Power System

Capacity of the Isotropic Fading Multiple Antenna

Downlink with Magnitude Feedback

Sudhir Srinivasa and Syed Ali Jafar

University of California Irvine

Irvine, CA 92697-2625

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

receiver may feedback only the norm (magnitude) of the

Abstract We determine the capacity region of a single-

cell isotropic fading multiple antenna downlink with perfect instantaneous channel vector with no information about its

channel knowledge at the mobiles and only the knowledge of direction. In this paper we determine the capacity region

the magnitudes of the users channel vectors at the base station.

of the vector Gaussian BC with perfect CSIR and only

Using a scalar upperbound we are able to prove the optimality of

channel magnitude information available to the transmitter. A

Gaussian codebooks on this vector broadcast channel. Numerical

scalar upperbound is used to prove optimality of Gaussian

results are provided to compare magnitude feedback against

opportunistic beamforming and antenna selection. codebooks. We also compare magnitude feedback with other

broadcast schemes such as opportunistic beamforming and

I. I NTRODUCTION

antenna selection.

One of the most intriguing aspects of multiple antenna

II. S YSTEM M ODEL

wireless systems is that the capacity bene ts of using mul-

tiple antennas can depend dramatically upon the assumptions In this paper we consider the capacity region of an isotropic

regarding the underlying time-varying channel model and how fading vector Gaussian BC with perfect CSIR and only chan-

well it can be tracked at the transmitter and the receiver [1]. nel magnitude information at the transmitter. By isotropic we

In practical systems, channel state information at the receiver mean the following: the transmitter has no knowledge of the

(CSIR) is obtained from pilot symbols and blind channel direction of any user s channel vector. In other words, all

estimation techniques. In the absence of delay constraints, directions are equivalent from the transmitter s standpoint. The

the receiver s task is simpli ed somewhat because unlike the precise system model is presented next.

transmitter, which needs real-time channel estimates, it can

wait until the end of transmission to form its channel estimate A. Broadcast Channel Model BC-V

based upon the entire received sequence of coded data symbols

Consider a fading vector broadcast channel with M transmit

as well as the pilot symbols. From an information theoretic

antennas at the base station and K users with a single receive

perspective, the assumption of perfect CSIR allows us to deal

antenna at each user given by the input/output relationship

in isolation with the issue of channel coding while ignoring

the issue of designing optimal training sequences and channel [1] [1] [1]

Yt = Ht Xt + Zt

estimation algorithms. [2] [2] [2]

Yt = Ht Xt + Zt (1)

Channel state information is hard to obtain at the transmitter.

.

In a time division duplexed system where the uplink and .

.

the downlink are using the same frequency spectrum, the [K ] [K ] [K ]

Yt = Ht Xt + Zt

reciprocity of wireless channels allows the transmitter to

obtain channel state information from the received uplink [k ]

where for user k at time instant t, Ht is the 1 M channel

transmissions. However, most current cellular systems as well [k ] [k ]

vector, Yt is the received scalar signal and Zt is additive

as those planned in the immediate future are frequency division

white Gaussian noise (AWGN). Xt is the M 1 complex

duplexed. For such systems, channel state information at the

vector symbol transmitted by the base station at time instant

transmitter is obtained only by means of a feedback channel

t. Let the average transmit power be P, so that

through which the receiver can send its current channel es-

E Tr(XX ) P.

timates to the transmitter. The task is even harder for vector (2)

broadcast channels because the number of channel coef cients

that each receiver needs to estimate and instantaneously feed- Figure 1 shows the channel model for BC-V with two

back is equal to the dimension of the channel vector. While, users. The channel fade and noise processes are ergodic and

even with perfect CSIR, it is impractical for the receiver stationary. We allow the channel fade process to have memory.

to feedback all the components of the instantaneous vector Thus, successive realizations of the channel and/or the noise

channel state to the transmitter, it may be possible to feedback may be correlated. For simplicity, the time index is suppressed.

partial information about the channel state. For example, the Isotropic fading [2] is de ned as follows:

Z [1] Z [1]

Transmit Power P

H[1] H[1]

Y [1] Y [1]

P

Transmit Power M

X X

Z [2] Z [2]

.

.

Y [2] H[2] Y [2]

H[2]

Fig. 2. BC-S

Fig. 1. BC-V

where the transmit power constraint is P instead of M andP

Isotropic Fading: We consider the class of channels that can [k ] [k ]

the users channels are h / M instead of h . To avoid

be described as confusion, in this paper we will use the system model as

H[k] = [k] h[k] (3) represented in Figure 2.

where [k] is a 1 M isotropically random complex unit The following theorem establishes the scalar upperbound

for the vector broadcast channel with magnitude feedback.

vector and

h[k] = H[k], Theorem 1: The capacity region of the vector fading

(4)

broadcast channel BC-V with perfect CSIR and magnitude

the norm of the instantaneous channel vector is a non-negative feedback is contained within the capacity region of the scalar

scalar random variable independent of [k] . It is important to fading broadcast channel BC-S with perfect CSIR and perfect

note that each users channel norm may have a completely CSIT.

different distribution. Proof of Theorem 1: The proof for the applicability of the

Recall that an isotropically random vector is one whose scalar upperbound is identical to that for Theorem 1 of [2]

distribution is not affected by multiplication with a unitary if we also assume magnitude feedback in the scalar fading

matrix. It is the mathematical way to capture the notion that broadcast system model BC-S. Thus, the capacity region of

the vector is equally likely to point in any direction in the the vector broadcast channel BC-V with magnitude feedback

M dimensional vector space. An example of a channel that is contained within the capacity region of the scalar broadcast

belongs to this class is the Rayleigh fading channel with channel BC-S with magnitude feedback. Interestingly, magni-

AWGN where each users channel vector H[k] consists of i.i.d. tude feedback for the scalar broadcast channel constitutes the

[k ] 2

complex Gaussian elements Hi N (0, k ), 1 i M . entire channel state information. Thus, we can equivalently

Speci cally, in terms of the system model BC-V, this means state that the capacity region of the vector broadcast channel

that h[k] is known to the transmitter, 1 k K . Perfect CSIR BC-V with perfect CSIR and magnitude feedback is contained

is still assumed. Next we introduce the scalar upperbound. within the capacity region of the scalar broadcast channel with

As the name implies, the scalar upperbound is an upper- perfect CSIR and perfect CSIT.

bound on the capacity region of the vector broadcast channel For our second result in this section we need the assumption

BC-V in terms of the capacity region of a scalar broadcast that the additive noise is white and Gaussian. Under this

channel BC-S. The new channel BC-S is described as follows: assumption, the capacity region of the scalar fading broadcast

channel with perfect CSIR and CSIT has already been found

B. Broadcast Channel Model BC-S

by Li and Goldsmith (Theorem 3.1 of [3]). From Li and

Associated with the broadcast channel BC-V, we de ne Goldsmith s result we can directly state the following:

another channel model, BC-S, with input/output relationship Theorem 2: The capacity region of the scalar fading Gaus-

sian broadcast channel BC-S with perfect CSIR and perfect

Y [1] = h[1] X + Z [1]

CSIT is

[2]

Y [2] = h X + Z [2] (5) C (P ) = P F CCD (P ), (7)

.

.

. where

Y [K ] = h[ K ] X + Z [ K ]

CCD (P ) = Rk Eh[ ] [log (1+

Notice that the input X is a scalar, and each users channel (h[k] )2 Pk (h[ ] )

is also a scalar equal to the norm of the corresponding original K [j ] 2 [ ] [j ] > h[ k ] )

1+ j =1 (h ) Pj (h )1(h

vector channel. The additive noise experienced by each user

1 k K } (8)

is the same as in the vector broadcast channel model BC-V.

is the achievable region with superposition coding and suc-

The new transmit power constraint is

cessive decoding with power allocation policy P . The joint

P channel state of all users is denoted by the vector h[ ] =

E[ X 2 ] . (6)

M {h[1], h[2],, h[K ] }. Pk (h[ ] ) is the transmit power allocated

to user k under power policy P . F is the set of all permis-

Figure 2 shows the channel model for BC-S with two users.

Of course, an alternate representation of the BC-S channel is sible power policies that satisfy the average power constraint

Pk (h[ ] ). 1 denotes the indicator function (1(x) = 1

K

degraded. If we do not assume (10) then the users are only

k=1

if x is true and 0 otherwise). Note that the way this theorem stochastically degraded.

is stated it assumes that no two users have the same channel Lemma 2: For the vector fading Gaussian broadcast

magnitude. As mentioned in [3] the case when, for example channel BC-V with a xed channel magnitude vec-

users i and j have identical channel magnitudes can easily tor h[ ] and a xed power allocation vector P =

{P1 (h[ ] ), P2 (h[ ] ),, PK (h[ ] )} the following rate tuples

be handled by viewing the two users as one user, applying

superposition coding and successive decoding to the resulting are achievable with superposition coding and successive

K 1 users. Time sharing is used for users i and j . Thus, for decoding

simplicity, we will assume henceforth in this section that all

(R[1], R[2],, R[K ] ) :

the users have distinct vector channel magnitudes.

Using Li and Goldsmith s result stated above we prove the

h[k] Pk (h[ ] )

following Theorem:

R[k] log 1 +,

K

h[j ] Pj (h[ ] )1(h[j ] > h[k] )

1+

Theorem 3: The capacity region of the isotropic vector j =1

fading Gaussian broadcast channel BC-V with perfect CSIR 1 k K } (11)

and only a knowledge of h[ ] at the transmitter (magnitude

By achievable we mean the rate tuple lies within the

feedback) is identical to the capacity region of the scalar fading

capacity region.

Gaussian broadcast channel BC-S described in Theorem 2.

Proof of Lemma 2: For a stochastically degraded channel the

Note that this theorem does not make any assumption regard-

capacity region is given by

ing the distribution of users magnitudes.

Proof of Theorem 3: In order to prove Theorem 3 we only C= p(U1,,UK 1,X):U1 UK 1 X (Y[1],,Y[K] )

need to prove achievability. The converse is already proved

(R[1],, R[K ] ) RK :

from the scalar upperbound. For achievability, we start by +

considering a xed channel magnitude vector h[ ] . Without loss

R[1] I (U1 ; Y [1] ),

of generality suppose the users xed magnitudes are ordered

R[k] I (Uk ; Y [k] U1,, Uk 1 ),

as

h[1]

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