For the conventional TH/DS system using N-ary BPPM in AWGN channel, to detect the i th data symbol of the desired user , the received signal is correlated with orthogonal template waveforms to obtain decision statistics as follows [3]:
(7)
The template waveform of the m th correlator is given by
(8)
where is the correlator output of the desired transmitted signal. Iconv is the correlator output coming from other K - 1 users' signals and is called MAI. The MAI can be described as , where is the MAI caused by the k th user. nconv is a Gaussian random variable with zero mean and variance . Based on the maximum likelihood decision rule for AWGN channel [5], the receiver of the desired user computes a bank of M correlators' outputs, in (7), and then chooses the index corresponding to the largest absolute value of the correlator's output as the estimate of the message symbol :
(9)
as well as
(10)
Consider the receiver structure of the proposed TH/DS system shown in Figure 3. The output of the m th correlator in the j th frame duration is
(11)
where nj,prois i.i.d. Gaussian noise with zero mean and variance . The template waveform of the m th correlator is expressed as
(12)
where is the th correlator's output; is the desired component corresponding to the data symbol , and where is the MAI from the k th user. Completing the combining process, the m th decision variable can then be acquired as
(13)
where npro is Gaussian random noise with zero mean and variance . is the desired component corresponding to the data symbol , and Ipro is the total MAI caused by the K - 1 interfering users,
(14)
Let τ
k
= α
k
T
f
+ Δ
k
and Δ
k
= β
k
T
p
+ γ
k
, where α
k
is the discrete uniformly distributed r.v. in {0, 1, ..., N
s
- 1}; β
k
is the discrete uniformly distributed r.v. in {0, 1, ..., M - 1}; and γ
k
is the continuous uniformly distributed r.v. in one pulse duration, i.e., 0 ≤ γ
k
< T
p
[4]. Hence, we can obtain as follows:
(15)
and can be rewritten as
(16)
where and . and are the discrete uniformly distributed r.v.s in {-1, +1} because the polarity codes a(k)and the message symbol n(k)of the user k are assumed to be random and equally likely. Γ
j
(Δ
k
) and can be expressed as follows:
(17)
(18)
where , and . The i th data symbol of the k th user is assumed to be an uniformly distributed r.v. in the range of and each element of the random TH code utilized by the k th user in the j th frame period is assumed to be an uniformly distributed r.v. with . Therefore, is an uniformly distributed r.v. with . The probability density function (PDF) of conditioned on γ
k
can be described as
(19)
where δD is the Dirac delta function. Therefore, the conditional CF of can be obtained as obtained as
(20)
The interferences are independent of each other because each element of the user's TH code c(k)is randomly and independently selected from the set G. Hence,
(21)
and we then obtain
(22)
(23)
It is worthy to note that the CF of the MAI component for each correlator's output of the proposed TH/DS receiver is di¤erent from that of the conventional TH/DS system which has been shown in [3].
3.1 Symbol error rate
Let the i th data symbol of the desired user be . According to our derived CF of the MAI component for the correlator's output of the proposed TH/DS receiver in (23), the average SER of the N-ary biorthogonal modulation has been expressed and calculated as [5]:
The decision statistics of the combining correlators' outputs are assumed to be independent [2, 3, 17–21]. Hence, the SER of the proposed system is
(24)
where is the CDF of r1. The first decision variable is r0 = A1N
s
R(0) + I + n, and the other M - 1 decision variables are . Therefore, (24) can be rewritten as
(25)
As the MAI and AWGN are assumed to be mutually independent, we obtain the CF of r1 as , where the CF of the AWGN is . Hence, the PDF of r1 can be acquired as
(26)
Applying the relationship between the CF and CDF [12–14], we have
(27)
If all M - 1 erroneous symbols are equally likely chosen, then the corresponding BER is P
b
= M · P
e
/[2 · (M - 1)] [5].