# Gaussian-Distributed Spread-Spectrum for Covert Communications
## 2 DSSS Modulation  
DSSS, or direct-sequence spread-spectrum, is a method of modulation in which the message bits are modulated by a pseudorandom bit sequence known as a spreading sequence. This sequence has a much higher rate than the original information rate, and a spreading factor \(N\) determines the number of spreading bits that map to a message bit. After spreading, the symbols to be transmitted are commonly referred to as chips. When a DSSS system maintains the same bit rate and energy per bit as before spreading, the signal bandwidth will be spread by a factor of \(N\), and the magnitude of the PSD of the signal will be reduced by a factor of \(N\). This reduction in PSD helps to mitigate interference from other signals. To spread the signal, the data are multiplied by a pseudo-noise (PN) sequence. This PN spreading sequence is unique to each transmitter and receiver pair and helps to ensure secure communication. A binary phase shift keying (BPSK) DSSS spreading process is illustrated in Figure 1.  
DSSS is an important technology for covert communications because it enables the system to operate under the thermal noise floor even at low SNR levels (i.e., much lower than 0 dB SNR). This means that the communication signal can be hidden in the noise, making it difficult for adversaries to detect and intercept the signal. The spreading process in DSSS spreads the signal across a wide bandwidth, which reduces the PSD of the signal. As a result, the signal is less susceptible to noise and interference, making it easier to detect at low SNR levels. This is particularly important for covert communications, where it is essential to maintain a low profile and avoid unfriendly detection.  
The bit error rate (BER) performance of a DSSS system using quadrature phase shift keying (QPSK) spreading is given by  
\[\text{BER}=0.5\:\text{erfc}\Big{(}(\sqrt{N(E_{s}/N_{0})/2}\Big{)} \tag{1}\]  
Figure 1: Spreading process of BPSK DSSS with repeating short PN sequence and with a long PN sequence.  
where erfc() is the complementary error function and \(E_{s}/N_{0}\) is the energy per channel symbol to noise power spectral density ratio. This measure will also be referred to as signal-to-noise ratio (SNR) in this paper.  
Examples for various spreading factors are shown in Figure 2. Figure 2 shows that the non-spread system achieves a target BER of \(10^{-5}\) at 12.5 dB, whereas the DSSS system with a spreading factor of 256 achieves the same BER at -11.5 dB. It is further shown that as the spreading length increases, so does the system's ability to operate deeper in the noise floor. However, an increase of the spreading factor also causes a reduction in the bit rate if the system bandwidth is maintained fixed.  
The standard DSSS system has several vulnerabilities and may not be ideal for covert communication. This system normally uses repeating patterns and has deterministic features that could be used for detection/interception. For example, Figure 3 shows the constellation of a typical DSSS-PN system with QPSK spreading. The points are color coded for future reference. As the amplitude and phase of the constellation points are fixed, they present deterministic features to the signal, making it easier for an adversary to detect the existence of the signal and possibly intercept the message using advanced signal processing methods [21]. The autocorrelation function (ACF) can be used to measure repeatable patterns of the PN sequences in a DSSS system. Some DSSS systems tackle the issue of predictable PN sequences by using a shared secret key in order to randomly generate the PN sequences [38]. However, implementation of such architectures for covert communication is difficult [39].  
Figure 2: Theoretical BER performance of a DSSS-QPSK system with various spreading factors.