Outage Probability Analysis of Single–User Decoding in the Multiple–Access Channel

Access Channel with Multipath Fading

In this section we consider three possible multiplexing strategies over multipath channels 1. Coding all users over all dimensions simultaneously (i.e. general CDMA)

2. A combination of spreading and coding (DS–CDMA )

3. Orthogonal multiplexing where each user’s signal occupies each part of the spectrum equally so as to exploit all the degrees of freedom the channel has to offer (pure TDMA or frequency–hopping).

We will assume single–user decoding.

In general, since TDMA uses the entire bandwidth the available diversity significantly reduces the outage probability. Moreover, by adding guard intervals which slightly lowers the average information rate, orthogonality can be maintained even in the presence of multipath and slight user asynchronism. It has the disadvantage that the peak–to–average power ratio is high since it is inherently bursty. FDMA transmits with a constant power but does not benefit as much from frequency–selectivity. Perfect or-thogonality is also difficult to achieve since the passband filters are never perfect and adjacent channel interference is inevitable. Some systems such as GSM combine FDMA and TDMA in order to lower the peak–to–average power ratio. In addition, one can perform slow–frequency hopping to take advantage of frequency–selectivity using the coding techniques we considered in Chapter 4 without losing orthog-onality. This is also shown in figure 7.4. The price to be paid for this approach is a slightly longer decoding delay. Orthogonal DS-CDMA suffers to a certain extent on wireless multiple–access channels since multipath and user asynchronism removes perfect orthogonality between the signals.

Here we examine only the case of single–user receivers without any channel state feedback. In order to simplify the analysis we will use the multi-tone signal model described in Chapters 2 and 3.

This allow us to capture several important characteristics of multiuser systems, most notably the reduced dimensionality of spread–spectrum waveforms and user asynchronism.

Recall that under the model we have

NS = WT

dimensions where signals are approximately time–limited to

[

^,

T= 2 ;T= 2]

seconds and strictly band-limited to

[

^,

W= 2 ;W= 2]

Hz. The multiuser case is simply a combination of (4.38) and (7.1) so that its input–output relationship for each dimension is

y s;n = ^L

^X,

¹

l=0 h l;s x l;s;n + z s;n ;

^(7.22)

where

l

^,

s

^and

n

are the user,sub-band and time indices respectively and the^f

z s;n

^gare i.i.d. zero–mean circular–symmetric complex Gaussian random variables with variance

N 0

. The power constraint for each user is

so that under the assumption of independent

x l;s;n

and that users transmit Gaussian signals, the mutual information conditioned on all of the channel realizations for user 0 is

I H = 1 NS

Now let us consider the DS–CDMA case, where we now choose a spreading factor

S

and spreading sequences of the form

g

l;n =

r

1 S

e ^j

^l;n;0

e ^j

^l;n;1

e ^j

^{l;n;S,1 (7.28)}

where

l;n;k

are independent uniformly distributed random variables between

[0 ; 2 )

. This is the fre-quency analog of classical DS–CDMA and is much simpler to analyze on multipath channels. The received signal for each time dimension is therefore of the form

y

n = ^L

^X,

¹

l=0 u l;n

^gh

l;n +

^z

n

^(7.29)

where^gh

l;n =

g l;0;n h l;0 g l;1;n h l;1

g l;S

^,

1;n h l;S

^,

1

.

Similarly to the simple case described section 7.1.2, the receiver decodes each user by first correlat-ing the received signal by the combined spreadcorrelat-ing sequence channel response (i.e. a frequency–domain RAKE receiver) for the user in question followed by conventional single–user decoding. We have for user 0

where

z 0;n

is a complex circular symmetric Gaussian random variable with mean zero and variance

1S

^P

S

^,

1

s=0

^j

h 0;s

^j

2

. We have, therefore, that the average mutual information conditioned on the channel realizations and the spreading sequences is

I H = 1 S ^{log 2 1 +}

WN P

0

Sk 0

1 + _WN ^P

₀

j 0

!

bits = dim

^(7.31)

where

k 0 = _1S

^P

^S _s=0

^,

¹

^j

h 0;s

^j

2

_and

j 0 =

^P

^L _l=1

^,

¹

^P

^S _s=0

^,

¹

^P

^S _s

^0,

₌₀ ¹ e ^j(

^l;s;n,

^0;s0;n

⁾ h

_0;s h

_l;s =k 0

. The basic dif-ference between (7.31) and (7.27) is the type of averaging which is performed. In the DS–CDMA case, both the averaging over fading (

k 0

) and over interference (

j 0

) are achieved by lowering the information rate in a rather brute–force fashion. Note the presence of

1 =S

outside the logarithm. The averaging effect is reduced since it is inside the logarithm. In the general low–rate coding case, the achievable code rate is raised/lowered by the extent of the fading and interference and not by an imposed by a particular coding scheme.

The information outage probabilities in both cases must be calculated by Monte Carlo integration.

This has been done using for the same system parameters as in Chapter 3 (i.e.

S = 128

^,

W = 1 : 2288

MHz, ETSI TU12 channel) for different numbers of users and an SNR of

P=WN 0 = 10

dB. The results are shown in Figure 7.10 where we plot the information outage probability vs. the total rate sum

LR

^{. We}

see that fairly low spectral efficiencies can be expected but that those where low–rate codes are used as opposed to PN spreading are significantly higher. We note that as the number of users increases, the total sum rate increases in the DS system. This is because the signal space is being used more efficiently as the number of users increases (i.e. the factor

1 =F

in front of the mutual information functional gradually disappears.)

Let us now use the IS-95 CDMA system as a means for comparison. The data rate is 9.6kb/s and the maximum number of users per 1.2288MHz is 64. With a voice activity factor of .5 (i.e. at any given time about half the users are actually transmitting and the signals for the others are squelched) this corresponds to a total rate sum spectral efficiency of 9.6*32/1229 = .25 bits/s/Hz. The information outage rate curve for 32 users crosses .25 bits/s/Hz at around

P out ( : 25) = 10

^,

²

which is an acceptable frame error rate for modern vocoders¹. This shows that our analysis is not that far off from reality.

We also plot the single–user information outage we computed in Chapter 3 which corresponds to the total rate sum for an orthogonal multiplexing system which uses the entire bandwidth. We see that it is noticeably lower (i.e. that higher spectral efficiencies can be expected.)

1the GSM vocoder is designed to operate at this speech frame error rate

10⁻² 10⁻¹ 10⁰ 10¹ 10⁻⁴

10⁻³ 10⁻² 10⁻¹ 10⁰

P out ( R )

LR

^bits/s/Hz

Orthogonal

L = 2

^(LRC)

L = 4

^(LRC)

L = 32

^(DS)

L = 4

^(DS)

L = 2

^(DS)

L = 32

^(LRC)

DS = Direct Sequence LRC =Low-Rate Coding

Figure 7.10: Comparison of Low–Rate Coding and Direct–Sequence CDMA schemes

For a single–cell system such as this we really must question the use of a non–orthogonal approach such as CDMA. Even if we could perform joint detection which would be a difficult task in a multipath fading environment, the amount that has to be gained to catch up to an othogonal scheme (with diversity) is huge. The real advantage of CDMA comes in cellular systems where co–channel interference is a significant problem [GJP

+

91]. Here the orthogonal scheme is at a real disadvantage since frequency–

reuse must be used which significantly reduces system capacity. The effect of co–channel interference in CDMA is simply a slight reduction of the signal–to–noise ratio so that it does not suffer to a great extent.

It has recently been shown [PC95][CKH97] that orthogonal schemes need not be limited by co–channel interference and that frequency–reuse may not be necessary. Moreover, they can offer system capacities comparable with CDMA, at the expense of increased decoding delay.

Dans le document Coding and multiple-access over fading channels (Page 183-187)