15)
The second part of the despreading operation occurs after the integration over the symbol or bit
time (Tb), resulting in the following despreader output signal (assuming perfect time synchronization,
):
(7.16)
Since the BPSK symbol is constant over the duration of the integration, it can be moved outside the
integration. We have assumed the absence of a channel.
(7.17)
What we notice is that the first integration on the RHS is the autocorrelation function evaluated at a
lag of zero. This was due to an earlier assumption of perfect time synchronization. The second summation
on the RHS is the cross-correlation of the two PN sequences evaluated at a lag of zero. Hence
we see the importance of the properties of the PN codes used in the system. The resulting signal contains
the desired plus interference plus noise components:
(7.18)
Notice, ideally, the second term would be zero if the PN codes were chosen to satisfy the orthogonality
principle.
7.3.2 Uplink Spreading Example
Let us consider the case when each user had a shifted version of the same PN sequence. For sake of
discussion, we have chosen User 2 to be spread with C1(t) but shifted in time by half the PN code
period (see Fig. 7.12). A major point to consider here is that this time difference should be larger than
the expected maximum delay spread of the channel.
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