1.1.Wireless Channels 5 sults in Section 2 to the fundamental limits of wireless communication channels described by random matrices.Section 3.1 deals with direct- sequence code-division multiple-access(DS-CDMA),with and without fading (both frequency-flat and frequency-selective)and with single and multiple receive antennas.Section 3.2 deals with multi-carrier code- division multiple access(MC-CDMA),which is the time-frequency dual of the model considered in Section 3.1.Channels with multiple receive and transmit antennas are reviewed in Section 3.3 using models that incorporate nonideal effects such as antenna correlation,polarization, and line-of-sight components. 1.1 Wireless Channels The last decade has witnessed a renaissance in the information theory of wireless communication channels.Two prime reasons for the strong level of activity in this field can be identified.The first is the grow- ing importance of the efficient use of bandwidth and power in view of the ever-increasing demand for wireless services.The second is the fact that some of the main challenges in the study of the capacity of wireless channels have only been successfully tackled recently.Fading, wideband,multiuser and multi-antenna are some of the key features that characterize wireless channels of contemporary interest.Most of the information theoretic literature that studies the effect of those fea- tures on channel capacity deals with linear vector memoryless channels of the form y=Hx+n (1.1) where x is the K-dimensional input vector,y is the N-dimensional output vector,and the N-dimensional vector n models the additive circularly symmetric Gaussian noise.All these quantities are,in gen- eral,complex-valued.In addition to input constraints,and the degree of knowledge of the channel at receiver and transmitter,(1.1)is char- acterized by the distribution of the N x K random channel matrix H whose entries are also complex-valued. The nature of the K and N dimensions depends on the actual ap- plication.For example,in the single-user narrowband channel with nr
1.1. Wireless Channels 5 sults in Section 2 to the fundamental limits of wireless communication channels described by random matrices. Section 3.1 deals with directsequence code-division multiple-access (DS-CDMA), with and without fading (both frequency-flat and frequency-selective) and with single and multiple receive antennas. Section 3.2 deals with multi-carrier codedivision multiple access (MC-CDMA), which is the time-frequency dual of the model considered in Section 3.1. Channels with multiple receive and transmit antennas are reviewed in Section 3.3 using models that incorporate nonideal effects such as antenna correlation, polarization, and line-of-sight components. 1.1 Wireless Channels The last decade has witnessed a renaissance in the information theory of wireless communication channels. Two prime reasons for the strong level of activity in this field can be identified. The first is the growing importance of the efficient use of bandwidth and power in view of the ever-increasing demand for wireless services. The second is the fact that some of the main challenges in the study of the capacity of wireless channels have only been successfully tackled recently. Fading, wideband, multiuser and multi-antenna are some of the key features that characterize wireless channels of contemporary interest. Most of the information theoretic literature that studies the effect of those features on channel capacity deals with linear vector memoryless channels of the form y = Hx + n (1.1) where x is the K-dimensional input vector, y is the N-dimensional output vector, and the N-dimensional vector n models the additive circularly symmetric Gaussian noise. All these quantities are, in general, complex-valued. In addition to input constraints, and the degree of knowledge of the channel at receiver and transmitter, (1.1) is characterized by the distribution of the N × K random channel matrix H whose entries are also complex-valued. The nature of the K and N dimensions depends on the actual application. For example, in the single-user narrowband channel with nT
6 Introduction and nr antennas at transmitter and receiver,respectively,we identify K=nr and N=nR;in the DS-CDMA channel,K is the number of users and N is the spreading gain. In the multi-antenna case,H models the propagation coefficients between each pair of transmit-receive antennas.In the synchronous DS- CDMA channel,in contrast,the entries of H depend on the received signature vectors (usually pseudo-noise sequences)and the fading coef- ficients seen by each user.For a channel with J users each transmitting with nr antennas using spread-spectrum with spreading gain G and a receiver with nR antennas,K =nrJ and N=nRG. Naturally,the simplest example is the one where H has i.i.d.entries, which constitutes the canonical model for the single-user narrowband multi-antenna channel.The same model applies to the randomly spread DS-CDMA channel not subject to fading.However,as we will see,in many cases of interest in wireless communications the entries of H are not i.i.d. 1.2 The Role of the Singular Values Assuming that the channel matrix H is completely known at the re- ceiver,the capacity of(1.1)under input power constraints depends on the distribution of the singular values of H.We focus in the simplest setting to illustrate this point as crisply as possible:suppose that the entries of the input vector x are i.i.d.For example,this is the case in a synchronous DS-CDMA multiaccess channel or for a single-user multi-antenna channel where the transmitter cannot track the channel. The empirical cumulative distribution function of the eigenvalues (also referred to as the spectrum or empirical distribution)of an n x n Hermitian matrix A is denoted by FA defined as! FA)=∑1A(A)≤， (1.2) i=1 where A1(A),...,An(A)are the eigenvalues of A and 1f.}is the indi- cator function 1IfF converges as n,then the corresponding limit(asymptotic empirical distribution or asymptotic spectrum)is simply denoted by FA(z)
6 Introduction and nR antennas at transmitter and receiver, respectively, we identify K = nT and N = nR; in the DS-CDMA channel, K is the number of users and N is the spreading gain. In the multi-antenna case, H models the propagation coefficients between each pair of transmit-receive antennas. In the synchronous DSCDMA channel, in contrast, the entries of H depend on the received signature vectors (usually pseudo-noise sequences) and the fading coef- ficients seen by each user. For a channel with J users each transmitting with nT antennas using spread-spectrum with spreading gain G and a receiver with nR antennas, K = nTJ and N = nRG. Naturally, the simplest example is the one where H has i.i.d. entries, which constitutes the canonical model for the single-user narrowband multi-antenna channel. The same model applies to the randomly spread DS-CDMA channel not subject to fading. However, as we will see, in many cases of interest in wireless communications the entries of H are not i.i.d. 1.2 The Role of the Singular Values Assuming that the channel matrix H is completely known at the receiver, the capacity of (1.1) under input power constraints depends on the distribution of the singular values of H. We focus in the simplest setting to illustrate this point as crisply as possible: suppose that the entries of the input vector x are i.i.d. For example, this is the case in a synchronous DS-CDMA multiaccess channel or for a single-user multi-antenna channel where the transmitter cannot track the channel. The empirical cumulative distribution function of the eigenvalues (also referred to as the spectrum or empirical distribution) of an n × n Hermitian matrix A is denoted by Fn A defined as1 Fn A(x) = 1 n n i=1 1{λi(A) ≤ x}, (1.2) where λ1(A),...,λn(A) are the eigenvalues of A and 1{·} is the indicator function. 1 If Fn A converges as n → ∞, then the corresponding limit (asymptotic empirical distribution or asymptotic spectrum) is simply denoted by FA(x)
1.2.The Role of the Singular Values 7 Now,consider an arbitrary N x K matrix H.Since the nonzero singular values of H and Ht are identical,we can write NF()-Nu(z)=KFRtH()-Ku(z) (1.3) where u(z)is the unit-step function (u(x)=0,x<0;u(x)=1,x>0). With an i.i.d.Gaussian input,the normalized input-output mutual information of (1.1)conditioned on H is2 ()logdet (I+SNR HH 1 1 (1.4) 片立s（1+sm) og（1+s5N到dF)回 (1.5) o with the transmitted signal-to-noise ratio (SNR) NE2] SNR= KE] (1.6) and with Ai(HHT)equal to the ith squared singular value of H. If the channel is known at the receiver and its variation over time is stationary and ergodic,then the expectation of(1.4)over the dis- tribution of H is the channel capacity (normalized to the number of receive antennas or the number of degrees of freedom per symbol in the CDMA channel).More generally,the distribution of the random variable (1.4)determines the outage capacity (e.g.[22]) Another important performance measure for (1.1)is the minimum mean-square-error (MMSE)achieved by a linear receiver,which deter- mines the maximum achievable output signal-to-interference-and-noise 2The celebrated log-det formula has a long history:In 1964,Pinsker [204]gave a general log-det formula for the mutual information between jointly Gaussian random vectors but did not particularize it to the linear model(1.1).Verdu [270]in 1986 gave the explicit form (1.4)as the capacity of the synchronous DS-CDMA channel as a function of the signature vectors.The 1991 textbook by Cover and Thomas [47]gives the log-det formula for the capacity of the power constrained vector Gaussian channel with arbitrary noise covariance matrix.In the mid 1990s,Foschini [77]and Telatar [250]gave (1.4)for the multi-antenna channel with i.i.d.Gaussian entries.Even prior to those works,the conventional analyses of Gaussian channels with memory via vector channels(e.g.[260,31])used the fact that the capacity can be expressed as the sum of the capacities of independent channels whose signal-to-noise ratios are governed by the singular values of the channel matrix
1.2. The Role of the Singular Values 7 Now, consider an arbitrary N × K matrix H. Since the nonzero singular values of H and H† are identical, we can write NFN HH† (x) − Nu(x) = KFK H†H(x) − Ku(x) (1.3) where u(x) is the unit-step function (u(x) = 0, x ≤ 0; u(x) = 1, x > 0). With an i.i.d. Gaussian input, the normalized input-output mutual information of (1.1) conditioned on H is2 1 N I(x; y|H) = 1 N log det I + SNR HH† (1.4) = 1 N N i=1 log 1 + SNR λi(HH† ) = ∞ 0 log (1 + SNR x) dFN HH† (x) (1.5) with the transmitted signal-to-noise ratio (SNR) SNR = NE[||x||2] KE[||n||2] , (1.6) and with λi(HH†) equal to the ith squared singular value of H. If the channel is known at the receiver and its variation over time is stationary and ergodic, then the expectation of (1.4) over the distribution of H is the channel capacity (normalized to the number of receive antennas or the number of degrees of freedom per symbol in the CDMA channel). More generally, the distribution of the random variable (1.4) determines the outage capacity (e.g. [22]). Another important performance measure for (1.1) is the minimum mean-square-error (MMSE) achieved by a linear receiver, which determines the maximum achievable output signal-to-interference-and-noise 2 The celebrated log-det formula has a long history: In 1964, Pinsker [204] gave a general log-det formula for the mutual information between jointly Gaussian random vectors but did not particularize it to the linear model (1.1). Verd´u [270] in 1986 gave the explicit form (1.4) as the capacity of the synchronous DS-CDMA channel as a function of the signature vectors. The 1991 textbook by Cover and Thomas [47] gives the log-det formula for the capacity of the power constrained vector Gaussian channel with arbitrary noise covariance matrix. In the mid 1990s, Foschini [77] and Telatar [250] gave (1.4) for the multi-antenna channel with i.i.d. Gaussian entries. Even prior to those works, the conventional analyses of Gaussian channels with memory via vector channels (e.g. [260, 31]) used the fact that the capacity can be expressed as the sum of the capacities of independent channels whose signal-to-noise ratios are governed by the singular values of the channel matrix.
8 Introduction ratio(SINR).For an i.i.d.input,the arithmetic mean over the users(or transmit antennas)of the MMSE is given,as function of H,by [271] (1.7) K 1 台I+SNR(H可 (1.8) 0 1+SNRT dF(z) N oo 1 dt(）- N-K K Jo 1+SNR K (1.9) where the expectation in(1.7)is over x and n while(1.9)follows from (1.3).Note,incidentally,that both performance measures as a function of sNR are coupled through d SNR- og et(+sHIr）=K-{在+H)} As we see in (1.5)and (1.9),both fundamental performance measures (capacity and MMSE)are dictated by the distribution of the empirical (squared)singular value distribution of the random channel matrix. In the simplest case of H having i.i.d.Gaussian entries,the density function corresponding to the expected value of Fcan be expressed explicitly in terms of the Laguerre polynomials.Although the integrals in (1.5)and (1.9)with respect to such a probability density function (p.d.f.)lead to explicit solutions,limited insight can be drawn from either the solutions or their numerical evaluation.Fortunately,much deeper insights can be obtained using the tools provided by asymptotic random matrix theory.Indeed,a rich body of results exists analyzing the asymptotic spectrum of H as the number of columns and rows goes to infinity while the aspect ratio of the matrix is kept constant. Before introducing the asymptotic spectrum results,some justifica- tion for their relevance to wireless communication problems is in order. In CDMA,channels with K and N between 32 and 64 would be fairly typical.In multi-antenna systems,arrays of 8 to 16 antennas would be
8 Introduction ratio (SINR). For an i.i.d. input, the arithmetic mean over the users (or transmit antennas) of the MMSE is given, as function of H, by [271] 1 K min M∈CK×N E ||x − My||2 = 1 K tr I + SNR H† H −1 (1.7) = 1 K K i=1 1 1 + SNR λi(H†H) (1.8) = ∞ 0 1 1 + SNR x dFK H†H(x) = N K ∞ 0 1 1 + SNR x dFN HH† (x) − N − K K (1.9) where the expectation in (1.7) is over x and n while (1.9) follows from (1.3). Note, incidentally, that both performance measures as a function of SNR are coupled through SNR d dSNR loge det I + SNR HH† = K − tr I + SNR H† H −1 . As we see in (1.5) and (1.9), both fundamental performance measures (capacity and MMSE) are dictated by the distribution of the empirical (squared) singular value distribution of the random channel matrix. In the simplest case of H having i.i.d. Gaussian entries, the density function corresponding to the expected value of FN HH† can be expressed explicitly in terms of the Laguerre polynomials. Although the integrals in (1.5) and (1.9) with respect to such a probability density function (p.d.f.) lead to explicit solutions, limited insight can be drawn from either the solutions or their numerical evaluation. Fortunately, much deeper insights can be obtained using the tools provided by asymptotic random matrix theory. Indeed, a rich body of results exists analyzing the asymptotic spectrum of H as the number of columns and rows goes to infinity while the aspect ratio of the matrix is kept constant. Before introducing the asymptotic spectrum results, some justification for their relevance to wireless communication problems is in order. In CDMA, channels with K and N between 32 and 64 would be fairly typical. In multi-antenna systems, arrays of 8 to 16 antennas would be
1.2.The Role of the Singular Values 9 at the forefront of what is envisioned to be feasible in the foreseeable fu- ture.Surprisingly,even quite smaller system sizes are large enough for the asymptotic limit to be an excellent approximation.Furthermore, not only do the averages of (1.4)and (1.9)converge to their limits surprisingly fast,but the randomness in those functionals due to the random outcome of H disappears extremely quickly.Naturally,such robustness has welcome consequences for the operational significance of the resulting formulas. 1.8 14 B=1 0.5 0 0.2 0.5 1.5 2.5 Fig.1.1 The Marcenko-Pastur density function (1.10)for B=1,0.5,0.2. As we will see in Section 2,a central result in random matrix theory states that when the entries of H are zero-mean i.i.d.with variance y, the empirical distribution of the eigenvalues of HH converges almost surely,asK,V一oo with→3，to the so-called Mar心enko-Pastur law whose density function is =(-动) x)+-Ot6-开 (1.10) 2π3x where (z)+=max(0,z)and a=(1-VE2 b=(1+V2. (1.11)
1.2. The Role of the Singular Values 9 at the forefront of what is envisioned to be feasible in the foreseeable future. Surprisingly, even quite smaller system sizes are large enough for the asymptotic limit to be an excellent approximation. Furthermore, not only do the averages of (1.4) and (1.9) converge to their limits surprisingly fast, but the randomness in those functionals due to the random outcome of H disappears extremely quickly. Naturally, such robustness has welcome consequences for the operational significance of the resulting formulas. 0 0.5 1 1.5 2 2.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 β= 0.2 0.5 1 Fig. 1.1 The Mar˘cenko-Pastur density function (1.10) for β = 1, 0.5, 0.2. As we will see in Section 2, a central result in random matrix theory states that when the entries of H are zero-mean i.i.d. with variance 1 N , the empirical distribution of the eigenvalues of H†H converges almost surely, as K, N → ∞ with K N → β, to the so-called Mar˘cenko-Pastur law whose density function is fβ(x) = 1 − 1 β + δ(x) + (x − a)+(b − x)+ 2πβx (1.10) where (z)+ = max (0, z) and a = (1 − β) 2 b = (1 + β) 2. (1.11)