Abstract
In
multiple accesses DS_CDMA system, it is important and difficult to achieve
minimum error bit rate. By regarding the multiple access interference as
Gaussian process of same PSD, we can decrease the BER through improve signal to
noise ratio (SNR). Although it is not necessary condition, maximizing SNR is a
sufficient way to improve error rate performance and SNR is good performance
measure, especially when the number of CDMA users is large. In [1], a modified
matched receiving filter is implemented to maximize SNR in compared with the
minimization of probability of error, under some constrains. It does not require
locking and any knowledge of other users’ spreading code. The experiments in
[1] are simulated in this project and the results are analyzed.
Introduction
In
DS_CDMA system, under only additive white Gaussian noise (AWGN), the optimal
receiver is the one that matched the spreading code of the user of interest.
But, when many users are present in the channel, the by-productive multiple
access noise will make very poor the performance of matched filter. The optimal
and many sub-optimal structures were proposed to overcome the MAI, but since
they need locking and de-spreading interfering signals, they are all
substantially complex.
A
practical and economical structure that can outperform the conventional ones is
introduced in [1]. In order to achieve simplicity, the author constrained the
receiver to operate bit-by-bit, and the decisions are based on observation of
the received waveform over one bit time without knowledge of other users’
signature, timing, and carrier phase.
Consider
a circumstance where there are many users accessing a channel almost
simultaneously. Since they are all independent each other, and according to the
Central Limited Theory, the sum of their interference is Gaussian. Noticing that
the interfering users are randomly delayed and their spreading codes are
pseudo-random, the multiple access noise is actually colored Gaussian process.
While in Gaussian noise circumstances, maximizing the SNR can minimize
the bit error rate. In [1], an SNR detection of one bit was studied, and several
receivers are analyzed, focusing on SNR maximizing. Such linear time invariant
filters can even adaptable to non-Gaussian signals.
The
difference between the proposed receiver and conventional ones is that there is
a noise-whitening filter at the beginning of it. Since it operate in the absence
of knowledge of the other user’s spreading codes, nor does require that
spreading codes be periodic in a bit time, it reject multiple access noise
mainly by taking advantaged of coloration in the power spectrum of them,
As
shown in [1], that the advantages of broader applicability and less complexity
than typical multi-user detections, are at the expense of poorer performance of
bit error rate.
This
project is based on the analysis and simulation of [1], and the report is
organized as follows: Part II system model; Part III infinite observation
interval with infinite frequency range; Part IV finite observation interval with
infinite signal band range; Part V under constrain of limited signal band. And
the simulations are given accordingly. Part VI gives the conclusions.
System Model
In
[1], both the transit and receiving model is given as follows
By analyzing, the above diagram can be rewrite as follows:
transmitter:
receiver:
where,
,
T = N * Tc,
represents inner production,
and * represents convolution
The judgement is sgn{G}
So, the
simpler expression is as follows
By
comparing with traditional matched filter receiver, we can see that the critical
is the filter q(t), which acts as the noise whitener, to make all interfering
signals stationary noise, so that the receiver can concentrate on the signal of
interest.
Following
is the analysis on what this filter’s structure and property is, and how it
overcome the colored Gaussian inter symbol interference. In [1], 3 different
constrains are added to the system, and different results are given. Among all
of them, only the second one can be simulatable. For the case of infinite
observation interval and band limited signaling, the theoretical results are
provided.
Infinite
Observation Interval
In
this case, we assume that the user of interest, say, user 1, just sends one
signal which lasts for T, while all other user transmit for infinite time. i.e.
and
Such
approximation is reasonable in a CDMA system, where the available bandwidth
typically exceeds the bit rate 1/T by the large processing gain. Under such
constrain, it is cited in[1], that the filter g(t) can be broken into two pats:
the whitening part Y(w) and matched part S*(w):
S*(w) is conjugation part of DFT of S(1)(t) . The receiver maximizes SNR for a signal s(t) being detected in stationary noise of PSD Sn(w) , regardless of whether the noise is Gaussian or not. Maximizing SNR is a sufficient way to improve error rate performance, although it is not necessary condition. But SNR is good performance measure, especially when the number of CDMA users is large. So,
and
,
where,
the DTFT of PTc(t) is
Combining all of the above, and by some computation,
we can get a solution in terms of autocorrelation function, which is coincide
with [1]. So, I write the expression used in [1]:
To interpret it in terms of matrix, R(l) is the summation of each symmetric diagonal in the autocorrelation matrix of a(1) i,j ,
{j-i = 0 ~ N-1}
Finite observation interval
Under infinite observation interval condition, the fitter q(t) can be of infinite duration in both directions. This is realizable only by accepting a very long delay. In order to derive a receiver that bases its decisions on observing the received wave form over a single bit time, so that to maximizing SNR of one bit duration, [1] gives the following receiver:
If the signal a*s(t), whith a = { ±1, Pr(a=+1)= 0.5}
is to be received in the presence of stationary colored Gaussian noise of
autocorrelation Kn(τ), and the receiver is constrained to observe the received waveform only
for t
[0,T], then the minimum-probability of error decision for d is d = sgn{G}, with:
where r(t) is the received waveform and q(t) is the solution to the integral equation
In order to resolve q(u), [1] presents a recursive
way. But there are many other methods. For example, the LMS or RLS are better
solution way, and we may even use the Weiner filter to find the q(t), which is
easy to do but needs more computations.
When the chip pulse is square save PTc(t),
,
Application to
band limited signaling
As
we discussed before, if h(t) has a sinc-function shape, the Q(w) resembles a
high-pass filter. In this case, it amplifies the high end of the spectrum,
especially when multiple access noise becomes much more significant than thermal
noise. So, when we use rectangular chip shape, we can take the advantage of
spectral energy well beyond the main lobe of the chip spectrum. But, quite
often, it is not the case, because many communication system of typically
limited to a specific spectral band. So, we need to look at the band limited
system model.
The
model is the same as we discussed before, for convenience, above is the rewrite
model. The only difference is that
H(
) is band limited : i. e.
for
some constant WH. Now, the chip wave form duration is TH=1/WH.
The corresponding filter is:
Although
Q(w) is independent to spreading sequnce, the SNR is not. As case of that most
spreading sequence have noise like autocorrelation function, it is reasonable to
consider the expected value of SNR over all random independent binary sepqences.
So, the yield is:
It
is proved in [1], that above equation has maximized vale when H(w) =
In
case of multiple access noise dominates thermal noise , which is a typicla
region of operation for CDMA system, we have, when Eb/No goes to infinity.
Where NH is the bandwidth ration here,
and so the processing gain.
Conclusion
It
can be seen that in the appropriate situation, significant improvement can be
make to DS-CDMA system. By using the property of spreading signal that looks
like random binary sequence, the
power spectral density of the multi access noise are mainly according to the psd
of them. In [1], the maximizing SNR is the main method of improve the bit error
probability.
In
this project, I did several simulation , but none of them is accurate according
to [1], so, I did not put them in. It may be due to some error of the codes. But
from theory, the adding a noise whitening filter to conventional matched filter
is workable.
Reference:
[1] A.
M. Monk, M. Davis, Laurence B. Milstein, and Carl W. Helstrom : “A noise
whitening approach to multiple access noise rejection – part I: theory and
background”, IEEE Journal on selected areas in communications, Vol.12, No. 5,
June 1994