BPSK Spreading - Receiver Statistics
Consider a modulator as shown in Figure 1. The data input
to this process is a sequence {xn}, the spreading
is a sequence {an}. The indices denotes periods
of the spreading sequence.
Figure 1. Direct sequence BPSK modulator.
The input sequence {x
n(k)} might be, as in the forward link,
repetitions of the code symbols from a convolutional coder after
application of the
orthogonal
cover, or it might be some other encoding of the data. The
important thing is that, in all cases, the input sequence {x
n}
is comprised of many chips per information bit. Or said another
way, the spreading (chip) rate is many times the data rate.
If the modulo-two addition of the spreading sequence can be
reversed in the receiver, then a summation of these "despread"
chips will recover an approximation of the original code symbol.
For this analysis we assume a channel model that consists only of attenuation
and phase shift, different for each user, . Additive white Gaussian noise
of one-sided spectral density N
0 Watts per Hertz accounts for
the receiver front end thermal noise. Possible jamming is represented by
q(t), which we assume to be wide-sense stationary.
Interesting though it is, multipath propagation is neglected. Multipath
with a delay spread that can be resolved by the receiver contributes some
diversity gain; within each correlator the distinct delay components behave
like other-user interference. Multipath with a delay spread
unresolved
by the receiver introduces inter-chip interference, which complicates the
analysis considerably, and it has little to do with the point we are trying
to make. Its effect is similar to the other-user noise, but with a much
smaller magnitude.
Figure 2. Channel model.
One possible demodulator suitable to remove the spreading
is shown in Figure 3. This structure is optimal if the noise is Gaussian
and uniform over the signal bandwidth. By analyzing the detection statistics
of a single chip in this demodulator, analysis of the actual receiver can
proceed according to the uses of the demodulator output {y
n}.
Figure 3. Direct sequence demodulator.
The decision amplitudes are further processed in accordance
with the nature of the coding and that preceded the spreading. Normally
this will include the creation of one or more detection statistics that
are linear combinations of the y
n. The detection statistics are
then processed by a decoder for the FEC code.
The received signal is
(1)
where the
hk are randomly phase-shifted
replicas of
h(t),
n(t) is the thermal noise of the receiver
front end, and
q(t) is jamming, if any.
All spreading sequences are assumed white and uncorrelated with one another.
The jamming q(t) is assumed wide-sense stationary, but otherwise arbitrary.
Normalization of the bandlimiting filter characteristic is assumed to be
(2)
This is a little unusual. It makes the impulse response
of the filter have dimensions of square-root of bandwidth. But is also conveniently
makes
the energy per chip for the k'th user.
The expectations of y
n for user k is, with this scaling and normalization
(3)
The other users contribute nothing to the expectation because
their spreading sequences are uncorrelated with that for user k. All interference,
noise, and jamming terms contribute nothing because none are correlated
to the spreading sequence.
Let the noise and interference part of y
n (k) be
. (4)
The variance of any y
n(k) is
(5)
where
(6)
The variance of the other-user interference contributes
a factor of
because of the assumption of random phase.
Using the well-known relationship between the autocorrelation functions
and their spectra, (5) is equivalent to
(7)
where J(f) is the one-sided power-spectral density of the
interference.
If we compare the AWGN term to the interference term, we see that
the
effect of the interference on the variance is the same as a white noise
level having the same total power in the passband:
(8)
Equations (7) and (8) represent the central conclusion from this calculation.
It underlies all the calculations relating to capacity and loading
that treat interference only in a total power sense. In calculations
of capacity we can replace N
0 by N
eff
= N
0 + Total other user energy per chip + J
eff.QPSK
SpreadingThe spreading in both directions in the real system
is not BPSK, but QPSK. How does this affect the mean and variance
of the detection amplitudes? To find out . . .
.
. . Continue to QPSK Receiver Statistics
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Copyright © 1996-1999 Arthur H. M. Ross, Ph.D., Limited
