Generalizing the Single–User Average Mutual Information for the Fading MAC . 179

We generalize the single user waveform channel from Chapter 3 to the

L

–user channel as

y n ( t ) = ^L

^X,

¹

l=0

Z

T=2

,

T=2 x n;l ( ) h n;l ( t; ) d + z n ( t ) ; t

²

[

^,

T o = 2 ;T o = 2]

^(7.32)

where

h n;l ( t; )

is now the response of user

l

’s channel at time

t

to an impulse at time

. The subscript

n

in all the signals denotes the

n ^th

realization or block of the corresponding process. We assume that the difference between the durations of the output signal and input signals

T o

^,

T

is larger than all the delays due to multipath propagation and user asynchronism. The noise process is again a zero–mean, circular symmetric, white complex Gaussian process with power spectral density

N 0

and the channel responses

are assumed to be stationary during the transmission of each block as

For the Gaussian MAC we are interested in the average mutual information functionals

I T;H;U = 1 T ^I( Y ( t ); B U ( t )

^j

B _U ( t ) ;

^f

H l ( t; ) = h l ( t; ) ;l = 0 ;

;L

^,

1

^g

)) ;

⁸

U

^2U

bits = s

We have assumed that the users signals are independent. In a mean–square sense we have with Gaussian signals that

As was the case in Chapter 4, in the limit of large

T

we obtain, via the Szeg ¨o eigenvalue distribution theorem [GS83], the following lower–bound for (7.34)

I T;H;U

I H;U =

and, under the assumption of bandlimited transmitted signals, the corresponding average power con-straint is

E

Z

W=2

,

W=2 S i;n ( f ) df = P i :

^(7.40)

The signals are band–limited to

[

^,

W= 2 ;W= 2]

^and

S i;n ( f )

is the power spectrum of user

i

^{in block}

n

^{. If}

the time–bandwidth product is large, as in the single–user case

I T;H

I H

^.

7.3.2 Systems Without Fast Power Control

Let us now compare two alternative systems. The first is orthogonal multiplexing where each user uses shares bandwidth/time equally. For FDMA each user would use a bandwidth

W=L

and transmit all the time. The

F

block average mutual information conditioned on the channel realizations for user

i

^with

FDMA is

where

h i

is the random variable describing the channel strength ^j

H i;n ( f )

^j

²

which is assumed to be identically distributed at each frequency and in each block and independent from user to user. This allows us to remove the integral in (7.41). Note that (7.41) holds for slow–frequency hopping as well as long as the statistics of the different frequency bands where the signal hops are identical. The TDMA case is also given by (7.42) since each user would use bandwidth

W

^{with power}

LP i

^{a fraction}

1 =L

^of

the time.

Now consider the general non–orthogonal approach which has the rate sum taken over a large number of blocks and using uniform power spectra

I _U;H ^opt = W E

^f

h

i

; i

²

U

^g

log 2 1 + 1 WN 0

X

i

²

U P i h i

!

bits = s :

^(7.43)

By virtue of the concavity of the logarithm and Jensen’s inequality [CT91] we have that

I _U;H ^orth

W

^j

U

^j

with strict inequality unless^j

U

^j

= L

^.

These important results are due to Gallager [Gal94] and show that the achievable rate region for orthogonal multiplexing (when users share the dimensions on an equal basis) always lies within that of an

optimal scheme which codes users over all the dimensions. Moreover, they are valid for TDMA,FDMA or slow–frequency hopping systems. Let us examine the implication of these results with a numerical example, by assuming that the fading process is Rayleigh distributed and

P i = P;

⁸

i

. This symmetric situation arises in practice if we have a perfect slow power controller which assures that the average received SNR remains constant and equal for every user. This is important since it assures an equal quality of service (on a long–term basis) for all users. In this case, all the channel strengths

h l

^are exponentially distributed as

f h ( ) = e

^,

;

0

^(7.46)

so that the rate sums are

I _U;H ^opt = W

^{Z 1}

The rate sums in (7.47) can be expressed in terms of the Meijer G-function[GR80] but are simpler to calculate via numerical integration. The orthogonal rates are just the single–user rates as in Chapter 3

I _U;H ^orth = W

^j

U

^j

In figure 7.11 we show the achievable rate regions for three two–user systems, the two we just described and an orthogonal scheme where we can adjust the proportion of the total dimensions allocated to each user. This was described in section 7.1.1 and for the fading case, we simply have

R 1

(1

^,

) W

We chose a signal to noise ratio

P= ( WN 0 ) = 10

dB. We see that there is a significant increase in spectral efficiency for the non–orthogonal scheme even at the equal rate point, which shows that the fading channel behaves quite differently than the non-fading channel. As pointed out by Gallager a rather peculiar result is that

L

users transmitting over

L

different channels can transmit more total average information than a single user transmitting over one channel using

L

times as much power. This

0 0.5 1 1.5 2 2.5 3 0

0.5 1 1.5 2 2.5 3

Optimal Orthogonal (variable sharing)

Orthogonal (equal sharing)

W

^,

¹ R 0

^bits/s/Hz

W

^,

¹ R 1

^bits/s/Hz

Figure 7.11: Two–user Rate Regions in Rayleigh Fading

is due to an averaging effect inside the logarithm in the multiuser case since the sum signal has a total received energy which has

L

degrees of freedom and not a single one as would be the case with one user.

This can be made more clear if we consider the special case where the bandwidth is proportional to the number of users, say

W = LW B

^{, where}

W B

can be thought of as the bandwidth per user. If we focus on the total rate sum (i.e. when

U =

^U), the orthogonal schemes with equal transmitted powers will have

I

^U

^orth _;H = LW B E h log ₂

1 + P

W B N 0 h

^(7.51)

and the optimal scheme will have

I

^U

^opt _;H = LW B E

L

log ₂

We note that for

L

^!1, the random variable

L =L

is 1 with probability 1 since the

h i

are independent and consequently the total rate sum tends to

L

times the Gaussian channel capacity given by

C G = LW B log ₂

If we now compute the difference in spectral efficiency on a per user level between the Gaussian channel capacity and orthogonal multiplexing on a Rayleigh fading channel, we have that [AS65]

( LW B )

^,

¹ ( C G

^,

I

^U

^orth _;H ) = log ₂

shows that the most that can be gained by using a non–orthogonal scheme such as CDMA on a multipath channel as opposed to any type of orthogonal multiplexing is .8327 bits/s/Hz. It assumes, of course, that we have a coding scheme with no delay constraint and that some optimal successive decoding scheme, such as onion peeling [RU96] can be used.

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