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Mutual Information of IID Complex Gaussian

Signals on Block Rayleigh-faded Channels

Fredrik Rusek Angel Lozano Nihar Jindal

Lund University Universitat Pompeu Fabra University of Minnesota

*** ** Lund, Sweden Barcelona 08003, Spain Minneapolis, MN 55455

Email: *******.*****@***.***.** Email: *****.******@***.*** Email: *****@***.***

become largely unfeasible in certain cases (in the low-

Abstract We present a method to compute, quickly and

ef ciently, the mutual information achieved by an IID (in- power regime, for example, they become unacceptably

dependent identically distributed) complex Gaussian signal peaky) and thus the capacity is sometimes less relevant

on a block Rayleigh-faded channel without side informa-

to system designers than the mutual information of IID

tion at the receiver. The method accommodates both scalar

complex Gaussian inputs.

and MIMO (multiple-input multiple-output) settings. Op-

This paper presents analytical expressions for the

erationally, this mutual information represents the highest

spectral ef ciency that can be attained using Gaussian output distributions of a block Rayleigh-faded channel

codebooks. Examples are provided that illustrate the loss fed with IID complex Gaussian inputs. Then, a simple

in spectral ef ciency caused by fast fading and how that

outer Monte-Carlo in conjunction with these distribu-

loss is ampli ed when multiple transmit antennas are used.

tions yields a semi-analytical method that allows eval-

These examples are further enriched by comparisons with

uating, quickly and ef ciently, the mutual information.

the channel capacity under perfect channel-state information

at the receiver, and with the spectral ef ciency attained by The method accommodates not only scalar channels, but

pilot-based transmission. also MIMO (multiple-input multiple-output) settings.

Altogether, this allows answering questions such as:

I. I NTRODUCTION

What is the impact of assuming side information?

IID (independent identically distributed) complex

How close to the true channel capacity (without

Gaussian inputs are highly relevant in channels impaired

side information) can IID complex Gaussian inputs

by Gaussian noise. Some of the arguments for this

operate?

relevance are that, with side information in the form of

How suboptimal are pilot-based schemes, i.e.,

perfect CSI (channel state information) at the receiver:

schemes that form explicit channel estimates on

These are the unique capacity-achieving inputs.

the basis of pilot observations at the receiver and

Their mutual information represents very well the

subsequently apply them to detect the data?

mutual information of proper complex discrete con-

At which power level is the power ef ciency maxi-

stellations (e.g., QAM) commonly used in wireless

mized (i.e., the energy per bit minimized)?

systems. (This holds up to some power level that

Not surprisingly, the answers to these questions end up

depends on the cardinality of the constellation [1].)

being a function of the fading rate and the numbers of

Indeed, expressions for the perfect-CSI capacity achieved

antennas. With the method presented, these relationships

by IID complex Gaussian inputs are available (cf. Section

can be precisely established, and some examples of this

IV) and thus such capacity can be easily evaluated.

are provided in the paper.

Remove now the side information. No expressions are

available for the mutual information achieved by IID

II. C HANNEL M ODEL

complex Gaussian inputs save for the very special case

of memoryless channels [2]. Moreover, straight Monte- Consider nT transmit and nR receive antennas and

let the nR nT matrix H represent the discrete-time

Carlo computation is not feasible because it would en-

tail large-dimensional histograms. Only bounds [3], [4] fading channel. Under block Rayleigh-fading, the chan-

and asymptotic low- and high-power expansions [5] nel entries are drawn from a zero-mean unit-variance

[8] are available for such mutual information. And yet, complex Gaussian distribution at the beginning of each

although no longer capacity-achieving without perfect fading block and they remain constant for the nb symbols

CSI [9], [10], IID complex Gaussian inputs remain highly within that block, where nb is the coherence time in sym-

relevant. Operationally, the mutual information they bols (or the coherence bandwidth if it is the frequency

achieve represents the highest spectral ef ciency that domain being modeled). This process is repeated for

can be attained using Gaussian codebooks. In fact, the every block in an IID fashion. There is no antenna cor-

capacity-achieving inputs in the absence of perfect-CSI relation and thus the entries of H are also independent.

Having expressed h(Y X ) in (3), we now turn our

Assembling into matrices the input, the output, and

the noise for the nb symbols within each block, their attention to h(Y ). The output Y is affected by a com-

relationship becomes bination of multiplicative and additive noise, and thus

Y is not Gaussian distributed. The unconditional out-

SNR

put distribution p(Y ), which had not been reported in

(1)

Y= HX + N

nT the literature to the best of our knowledge, constitutes

where the input X is an nT nb matrix while the output the central result in this paper. While the formula we

Y and the noise N are nR nb matrices. Both X and obtain appears too involved to allow for a closed-form

N have IID zero-mean unit-variance complex Gaussian expression of h(Y ), the formula is easy to evaluate and

entries. With that, SNR indicates the average signal-to- it thereby enables computing

noise ratio per receive antenna.

h(Y ) =

Each codeword spans a large number of fading blocks, (5)

p(Y ) log2 p(Y )dY

in the time and/or frequency domains, which endows

= E [log2 p(Y )] . (6)

ergodic quantities with operational meaning.

Although a block-fading structure admittedly repre- through straight Monte-Carlo averaging. Our formula

sents a drastic simpli cation of reality, it does capture for p(Y ) is given in the next proposition, where we use

the essential nature of fading and generally yields re- the standard notation [z ]+ = max{z, 0}.

sults that are remarkably similar to those obtained with

continuous-fading models [11]. In fact, for a rectangular Proposition 2 For 1 k min(nR, nT ), de ne the func-

Doppler spectrum, an exact correspondence in terms of tions

the estimation of H can be established between block- +

z k 1+[nT nR ]

x SNR z

and continuous-fading models [12] whereby z

fk (x) = exp n +1 nR dz.

z SNR + nT (z SNR/nT +1) b

(2) (7)

nb =

2fm Ts

Let d = [d1, . . ., dnR ] be the eigenvalues of Y Y and de ne

where fm and Ts are the maximum Doppler frequency the nR nR matrix Z with entries

and the symbol period, respectively. Typically, fm = j 1

1 i nR, 1 j s

(v/c)fc with v the velocity and fc the carrier frequency. fj (di ),

Zij = SNR

The mapping in (2) is in terms of the minimum mean- j nT 1

1 i nR, s + 1 j nR

di,

square error in the estimation of H, and thus it is

(8)

exact for pilot-based schemes that rely on such explicit

where s = min{nT, nR }. Then

estimation, but more broadly we take it as indicative of

the fading rate represented by a given value of nb . 2

nb nR e Y

detZ . (9)

p(Y ) = nT 1

di )

1 i,j nR (dj k!

III. C OMPUTATION OF THE M UTUAL I NFORMATION k=[nT nR ]+

The mutual information under investigation can be

Proof: See Appendix.

expressed as

1 In the special case of memoryless channels, i.e., for

I= [h(Y ) h(Y X )] (3)

nb nb = 1, the solution in Proposition 2 reduces to the one

in [2].

where h denotes the differential entropy operator.

We also note that, due to the rotational invariance

Our rst result leverages the derivations in [13] to

of H, only the eigenvalues of Y Y are relevant to the

obtain a closed-form expression for h(Y X ).

distribution in (9).

Using (4) and (9), an algorithm to compute I can be

Proposition 1 Let Eq denote the exponential integral, i.e.,

put forth as follows.

Eq = 1 t q e t dt. Then,

Algorithm 1: Evaluation of I .

nT 1 2j

2i 2j

h(Y X ) = nR log2 (e) enT /SNR 1. Pre-compute fk (x), 1 k nR, on a discrete set

i j

i=0 j =0 =0 X with a suitable stepsize x = xk xk 1 .

( 1) (2j )! (nb nT + )!

2j + 2nb 2nT 2. Generate a suf ciently large number of input and

2j 22i j ! ! (nb nT + j )! output vectors according to (1).

3. For each input and output pair, apply (9) to

nb nT +

nT obtain p(Y ).

(4)

Eq+1 + nR nb log2 ( e)

SNR

4. Compute the sample average of log2 p(Y ) via

q =0

Monte Carlo, thereby obtaining h(Y ).

5. Compute h(Y X ) from (4) and apply (3).

Proof: See Appendix.

During the transmission of pilot symbols,

The accuracy can be made as high as desired by

SNR

averaging over more input/output sample pairs and (12)

Yp= HP + N p

by increasing the precision in Step 1. For the results

presented in Section V, the number of samples and where the output, Y p, and the noise, N p, are nR np

the value of x were chosen such that two decimal matrices. The entries of N p are IID zero-mean unit-

digits are correct with 90% probability. With a standard variance complex Gaussian while P is deterministic and

workstation, the entire computation process is a matter satis es P P = np I [17].

of seconds. During the transmission of data symbols, in turn, (1)

As a nal remark, we mention that Proposition 2 is applies with X and N of dimension nT (nb np ) and

easily extendable to include all input distributions X nR (nb np ), respectively.

that are rotationally invariant and where the eigenvalue The value of np, which can be optimized by solving

distribution of X X (or of XX ) is of the form a convex problem, depends on SNR, nb and nT . This op-

timization, and the ensuing spectral ef ciency, has been

p = det2 V studied extensively, e.g., [3], [5], [17] [22]. In bits/s/Hz,

gk ( k ),

such spectral ef ciency equals

k=1

where V denotes a Vandermonde matrix and the 2

np np /nT

SNR

1 (13)

max C

functions gk are arbitrary. Further details are given in nb 1 + SNR (1 + np /nT )

np :1 np nb

the Appendix.

where C is the perfect-CSI capacity in (10).

If the pilot and data symbols are not required to have

IV. B ASELINES

the same power, i.e., if pilot power-boosting is allowed,

Before exemplifying the method described in Section

then it is optimal to set np = nT and to optimize only

III, we introduce the perfect-CSI capacity, a lower bound

over the relative powers of pilots and data. This results

to I, and the spectral ef ciency achievable with pilot-

in a different convex optimization, which in this case can

based communication, all of which serve as baselines.

be solved explicitly [17] leading to1

A. Capacity with Perfect CSI 2

nT nb SNR

1 1 (14)

If the receiver is provided with perfect CSI on the side, nb 2 nT

the ergodic capacity, in bits/s/Hz, equals

in bits/s/Hz, and with

SNR nb SNR + nT

HH (10)

C (SNR) = E log2 det I + (15)

= 2 nT .

nT nb SNR nbb nT

closed forms for which can be found in [13], [14]. The spectral ef ciency in (14) is superior to that in

(13). However, pilot power boosting increases the peak-

B. Mutual Information Lower Bound

iness of the overall signal distribution, rendering it less

A simple application of Jensen s inequality to the amenable to ef cient ampli cation.

bound in [15, Theorem 2] yields the following.

V. S OME E XAMPLES

Proposition 3 The mutual information achieved by IID Recalling (2), and in order to calibrate the relevant

Gaussian inputs satis es I (SNR) Ilower (SNR) with values of nb, the following observations can be made

in the context of emerging systems such as 3GPP LTE

nT nR nb

Ilower (SNR) = C (SNR) [23] or IEEE 802.16 WiMAX [24]:

(11)

log2 1 + SNR .

nb nT

The carrier frequency fc typically lies between 1 and

C. Pilot-Based Communication 5 GHz.

The symbol period is Ts 100 s. However, it could

In pilot-based communication, np pilot symbols are

be shortened to Ts 10-20 s and the at-faded

inserted within each fading block, leaving nb np sym-

model in (1) would still apply. (For wider band-

bols available for data. The channel is estimated on

widths, a frequency-selective model would be re-

the basis of the pilot observations at the receiver, and

quired and the computation algorithm would have

this estimate is subsequently utilized to detect the data.

to be modi ed accordingly.)

We analyze here the spectral ef ciency achievable with

Vehicular velocities up to v 120 Km/h are of

separate processing of the pilots and the data symbols,

interest, and for high-speed trains this extends to

which refers to estimating the channel on the basis of

v 300 Km/h.

only the received pilots and then decoding the data

(through nearest neighbor decoding) as if that estimate 1 Eq. (14) requires that n > 2 n ; variations thereof are also available

b T

was perfect [16]. for nb 2 nT [17].

Fig. 1. In solid, I (SNR) for nT = nR = 1 with nb = 100. In dashed, Fig. 4. Optimum nT for nR = 4 as function of SNR and nb .

the perfect-CSI capacity.

Fig. 5. In solid, I (SNR) for nT = nR = 2, with nb = 10 and with

Fig. 2. In solid, I (SNR) for nT = nR = 1 with nb = 10. Also in solid, nb = 4. In dashed, the corresponding perfect-CSI capacity.

spectral ef ciencies achieved by pilot-based communication, with and

without pilot power boosting. In dashed, the perfect-CSI capacity.

Let us now turn our attention to MIMO settings. A

well-known feature of the perfect-CSI capacity is that

With all of this taken into account, nb can take values

it always increases with additional antennas, be it at

ranging from just over unity to several hundred. As the

the transmitter or at the receiver. However, [5] and [17]

following example evidences, for large nb the perfect-

suggest that, without perfect CSI, activating too many

CSI capacity accurately represents the achievable mutual

transmit antennas would be detrimental at suf ciently

information.

high SNR. This is indeed the case, and the SNR above

which a speci c nT becomes optimal depends on nb as

Example 1 Let nT = nR = 1 and let nb = 100. Shown in

the following example illustrates.

Fig. 1 are the mutual information and the perfect-CSI capacity

as function of SNR.

Example 4 Let nR = 4. Shown in Fig. 4 is the optimum

number of transmit antennas as function of both SNR and nb .

For the remainder of this section, we shall thus focus

on scenarios where nb is small. Speci cally, we shall use

Next, we see the impact of varying nb and/or SNR with

nb = 10 and nb = 4. These will tend to correspond

xed nT and nR .

to vehicular and high-speed-train velocities, possibly in

conjunction with relatively long symbol periods and

Example 5 Let nT = nR = 2. Shown in Fig. 5 is the mutual

relatively high carrier frequencies.

information as function of SNR with nb = 10 and with nb = 4.

Also shown is the corresponding perfect-CSI capacity.

Example 2 Let nT = nR = 1 and let nb = 10. Shown in

Fig. 2 is the mutual information as function of SNR. Also Comparing Example 5 with Examples 2 and 3, notice

shown are the spectral ef ciencies achieved by pilot-based how, at each fading rate, MIMO transmission suffers

communication, with and without pilot power boosting, and a more drastic loss relative to the perfect-CSI capacity.

the perfect-CSI capacity. However, even with nb = 4, the mutual information for

nT = nR = 2 is larger than for nT = nR = 1 (Example

We observe that a hefty share of the perfect-CSI

3). Thus, although additional transmit antennas should

capacity is achieved at high SNR, although this share

be activated only for suf ciently long nb, additional

diminishes with the SNR. We further observe that, by

transmit-receive pairs should be activated even for short

optimizing the pilot overhead or the pilot power boost

nb .

at every SNR, pilot-based communication schemes can

As a nal and very illuminating example, we examine

perform remarkably close to the fundamental communi-

the scaling of the mutual information with the number

cation limit of IID complex Gaussian inputs in this case.

of antennas for nT = nR .

Example 3 Shown in Fig. 3 is a re-evaluation of Example 2

Example 6 Let nT = nR . Shown in Fig. 6 is the mutual

with nb = 4.

information for SNR = 3 dB as function of nT = nR with nb =

10 and nb = 100. Also shown is the perfect-CSI capacity.

In this case, the relative gap between the perfect-

CSI capacity and the achievable mutual information is

The linear scaling of the perfect-CSI capacity with

very substantial. (At 0 dB, less than half the perfect-

nT = nR is what fueled the early interest in MIMO.

CSI capacity can actually be achieved by IID complex

Without perfect CSI, the linear scaling is upheld approx-

Gaussian inputs.) The spectral ef ciency of pilot-based

imately as long as the number of antennas is suf ciently

schemes is similarly affected. Remarkably though, the

small relative to nb, but not otherwise. This is not

performance of these schemes relative to the mutual

just a limitation associated with the suboptimality of

information limit is essentially unaffected.

Gaussian inputs in the absence of perfect CSI, but rather

a fundamental issue caused by channel uncertainty [9].

VI. T HE L OW-SNR R EGIME

Fig. 3. In solid, I (SNR) for nT = nR = 1 with nb = 4. Also in solid,

In power-limited conditions, power ef ciency becomes

spectral ef ciencies achieved by pilot-based communication, with and

without pilot power boosting. In dashed, the perfect-CSI capacity. relevant and the gure of merit that quanti es such

Fig. 6. In solid, I (SNR) for SNR = 3 dB as function of nT = nR Fig. 7. Eb /N0 as function of SNR for nT = nR = 1 with nb = 10.

with nb = 10 and nb = 100. In dashed, the corresponding perfect-CSI In solid, values obtained from I (SNR) and also values corresponding

capacity. to pilot-based communication, with and without pilot power boosting.

r r

For each curve, (Eb /N0 )min is explicitly indicated. In dashed, Eb /N0

with perfect CSI, i.e., obtained from C (SNR).

ef ciency is the energy per bit normalized by the noise

spectral density. Measured at the receiver, this gure of

merit equals Fig. 8. SNR at which (Eb /N0 )min is achieved as function of nb

r for nT = nR = 1. Values obtained from I (SNR) and also values

Eb SNR

(16)

= nR corresponding to pilot-based communication, with and without pilot

N0 R/B power boosting.

where R/B is the spectral ef ciency, i.e., C (SNR) with

perfect CSI or I (SNR) without it.

Er nb . This characterization is presented in Figs. 8 and 9

With perfect CSI, it is known that Nb is minimized for

for nT = nR = 1, and similar results can be readily

SNR 0 and that such minimum equals [25]

obtained for MIMO. For typical vehicular scenarios, the

Eb 1 operating points that maximize the power ef ciency

(17)

N0 min log2 e are substantially higher than what one might anticipate

from a perfect-CSI analysis, and the corresponding Nb min

which equals 1.59 dB. Without perfect CSI on the 0

levels are markedly above the 1.59-dB oor.

side, I (SNR) with IID complex Gaussian inputs is convex

below some (low) SNR and concave above it [5] [8].

Er VII. T HE H IGH -SNR R EGIME

It follows that Nb is minimized at some nite SNR.

However, this minimizing SNR, and the corresponding A. Case nb nR + min{nT, nR }

N0 min, cannot be obtained using the low-SNR expansions

For such nb, it is shown in [5] that the high-SNR slope

available in the literature because only the convex behav-

of the true channel capacity (without side information at

ior of I (SNR) is captured therein. Speci cally, the most

the receiver) is

re ned low-SNR expansion available is [7]

nb nR min{nT, nR }

2 2

I (SNR) = (18)

SNR + o(SNR ) min{nT, nR } 1 (19)

2 nT nb

based on which Nb min indeed cannot be obtained.2 in bits/s/Hz/(3 dB). By activating min{nT, nR } transmit

Applying the method presented in Section III, the cor- antennas and min{nT, nR } receive antennas, a straight-

forward computation shows that the lower bound in

rect Nb min and the corresponding SNR can be calculated.

Proposition 3 achieves the same high-SNR slope. Since

In pilot-based communication, the spectral ef ciency

I (SNR) increases with the number of receive antennas

is also a convex function of SNR below some SNR and

(by the chain rule, additional outputs are never harmful),

concave above it [17]. Thus, Nb min is also achieved at

this implies that the optimal high-SNR slope is achieved if

some nite SNR, which can be calculated numerically

IID Gaussian inputs are sent from min{nT, nR } transmit

by solving for the spectral ef ciencies in (13) and (14)

antennas and all available receive antennas are used.

and then using those results to minimize (16). Such

Based upon (13), it is also straightforward to con rm that

calculations are conducted in [26].

pilot-based communication achieves the optimal high-

SNR slope if min{nT, nR } transmit antennas are used with

Example 7 Let nT = 1 = nR = 1 and let nb = 10. Shown

E one pilot per antenna.

in Fig. 7 is the Nb as function of SNR.

Thus, the slope is not a de ning feature at high SNR.

At this fading rate (nb = 10), the most power-ef ciency Rather, it is the power offset [27] that determines the

operating point is SNR = 2.4 dB. The corresponding performance in this regime, and the method in Section

N0 min equals 2.1 dB, almost 4 dB above what a perfect- III can used to quantify it for IID Gaussian inputs.

CSI analysis would indicate. With pilot-based transmis-

sion, an additional penalty of over 1 dB in Nb min is Example 8 Let nT = nR = 2 and let nb = 10. Shown in

suffered. Fig. 10 is the high-SNR mutual information. Also shown are

More generally, the method in Section III allows char-

acterizing the power-ef cient operating point and the

corresponding Nb min as a function of the fading rate,

Fig. 9. (Eb /N0 )min as function of nb for nT = nR = 1. Values

2 Eq. obtained from I (SNR) and also values corresponding to pilot-based

(18) would indicate that Nb is achieved for SNR, but

0 min

communication, with and without pilot power boosting.

(18) does not apply beyond the low-SNR regime.

A CKNOWLEDGMENTS

Fig. 10. In solid, I (SNR) for nT = nR = 2 and nb = 10. In dashed, the

The authors thank Prof. Babak Hassibi (Caltech,

corresponding perfect-CSI capacity. In circles, the high-SNR expansion

USA) for pointing out some valuable references. The

of the true capacity as per (20).

work of A. Lozano is supported by the Spanish

Ministry of Science and Innovation (Refs. TEC2009-

13000 and CONSOLIDER-INGENIO CSD2008-00010

the perfect-CSI capacity and the high-SNR expansion of the

COMONSENS ).

true capacity, which for nT = nR is given in [5] as

nT nb 1 A PPENDIX

1 C (SNR) + nT log2 + log2 G(nb, nT )+o(1)

nb e nb

A. Preliminaries

(20)

with For subsequent use, we present four relevant identi-

ties. The rst one, easily veri ed, is

2 i

(i 1)!

i=t n+1

(21)

G(t, n) = . exp{ x2 A + xB } dx = exp (23)

4A A

2 i

(i 1)!

The second one is an integral due to Itzykson and

i=1

Zuber [29]. Given an M M diagonal matrix B with

Although for nb nR + min{nT, nR } the input X that

diagonal entries b, an arbitrary M M matrix D with

achieves the true capacity for SNR is an isotropically

eigenvalues d, and an M M isotropically distributed

random unitary matrix [9], [28], IID Gaussian inputs

unitary random matrix U,

seem to perform very well at high SNR, even in relatively

fast fading. Non-asymptotically in the SNR, the optimum M

(m 1)! detE (d, b)

eTr{U DU Z}

X is no longer just a unitary matrix but rather the (24)

p(U ) dU =

detV (d) detV (b)

product of a unitary matrix and a nonnegative real m=1

diagonal matrix. No expressions are then available for

where the (i, j )th entry of the M M matrix E (d, b)

the true capacity.

equals

Eij = exp{di bj } (25)

B. Case nb

In this case, the optimum X is again the product while V denotes a Vandermonde matrix, i.e., such that

of a unitary matrix and a nonnegative real diagonal

(dj di ). (26)

detV (d) =

matrix, even for SNR . The high-SNR slope of the

true capacity equals [5] 1 i N,

detE (d, b)

but no further expressions are available for the true ca- lim

bN +1 bM 0 detV (b)

pacity. Our method to compute the mutual information

( 1)(M N )(M N 1)/2

of IID Gaussian inputs continues to apply. detE (d, b)

(27)

= N M N

detV (b[1...N ] )

(k 1)!

k=1 bk

VIII. C ONCLUSION

where the (i, j )th entry of E (d, z ) equals exp{di zj } for

We have presented a method (part analytical, part

j N and di N 1 for N + 1 j M .

Monte Carlo) to compute the mutual information

achieved by IID complex Gaussian inputs on block The nal identity was proved in [30] by Chiani, Win

and Zanella. Given two arbitrary M M matrices (x)

Rayleigh-faded channels, both scalar and MIMO. This

mutual information is highly relevant as it represents and (x) with (i, j )th entries i (xj ) and i (xj ), respec-

tively, and an arbitrary function,

the highest spectral ef ciency attainable with Gaussian

codebooks. M

The method presented may be of further interest to

det (x) det (x) (xm )dx

other multivariate problems involving combinations of Dord m=1

multiplicative and additive Gaussian noise, either with

respect to the mutual information or to the constituting

(28)

= det i (x) j (x) (x)dx

differential entropies. a

i,j =1...M

A software routine that implements the described

where the multiple integral is over the domain Dord =

method in Matlab code is available for download at

http://www.dtic.upf.edu/ alozano/software. {b x1 x2 . . . xM a}.

B. Proof of Proposition 1 Applying (23) to each variable xt,k in (34) gives

Conditioned on X, the output Y is complex Gaussian.

Furthermore, the rows of Y are IID conditioned on X . 2

Re{y u }

nb s

Hence, to obtain h(Y X ) it suf ces to evaluate its value exp Y

k tk

p(Y ) = exp

for an arbitrary row of Y and then scale it by the number nR nb 1

k +

t=1 k=1

of rows, i.e., by nR .

Let y be an arbitrary row of Y . The conditional co-

Im{y uk }

k 1

variance of the nb -dimensional column vector y equals t

exp (36)

k + 1

k +

SNR

E y y X = I + (29)

nT s

k u Y Y uk

exp Y k

= exp

nR nb

and thus ( k + 1)

k=1

h(y X ) = h(y X ) (30) (37)

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