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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 {xn(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 {xn} 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 N0 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 {yn}.
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 yn. 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 makesthe energy per chip for the k'th user.

The expectations of yn 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 yn (k) be
. (4)
The variance of any yn(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 N0 by Neff = N0 + Total other user energy per chip + Jeff.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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