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Direct Sequence Spread Spectrum

In this page we show how direct sequence spectrum spreading alters the detection statistics of a communication system. The major result is the interference averaging property that we assume when doing estimates of system capacity. The intent here is to do the math correctly, or at least to outline the steps of the analysis.

The commercial CDMA spreading is quadrature in both forward and reverse channels. As a prelude to analysis of that system we first consider BPSK spreading. Extension to QPSK case is easy once we have the BPSK results.

Pseudo-Random Spreading Sequences

In this section we approximate the actual binary PN spreading sequences by an ideal Bernoulli (coin-tossing) sequence with equiprobable outcomes. That is Pr[0] = Pr[1] = 1/2, with all trials independent. Mapping the (0, 1) sequence to a (+1, -1) discrete modulation sequence {an}, the autocorrelation of the latter is a Kronecker delta function, that is

The actual sequences have off-time correlations of the order of 1/N, where N is the length of the sequence. This approximation is well justified in practice because the random relative RF phases of the interferers tend to remove the small bias that the approximation might otherwise introduce.

Alternative representations

In transmitters the most convenient way to impose spreading on data is usually modulo-two addition (exclusive OR) in conventional binary-valued logic. In the analog world those binary values are represented by bipolar signals. The modulo-two binary addition is equivalent to analog multiplication by ±1, provided binary 1 maps to bipolar -1 and binary 0 maps to +1.

We also will refer to impulse modulators. The impulse modulator accepts a binary-valued input, but produces a bipolar impulse output.
These alternative representations should not cause any confusion. They come up often enough that we will not specifically call attention to them.


For the purposes of this analysis we assume that time and frequency synchronization have been achieved by means we don't worry about for now.

A Note on the Differences between the models analyzed here and the real thing ...

Forward CDMA Channel The QPSK model that we consider here is similar to the Forward CDMA Channel except we neglect the orthogonal channelization. We assume here that the spreading sequences are completely uncorrelated between users. There are two consequences of this assumption. First, it means that the users active in the various channels of one base station interfere with one another just as though they would if they came from different stations. Second, it means that the expectations of the chip detection amplitudes depend only on the user being addressed, and have no contributions from the other users.

Neither of these is true in the real Forward CDMA Channel. First, not only are the spreading sequences correlated, they are specifically designed to be rigorously orthogonal over the span of 64 chips, which is the span of an FEC code symbol. This means that the mutual interference terms are correlated in such a way that when the amplitudes are summed to make a soft code symbol, they the other-channel interference terms rigorously cancel. Second, the there is a contribution to the mean detection amplitude from all the code channels. It is precisely that property that lets us separate the code channels in the receiver by selectively de-covering with the desired code channel.

The effect of the orthogonal channelization is to reduce the mutual interference between users. While the cancellation is not perfect in a real system due to unavoidable multipath, it does help, and contributes somewhat to the forward capacity.

Reverse CDMA Channel

The QPSK model that we consider here is very similar to the Reverse CDMA Channel except for the offset modulation. This does not affect the primary conclusion about interference averaging in any significant way.


The calculations we're doing here pertain only to the second-order (i.e., mean and variance) of a single chip from the DSSS demodulator. The overall system performance depends on the coding that takes place outside the domain of the spreading-despreading operations. The forward link, for purposes of calculating symbol energy-to-noise ratios, is essentially repetition coded. That is, each FEC code symbol is repeated 64 times. The SNR of its detection statistic is approximately 64 times (18 dB greater than) the per-chip SNR because the 64 chips sum coherently, while the variances sum rms fashion. Subsequent to the soft decision detection statistic, one must consider the performance of the Viterbi decoder to ascertain the overall SNR (Eb/N0 performance. See, for example, Viterbi's CDMA book.

The reverse link is somewhat more difficult because the receiver must form 64 decision metrics from 256 chips, and the noise contributions to those 64 metrics are correlated. An analysis of this situation can be found (until we get around to writing it here) in Proakis, pp. 807-810.

Our purpose here is to show how the effects of multiple access interference and possible jamming and interference can be accounted for in our capacity calculations and link budgets by a simple calculation of an effective noise power spectral density. The unwanted signals can be modeled, as far as the communication link is concerned, like thermal noise. within the spreading bandwidth.If you want to see all the details ...

. . . Continue to BPSK Receiver Statistics

... or if you just want the main point, it's the interference averaging property.The primary result of all the mathematical gore is that the effect of mutual interference and jamming is, for most purposes, the same as an effective total noise level of
where N0 is the thermal noise level of the receiver, and P interf is the total in-band (within the spreading band of width W = 1.25 MHz) interference power.

This interference averaging property is the primary direct benefit of the use of CDMA.

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Copyright © 1996-1999 Arthur H. M. Ross, Ph.D., Limited