In wireless communications, the bit error rate (BER) is an important metric which is used to gauge and compare the system performance. Since this noise modulated covert communications system is a new architecture, the theoretical BER performance in an additive white Gaussian noise channel is derived and compared with simulation results in this section. Unlike other single-channel spread spectrum systems, the low-pass equivalent model can directly be used to model the system behavior in the Gaussian channel. The spreading and dispreading process of our system is accomplished at the RF front-end. The noise floor at the antenna output is not the same as that at the output of the first mixer, and the noise terms within the system are generated by mixing of two zero mean independent Gaussian random variables. Thus, the system behavior needs to be modeled based upon the relationship between the SNR at the output of receiver antenna and the probability of bit error. In this section, we will demonstrate that the mixed noise can be approximated as Gaussian after passing through a narrow-band filter, and the BER equation can be expressed using the *Q*-function. The bandwidths of the signal, antenna, low-pass filter, and the SNR at the output of receiver's antenna are the parameters which dominate the BER when the bit rate is fixed.

To simplify the analysis, we assume that the delay term is set to zero in both the transmitter and the receiver. This simplification will not affect the BER analysis. In an additive white Gaussian noise channel, the actual received signal from the vertically polarized antenna and the horizontally polarized antenna can be written, respectively, as

The and terms are independent zero-mean band-limited Gaussian noise in the vertical and horizontal polarization channels, and these terms are also independent of and . Their analytical forms are similar to as shown in (1), that is,

where and are the polarization dependent random Rayleigh-distributed amplitude and uniformly-distributed phase terms, respectively. The power of and is equal to their variance since they are zero-mean random variables and these are denoted as and , respectively. We further assume that the powers of and , both of which are zero-mean band-limited Gaussian processes, are the same, and each is denoted as . The corresponding SNR values at the output of vertical and horizontal polarized antennas are and respectively, and are denoted as and . In reality, the bandwidth of is slightly greater than that of due to the modulation induced on it. However, the bandwidth of is very small compared with . We assume that the signal bandwidth of and (hence the bandwidth of and ) is , and that the bandwidth of and is (equal to the receive antenna bandwidth). Usually, is almost the same as in order to avoid receiving additional interference.

Down the receiver chain, the noisy signals and are mixed together, and the mixed signal contains the desired signal term (first term below) and three interference cross-terms given by

In the real system implementation, the bandpass filter is used to capture just the sum frequency signal centered at (3 GHz) containing the information message, while discarding all difference frequency signals contained in is discarded as noise. Let denote the bandpass filtered output of the signal The bandpass filtered noise signals are denoted as , and , where , and . Generally, the probability density function of the noise needs to be found in order to calculate the BER. Since the probability density function of the product of two independent zero-mean normal distributions is approximated by a modified Bessel function of the second kind, the closed form probability density function for the sum is extremely difficult to derive. Because the bandwidth of filtered noise is much smaller than before filtering, the noise spectrum following the filter is relatively flat compared to the sum frequency noise. Thus, we can approximate the filtered noise as a Gaussian variable. For convenience, we assume that the bandwidth of the bandpass filter is twice that of the low-pass filter following the second down-conversion, since the low-pass filter is the key component dominating the received noise spectrum before the decision circuit. Later in this section, we will compare the theoretical results with simulation results to show that our derivation by applying this assumption also works when the bandwidth of bandpass filter is much greater than bandwidth of low-pass filter.

Based on our simulation analysis, a cumulative distribution function comparison between (a representative interference term) and a zero-mean band-limited Gaussian with the same power and frequency range is shown in Figure 4. In the simulation, the bandwidth of bandpass filter is 40 MHz ( MHz), the bandwidth of signal is 970 MHz, and the bandwidth of the channel noise is 980 MHz. We note that the two cumulative distribution function plots are very close. Thus, these results validate our assumption that the filtered sum frequency noise terms can be approximated as Gaussian.

After realizing that the filtered noise terms can be approximated as Gaussians, their means and variances need to be found for calculating the BER. The mean value of is found as zero, as seen from

where is impulse response of bandpass filter [19]. Similarly, the mean values of and are both zero.

The next step is to calculate the variance of the filtered noise, which is equal to its power. Clearly, the power of , and can be calculated by integrating the power spectrum of the sum frequency noise of , and within the bandpass filter frequency range.

Let the power spectral density of the sum frequency noise of be denoted as . The average power of the sum frequency noise needs to be found first in order to find the mathematical expression for . We know that for a given ergodic random process , its autocorrelation function and its power spectral density form a Fourier transform pair, that is, . Furthermore, the average power of such a random process is the value of the autocorrelation function at zero lag, that is, equal to .

The sum frequency noise of , noting that , can be expressed as

The average power of can be determined from its autocorrelation function with the lag set equal to zero and can be expressed as

Recognizing that and are independent Rayleigh distributed random variables. Furthermore, the *k* th moment of a Rayleigh distributed random variable is noted as [19]

where is the mean. For , that is, , we have and . We therefore have

Thus, the value of the corresponding power spectral density of the sum frequency noise integrated over frequency is . Since the sum frequency noise is the product of two band-limited rectangular spectra centered at with bandwidths and (), respectively, has an isosceles triangle shape centered also at with an overall bandwidth equal to . Therefore, can be expressed as

The power of contained within the low-pass filter bandwidth can be finally found from

where is given by

In a similar manner, and can be derived as and , respectively, where is given by

The summation of , , and , representing the total interference component, is also a zero-mean band-limited Gaussian random variable and we denote it as . The variance of is equal to its average power and is given by

Since , and are uncorrelated zero-mean Gaussian distributions, the covariance terms are zero, and therefore, the interference power is obtained as

The term is mixed with the 3-GHz carrier and down to the baseband with a power that is equal to . Since the baseband noise is zero-mean Gaussian and binary modulation is used, the BER equation for the optimal receiver can be expressed by the *Q*-function with two parameters: the spectrum magnitude of the noise and the bit energy [20, 21].

From (7), when there is no low-pass filter truncating the signal spectrum, the average power of received baseband signal can be found using the fourth moment of and is shown to be

Since the term in (7) will spread out the baseband signal power over a frequency range wider than the low-pass filter bandwidth, the low-pass filter at the receiver will truncate the signal spectrum, and the received power will be lower than the value obtained in (22). Therefore, the bit energy at the output of low-pass filter can be expressed as when bit duration time is . The is the power loss factor due to the filtering, defined as the ratio between the truncated baseband signal power after the low-pass filter to the untruncated baseband signal. Clearly, the loss factor satisfies . From above discussion, the BER of the noise modulated covert communication system with a two-sided spectrum can be mathematically expressed as

The well-known function is shown below for reference as

Equation (23) can be also expressed using and as follows:

A full system simulation in an additive white Gaussian noise channel was done to validate the theoretical results in (25), and the results are shown in Figures 5 and 6. In the simulation, both the and the terms are equal, and the bandwidth of the antenna is 10 MHz wider than the bandwidth of the transmitted signal in order to avoid truncation of the wider spectrum caused by the modulation. The bandpass filter has a bandwidth of 100 MHz and is centered at 3 GHz. In Figure 5, a low-pass filter bandwidth of 10 MHz is used for the simulation. The value of depends on the bit rate and the low-pass filter bandwidth. From our independent simulation result, for a bit rate of 5 Mbps, the value of was determined to be approximately 0.487 when the transmitted signal bandwidth is 970 MHz and approximately 0.5 when the transmitted signal bandwidth is 500 MHz. In Figure 6, the low-pass filter bandwidth is 20 MHz, and the signal bandwidth is 970 MHz bandwidth in the simulation. The value of was determined to be 0.49, 0.5, and 0.518 when the bit rate is 10 Mbps, 5 Mbps, and 2 Mbps, respectively. From Figures 5 and 6, we note that the maximum deviation between the simulation results and theoretical results is 0.5 dB. Thus, the system behavior of this ultrawideband communication system is properly modeled. As the bandwidth of and is increased, the noise power will be dispersed into larger frequency ranges after the mixing process, and the system performance will improve because the processing gain will increase.