2.1. Principle of MCSS
Each symbol in the M-ary Spread Spectrum (MSS) group of pseudorandom codes can represent M bits and has a quantity of
(
). Cyclic Shift Keying (CSK) is a communication method in which the pseudonoise (PN) sequence is cyclic shifted and the shifted information is demodulated through the peak. Depending on the amount of cyclic shift, the bit that can be expressed is defined. For example, a cyclic shift of
is required to represent N bits [
25]. However, since demodulation is performed through the peak position, it is highly prone to Doppler spread. Both are much more efficient in transmitting data than the conventional direct sequence spread spectrum. Via MSS communication, we can express M bits, N bits with CSK communication, and (M + N) bits in a single symbol. As a result, we transmit MSS with
PN sequences and CSK with 1
PN sequence, resulting in
PN sequences. We created a very long
PN sequence (
, where
k is a very large constant) and then cut it to the appropriate length, then used it to generate the
PN sequence. The very long
PN sequence is expressed as:
where
is the
matrix and
L is the length of a
PN sequence used for one symbol. The first row in the
matrix is
and is expressed as
. As a result, the
matrix has the form [
].
One of the most significant advantages of MSS is that these multiple code sequences are suitable for communication with a low probability of intercept (LPI). Below,
Figure 1 is a four-by-four matrix. The image in the top left, for example, is autocorrelated with the first
PN sequence, and the figure to the right of it is a picture of cross-correlating the first
PN sequence with the second
PN sequence, and so on.
The proposed method represents M data in the MSS method and N data in the CSK method. A decimal representation of several bits that make up a single symbol is N and M. If three consecutive bits are of the form ‘101’, for example,
is 5. The nonshifted
PN sequence and the shifted
PN sequence are combined to form one symbol. The proposed method applies a
PN sequence cyclic shift method, but unlike the conventional CSK method, the MSS and cyclically shifted
PN sequences overlap, as shown in
Figure 2. The
PN sequence used for CSK is denoted by
, and the
PN sequence used for MSS is not cyclically shifted. As much as
of the cyclic shift is performed. Here,
(
) represents the minimum distance, and the reason for using the minimum distance is that the cyclic shift of one chip does not have discriminating power in demodulation with the peak-to-peak distance. As a result, multiplying by m generates and transmits an adequate distance. If m is large, however,
becomes very large and the orthogonality may be lost. As a result of comparing BER performance according to
in simulation, it was confirmed that the best performance was achieved when
.
The PN sequences for MSS and CSK are merged to create a single sign. An encoder is used to convert a binary number to a decimal number, and then the cyclically shifted PN sequence and the noncyclically shifted PN sequence are combined to create an MCSS signal. The proposed method is unique in that the receiver finds M through a cross-collation of PN sequences in one symbol and then cross-collations from the symbol to find as a result of using MCSS by synthesizing MSS and CSK. The main advantage of MCSS is that, unlike conventional CSK, the peak position in MSS and the peak position of the cross-collation can both be found within one symbol, making it more resistant to Doppler spread.
One symbol is modulated per sequence in conventional DSSS systems. The
i-th signal is as follows:
The
i-th transmitted symbol is
, and the spreading sequence is
. The cyclic shift matrix can be written as:
where
denotes the
identity matrix and
is the length of the
PN sequence. As shown in
Figure 2, cyclically shifted
and noncyclically shifted
are combined to form one symbol, where
[1, 2...,
].
represents M bits of information and
represents N bits of information. In the
i-th symbol
, M bits indicated by
are
, and N bits indicated by n are
. As a result, the
i-th symbol represents (M+N) bit information. The symbol
can be expressed as:
where the symbol
is an
L matrix and the passband is expressed as:
where
are the symbol and chip duration time,
is the carrier frequency, and
is the length of the symbol.
is the impulse response of the pulse-shaping filter. Equation (6) means the convolution between the pulse-shaping filter and the symbol. A raised-cosine filter was utilized with a roll factor of 0.5.
2.2. Estimation of the Approximate Doppler Shift Based on Two Peaks
The signal is compressed or extended as a result of Doppler distortion in the time domain. As a result of the Doppler’s influence, UWA signals are frequently modeled in the following way.
where
is the received signal and can be expressed as the sum of each symbol.
is the Doppler scale, which is defined as the ratio of the relative velocity between the source and receiver to the propagation wave velocity. The received signal with a Doppler scale (
) is equivalent to a scaling of the sampling period in a discrete-time processing system with interpolation.
The sampling frequency and the carrier frequency are both affected by the same ratio of Doppler spread and
indicates the sampling frequency. The underwater vehicle’s top speed is approximately 45 knots (12.86 m/s). The sound speed is 1500 m/s, with a range of
0.0085 to 0.0085. As a result, if
= 2.5 kHz, the Doppler frequency range is −21.4 Hz to 21.4 Hz. We want to find Doppler by the approximate sample difference of the chip in the sample area since the Doppler spread is highly unpredictable.
Here, indicates the number of samples that one chip has. The symbol rate (symbol per second, sps) of the transmission signal is 10 and one symbol includes 127 chips. As a result, the chip duration is 0.787 ms and each chip has 78 samples when the sampling frequency is set to 100 k. The number of samples that can be changed by Doppler ranges from −84 to 84 when 0.0085 to 0.0085 and one symbol has 9906 samples. Because there is a sample change greater than the number of samples on a single chip, a stable system can be achieved by setting the minimum distance m to 3 or more.
Signals such as Equation (11) are generated to determine the
sequence with the highest peak through MSS, and Equation (12) can be written as a reference signal to find the difference between the peaks of CSK as follows:
As shown in
Figure 3, the sequence with the largest amplitude can be found by correlating
, and the correlation result of
(
[1, 2..., Q]) is shown. It was also possible to check how much cyclic shift there was by correlating
in the same symbol. Each chip contains approximately 78 samples, as previously stated. The desired value of
can be calculated because the minimum distance
is determined when transmitting. The peak-to-peak distance in
Figure 3 differs by 1186 samples. Each chip has 78 samples and the cyclic shift is as much as the product of the minimum distance
and bit information
. As a result, 1186 divided by 78
equals
.
The peak-to-peak distance can be used to determine the data and Doppler.
and
are used to correlate the received signal, where
is the measured correlation. The formula for determining the position in the result with the highest value is as follows:
denotes the
i-th symbol transmitted and
indicates the position of the sample with the maximum value when correlated with
. When correlated with
, the position of the sample showing the maximum value is given as
. Doppler compensation is applied to the first symbol (when
) in the transmission signal design by adding a preamble signal in front of the payload, and the peak difference is determined using the premise that the peak floats in the middle due to Doppler compensation.
MSS has already demodulated
using Equation (13), and after demodulating
q, the maximum value of MSS is used as a reference and the peak of CSK is found to obtain
. Additionally, using the variables resulting from Equations (15) and (16), the Doppler scale (
) can be approximated using the following equation:
where
means rounding
to the nearest integer. Equation 16 considers that Doppler occurs in proportion to the number of samples shifted and calculates approximate Doppler based on that number. That is, the Doppler frequency is roughly estimated in proportion to the number of samples between the peaks in one symbol. Although the number of samples is insufficient to perfectly estimate the Doppler frequency, an approximate value can be estimated, which reduces the amount of computation when using a Doppler bank.
2.3. Symbol Synchronization and Doppler estimation with MSS
The method proposed in this paper is the calculation of the sample change at the peak position, the calculation of the approximate Doppler based on the change amount, and the calculation of
accurately within the range. Within the approximately calculated Doppler range (
~
), the step size is established and the (
) with the highest correlation result is found and compensated. It compensates
and compensates for the starting point of the next symbol as follows:
The number of samples per
i-th symbol is denoted by
. We can compensate for the start synchronization of the next symbol for each of the symbols by adjusting the number of samples after Doppler compensation [
26].
Following the MSS method, the step size is determined within the range of −
to +
using the previously determined
, and the largest value is found to compensate for precise Doppler, as shown in
Figure 4. The peak is then found using the CSK’s correlation in the Doppler-compensated symbol, and the value of n is determined by comparing it to the peak position of the
correlation. The derived
and n are then decoded into binary numbers.
In the conventional M-ary cyclic shift keying (MCSK) method, if sequences are used for MSS and N cyclic shifts are performed on the original MCSK signal, ( + ) bits can be expressed in one symbol. The MCSK method, on the other hand, creates a number of cases by cyclic shifting each MSS sequence, whereas the MCSS method proposed by us overlaps the MSS sequences and the CSK sequence so that the receiver can demodulate it using only MF. Furthermore, because MSS and CSK overlap, MSS serves as a reference for the midpoint of CSK’s cyclic shift, allowing the cyclic shift value to be demodulated more consistently.