performance, compared
`with the coherent version of the Alamouti code, which assumes knowledge of the
`channel state information at the receiver.
`Space was also discussed in the context of space-division multiple-access
`(SDMA), the mechanization of which relies on the use of highly directional antennas.
`SDMA improves system capacity by allowing a greater reuse of the available spectrum
`through a combination of two approaches: minimization of the effects of interference
`and increased signal strength for both the user terminal and the base station. Advanced
`techniques such as phased-array antennas and adaptive antennas, which have been
`
`
`
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`SAMSUNG EXHIBIT 1010 (Part 4 of 4)
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`

`433
`
`Chapter 6 Diversity, Capacity and Space-Division Multiple Access
`
`researched extensively under the umbrellas of signal processing and radar for more
`than three decades, are well suited for implementing the practical requirements of both
`approaches.
`Underthe three theme examples, we discussed three different BLAST architec-
`tures issues relating to antenna diversity, spectral efficiency, as well as keyhole chan-
`nels. Each of the BLAST architectures, namely, diagonal-BLAST, vertical-BLAST,
`and Turbo-BLAST,offers distinct features of its own. Diagonal-BLAST (D-BLAST)
`makesit possible to closely approximate the ergodic channelcapacity in a rich scatter-
`ing environment and maytherefore be viewed as the benchmark BLASTarchitecture.
`Butit is impractical, as it suffers from a serious space-time edge wastage. Vertical-
`BLAST (V-BLAST) mitigates the computational difficulty problem of D-BLASTat
`the expense of a reduced channel capacity. Turbo-BLAST uses a random layered
`space-time code at the transmitter and incorporates the turbo coding principle in
`designing an iterative receiver. In so doing, Turbo-BLAST offers a significant
`improvementin spectral efficiency over V-BLAST,yet the computational complexity
`is maintained at a manageable level. In terms of performance, Turbo-BLAST outper-
`forms V-BLASTfor a prescribed (N,, N,) antenna configuration, but does not perform
`as well as D-BLAST.
`The different BLAST architectures were discussed in Theme Example 1. The
`material presented in Theme Example 2 taughtus the following:
`e The twobasic formsofdiversity, namely, transmit diversity and receive diversity,
`play complementaryroles, with both of them being locatedat the basestation.
`e For low SNR and fixed spectral efficiency, V-BLAST outperforms space-time
`block codes (STBCs) on {N,, N,} antenna configurations with N,.>N,.
`e Assuming the use of forward error-correction channel codes, a two-by-two
`STBC system could provide an adequate performance for wireless communica-
`tions at low SNR.
`e Diversity order is determined experimentally by measuring the asymptotic slope
`of the average frameerror rate (or average symbolerror rate) plotted versus the
`signal-to-noise ratio on a log-log scale.
`° MIMOsystemsprovide a trade-off between outage capacity and diversity order,
`depending on howthe system is configured.
`The degenerate occurrence of keyhole channels, discussed in Theme Example 3
`arises when the rank of the channel matrix is reduced to unity, in which case the capa-
`city of the MIMOlinkis equivalent to that of a single-input, single-output link operat-
`ing at the samesignal-to-noise ratio, Fortunately, the physical occurrence of keyhole
`channels is a rare phenomenon.
`Onelast commentis in order: the discussion of channel capacity presented in the
`chapter focused on single-user MIMOlinks. Although,indeed, wireless systems in cur-
`rent use cater to the needs of multiple users, the focus on single users may bejustified
`on the following grounds:
`e The derivation of MIMO channelcapacity is much easier to undertakefor single
`users than multiple users.
`
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`

`

`Notes and References
`
`439
`
`e Capacity formulas are known for many single-user MIMOcases, whereas the
`corresponding multiuser ones are unsolved.
`Simply put, very little is known about the channel capacity of multiuser MIMOlinks,
`unless the channel state is knownat both the transmitter and receiver.”°
`
`NOTES AND REFERENCES
`
`| For detailed discussions of the receive diversity techniquesofselection combining, max-
`imal-ratio combining, and square-law combining, see Schwartzetal. (1966), Chapter 10.
`? The term “maximal-ratio combiner” was coined inaclassic paperon linear diversity
`combining techniques by Brennan (1959).
`3 The three-point exposition presented in Section 6.2.3 on maximal-ratio combining
`follows S. Stein in Schwartz (1966), pp. 653-654.
`‘For expository discussions of the many facets of MIMO wireless communications,
`see the papers by Gesbert et al. (2003), Diggavi et al. (2003), and Goldsmith et al.
`(2003). The paper by Diggaviet al. includes an exhaustive list of references to MIMO
`wireless communications and related issues. For books on wireless communications
`using multiple antennas, see Hottinen et al. (2003) and Vucetic and Yuan (2003).
`Impulsive noise due to human-made electromagnetic interference is discussed in
`Blackard and Rappaport (1993), and Wang and Poor(1999); see also Chapter 2.
`©The formula of Eq. (6.56), defining the ergodic capacity of a flat-fading channel,is
`derived in Ericson (1970).
`1 The log-det capacity formula of Eq. (6.59) for MIMOwireless links operating in rich
`scattering environments was derived independently by Teletar (1995) and Foschini
`(1996); Teletar’s report was published subsequently as a journal article (1999). For a
`detailed derivation of the log-det capacity formula, see Appendix G.
`8 The Gaussian approximation of the probability density function of the instantaneous
`channel capacity of a MIMOwirelesslink, which is governed by the log-det formula,is
`discussed in detail in Hochwald etal. (2003).
`° The result that at high signal-to-noise ratios the outage probability and frame (burst)
`error probability are the sameis derived in Zheng and Tse (2002).
`!0MIMOwireless communications systems incorporating the use of feedback chan-
`nels are discussed in Vishwanath et al. (2001), Simon and Moustakas (2003), and
`Hochwald et al. (2003). The latter paper introduces the notion of rate feedback by
`quantizing the instantaneous channel capacity of the MIMO link.
`' The effect of correlation fading on the channel capacity of MIMO wireless commu-
`nicationsis discussed in Shiu et al. (2000) and Smith et al. (2003).
`i2 Space-timetrellis codes are discussed in Tarokhet al. (1998).
`The Alamouti code was pioneered by Siavash Alamouti (1998); the code has been
`adopted in third-generation (3G) wireless systems, in which it is known as space-time
`transmit diversity (STTD).
`
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`

`440
`
`Chapter 6 Diversity, Capacity and Space-Division Multiple Access
`
`4 The generalized space-time orthogonal codes were originated by Tarokh etal.
`(1999a,b).
`15 The decoding algorithms (written in MATLAB)for the Alamouti code S, and the
`orthogonal space-time codes G3, Gy, H3, and Hy due to Tarokhet al. are presented
`in the Solutions Manual to this book. It should, however, be noted that there are
`minor errors in the original decoding algorithms for Hz, and Hy listed in the Appen-
`dix to the paper by Tarokhetal. (1998). These errors have been corrected in the per-
`tinent MATLABcodes.
`16 Differential space-time block coding, based on the Alamouti code, was first
`described by Tarokh and Jafarkhani (2000). See also the article by Diggavi et al.
`(2002), which combines this form of differential coding with orthogonal frequency-
`division multiplexing (OFDM)for signal transmission over fading frequency-selective
`channels; OFDM wasdiscussed in Chapter3.
`'7 Chapter 3 of Liberti and Rappaport (1999) describes more general models for
`phased arrays other than linear and wheregain in elevation angle as well as azimuthis
`of interest. Chapter 8 of the same book describes various algorithms for adapting the
`weighting vector, depending upon the direction ofarrival of the signal.
`18-The circular model for effective scatterers was proposed in Lee (1982).
`'° Tn Chapter 7 of Liberti and Rappaport (1999), the single-bounceelliptical modelis
`described in greater detail. Note that the model does not take into accountthe effects
`of diffraction.
`
`20 The D-BLASTarchitecture was pioneered by Foschini (1996) and discussed further
`in the papers by Foschini and Gans (1999) and Foschinietal. (2003).
`? Thefirst experimentalresults in the V-BLASTarchitecture wereoriginally reported
`in the article by Golden et al. (1999); see also the paper by Foschini et al. (2003), in
`which this particular form of BLAST is referred to as horizontal-BLAST, or
`H-BLAST.
`
`22The Turbo-BLASTarchitecture wasfirst described by Sellathurai and Haykin
`(1998), with additional results reported subsequently in the papers by the same
`authors (2000, 2002, 2003).
`3 The experimental results presented in Figs. 6.40 through 6.42 are reproduced from
`the paper by Sellathurai and Haykin (2002) with permission of the IEEE.
`74 According to deHaas (1927, 1928), the possibility of using antenna diversity for mit-
`igating short-term fading effects in radio communications was apparently first discov-
`ered in experiments with spaced receiving antennas operating in the high-frequency
`(HF) band. For additional historical notes, see Chapter 10 by Seymour Stein in
`Schwartz et al. (1966).
`25 For definitions of the diversity order and multiplexing gain of MIMO wireless com-
`munication systems and the implications of these definitions in terms of system beha-
`vior, see Digavi (2003).
`
`
`
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`
`

`

`Additional Problems
`
`441
`
`°6 Keyhole channels, also dubbed pinhole channels, were described independently by
`Gesbertet al. (2002) and Chizhik et al. (2002).
`27 The GBGP model for MIMOwirelesslinks is described in Gesbertet al. (2002).
`78 Multiuser MIMOwireless systems are discussed in Diggaviet al. (2003) and Gold-
`smith etal. (2003).
`
`ADDITIONAL PROBLEMS
`
`Diversity-on-receive techniques
`
`Problem 6.21 A receive-diversity system uses a selection combiner with two diversity paths.
`An outage occurs when the instantaneoussignal-to-noise ratio y drops below 0.25 y,,,, where y,,
`is the average signal-to-noise ratio. Determine the probability of outage experienced by the
`receiver.
`
`Problem 6.22 Theaveragesignal-to-noise ratio in a selection combiner is 20 dB. Compute
`the probability that the instantaneous signal-to-noise ratio of the device drops below y= 10 dB
`for the following number of receive antennas:
`
`(a) N,=1
`(b) N,=2
`(c) N,=3
`(d) N,=4
`Comment on yourresults.
`
`Problem 6.23 Repeat Problem 6.22 for y= 15 dB.
`
`Problem 6.24 In Section 6.2.2, we derived the optimum values of Eq. (6.18) for complex
`weighting factors of the maximal-ratio combiner using the Cauchy—Schwartz inequality. This
`problem addresses the sameissue, using the standard maximization procedure. To simplify mat-
`ters, the numberN, of diversity pathsis restricted to two, with the complex weighting parameters
`denoted by a, and ap.
`Let
`
`Then the complex derivative with respect to a, is defined by
`
`ay = XyetjVy
`
`k= 1,2
`
`FUP2)
`da,
`2\dx,
`“OY,
`
`pars
`
`Applying this formula to the combiner’s output signal-to-noise ratio y, of Eq. (6.14), derive
`Eq. (6.18).
`
`In this problem, we develop an approximate formula for the probability of
`Problem 6.25
`error, P,, produced by a maximal-ratio combiner for coherent FSK. We start with Eq. (6.25), and
`for small ¥);., we may use the following approximation for the probability density function:
`
`1
`N,-1
`Sré mre) = VY Ymre
`¥3(N,—1)!
`
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`
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`
`

`

`442
`
`Chapter 6 Diversity, Capacity and Space-Division Multiple Access
`
`(a) Using the conditional probability of error for coherent BFSK,thatis,
`
`Prob(error|¥pyrc) = yerfe( +Yaxc)
`
`1
`
`derive the approximation
`
`Ee
`
`1
`N,
`aftan) (N,-1)!
`
`1
`
`Sa
`J erfe(/y)y
`0
`
`Nill
`
`dy
`
`1
`where yp = 5 Vinre"
`Integrating the definite integral by parts and using the definition of the complementary
`error function, show that
`
`(b)
`
`1
`
`—
`
`Pe=§—rey ow
`2d(5Ye) N,!
`
`(c) Finally, using the definite integral
`
`obtain the desired approximation
`
`Pé
`
`*
`
`1
`2Je(5%)
`
`WN,
`
`A
`
`Problem 6.26
`
`(a) Using the approximation for /,(y,,,.) given in Problem 6.25, determine the probability of
`symbol error for a maximal-ratio combiner that uses noncoherent BFSK.
`(b) Compare yourresult of part(a) with that of Problem 6.25 for coherent BFSK.
`
`Problem 6.27
`
`(a) Continuing the approximation to f-(Ym;,), determine the probability of symbol error for
`a maximal-ratio combiner that uses coherent BPSK.
`
`(b) Compare yourresult of part(a) with that of Problem 6.25 for coherent BFSK.
`
`Problem 6.28 Asdiscussed in Section 6.2.3, an equal-gain combineris a special form of the
`maximal-ratio combiner for which the weighting factors are all equal. For convenience of pre-
`sentation, the weighting parameters are set to unity. Assuming that the instantaneoussignal-to-
`noise ratio yis small compared with the average signal-to-noise ratio y,,, derive an approximate
`formula for the probability density function of y.
`
`Problem 6.29 Compare the performances of the following linear diversity-on-receive
`techniques:
`
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`
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`
`

`

`
`
`Additional Problems
`
`443
`
`(a) Selection combiner
`(b) Maximal-ratio combiner
`(c) Equal-gain combiner
`Base the comparison on signal-to-noise improvement, expressed in dB, for N, = 2,3, 4,5, and 6
`diversity branches.
`
`Show that the maximum-likelihood decision rule for the maximal-ratio
`Problem 6.30
`combiner may be formulated in the following equivalent forms:
`(a) Choose symbols; over s, if and only if
`
`
`
`
`2, 2 a BioyoiBo 2 et '
`
`(a, + 05)]s;| ~¥18} —Y15; < (G4 + O})| 54 — VS -ViSy
`k#i
`(b) Choose symbol 5; over s, if and only if
`.
`2.
`2
`Qin.
`oD
`2
`2
`i.
`k#i
`(a+ 05 -1)|5,|" +d" (11, 8,) < (a + a -1) 5,| +4 (py 5)
`
`Here, Hy, .§;) denotes the squared Euclidean distance betweenthe received signal y, and
`constellation points s;.
`
`It may be argued that,in a rather loose sense, transmit-diversity and receive-
`Problem 6.31
`diversity antenna configurationsare the dual of each other, asillustrated in Fig. 6.46.
`(a) Taking a general viewpoint,justify the mathematicalbasis for this duality.
`
`Transmit
`antenna
`
`Me,
`
`>——— 1
`
`&
`
`:
`:
`Diversity 7
`aths
`/”
`E
`-
`x
`vy et——
`fa
`x
`
`Rg.
`
`Ny~~
`
` ——«—<.
`Multiple
`receive
`antennas
`
`i. a ‘
`
`NN
`
`Ng
`2————<..
`wo \
`Ss
`Receive
`ft
`“
`antenna
`4
`/ Diversity paths
`
`“
`
`N..=-——————<
`Multiple
`transmit
`antennas
`
`FIGURE 6.46 Diagram for Problem 6.31.
`
`
`
`
`
`Page 382 of 474
`
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`
`

`

`daa
`
`Chapter 6 Diversity, Capacity and Space-Division Multiple Access
`
`(b) However, we may cite the example of frequency-division diplexing (FDD), in which, in a
`strict sense, the duality depicted in Fig. 6.44 is violated. How is it possible for the violation
`to arise in this example?
`
`MIMO Channel Capacity
`
`Problem 6.32
`
`In this problem, we continue with the solution to Problem 6.9, namely,
`as N—
`log ,2
`where N, = N, = N andA,,is the average eigenvalue of HH’ = H'H,
`
` Aay3 }
`
`(a) Justify the asymptotic result given in Eq. (6.61)—thatis,
`
`c 2 constant
`N
`
`Whatis the value of the constant?
`
`(b) What conclusion can you draw from the asymptotic result?
`
`Byandlarge, the treatment of the ergodic capacity of a MIMO channel, as
`Problem 6.33
`presented in Sections 6.3 and 6.5, focused on the assumption that the channel is Rayleigh dis-
`tributed. In this problem, we expand on that assumption by considering the channel to be Rician
`distributed. In such an environment, we may express the channel matrix as
`
`H = oH,,+H,,
`where H.,, and H,, denote the specular and scattered components, respectively. To be consistent
`with the MIMO modeldescribedin Section 6.3, the entries of both H,, and H,, have unit ampli-
`tude variance, with H,, being deterministic and H,, consisting of iid complex Gaussian-distributed
`variables with zero mean. The scaling parameter a is related to the Rice K-factor by the formula
`K = 10log,)a°dB
`(a) Considering the case of a pureline of sight (LOS), show that the MIMO channelhas the
`deterministic capacity
`
`C=log,(1 + N,a’p)_ bits/s/Hz
`where N,.is the numberof receive antennas andpis the total signal-to-noise ratio at each
`receiver input.
`(b) Compare the result obtained in part (a) with that pertaining to the pure Rayleigh distri-
`buted MIMOchannel.
`(c) Explore the more general situation, involving the combined presence of both the specular
`and scattered components in the channel matrix H.
`
`Problem 6.34 Suppose that an additive, temporally stationary Gaussian interference v(¢)
`corrupts the basic channel modelof Eq.(6.48). The interference v(t) has zero mean and correla-
`tion matrix R,, Evaluate the effect of v(t) on the ergodic capacity of the MIMOlink.
`
`Problem 6.35 Consider a MIMOlink for which the channel may be considered to be essen-
`tially constant for tT uses of the channel.
`
`
`
`Page 383 of 474
`
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`
`

`

`
`
`Additional Problems
`
`445
`
`(a) Starting with the basic channel model of Eq. (6.48), formulate the input-output relation-
`ship ofthis link, with the input described by the N,-by-t matrix
`S = [S1, S85, wees s,]
`(b) Howis the log-det capacity formula of the link correspondingly modified?
`
`Orthogonal Space-Time Block Codes
`
`Problem 6.36 The objective of this problem is to fill in the mathematical details that lie
`behind the formulas of Eqs. (6.104) and (6.105) for the maximum-likelihood estimates $1 and $9.
`(a) Starting with Eq. (6.102) for the combiner output yx and using Eq. (6.103) for the proba-
`bility density function of the additive complex Gaussian noise %;> formulate the expres-
`sion for the likelihood function of transmitted symbol sj; & = 1,2.
`(b) Hence, using the result of part (a), derive the formulas of Eqs. (6.103) and (6.104).
`Problem 6.37
`Figure 6.47 shows the extension of orthogonal space-time codes to the
`Alamouti code, using two antennas on transmit and receive. The sequence ofsignal encoding
`and transmissionsis identical to that of the single-receiver case of Fig. 6.18. Table 6.5(a) defines
`the channels between the transmit and receive antennas. Table 6.5(b) defines the outputs of the
`receive antennasat times f’ and f’ + 7’, where T is the symbol duration.
`
`51.
`id
`~ $2
`
`.
`
`Transmit
`antenna 1
`
`82,
`ie
`5]
`
`a
`
`.
`
`I Tipaaca
`|
`anten
`
`hy
`
`hy
`
`hy
`
`hg
`
`Receive
`Receive
`antenna 2
`antenna 1
`
`Interference
`
`
`@) mone
`a) xan noise
`
`
`() and noise It
`
`FIGURE 6.47 Diagram for Problem 6.37.
`
`
`
`Page 384 of 474
`
`
`
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`
`

`

`446
`
`Chapter 6 Diversity, Capacity and Space-Division Multiple Access
`
`TABLE 6.5
`
`‘Table for Problem 6.36.
`
`(a)
`Receive antenna |
`Receive antenna 2
`
`
`Transmit antenna 1
`hy
`hy
`
`Transmit antenna 2
`hy
`hg
`
`(b)
`
`Receive antenna 2
`Receive antenna 1
`
`¥3
`X,
`
`x5 4
`
`Time
`Time (+7
`
`(a) Derive expressions for the received signals ¥1, 2, ¥3, and X4, including the respective
`additive noise components, in termsof the transmitted symbols.
`(b) Derive expressionsfor the line of combined outputs in termsof the received signals.
`(c) Derive the maximum-likelihood decision rule for the estimates 5, and 5.
`
`Problem 6.38 This problem explores a newinterpretation of the Alamouti code. Let
`5; = sh) js?)
`B= 1y2
`where a and 5?) are both real numbers. The complex entry &; in the two-by-two Alamouti code
`is represented by the two-by-two real orthogonal matrix
`
`(2)
`Ss;
`:
`(4)
`(2)
`Uy
`Likewise, the complex-conjugated entry §; is represented by the two-by-two real orthogonal
`matrix
`
`i=1,2
`
`(1)
`S:
`,
`
`(1)
`(1)
`Se
`—s;
`f
`:
`2I
`
`i= 12
`
`(a) Show that the two-by-two complex Alamouti code§ is equivalent to the four-by-four real
`transmission matrix
`
`A 3) te Ww
`|
`(2)
`(1)
`(2)
`(1)
`“Sp
`87°
`| 8)
`89
`Sy base a —_——
`I
`6) $2 |2)
`I
`2
`1)
`|
`2
`1
`52)| ne ff)
`
`Page 385 of 474
`
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`
`

`

`
`
`Additional Problems
`
`447
`
`(b) Show that S, is an orthogonal matrix.
`(c) Whatis the advantage of the complex code S$ over the real code $4?
`
`Problem 6.39
`(a) Show that the generalized complex orthogonal space-time codes of Eqs. (6.107) and
`(6.108) satisfy the temporal orthogonality condition
`
`c'eé =1
`
`where the superscript t denotes Hermitian transposition and I denotes the identity
`matrix.
`
`(b) Likewise, show that the sporadic complex orthogonal space-time codes of Eqs. (6.109)
`and (6.110) satisfy the temporal orthogonality condition
`
`HH =I
`
`Problem 6.40 Applying the maximum-likelihood decoding rule, derive the optimum
`receivers for the generalized complex orthogonal space-time codes of Eqs. (6.107) and (6.108).
`
`Problem 6.41 Repeat Problem 6.40 for the sporadic complex orthogonal space-time codes
`of Eqs. (6.109) and (6.110).
`
`Show that the channel capacity of the Alamouti code is equal to the sum of
`Problem 6.42
`the channelcapacities of two single-input, single-output systems.
`
`Differential Space-Time Block Coding
`
`Problem 6.43 Equation (6.116) defines the input-output matrix relationship of the differ-
`ential space-time block coding system described in Section 6.7. Starting with Eqs. (6.98) and
`(6.99), derive Eq. (6.116).
`
`Problem 6.44 The constellation expansionillustrated in Fig. 6.44 is based on the polar base-
`band representation {-1, +1} for BPSK transmissions of the Alamouti code on antennas 1 and 2.
`Explorethe constellation expansion property of differential spacetime codingfor the following
`two situations:
`
`(a) Frameof reference: dibit 00
`
`(b) Frameof reference: dibit 11
`
`Comment on yourresults.
`
`In this problem, we investigate the use of QPSK for transmission of the
`Problem 6.45
`Alamouti code on antennas 1 and 2. The corresponding inputblock of data will be in the form of
`quadbits(i.e., 4-bit blocks). Perform the investigation for each of the two OPSK constellations
`depicted in Fig. 6.48. Use 0000 as the frameof reference.
`
`
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`448
`
`Chapter 6 Diversity, Capacity and Space-Division Multiple Access
`
`Imaginary
`
`Real
`Real
`
`Imaginary
`
`(b)
`
`FIGURE 6.48 Diagram for Problem 6.44.
`
`Problem 6.46 Repeat Problem 6.45 for the frame of reference 1111.
`
`Problem 6.47 In the analytic study of differential space-time block coding presented in
`Section 6.7, we ignored the presence of channel noise. This problem addresses the extension of
`Eg. (6.116) by including the effect of channel noise.
`
`(a) Starting with Eq. (6.101), expand the formulas of Eqs. (6.116) and (6.117) by including the
`effect of channel noise modeled as additive white Gaussian noise.
`
`(b) Using the result derived in part (a), expand the formula of Eq. (6.121) by including the
`effect of channel noise, which consists of the following components:
`
`(i) Two signal-dependent noise terms
`(ii) A multiplicative noise term consisting of the product of two additive white Gauss-
`ian noise terms
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`
`
`449
`
`(c) Show that, when the signal-to-noise ratio is high, the noise term (ii) of part (b) may be
`ignored, with the result that the remaining two signal-dependent noise terms (i) double
`the average power of noise compared with that experienced in the coherent detection of
`the Alamouti code.
`
`Theme Examples
`
`Problem 6.48 In this problem, we repeat Experiment 1 of Section 6.10, but this time we
`investigate the effect of increasing signal-to-noise ratio (SNR) on the symbol error rate (SER)
`for a prescribed modulation scheme,still operating in a Rayleigh fading environment.
`
`(a) Using 4-PSK for both STBC and V-BLAST,plot the SER versus SNR for the following
`antenna configurations:
`
`(i) N,=2,N,=2
`(ii) N,=2,N,=4
`
`(b) What conclusions do you draw from the experimental results of part (a)?
`
`Problem 6.49 Continuing with Problem 6.48, suppose the STBC and V-BLASTsystemsuse
`4-PSK. This time, however, we wish to display the spectral efficiency in bits/s/Hz versus the
`SNR. How would you expect the performance curve of STBC to compare against that of V-
`BLAST? Explain.
`
`Problem 6.50 Compare the relative merits of STBC systems versus BLAST systems in
`terms of the following issues:
`
`® Capacity
`e Diversity order
`e Multiplexing gain
`*« Computational complexity
`
`Additional Problems
`
`
`
`In Chapter 2, we discussed the reciprocity theorem in the context of a
`Problem 6.51
`single-input, single-output wireless communication link. Show that the theorem also applies
`to Eq. (6.146); that is, show that the channel matrix H of the MIMO linksatisfies the Hermitian
`property.
`
`
`
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`APPENDIX A
`
`Fourier Theory
`
`A.1 THE FOURIER TRANSFORM’
`
`Let g(t) denote a nonperiodic deterministic signal, expressed as some function of timet.
`By definition, the Fourier transform of the signal g(t) is given by the integral
`
`(A.1)
`Gf) = J g(dexp(j2afrar
`where j = ./—1, and the variable f denotes frequency. Given the Fourier transform
`G(f), the original signal g(¢) is recovered exactly using the formulafor the inverse Fou-
`rier transform:
`
`(A.2)
`a(t) = | G(fexp(j2afiaf
`Note that in Eqs. (A.1) and (A.2) we have used a lowercase letter to denote the time
`function and an uppercaseletter to denote the corresponding frequency function. The
`functions g(t) and G(f) are said to constitute a Fourier-transform pair.
`For the Fourier transform of a signal g(f) to exist, it is sufficient, but not neces-
`sary, that g(¢) satisfy three conditions, knowncollectively as Dirichlet’s conditions:
`
`1. The function g(t) is single valued, with a finite number of maxima and minima in
`any finite time interval.
`2. The function g(t) has a finite numberof discontinuities in any finite timeinterval.
`3. The function g(t) is absolutely integrable; thatis,
`
`[- le@lat<e
`
`We may safely ignore the question of the existence of the Fourier transform of a time
`function g(t) when g(t) is an accurately specified description of a physically realizable
`signal. In other words, physical realizability is a sufficient condition for the existence of
`a Fourier transform. Indeed, we may go one step further and state that all finite-
`energy signals are Fourier transformable.
`
`479
`
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`480
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`Appendix A Fourier Theory
`
`The absolute value of the Fourier transform G(f), plotted as a function of fre-
`quencyf, is referred to as the amplitude spectrum or magnitude spectrum of the signal
`g(t). By the same token, the argumentof the Fourier transform,plotted as a function
`of frequencyf, is referred to as the phase spectrum of the signal g(t). The amplitude
`spectrum is denoted by |G(/)| and the phase spectrum is denoted by 6(f). When g(*) is
`a real-valued function of time f, the amplitude spectrum IG(f)| is symmetrical about
`the origin f= 0, whereas the phase spectrum @(f ) is antisymmetrical about f= 0.
`Strictly speaking, the theory of the Fourier transform is applicable only to time
`functions that satisfy the Dirichlet conditions. (Among such functions are energy sig-
`nals.) However,it would be highly desirable to extendthis theory in two ways to include
`powersignals(i.c., signals whose average poweris finite). It turns out that this objective
`can be met through the “proper use”of the Dirac delta function, or unit impulse.
`The Dirac delta function, denoted by 6(¢), is defined as having zero amplitude
`everywhere except at t=0, whereitis infinitely large in such a way thatit contains unit
`area underits curve; thatis,
`
`a(t) = 0
`
`t#0
`
`(A.3)
`
`and
`
`(A.A)
`Pr &(A)dt = 1
`An implication of this pair of relations is that the delta function is an even func-
`tion oftime; that is, 6(-f) = 6(¢). Another important property of the Delta functionis
`the replication property described by
`
`(A.5)
`[- g(t)d(t-T)dt = g(t)
`which states that the convolution of any function with the delta function leaves that
`function unchanged.
`Tables A.1 and A.2 build on the formulas of Eqs. (A.1) through (A.5). In partic-
`ular, Table A.1 summarizes the properties of the Fourier transform, while Table A.2
`lists a set of Fourier-transform pairs.
`In the time domain, a linear system (e.g., filter) is described in termsofits impulse
`response, defined as the response of the system (with zero initial conditions) to a unit
`impulse or delta function 6(t) applied to the input of the system at time t =0.If the sys-
`tem is time invariant, then the shape of the impulse response is the same, no matter
`when the unit impulse is applied to the system. Thus, assuming that the unit impulse or
`delta function is applied at time t=0, we may denote the impulse responseof a linear
`time-invariant system by h(¢). Let this system be subjected to an arbitrary excitation
`x(t), as in Fig. A.1(a). Then the responsey(t) of the system is determined by the formula
`y(t) = i x(T)A(t—t)dt
`
`= r A(t)x(t—t)dt
`
`(A.6)
`
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` —sagtscetensunsbeteteaesortatetdy
`
`
`
`ieee StabiaadGocastoradsoucede
`
` ESSERESSSYESEEETR
`
`11. Multiplication in the time domain
`
`(ant) = [GANG(f- Ada
`
`12. Convolution in the time domain
`
`fo sigg(t- dat = G(f)G/)
`
`sghagnatene
`set
`
`13. Correlation theorem
`
`[aioes dt = GNGY)
`
`i4. Rayleigh’s energy theorem
`
`
`[lear = J" iecnPar
`
`The formula of Eq. (A.6)is called the convolution integral. Three different time
`scales are involved init: the excitation time t response time t, and system-memory time
`¢—t, Equation (A.6)is the basis of the time-domain analysis of linear time-invariant
`
`|
`
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`
`
`Section A.1 The Fourier Transform
`
`481
`
`
`
`TABLE A.1 Summary of Properties of the Fourier Transform.
`
`Property
`
`1. Linearity
`
`2. Time scaling
`
`3. Duality
`
`4. Timeshifting
`
`Mathematical Description
`
`
`ag (t)+ bgo(t) == aG,(f)+bG,(f )
`where @ and b are constants
`
`gaye taf)
`
`\a
`lal"
`where ais a constant
`
`it
`
`g(t} = GS),
`
`then G(t) => g(-/)
`
`B(t~ty) == G(flexp (-/2nfiq)
`
`3. Frequency shifting
`
`exp(j2af,oe(t) == Gf-f,)
`
`
`
`6. Area underg(r)
`
`7, Area under G(f)
`
`8. Differentiation in the time domain
`
`9, Integration in the time domain
`
`rr git)dt = G(0)
`
`(0) = [" GNaF
`
`i.
`d
`aol) = JemfGtys)
`
`t 1 G(0)
`Od = GN) + HS)
`
`10. Conjugate