Figure 1 presents the block scheme of a communication system for physical layer using Offset Quadrature Phase Shift Keying (OQPSK) modulation and spreading of the signals as used in wireless sensor networks. When only Gaussian noise is present in the channel the fading coefficient is set to be one,
i.e.,
α = 1. In order to develop theoretical model of the whole system with
N correlators, the analysis will start with the case when the symbols are generating at the source output in binary form and detected at the receiver side using one correlator. Generally, for the system with
N-correlators, the source generates message bits which are converted into symbols
bjn(
k). To each symbol a spreading sequences
cin(
k) is assigned to obtain a chip sequence
m(
k) =
bjn(
k)
cin(
k) that comes to the input of a multiplexer. If a symbol is in binary form,
bjn(
k) = ±1, then the system represents a direct sequence spread spectrum system (DSSSS) and the receiver has one correlator as it was said before. If a combination of
K bits represents a symbol then the number of sequences sent is
N = 2
K, which is equal to the number of required correlators at the receiver side. Thus, in the following sections, two cases, a single and
N-correlator receiver, will be analyzed.
2.1. Single-Correlator Receiver
In this case the spreader is represented by a multiplier, as shown on the left hand upper side of
Figure 1, and a single correlator with output
w1 at the receiver side. The chip sequence
m(
k) is split into in-phase and quadrature sequences using the demultiplexer block (DEMUX) in such a way that the even-indexed chip sequence
mI(
k) modulates the in-phase carrier
and odd-indexed chip sequence
mQ(
k) modulates the quadrature carrier
, where
Ec is the energy per chip and
M is the number of interpolated samples contained in one chip interval. Therefore, the transmitted signal can be defined as
where
mI(
k) and
mQ(
k) are in-phase and quadrature chip sequences expressed in discrete time domain respectively and
Ωc is the frequency of the carrier. It is important to note that
k is a discrete time variable.
The noise is to be generated at discrete time instants defined by
k. Thus, the band-limited pass-band noise is expressed as
where the energy of the noise samples inside a chip interval is
EN =
N0/2 and
nI and
nQ are in-phase and quadrature noise samples of zero mean and unit variance. The block schematic of this noise generator is presented inside
Figure 1. The noise is expressed in this form to comply with the applied signal processing demodulation procedure of both the signal and noise at the receiver side. Namely, in simulation, it is important to achieve that the power of the noise in respect to the power of the signals are controlled at all times at the transmitter side.
Figure 1.
Block schematic of communication system.
Figure 1.
Block schematic of communication system.
The received noisy signal
sR(
k)=
s(
k) =
n(
k) is demodulated using a correlator that consists of a multiplier and an adder. It is sufficient to present the processing of the signal in one branch of the demodulator because the processing in both branches is equivalent. The signal at output of the receiver multiplier is
The samples of this signal are added in the chip interval (corresponds to integration in continuous time systems). Because, in the case of a single-correlator receiver, the source generates binary signal the output of the transmitting spreader was the first spreading sequence
ci1(
k),
i.e.,
m(
k) =
ci1(
k), and a random sample
zi of a random process
Zi in
I branch is obtained,
i.e.,
and in
Q branch as
A multiplexer (MUX) is used to combine in-phase and quadrature sequences back into a 2
β-chip sequence
zi
where
ni are samples of the in-phase and quadrature baseband noise having zero mean and unit variance. In the correlator block, a locally generated reference chip sequences (
ci1,
i 1,2,3,...,2
β) is multiplied with the incoming
zi random sequence and then the products are added inside the bit interval. The resulting sum for the first positive message bit sent is
This value is a random sample of a random variable defined for the first bit received. If the source generates binary bits from the alphabet ±1, the threshold value in the decision circuit needs to be set to zero, and the optimum decision need to be made according to this rule
Due to the central limit theorem (CLT) the random variable
w1 can be approximated by the Gaussian random variable, with its mean
Due to the statistical independence of the noise and the spreading sequence the second term in Equation (9) is zero. Assuming that the powers of all chips are identical and equal to
Pc, we may have
The related variance is
Due to the statistical independence of noise variables, the second term is zero. The first and third terms are
The powers for all chips are equal and the power of noise is equal to the noise variance. Therefore, the variance of
w1 can be expressed in this general form
According to the CLT the density function of variable
w1 is Gaussian and can be expressed as
Then, the probability of error can be calculated according to this expression
By inserting Equations (10) and (13) into (14) we may find the expression for the probability of bit error in this closed form
which can be simplified to
where expression
Ψ =
represents random variability of the spreading sequence. The energy of the noise is equivalent to the power spectral density of the two sided noise spectrum. As it was said before, this system is analyzed assuming that the signals are generated in discrete time domain. Each chip and related noise sample are generated once for each chip interval and then repeated (interpolated)
M times in that chip interval to allow the discrete time carrier modulation. For
M repeated samples of noise in a chip interval the energy is
EN =
Mσ
2. For variance σ
2 =
BN0 and the bandwidth
B=1/2
Tc=1/2
M, the energy is calculated to be
EN =
N0/2. If the source generates binary bits and the spreading sequence is in binary form, we may have
where the energy of a bit is related to the energy of a chip as
Eb = 2
βEc.
2.2. N-Correlator Receiver
In this case, the source can generate
K-bit symbols [
1]. Thus, the number of required sequences will be
N = 2
K. The modulation and demodulation will be the same as in the case of a single-correlator receiver. However, the correlation for each sequence must be performed in its own receiver correlator. Therefore, at the receiver, a bank of
N correlators needs to be implemented, as shown in
Figure 1.
The receiver for binary symbols transmission is analyzed in previous section. In that case it was sufficient to have the first correlator in the receiver and the symbol to sequence conversion is performed by a multiplier as shown separately in
Figure 1. For that case the random variable
w1 at the output of the first correlator is calculated in Equation (7) and the related mean, variance and probability of error in Equations (9), (12) and (16).
If the source generates
N symbols, which correspond to combinations of message bits, the receiver will use all
N correlators in
Figure 1. The decision circuit now is a comparator of correlator outputs. The order of these outputs corresponds to the order of the spreading sequences that could be sent at the transmitter side. The decision circuit decides according to this decision rule: the sequence/symbol received corresponds to the maximum output value of the decision circuit. This, if the maximum is at the output three the decision sequence will say the third symbol was sent.
Suppose that the first sequence is sent. Following the procedure of a single-correlator receiver modeling, the output of the
n-th correlator can be expressed as
The first term is inter-sequence interference and the second term is noise term. The value
wn is a random sample of a random variable defined for the
n-th correlator output. Due to the central limit theorem (CLT) this random variable can be approximated by the Gaussian random variable with zero mean
and variance
In the case of binary chip transmission this variance is
The density function of this random variable can be approximated by Gaussian density expressed as
Suppose the threshold value inside the decision circuit is
w1 =
w. The probability of correct decision is equal to the conditional probability of that all outputs are less than this threshold value for the given value
w1 =
w,
i.e.,
Inserting Equation (21), and having in mind statistical independence between variables
wn, this expression becomes
This expression can be simplified, by applying the error complementary function and binomial theorem, and expressed in this form
This expression gives one for
i = 0. If (−1) is taken out of the brackets of
erfc, the expression becomes
The mean value of this random function over all
w1 values is the probability of that the first symbol is correctly transmitted. It can be expressed as
The probability of symbol error is
If one symbol is represented by
K bits then the number of spreading sentences is
N = 2
K. By inserting Equation (21) the bit error probability can be calculated as