The power-spectrum sub-band energy ratio (PSER) has been applied in a variety of fields, but reports on its statistical properties and application in signal detection have been limited. Therefore, the statistical characteristics of the PSER were investigated and a signal detection method based on the PSER was created in this paper. By analyzing the probability and independence of power spectrum bins, as well as the relationship between F and beta distributions, we developed a probability distribution for the PSER. Our results showed that in a case of pure noise, the PSER follows beta distribution. In addition, the probability density function exhibited no relationship with the noise variance—only with the number of bins in the power spectrum. When Gaussian white noise was mixed with the signal, the resulting PSER followed a doubly non-central beta distribution. In this case, the probability density and cumulative distribution functions were represented by infinite double series. Under the constant false alarm strategy, we established a signal detector based on the PSER and derived the false alarm probability and detection probability of the PSER. The main advantage of this detector is that it did not need to estimate noise variance. Compared with time-domain energy detection and local spectral energy detection, we found that the PSER had better robustness under noise uncertainty. Finally, the results in the simulation and real signal showed that this detection method was valid.

In this paper, we present a novel blind signal detector based on the entropy of the power spectrum subband energy ratio (PSER), the detection performance of which is significantly better than that of the classical energy detector. This detector is a full power spectrum detection method, and does not require the noise variance or prior information about the signal to be detected. According to the analysis of the statistical characteristics of the power spectrum subband energy ratio, this paper proposes concepts such as interval probability, interval entropy, sample entropy, joint interval entropy, PSER entropy, and sample entropy variance. Based on the multinomial distribution, in this paper the formulas for calculating the PSER entropy and the variance of sample entropy in the case of pure noise are derived. Based on the mixture multinomial distribution, the formulas for calculating the PSER entropy and the variance of sample entropy in the case of the signals mixed with noise are also derived. Under the constant false alarm strategy, the detector based on the entropy of the power spectrum subband energy ratio is derived. The experimental results for the primary signal detection are consistent with the theoretical calculation results, which proves that the detection method is correct.

Since the cumulative distribution function(CDF) of the doubly non-central beta distribution can be represented by the infinite double series, therefore, truncating the summation of the series can obtain the approximate value of CDF. The existing numerical computation methods must specify the truncating range and cannot estimate the calculation error. In this paper, two new numerical computation methods based on the division on the infinite double series, named DIV1 and DIV2, are proposed. Both methods can perform automated calculations after configuring the error control lines, the truncating range does not need to be determined, and their computation times are comparable. After detailed derivation, the upper error bounds of both methods are obtained, which means that the accuracy of the two methods can be determined precisely. Therefore, this paper recommends two methods to calculate the CDF of the doubly non-central beta distribution.

The cumulative distribution function (CDF) of the doubly non-central beta distribution can be expressed as an infinite double series. By truncating the sum of this series, one can obtain an approximate value of the CDF. Although numerous methods exist for calculating the non-central beta distribution, which allow for the control of the truncation range and estimation of the computational error, no such methods have been developed for the doubly non-central beta distribution. In this paper, we propose two new numerical computation methods based on the segmentation of the infinite double series, termed DIV1 and DIV2. Both methods enable automated calculations once the error control parameters are set; there is no need to predetermine the truncation range, and their computational times are comparable. Following detailed derivations, we have established the upper bounds of the errors for both methods, thus ensuring the determinability of the precision.

The doubly non-central F distribution is a widely applied probability distribution. The cumulative distribution function (CDF) of the doubly non-central F distribution can be expressed using the infinite doubly series. In this paper, a CDF calculation method that allows for controlled precision is derived through the study of the infinite doubly series. By setting parameters, this method can provide an upper limit of error for the computed results, thereby allowing for precision control in the calculations. Experimental results demonstrate that this method achieves a good balance between computational precision and speed.