Wireless Communications over MIMO Channels phần 9 pps

Figure 6.4 Structure of transmit diversity system withNR receive antennas Combining allL vectors x[k], y[k], and n[k] within one coded frame as column vectors into the matrices X, Y, and

independent ofNRand is equally distributed onto the diversity paths so that only theNRthpart of Eb can be exploited at each receive antenna In this scenario, the gain obtainedsolely by diversity can be observed On the contrary, Figure b depicts the error rate versusthe average Es/N0 at each receive antenna Therefore, the total transmit power increaseslinearly with NR and the entire SNR after maximum ratio combining becomesNR timeslarger, indicating the additional array gain Comparing the difference between adjacentcurves in both the plots, we recognize a difference of 3 dB that exactly represents the gainobtained by doubling the number of receive antennas.

We can conclude that receive diversity is an efficient and simple possibility to increasethe link reliability However, its applicability becomes immediately limited if the size ofthe receiving terminal is very small Cell phones for mobile radio communications havebecome smaller and smaller in recent years so that it is a difficult task to place severalantennas on such small devices Even if we succeed, it is questionable whether the spacingwould be large enough to guarantee uncorrelated channels Although different polarizationsrepresent a further dimension to obtain diversity, the decoupling is generally imperfect,leading to cross talk In this situation, the question arises whether diversity can also beexploited with multiple antennas at the transmitter

6.2.2 Performance Analysis of Space–Time Codes

In this subsection, the general concept of space–time transmit diversity is addressed, that

is, using multiple antennas at the transmitter A straightforward implementation where asignalx[
] is transmitted simultaneously over several antennas will not provide the desired

diversity gain Looking at the received signal

dimensions space and time leading to the name space–time codes First, this subsection

dis-cusses the potential of STCs and derives some guidelines concerning the code construction

In the next two subsections, specific codes, namely, orthogonal space–time block codes(oSTBCs) and space–time trellis codess (STTCs) are introduced

The general structure of the considered system is depicted in Figure 6.4 The data bits

d[i] are fed into the space–time encoder that outputs L vectors x[k]=x1[k]· · · x NT[k]T

of lengthNT They are transmitted over a MIMO channel according to (6.1) The channelcoefficientsh µ,ν[k]= h µ,ν are assumed to be constant during one encoded frame so thatthe received signal becomes

1 Note that the total transmit power has been normalized according to the agreement on page 289 so that it is independent of the number of antennas.

Figure 6.4 Structure of transmit diversity system withNR receive antennas

Combining allL vectors x[k], y[k], and n[k] within one coded frame as column vectors

into the matrices X, Y, and N, respectively, results in

F of both codewords instead

N = N0/Ts for complex-valued signalsPr

The complementary error function can be upper bounded by erfc(√

Obviously, the matrix A= (B − ˜B)(B − ˜B) H is Hermitian and its rank r equals that of

B− ˜B Moreover, it is positive semidefinite and its r nonzero eigenvalues λ ν obtained

by an eigenvalue decomposition A= UU H are real and positive The pairwise errorprobability can now be expressed as

wherer denotes the rank of A, that is, the number of nonzero eigenvalues A further upper

bound that is tight for large SNRs is obtained by dropping the +1 in the denominator

Rewriting (6.13) finally leads to the expression

the diversity gain Hence, in order to achieve the maximum possible diversity degree, the

minimum rankr among all pairwise differences B− ˜B should be maximized, leading to

the diversity gain

We obtain the code design criteria according to (Tarokh et al 1998):

• rank criterion: In order to obtain the maximum diversity gain, the first design goal is

to maximize the minimum rankr of all matrices X− ˜X The diversity degree equals

rNR; its maximum isNTNR

• determinant criterion: For a diversity gain of rNR, the coding gain is maximized ifthe minimum of(r

ν=1λ ν )1/r is maximized over all codeword pairs

A code optimization according to these criteria cannot be performed analytically buthas to be carried out as a computer-based code search The next two subsections introduceexamples for space–time coding schemes First, orthogonal STBCs are presented Sincetheir codewords are obtained by orthogonal matrix design, the determinant is constant and

no coding gain is obtained However, full diversity gains are achievable and the receiverstructures are very simple Second, space–time trellis codes are briefly described, providingadditional coding gains at the expense of much higher decoding complexity

6.2.3 Orthogonal Space–Time Block Codes

Figure 6.5 shows the principle structure of a space–time block coding system forNR= 1receive antenna The subsequent derivation includes more generally the application of anarbitrary number of receive antennas As a variation from the general concept of space–timecoding depicted in Figure 6.4, the signal mapper and space–time encoder are separated.First, the data bits are mapped onto symbols a[
] that are elements of a finite signal

Figure 6.5 System structure for space–time block codes with NR= 1 receive antenna

constellation according to the linear modulation schemes presented in Section 1.4 Next,the space–time block encoder collects a block ofK successive symbols a[
] and maps them

onto a sequence ofL consecutive vectors x[k]=x1[k]· · · x NT[k]T

Alamouti’s Scheme

In order to illustrate how oSTBCs work, a simple example introduced by Alamouti (1998)

is used Originally, it employs NT= 2 transmit antennas and NR= 1 receive antenna.However, it can be easily extended to more receive antennas To be precise, we have

to consider blocks of K = 2 consecutive symbols, say a1 = a[2
] and a2 = a[2
+ 1].

These two symbols are now encoded in the following way At time instant 2k= 2
,

sym-bolx1[2k]= a1/√

2 is transmitted at the first antenna and x2[2k]= a2/√

2 at the secondantenna At the next time instant 2k+ 1, the symbols are flipped and x1[2k+ 1] = −a∗

2/√2

as well asx2[2k+ 1] = a∗

1/√

2 hold The whole codeword arranged in space and time can

be described using vector notations

X2=x[2k] x[2k+ 1]=√1

2 ·

*

a1 −a∗ 2

a2 a1∗

+

(6.19)

where the factor 1/√

2 ensures that the total average transmit power per symbol equals

Es/Ts The entire set of codewords is denoted by X2 The columns comprise the bols transmitted at a certain time instant, while the rows represent the symbols transmittedover a certain antenna Since K = 2 symbols a1 and a2 are transmitted during L= 2time instants, the rate of this code is Rc= K/L = 1 It is important to mention that

sym-the columns in X2 are orthogonal and so Alamouti’s scheme does not provide a ing gain

cod-A different implementation was chosen in the UMTS standard (3GPP 1999) withoutchanging the achievable diversity gain Here, the code matrix has the form

The corresponding two received symbols can be expressed by

into vectors y=y[2k] y[2k+ 1]T

channel h into an equivalent MIMO channel H[X2] The matrix describing this lent channel has orthogonal columns In this case, we already know from Chapter 4 thatthe matched filter represents the optimum detector according to the maximum likelihoodprinciple The matched filter output becomes

NT= 2 that can be achieved with two transmit antennas Moreover, no interference between

a1 anda2 disturbs the transmission because HH[X2]H[X2] is a diagonal matrix Owing tothis reason and the fact that the noise remains white when multiplied by a matrix consisting

of orthogonal columns, the ML decision with respect to the vector a can be split into

element-wise decisions

ˆa µ = argmin

˜a ˜r µ − (|h1|2+ |h2|2) ˜a 2

Although (6.24) looks similar to the result of simple receive diversity, there exists amajor difference Indeed, the diversity gain is exactly the same for receive and transmitdiversity concepts However, the factor 1/√

2 in (6.24) leads to an SNR loss of 3 dB.The reason is that the receiver was assumed to have perfect channel knowledge so thatbeamforming with an antenna gain of 10 log10(NR)≈ 3 dB is possible On the contrary,

we have no channel knowledge at the transmitter so that space–time transmit diversitytechniques do not achieve any antenna gain

As all space–time coding schemes, the Alamouti scheme can be easily combined withmultiple receive antennas According to (6.23), we obtain a vector

Extension to More than Two Transmit Antennas

Using some basic results from matrix theory, one can show that Alamouti’s scheme is theonly orthogonal space–time code with rate 1 For more than two transmit antennas, severalorthogonal codes have been found with lower rates, so that spectral efficiency is lost The

code matrix XNT generally consists ofNTrows andL columns and contains the symbols

a1, , a K as well as the conjugate complex counterparts a1∗, , a K∗ The construction

of XNT has to be performed such that XNT has orthogonal rows, that is,

holds, where P is a constant depending on the symbol powers that will be discussed on

page 289 In the following part, all codeword matrices are presented without normalization

In Tarokh et al (1999a), it is shown that there exist half-rate codes for an arbitrarynumber of transmit antennas The code matrices for NT= 3 and NT= 4 are presented asexamples ForNT= 3, we obtain

providing a diversity degree ofD = NT= 3 Obviously, X3 consists ofL= 8 columns and

K = 4 different symbols a1, , a4 are encoded, leading to the rate Rc= K/L = 1/2.

Each symbol a µ occurs six times with full energy in X From (6.30), we can write the

We observe in (6.31) that the last four symbols in y only depend on the conjugate

com-plex transmit symbols Hence, conjugating the last four rows similar to the procedure forAlamouti’s scheme in (6.23) results in

Obviously, (6.32) uses only the original symbols a= [a1 · · · a4]T and not their conjugate

complex versions Moreover, the columns in H[X3] are orthogonal so that

ForNT= 4, a diversity gain of D = NT= 4 is achieved with the code matrix

Equivalent to the case ofNT= 3, we obtain a received vector y according to

Again, the columns of H[X4] are mutually orthogonal and estimates ˆa are obtained by

multiplying ˜y with HH[X4] and appropriate scaling

Looking at higher spectral efficiencies, only two codes withNT= 3 and NT= 4 havebeen found forRc> 1/2 (Tarokh et al 1999a,b) In order to distinguish them from the codes

presented so far, we use the notationsT3 andT4 ForNT= 3, the orthogonal space–timecodeword is

T3=



 2 1 −2a

∗ 2

Since it comprises four time instants for transmitting three symbols, the code rate amounts

toRc= 3/4 Using (6.37), the received vector can be written as

h3

√ 2

h1−h2√2

Unfortunately, the channel matrix in (6.38) does not have the block diagonal structure

so that a separation into rows associated only with the original symbols a1, , a3 andthose associated with their complex conjugate versions is not possible Hence, a direct

construction of an equivalent matrix H[T3] containing the complex channel coefficients isnot possible However, we can separate real and imaginary parts of all components andstack them into vectors and matrices similar to the approach applied to linear multiuser

detectors for real-valued modulation schemes discussed in Sections 5.2.1, 5.2.2, and 5.4.2.Denoting the real part of a complex symboly with y
and the imaginary part withy

, wedefine the real-valued vectors

h
2 −h

1

h
3

√ 2

√ 2

, real and imaginary parts of each symbol experience

a diversity gain of NT For multiamplitude modulation, they have to be normalized andcombined into a complex symbol again to allow the demodulation

Finally, a space–time coding scheme withNT= 4 transmit antennas shall be presented.The space–time codeword is

Again, three symbols are transmitted within a block covering four time instants, leading to

Rc= 3/4 The received vector can be described using (6.41) yielding

h1−h2√2

The channel matrix for the real-valued received vector can now be expressed as

h

1+h
2

√ 2

STBCs constructed with real-valued notations are called linear dispersion codes (Hassibi

and Hochwald 2002) and are addressed in Section 6.5

As already explained for Alamouti’s scheme, each of the discussed STBCs can becombined with several receive antennas In this case, we obtain several equivalent channelmatrices which are stacked into a large matrix according to (6.27) The receiver consists

of a bank ofNR matched filters and simply sums their outputs This leads to an overalldiversity degree ofD = NT· NR

Although oSTBCs do not provide a coding gain, they have the great advantage thatdecoding simply requires some linear combinations of the received symbols Moreover,they provide the full diversity degree achievable with a certain number of transmit andreceive antennas

Normalizing the Transmit Power

We now have to consider the transmit power of the presented STBCs in more detail.Certainly, there exist several possibilities for normalizing the transmit power From thechannel coding perspective, we know to distinguishEsandEb In the context of space–timecoding, we have the possibility of fixing the average SNR per channel use, that is, pertime instant In this case, the constant P in (6.29) grows linearly with the length L of a

space–time codeword and we obtain

In order to draw a fair comparison among the discussed STC approaches, we can also

fix the average power spent per data symbol toEs/Ts, leading to

K = 1/√2 as already used on page 283 For the codes X3 and X4, eachsymbol is transmitted six and eight times, respectively Hence, we obtain the factors 1/√

6and 1/√

8 In relation to T3 and T4, the scaling factors before the codeword matricesamount to 1/2 With this normalization, the error rate is depicted against Es/N0

Finally, a comparison of schemes with different spectral efficiencies is generally drawnwith respect toEb/N0 instead ofEs/N0 Normalizing to the number of receive antennas sothat no array gain is measured, we obtain the following relationship between the averageenergyEb per information bit and the symbol energyEs

Es= m · Rc

NR · Eb= m · K

wherem denotes the number of bits per symbol Alternatively, the SNR at each receive

antenna can also be used so that the array gain of the receiver becomes obvious However,this must be explicitly mentioned

Figure 6.6 Bit error rate of Alamouti’s scheme for different modulation types and number

of receive antennas, (solid bold line: AWGN channel, solid dashed line: Rayleigh fadingchannel without diversity)

a) error rate versusEs/N0 b) error rate versusEb/N0

Figure 6.7 Bit error rate for different orthogonal STBCs, BPSK, and NR= 1 receiveantenna

different number of receive antennas Since X2 provides a diversity degree of D= 2,additional receive antennas multiply this degree, leading toD = 4, D = 6 and D = 8 for

NR= 2, NR= 3, and NR= 4, respectively A comparison between theoretical results fromSection 1.5 (lines) and simulation results (symbols) illustrates that both coincide perfectly.Hence, as long as the channel is ideally known to the receiver, optimum diversity per-formance is achieved Figure 6.6b shows the performance of X2 for different modulationschemes Both quaternary phase shift keying (QPSK) and 8-PSK profit by an increaseddiversity degree

Next, we compare space–time coding schemes for binary phase shift keying (BPSK)and a single receive antenna From Figure 6.7a, it becomes obvious thatX3 andT3 haveidentical diversity degrees in addition to X4 andT4 The results are identical with thoseobtained from Section 1.5 However, the codes have different ratesRc, leading to differentspectral efficiencies Therefore, we have to depict the error rates againstEb/N0 instead of

Es/N0 Figure 6.7b shows the corresponding relations The slopes of all curves are stillthe same as shown in Figure 6.7a but those ofX3,X4,T3, andT4 are shifted horizontally

by 10 log10(Rc) The half-rate codes X3 andX4 perform worse especially at small SNRscompared to T3 and T4 Despite its higher diversity degree, X3 outperforms Alamouti’sscheme only for SNRs above 15 dB Similar intersections exist forX4 andT3

A fair comparison between different space–time coding schemes can be guaranteed

if it is drawn for identical spectral efficiencies This can be achieved by choosing anappropriate modulation scheme for each STC Table 6.1 summarizes some constellationsconsidered here Forη= 2 bits/s/Hz, Alamouti’s scheme employs a QPSK while X3 and

X4 have to use a 16-QAM or 16-PSK because of their lower code rate of Rc= 1/2 For

η= 3 bits/s/Hz, we use the 8-PSK for X2 and 16-QAM forT3 andT4

The results for η= 1 bit/s/Hz are depicted in Figure 6.8a Since BPSK and QPSKshow the same bit error rate (BER) performance against Eb/N0, X3 andX4 do not suf-fer from a higher sensitivity of the modulation scheme and can fully exploit the larger

Table 6.1 Combinations of space–time codes and modulation

schemes for different overall spectral efficiencies

dif-in Figure b) both for 16-QAM)

diversity degree In Figure 6.8b, we observe different results forη= 2 bit/s/Hz QPSK ismuch more robust than 16-QAM against the influence of noise Hence, the higher diversitydegree becomes obvious only for high SNRs At low SNRs, Alamouti’s scheme with QPSKstill performs best

Finally, Figure 6.9 illustrates the results obtained for a spectral efficiency of η=

3 bit/s/Hz Because of the relative high code rate of Rc= 0.75, we have to just switch

between 8-PSK and 16-QAM However, 16-QAM performs nearly as good as 8-PSKbecause it exploits the signal space more efficiently (cf Section 1.4) Therefore, the lossobtained by changing from 8-PSK to 16-QAM is rather low and the diversity gain dominatesthe bit error rate forT3 andT4

The following conclusion can be drawn in relation to the trade-off between diversitydegree and modulation type for a fixed spectral efficiency η In the high SNR regime,

diversity is most important and overcompensates the larger sensitivity of high-order lation schemes At low SNRs, robust modulation schemes such as QPSK should be preferredbecause the diversity gain is smaller than the loss associated with a change of the modulationscheme

6.2.4 Space–Time Trellis Codes

Contrary to the previously presented oSTBCs, STTCs can also provide a coding gain First,optimization criteria and some handmade codes have been presented in Seshadri et al.(1997), Tarokh et al (1997, 1998) Results of a systematic computer-based code search can

be found in B¨aro et al (2000a,b) and some implementation aspects in Naguib et al (1997,1998)

Figure 6.10 shows the general structure of an encoder withNT= 2 transmit antennas.Obviously, STTCs are related to convolutional codes explained in Section 3.3 At each timeinstant
, a vector d[
]=d1[
]· · · d K[
]T

is fed into the linear shift register consisting

ofLcblocks each comprisingK bits The old content is shifted by K positions to the right.

Hence, the total length of the register isLcK bits and Lc represents the constraint length

as for convolutional codes The variableQ = Lc− 1 denotes the memory of the register.The major difference compared to binary convolutional codes is the way in which theregister content

TheNTintegersb µ[
]∈ {0, · · · M − 1} are then mapped onto M-ary phase shift keying (PSK)

or quadrature amplitude modulation (QAM) symbols byNTindependent signal mappers

In Tarokh et al (1998), it is shown that the maximumK is restricted by the modulation

scheme if maximum diversity degree ofNTNR should be achieved Hence, K= log2(M)

holds forM-ary modulation schemes The number of states naturally depends on the

mem-ory of the register However, it may happen that the left-most and the right-most bit tuples

d[
] and d[
− Q] are not fully connected to the generators Assuming that the last τ

ele-ments of a[
] are not connected to the generators, only QK− τ memory elements are used

and the number of states reduces to 2QK −τ In this case, the generator matrix is not fullyloaded (Blum 2000)

Similar to convolutional codes, STTCs can also be graphically described with a lis diagram An example with four states and NT= 2 transmit antennas is depicted inFigure 6.11 whereK = 2 and Lc= 2 hold, resulting in 22= 4 states At each time instant,two input bits d1[
] and d2[
] are encoded in a register with memory Q= 1, resulting infour branches leaving each state On the left-hand side, the binary representation of eachstate, that is, the register content [q3[
]q4[
]], is depicted On the right-hand side, the outputsymbolsx1[
] and x2[
] belonging to different branches are listed, wherein the first symbolpair belongs to the uppermost branch leaving a state and the last belongs to the lowestbranch Generally, natural mapping (see Section 1.4) is applied as can be seen from thesignal space of QPSK

trel-Decoding Space–Time Trellis Codes

Owing to the equivalence between convolutional codes and STTCs, we can use the Viterbialgorithm for decoding However, there exists a major difference In the case of binary

00, 02, 22, 20

01, 03, 23, 21

10, 12, 32, 30

31, 33, 13, 1100

holds This modification has to be considered when calculating the incremental metrics

γ (s
→s)[
] given in (3.32) All interfering symbols at time instant
originate from the same

states
The incremental metric between the statess
ands becomes

ν denotes the hypothesis of the symbol transmitted over antenna ν for the

transition between statess
ands Consequently, z (s
→s)comprises allNThypotheses Theremaining parts of the Viterbi algorithm are identical to that of convolutional codes

Examples for Space–Time Trellis Codes

In the following part some codes, derived by Wittneben, Tarokh, Yan, and Bro (Tarokh

et al 1998; Wittneben 1991, 1993; Yan and Blum 2000), are presented This list doesnot claim to be comprehensive In order to distinguish the codes, the following notation

is used The codes from Wittneben, Tarokh, Yan, and Bro are denoted by W(M, Z, NT), T(M, Z, NT), Y(M, Z, NT), and B(M, Z, NT), respectively The three parameters describe

the constellation sizeM of the linear modulation, the number of states Z in the trellis, and

the number of transmit antennasNT All codes achieve the maximum diversity gainNTNR

so that only the coding gain has to be considered

Delay diversity by Wittneben

The delay diversity scheme proposed by Wittneben (1991, 1993) represents an exceptionbecause it provides no coding gain However, it can be interpreted as the simplest STTCand is illustrated in Figure 6.12 For the general case ofM-ary modulation schemes, K=log2(M) bits are fed into the shift register at each time instant In the example, QPSK,

resulting in K= 2 is used The number of transmit antennas equals the constraint length

Lc= NT because each K bit block is connected to a mapper of only one antenna This

leads to the general structure of the generator matrix

We recognize that (6.54) describes the convolution of a sequencex1[
] with a

frequency-selective channel hµ = [h µ,1 · · · h µ,NT] Therefore, the flat MISO channel is transformed

by the delay diversity scheme into a frequency-selective single-input single-output channelproviding the full diversity degree of D = NT= Lc Decoding is identical to the equal-ization of intersymbol interference channels and can be performed by a Viterbi equalizer(Kammeyer 2004; Proakis 2001)

For the example W(4, 4, 2) of NT= 2 transmit antennas, QPSK, and four states, weobtain the trellis segment depicted in Figure 6.13 While the first antenna always transmits

symbolν in the νth state, the second antenna transmits a symbol µ identifying the successive

states = µ The generator matrix has the form

Space–time trellis codes with NT= 2 transmit antennas

Next, we focus on schemes providing a coding gain gc with only two transmit nas Table 6.2 lists the codes T(4, Z, 2) and Y(4, Z, 2) by Tarokh et al (1998), Yan and

anten-Blum (2000) for Z states and QPSK, Table 6.3 the codes B(4, Z, 2) by Bro (B¨aro et al.

Table 6.2 List of space–time trellis codes taken from Tarokh et al (1998),Yan and Blum (2000) for NT= 2, QPSK, η = 2 bits/s/Hz and diversity

*

+ √12

*

+ √40