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Explain Pseudorandom spreading sequences with equation.
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Solution:

Pseudorandom spreading sequences A common approach is to employ spreading sequences that are either aperiodic or periodic with periods much longer than the processing gain N.

In this case, the section of the spreading sequence corresponding to a single symbol is well modeled as consisting of randomly chosen, i.i.d., elements (e.g., $\pm 1$ with equal probability for binary spreading sequences).

We term this the random spreading sequence model. A convenient way to generate sequences of very long periods is to use maximum length shift register (MLSR) sequences: a shift register of length m, together with some combinatorial logic, can be used to generate a periodic sequence of period $2^m-1$ (the IS-95 digital cellular standard employs m=42, which generates a sequence of period $2^{42}-1$ in excess of 4 billion).

Such sequences are often termed pseudorandom, or pseudo-noise ( $\mathrm{PN}$ ) sequences. See Problem 8.19 for a simple example of an MLSR sequence.

The CDMA-based digital cellular standard IS-95 employs long spreading waveforms, with the spreading waveforms for multiple users generated from a single PN sequence of very long periods simply by assigning differently delayed versions of the sequence to different users.

For the random spreading sequence model, the crosscorrelation function between two pseudorandom sequences u and v of length N can be modeled as follows:

$R_{u, v}[n]=\sum_l u[l] v^*[l-n]=\sum_{l=n}^{N-1} u[l] v^*[l-n] .\\$

When $\{u[l]\},\{v[l]\}$ are modeled as i.i.d., symmetric Bernoulli, $R_{u, v}[n]$ can be modeled as a sum of N-n i.i.d., symmetric Bernoulli random variables.

Note that PN sequences of the shorter periods (e.g., period 31 corresponding to m=5 ), and sequences constructed from PN sequences such as Gold sequences, can be useful for CDMA systems employing short spreading waveforms.