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### Spectral Efficiency of CDMAwith Random Spreading

Siam M. Hossain

Outline

- Introduction
- Crosscorrelations of Random Sequences
- Optimal Decoding
- Single-user Matched Filter
- Decorrelator
- Linear MMSE Receiver
- Summary of Results
- Conclusion

Introduction

The main purpose was to evaluate the spectral efficiency of CDMA systems where signature waveforms were assigned randomly.

Each signature can be viewed as a unit-norm vector in an N-dimensional signal space, where N is the spreading gain or number of chips per symbol. K is the number of users who linearly modulate their signatures.

The analysis considers a white Gaussian channel with users constrained to have identical average received powers.

The following receivers (linear multiuser detector front-ends) are analyzed:

a) Optimal joint processing

b) Single-user matched filtering

c) Decorrelation

d) MMSE linear processing.

Introduction -spectral efficiency

Spectral efficiency C, is defined as the total number of bits per chip that can be transmitted reliably arbitrarily.

The maximum spectral efficiency in the absence of spreading, is

Crosscorrelations of Random Sequences

The kth user sends the codeword

by transmitting

Sk has duration-T , unit energy, and lives in an N-dimensional space.

Where Ck is the spreading code assigned to the kth user.

The crosscorrelations between the

signature waveforms are denoted by

Optimal Decoding

A. Two-User Channel

The gain due to dynamic power assignment is very minute in this case. The maximum difference occurs for asymptotically high and that’s 0.03 bit/chip. For all but very low , the maximum relative gain is also very small 1.125.

B. K-User Channel

For larger number of users the gain realized by dynamic power assignment is even smaller. The reason is that the likelihood of atypically bad/good crosscorrelation matrices decreases with K (and also with N).

Optimal Decoding

C.

The complexity of analytical results on spectral efficiency quickly grows with the number of users.

Here, the analytical results become feasible and also the randomness of spectral efficiency due to the random choice of signatures, vanishes.

Single-user Matched Filter

The output of the matched filter of user 1 is the following discrete-time process:

The single-user channel equation is non-Gaussian, and its capacity depends on the crosscorrelations.

Achieving the capacity of the equation, requires that the receiver of user 1 knows the crosscorrelations and input distributions of all the interferers.

However, information becomes useless as the number of users grows without bound. Furthermore, the dependence of the capacity on the actual realization of signature waveforms vanishes asymptotically.

Single-user Matched Filter

If , ,orthogonal sequences are optimal, and if K=mN, they remain optimal provided each sequence is assigned to m users.

The ratio of spectral efficiencies is monotonically decreasing with

When <-1.6 db, the ratio is 1/3 at K=N, and higher for other K/N.

Decorrelator

In contrast to the single-user matched filter, the decorrelator correlates the received signal with respect to the projection of the signature waveform (Sk) on the subspace orthogonal to the space spanned by the interfering waveforms.

Such a transformation succeeds in completely eliminating any interference from other users, and the decoder sees a singleuser memoryless channel.

For reliable communication, the minimum = 1.6 dB + noise enhancement factor.

Therefore, for any given , the spectral efficiency of the decorrelator becomes zero for a value of that is strictly smaller than 1.

Decorrelator

Comparing this figure to Fig. 5, we can see that for high and low the decorrelator almost achieves optimal spectral efficiency.

In contrast to the single-user matched filter, the suboptimality of the random choice decreases with .

Linear MMSE Receiver

If all the received SNRs are identical, the MMSE receiver for the kth user correlates the incoming signal with

It has been observed that, the Gaussian approximation for the output of the MMSE transformation is excellent even if there are very few binary-valued interferers.

And also that the spectral efficiency is not affected by the distribution of the symbols transmitted by the interferers.

Linear MMSE Receiver

- For K=3N, the MMSE receiver suffers substantial losses, unlike the optimal receiver,.
- For K=N, and in the range of considered in Fig. 8, a random choice achieves around 40% of the spectral efficiency achieved by orthogonal sequences.
- As the spreading gain N increases, the MMSE detector loss is more important at low and approaches that in Figs. 5 and 7 for large .
- The deleterious effect of low on the decorrelator is not suffered by the MMSE receiver.
- Relative to Fig. 6, we see that at low the MMSE and single-user matched filter behave similarly; at high the comparison depends heavily on .

Conclusion

With large K/N, random CDMA incurs negligible spectral efficiency loss relative to no-spreading if an optimum receiver is used.

The optimum coding–spreading tradeoff favors negligible spreading for either optimum or single-user matched-filter processing. In contrast, non-negligible spreading is optimum for linear multiuser detectors such as the decorrelator and the MMSE receiver.

With an optimal choice of spreading factor, the spectral efficiencies of the decorrelator and MMSE receivers grow without bound as increases, in contrast to the single-user matched filter, where large SNR offer little incentive.

For the optimal receiver, the single-user matched filter, and the decorrelator, the maximal loss occurs at K=N.

For low K/N systems, either the decorrelator or the MMSE are excellent choices and little inefficiency results from random rather than orthogonal signatures.

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