Section 6.12 Summary and Discussion
`
`437
`
`referred to as a multiple-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 MlMO system is that, in a rich Rayleigh scattering
`environment, it can provide a high spectral efficiency, which may be explained as fol-
`lows: The signals radiated simultaneously by the transmit antennas arrive at the input
`of each receive antenna in an uncorrelated manner due to 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 number of transmit or receive antennas, whichever
`is less. This result assumes that the receiver has knowledge of channel state informa-
`tion.The spectral efficiency of the MIMO system can be further enhanced by including
`a feedback channel from the transmitter to the receiver, whereby the channel state 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 MIMO transmission schemes. Another important motivation is
`the development of 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-lime 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 code satisfies the condition
`for complex orthogonality or unitafity 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 spaceitime codes, including the Alamouti code and gen-
`eralized forms, permit the use of linear receivers.
`
`The complex orthogonal property of the Alamouti code is exploited in the devel-
`opment of 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
`
`
`
`.
`
`Page 376 of 474
`
`SAMSUNG EXHIBIT 1010 (Part 4 of 4)
`
`Page 376 of 474
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`
`

`

`438
`
`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.
`Under the 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 TurbOHBLAST, offers distinct features of its own. Diagonal-BLAST (D-BLAST)
`makes it possible to closely approximate the ergodic channel capacity in a rich scatter-
`ing environment and may therefore be viewed as the benchmark BLAST architecture.
`But it is impractical, as it suffers from a serious space—time edge wastage. Vertical-
`BLAST (V—BLAST) mitigates the computational difficulty problem of D-BLAST at
`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
`improvement in spectral efficiency over V—BLAST, yet the computational complexity
`is maintained at a manageable level. In terms of performance, Turbo—BLAST outper—
`forms V—BLAST for a prescribed (NI, Nr) 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 taught us the following:
`
`. The two basic forms of diversity, namely, transmit diversity and receive diversity,
`play complementary roles, with both of them being located at the base station.
`0 For low SNR and fixed spectral efficiency, V—BLAST outperforms space—time
`block codes (STBCs) on {NV NF} antenna configurations with N, > N:-
`
`0 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.
`
`0 Diversity order is determined experimentally by measuring the asymptotic slope
`of the average frame error rate (or average symbol error rate) plotted versus the
`signal-to-noise ratio on a log-log scale.
`0 MIMO systems provide a trade-off between outage capacity and diversity order,
`depending on how the 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 MIMO link is equivalent to that of a single-input, single—output link operat-
`ing at the same signal—to-noise ratio. Fortunately, the physical occurrence of keyhole
`channels is a rare phenomenon.
`One last comment is in order: the discussion of channel capacity presented in the
`chapter focused on single-user MIMO links. Although, indeed, wireless systems in cur—
`rent use cater to the needs of multiple users, the focus on single users may be justified
`on the following grounds:
`
`0 The derivation of MIMO channel capacity is much easier to undertake for single
`users than multiple users.
`
`Page 377 of 474
`
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`
`

`

`Notes and References
`
`439
`
`' Capacity formulas are known for many single-user MIMO cases, whereas the
`corresponding multiuser ones are unsolved.
`
`Simply put, very little is known about the channel capacity of multiuser MIMO links.
`unless the channel state is known at both the transmitter and receiver.28
`
`NOTES AND REFERENCES
`
`1 For detailed discussions of the receive diversity techniques of selection combining, max-
`imal-ratio combining, and square-law combining, see Schwartz et a1. (1966), Chapter 10.
`21116 term “maximal-ratio combiner" was coined in a classic paper on linear diversity
`combining techniques by Brennan (1959).
`3The three-point exposition presented in Section 6.2.3 on maximal-ratio combining
`follows S. Stein in Schwartz (1966), pp. 653—654.
`4For expository discussions of the many facets of MIMO wireless communications.
`see the papers by Gesbert et al. (2003), Diggavi et a1. (2003), and Goldsmith et a1.
`(2003). The paper by Diggavi et 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 a1. (2003) and Vucetic and Yuan (2003).
`5 Impulsive noise due to human-made electromagnetic interference is discussed in
`Blackard and Rappaport (1993), and Wang and Poor (1999); see also Chapter 2.
`6The formula of Eq. (6.56), defining the ergodic capacity of a flat-fading channel, is
`derived in Ericson (1970).
`7The log-det capacity formula of Eq. (6.59) for MIMO wireless 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.
`8The Gaussian approximation of the probability density function of the instantaneous
`channel capacity of a MIMO wireless link, which is governed by the log-det formula, is
`discussed in detail in Hochwald et al. (2003).
`9The result that at high signal—to-noise ratios the outage probability and frame (burst)
`errOr probability are the Same is derived in Zheng and Tse (2002).
`10 MIMO wireless 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.
`11 The effect of correlation fading on the channel capacity of MIMO wireless commu-
`nications is discussed in Shiu et al. (2000) and Smith et al. (2003).
`12 Space-time trellis codes are discussed in Tarokh et al. (1998).
`13 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).
`
`Page 378 of 474
`
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`
`

`

`440
`
`Chapter 6 Diversity, Capacity and SpacewDivision Multiple Access
`
`14The generalized space—time orthogonal codes were originated by Tarokh et al.
`(1999a,b).
`
`15rbe decoding algorithms (written in MATLAB) for the Alamouti code 5, and the
`orthogonal space—time codes G3, G4, H3, and H4 due to Tarokh et 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 H3, and H4 listed in the Appen-
`dix to the paper by Tarokh et al. (1998). These errors have been corrected in the per-
`tinent MATLAB codes.
`
`16Differential 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 was discussed in Chapter 3.
`
`17 Chapter 3 of Liberti and Rappaport (1999) describes more general models for
`phased arrays other than linear and where gain in elevation angle as well as azimuth is
`of interest. Chapter 8 of the same book describes various algorithms for adapting the
`weighting vector, depending upon the direction of arrival of the signal.
`
`18 The circular model for effective scatterers was proposed in Lee (1982).
`
`19 In Chapter 7 of Liberti and Rappaport (1999), the single-bounce elliptical model is
`described in greater detail. Note that the model does not take into account the effects
`of diffraction.
`
`20 The D—BLAST architecture was pioneered by Foschini (1996) and discussed further
`in the papers by Foschini and Gans (1999) and Foschini et a1. (2003).
`
`21 The first experimental results in the V—BLAST architecture were originally reported
`in the article by Golden et a1. (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-BLAST architecture was first described by Sellathurai and Haykin
`(1998), with additional results reported subsequently in the papers by the same
`authors (2000, 2002, 2003).
`
`23 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.
`
`24 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) hand. For additional historical notes, see Chapter 10 by Seymour Stein in
`Schwartz et a1. (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).
`
`
`
`Page 379 of 474
`
`Page 379 of 474
`
`

`

`Additional Problems
`
`441
`
`26 Keyhole channels, also dubbed pinhole channels, were described independently by
`Gesbert et al. (2002) and Chizhik et a1. (2002).
`
`27 The GBGP model for MIMO wireless links is described in Gesbert et al. (2002).
`28 Multiuser MIMO wireless systems are discussed in Diggavi et al. (2003) and Gold-
`smith et a1. (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 instantaneous signal-to-noise ratio ydrops below 0.25 yaw where y“
`is the average signaluto—noise ratio. Determine the probability of outage experienced by the
`receiver.
`
`Problem 6.22 The average signal-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 7: 10 dB
`for the following number of receive antennas:
`
`(a) N,:1
`
`(b) N,:2
`(c) N,:3
`(d) N,=4
`Comment on your results.
`
`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 same issue, using the standard maximization procedure. To simplify mat-
`ters, the number N, of diversity paths is restricted to two, with the complex weighting parameters
`denoted by (11 and (12.
`Let
`
`ck : xk +jyk
`
`It = 1, 2
`
`Then the complex derivative with respect to ak is defined by
`
`3
`1(8
`. a j
`at: = _ — +.)'—
`dark
`2 axk
`ayk
`
`k 3 112
`
`Applying this formula to the combiner’s output signal-townoise ratio ye of Eq. (6.14), derive
`Eq. (6.18).
`
`In this problem, we develop an approximate formula for the probability of
`Problem 6.25
`error, Pg, produced by a maximal-ratio combiner for coherent FSK. We start with Eq. (6.25), and
`for small ymrc, we may use the following approximation for the probability density function:
`
`Nr—l
`1
`fru’mrc) = 'Nr—ymrc
`yav(N,.— l)!
`
`Page 380 of 474
`
`Page 380 of 474
`
`

`

`442
`
`Chapter 6 Diversity, Capacity and Space-Division Multiple Access
`
`(:1) Using the conditional probability of error for coherent BFSK, that is,
`
`Prob(errorlymrc} = éerfc[ éynm)
`
`derive the approximation
`
`—1
`PE a éjmerfc(Jj))yN dy
`21 MN 1.0
`(gr...)
`{
`,— ).
`
`(1))
`
`Wherey = 27mm
`Integrating the definite integral by parts and using the definition of the complementary
`error function, show that
`
`Pg:—'[:ein)1
`2&[%:averN!
`
`(0) Finally, using the definite integral
`
`obtain the desired approximation
`
`PB:
`
`—N_fwea
`
`Problem 6.26
`
`(:1) Using the approximation for jflymm) given in Problem 6.25, determine the probability of
`symbol error for a maximal-ratio combiner that uses noncoherent BFSK.
`(b) Compare your result of part(a) with that of Problem 6.25 for coherent BFSK.
`
`Problem 6.27
`
`(a) Continuing the approximation to fl—(ymrc), determine the probability of symbol error for
`a maximal-ratio combiner that uses coherent BPSK.
`
`(b) Compare your result of part(a) with that of Problem 6.25 for coherent BFSK.
`
`Problem 6.28 As discussed in Section 6.2.3, an equal-gain combiner is 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 instantaneous signal-to-
`noise ratio ’yis small compared with the average Signal-tonoise ratio yaw derive an approximate
`formula for the probability density function of 9!.
`
`Problem 6.29 Compare the performances of the following linear diversity-on-receive
`techniques:
`
`Page 381 of 474
`
`Page 381 of 474
`
`

`

`
`
`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 Nr 2 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 symbol s,- over 3k if and only if
`
`.
`.5.
`2
`2
`2
`,.
`2
`2
`2
`(a1+a’2)|sf| —ylsi‘—y{s£<(061+a2)iskl “y13k_yi5k
`
`.
`kit
`
`(b) Choose symbol si over sk if and only if
`2
`2
`2
`2
`2
`2
`2
`2
`(al+a2a1)|sf| +d (y1,si)<(a1+a2—1)sk| +d (3235,?)
`
`
`.
`kit
`
`Here, £12021 ,si) denotes the squared Euclidean distance between the received signal yl and
`constellation points sf.
`
`It may be argued that, in a rather loose sense, transmit—diversity and receive—
`Problem 6.31
`diversity antenna configurations are the dual of each other, as illustrated in Fig. 6.46.
`
`(3) Taking a general Viewpoint, justify the mathematical basis for this duality.
`
`. >—— 1
`.
`Diver51ty /
`aths
`//
`p
`/
`
`/
`x/ //”>—_‘
`////
`fi—<*’<
`Transmit
`antenna
`
`\\
`
`\\
`
`2
`
`\\\\
`
`‘>— Nr
`Multiple
`receive
`antennas
`
`1 *—< \ \ \
`
`\
`
`\\
`2 ——<‘\“~c \\
`
`/
`
`3" >—.
`Recewe
`antenna
`///
`/’ Diversity paths
`
`/
`
`N, ——<’
`Multiple
`transmit
`antennas
`
`FIGURE 6.46 Diagram for Problem 6.31.
`
`
`
`
`
`Page 382 of 474
`
`
`
`Page 382 of 474
`
`

`

`444
`
`Chapter 6 Diversity, Capacity and Space-Division Multiple Access
`
`(1]) However, we may cite the example of frequency~divisi0n 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,
`
`lav
`C—>[10 ij
`g2
`
`asNeoo
`
`where NI = N, = N and 1a,, is the average eigenvalue of HHI : HIH,
`
`(a) Justify the asymptotic result given in Eq. (6.61)7that is,
`
`What is the value of the constant?
`
`E 2 constant
`N
`
`(b) What conclusion can you draw from the asymptotic result?
`
`Problem 6.33 By and large, the treatment of the ergodic capacity of a MIMO channel, as
`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 = ans}, +HSC
`
`where HEP and HSC denote the specular and scattered components, respectively To be consistent
`with the MIMO model described in Section 6.3, the entries of both HSp and HSC have unit amplia
`tude variance, with HSp being deterministic and H35 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 = 1010g10a2dB
`
`(:1) Considering the case of a pure line of sight (LOS), Show that the MIMO channel has the
`deterministic capacity
`
`C = log2(1 + Maip) bits/s/Hz
`
`Where Nr is the number of receive antennas and p is the total signalutornoise ratio at each
`receiver input.
`
`(b) Compare the result obtained in part (a) with that pertaining to the pure Rayleigh distria
`buted MIMO channel.
`
`(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, temporaliy stationary Gaussian interference v(t)
`corrupts the basic channel model of Eq. (6.48). The interference v(r) has zero mean and correla—
`tion matrix RV. Evaluate the effect of v(t) on the ergodic capacity of the MIMO link.
`
`Problem 6.35 Consider a MIMO iink for which the channel may be considered to be essenv
`tially constant for ’6 uses of the channel.
`
`
`
`Page 383 of 474
`
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`
`

`

`
`
`Additional Problems
`
`445
`
`(3) Starting with the basic channel model of Eq. (6.48), formulate the input—output relation-
`ship of this link, with the input described by the Nr-by-i: matrix
`
`(b) How is the log-det capacity formula of the link correspondingly modified?
`
`Orthogonal Space-Time Block Codes
`
`= [51, 529 “'5 ST]
`
`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 s1 and s2.
`(a) Starting with Eq. (6.102) for the combiner output 51k and using Eq. (6.103) for the proba-
`bility density function of the additive complex Gaussian noise \7k, formulate the expres
`sion for the likelihood function of transmitted symbol Sk , k : 1,2.
`(h) Hence, using the result of part (a), derive the formulas of Eqs. (6.103) and (6.104).
`
`Problem 637 Figure 6.47 shows the extension of orthogonal spacektime codes to the
`Alamouti code, using two antennas on transmit and receive. The sequence of signal encoding
`and transmissions is 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. Tabie 6.5(b) defines the outputs of the
`receive antennas at times t’ and t' + T, where T is the symbol duration.
`
`Er.
`is:
`
`.
`Transmit
`antenna 1
`
`i2,
`‘s?
`,
`ant
`* Transmit
`l
`‘
`enna 2
`
`kl
`
`kg
`
`{13
`
`m
`
`Receive
`antenna 1
`
`Receive
`antenna 2
`
`
`
`
`
`Interference
`
`
`@lnterference
`104
`93nd noise
`@and noise
`
`
`
`”and
`
`
`e§?:1:;t?:l‘_ Linear
`estimator
`combiner
`
`
`
`
`
`
`
`Maximum-likelihood decoder
`
`
`
`
`a,
`
`3,
`
`FIGURE 6.47 Diagram for Problem 637.
`
`
`
`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.
`
`
`(3)
`
`Receive antenna 1
`Receive antenna 2
`
`
`Transmit antenna 1
`I11
`I13
`
`Transmit antenna 2
`hg
`114
`
`(13)
`
`Receive antenna 1
`
`
`Receive antenna 2
`
`Time f
`
`5r]
`
`3C3
`
`Time I + T
`
`
`
`5&2 3:4
`
`(a) Derive expressions for the received signals 5:1, 17:2,}:3, and 564, including the respective
`additive noise components, in terms of the transmitted symbols.
`
`0)) Derive expressions for the line of combined outputs in terms of the received signals.
`(c) Derive the maximum-likelihood decision rule for the estimates E1 and E2.
`
`Problem 6.38 This problem explores a new interpretation of the Alamouti code. Let
`
`3;- = SE1) +13?)
`1': 1,2
`w .
`2
`.
`where SE“ and SE ) are both real numbersThe complex entry S:- in the twoeby-two Alamouti code
`is represented by the two-by-two real orthogonal matrix
`
`(1)
`Si
`
`(2)
`_Si
`
`(2)
`Si
`so)
`1
`
`: = 1, 2
`
`Likewise, the complex~conjugated entry E? is represented by the two-by-two real orthogonal
`matrix
`
`(1)
`SI"
`i
`3‘.”
`
`(1)
`Si
`s?)
`
`i = 1, 2
`
`(a) Show that the two-by—two complex Alamouti code S is equivalent to the four-by—four real
`transmission matrix
`
`551)
`
`$52)
`
`:Sgll
`l
`
`S(22)
`
`S4 = ______ :_ _____
`I
`-39) 532)
`.ng 1552)
`l'
`2
`l
`l
`2
`1
`elf—3;) ‘ 5(1)
`
`Si
`
`)
`
`Page 385 of 474
`
`Page 385 of 474
`
`

`

`
`
`Additional Problems
`
`447
`
`(b) Show that S4 is an orthogonal matrix.
`
`(c) What is the advantage of the complex code S over the real code S4?
`
`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
`
`GTG=I
`
`where the superscript 1' denotes Hermitian transposition and 1 denotes the identity
`matrix.
`
`(1)) Likewise, show that the sporadic complex orthogonal space-time codes of Eqs. (6.109)
`and (6.110) satisfy the temporal orthogonality condition
`
`H+H=I
`
`Problem 5.40 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 channel capacities of two single-input, single-output systems.
`
`Differential Space—Time Block Coding
`
`Problem 6.43 Equation (6.116) defines the inputioutput 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 expansion illustrated 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.
`Explore the constellation expansion property of differential spaceitime coding for the following
`two situations:
`
`(a) Frame of reference: dibit 00
`
`(b) Frame of reference: dibit 11
`
`Comment on your results.
`
`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 input block of data will be in the form of
`quadbits (i.e., 4—bit blocks). Perform the investigation for each of the two QPSK constellations
`depicted in Fig. 6.48. Use 0000 as the frame of reference.
`
`
`
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`

`

`Imaginary
`
`Imaginary
`
`Chapter 6 Diversity, Capacity and Space-Division Multiple Access
`
`Real
`Real
`
`(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
`Eq. (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
`
`448
`
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`

`

`
`
`Additional Problems
`
`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 5.48 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 preseribed modulation scheme, still operating in a Rayleigh fading environment.
`
`(21) Using 4-PSK for both STBC and V-BLAST, plot the SER versus SNR for the following
`antenna configurations:
`
`filelM=2
`m)M:am: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~BLAST systems use
`4-PSK. This time, however, we wish to cliSplay 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
`0 Diversity order
`. Multiplexing gain
`- Computational complexity
`
`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 link satisfies the Hermitian
`property.
`
`
`
`
`
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`

`

`APPENDEXA
`
`Fourier Theory
`
`A.1 THE FOURIER TRANSFORMI
`
`Let g(t) denote a nonperiodic deterministic signal, expressed as some function of time t.
`By definition, the Fourier transform of the signal g(t) is given by the integral
`
`G0) = f g(t)eXp(—127rfi)dt
`
`(A.1)
`
`where j = fl, and the variable f denotes frequency. Given the Fourier transform
`GU), the original signal g(t) is recovered exactly using the formula for the inverse Fou-
`rier transform:
`
`gm = f Gmexpoznfiidr
`
`(A2)
`
`Note that in Eqs. (A.1) and (A2) we have used a lowercase letter to denote the time
`function and an uppercase letter to denote the corresponding frequency function. The
`functions g(t) and GU" ) are said to constitute a Fourier-transform pair.
`For the Fourier transform of a signal g(t) to exist, it is sufficient, but not neces-
`sary, that g(r) satisfy three conditions, known collectively 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(i‘) has a finite number of discontinuities in any finite time interval.
`
`3. The function g0) is absolutely integrable; that is,
`
`f” Igmldzw
`
`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
`
`Appendix A Fourier Theory
`
`The absolute value of the Fourier transform G(f ), plotted as a function of fre-
`quency f, is referred to as the amplitude spectrum or magnitude spectrum of the signal
`g(t). By the same token, the argument of the Fourier transform, plotted as a function
`of frequency f; is referred to as the phase spectrum of the signal 5(1). The amplitude
`spectrum is denoted by lG(f )l and the phase spectrum is denoted by 9(f ).When g(t) is
`a realuvalued function of time I, the amplitude spectrum |G(f)l is symmetrical about
`the origin f: 0, whereas the phase spectrum 9(f ) is antisymmetrical aboutf: 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 extend this theory in two ways to include
`power signals (i.e., signals whose average power is 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(t), is defined as having zero amplitude
`everywhere except at t: 0, Where it is infinitely large in such a way that it contains unit
`area under its curve; that is,
`
`and
`
`6(t) = 0
`
`t¢0
`
`f” 5(t)dt = 1
`
`(A3)
`
`(A4)
`
`An implication of this pair of relations is that the delta function is an even fanc—
`tion of time; that is, 6(—t) : 6(1). Another important property of the Delta function is
`the replication property described by
`
`I” g(f)5(t—’C)d‘c = g(t)
`
`(A5)
`
`which states that the convolution of any function with the delta function leaves that
`function unchanged.
`Tables A.1 and A2 build on the formulas of Eqs. (A1) through (A5). In partic_
`ular, Table A1 summarizes the properties of the Fourier transform, while Table A2
`lists a set of Fourier-transform pairs.
`In the time domain, a linear system (e.g., filter) is described in terms of its impulse
`response, defined as the response of the system (with zero initial conditions) to a unit
`impulse or delta function 5(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 0r
`delta function is applied at time t=0, we may denote the impulse response of a linear
`time—invariant system by Mr). Let this system be subjected to an arbitrary excitation
`x(r), as in Fig. A.1(a). Then the response y(t) of the system is determined by the formula
`
`y(t) = I“ ante—oer
`
`= I” hump new
`
`(A6)
`
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`

`

`
`
`Section A.1 The Fourier Transform
`
`481
`
`
`
`TAB LE A.1 Summary of Properties of the Fourier Transform.
`
`Mathematical Description
`
`
`Property
`
`1. Linearity
`
`2. Time scaling
`
`3. Duality
`
`4. Time shifting
`
`5. Frequency shifting
`
`6. Area under g0)
`
`7. Area under GO")
`
`
`
`92:”‘1t.
`l¥mfis‘3§2¥§&fi;§m‘i39122}?
`
`
`rWe!»X.
`
`
`
` $sts42s”<“WWWAW".amssssm.*v.
`
`dart!) + £3320") : aG1(f} + bGztf)
`where a and b are constants
`
`gums? igff]
`ial
`a
`where a is a constant
`
`if
`
`g(t) :Gif),
`
`then
`
`G (I) saw—3: g(—f)
`
`gusto) :«f G(f)eXp(-j2fift0)
`
`expijMfctlgm : (Hf-fa)
`
`f” and: = Gm)
`
`gm) = [:06de
`
`digit) «are jznme
`
`
`
`
`
`
`8. Differentiation in the time domain
`
`9. Integration in the time domain
`
`I
`_..
`1
`G(O)
`Legion ‘— W00") +760)
`
`10. Con§ugate functions
`
`If
`
`at!) $3“ th).
`
`then
`
`g*(i) {:3 G*(wf)
`
`11. Multiplication in the time domain
`
`g1(t)g2(tl :