Digital Interference
Now let us consider a system with one spread-spectrum (SS)
signal and one narrowband binary signal in an otherwise additive white Gaussian
noise (AWGN) channel. We assume for now that the narrowband signal is
synchronized with the SS signal. Furthermore, we assume a relationship between
the data rates of the two users (i.e., m bits of
the narrowband user occur for each bit of the SS user). (Given the typical data
rates employed in many wireless systems, it is often reasonable to assume an
integer relationship between the bit rates. Situations in which this is not true
are considered in [57].) As shown in Fig. 7.11, the narrowband digital signal can be regarded
as m virtual users, each with its virtual
signature sequence. The first virtual user's signature sequence equals 1 during
the first narrowband user's bit interval (i.e., a virtual chip interval) and
zero everywhere else. Similarly, each other narrowband user's bit can be thought
of as a signal arising from a virtual user with a signature sequence with only
one nonzero entry. It is obvious from this construction that the signature
waveforms of the virtual users are orthogonal to each other. However, in
general, the kth virtual user has some
cross-correlation with the spread-spectrum user. If we use r to denote the vectors
formed by the cross-correlations, defined explicitly in (7.108), the cross-correlation matrix R of this virtual multiuser system has the
following simple structure (note that the SS user is numbered 0, and the m virtual users are numbered from 1 to m):
Equation 7.107
We have assumed that the narrowband user had a faster data rate
than the SS user (but this rate is still much slower than the chip rate). The
opposite case can also hold, and our analysis applies to it as well, although we
do not discuss that case explicitly. The covariance matrix of the system in that
case has the same structure as (7.107).
Let T be the bit duration of the
SS user, so that T/m is the bit duration of the
narrowband user. Similarly, let N be the
processing gain of the SS signal, so that the chip interval has length T/N. By our assumption that the interferer is
narrowband, we have N >> m. Let s(t) be the
normalized signature waveform of the SS user [i.e., s(t) is zero outside the interval [0,T] and has unity energy]. Similarly, let p(t) be the normalized bit waveform of the narrowband
user [i.e., p(t) is zero outside the interval
[0,T/m] and has unity energy]. Then the
normalized signature waveform of the kth virtual
user is pk(t) = p(t–(k–1)T/m). The
cross-correlation vector mentioned earlier is r = [r1 r2 ... rm]T,
where rk is
the cross-correlation between the kth virtual
user and the SS user, defined as
Equation 7.108
where the inner product notation denotes 
We assume that the SS user and the narrowband user are sending
digital data through the same channel characterized by AWGN with power spectral
density s2.
Let AI
be the received amplitude of the narrowband signal and A be the received amplitude of the SS signal. We use
the notation that the narrowband user data bits during the interval (0,T) are d1,
d2,..., dm, and the
SS bit is b. When the users are synchronous, it
is sufficient to consider the one-shot version of the received signal:
Equation 7.109
where n(t) is the white Gaussian
noise with power spectral density s2.
The linear MMSE detector for user 0 (i.e., the SS user) is
characterized by the impulse response w 
L2[0,
T], such that the decision on b0 is
Equation 7.110
A closed-form expression for w
is given by [520]
as
Equation 7.111
where wT = [w0, w1, ..., wm] is the
first row of the matrix
Equation 7.112
and A =
diag{A, AI, ... AI}.
Substituting (7.107) into (7.112), we have
Equation 7.113
The following matrix identity can be easily verified,
Equation 7.114
where g
= 1–rT r/ab.
Now on defining a = 1 + s2/A2 and b = 1 + s2/A12, the first row of C in (7.113) is then given by
Equation 7.115
Substituting (7.115)
into (7.111) we get an expression for
the linear MMSE detector for the SS user:
Equation 7.116
Using (7.74), the SINR
at the output of the linear MMSE detector w(t) becomes
Equation 7.117
That is, the SINR is the ratio of the desired SS signal power
to the sum of the powers due to narrowband interference and noise at the output
of the filter w(t). Substituting (7.116) into (7.117), we obtain
Equation 7.118
Figure 7.12 illustrates
the virtual multiuser system for the asynchronous case. Let t0 be the fixed time lag between the
spread-spectrum bit and the nearest previous start of a narrowband bit (i.e., 0
t0
T/m). We see that because of the time lag t0, the virtual user 1 in Fig. 7.11 effectively contributes two interference signals
during an SS bit interval: at the beginning and end of the SS bit interval,
respectively. We can therefore treat the asynchronous system as a synchronous
system with one additional virtual user (i.e., a synchronous system with one SS
user) and m + 1 virtual users. The preceding
analysis therefore holds in the asynchronous case as well, with only minor
modification.
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