A review on spectrum sensing for cognitive radio: challenges and solutions <s> Noise Power Uncertainty and Estimation <s> The standard analysis of the radiometric detectability of a spread-spectrum signal assumes a background of stationary, white Gaussian noise whose power spectral density can be measured very accurately. This assumption yields a fairly high probability of interception, even for signals of short duration. By explicitly considering the effect of uncertain knowledge of the noise power density, it is demonstrated that detection of these signals by a wideband radiometer can be considerably more difficult in practice than is indicated by the standard result. Worst-case performance bounds are provided as a function of input signal-to-noise ratio (SNR), time-bandwidth (TW) product and peak-to-peak noise uncertainty. The results are illustrated graphically for a number of situations of interest. It is also shown that asymptotically, as the TW product becomes large, the SNR required for detection becomes a function of noise uncertainty only and is independent of the detection parameters and the observation interval. > <s> BIB001 </s> A review on spectrum sensing for cognitive radio: challenges and solutions <s> Noise Power Uncertainty and Estimation <s> In this paper we consider the problem of detecting whether a frequency band is being used by a known primary user. We derive fundamental bounds on detection performance in low SNR in the presence of noise uncertainty - the noise is assumed to be white, but we know its distribution only to within a particular set. For clarity of analysis, we focus on primary transmissions that are BPSK-modulated random data without any pilot tones or training sequences. The results should all generalize to more general primary transmissions as long as no deterministic component is present. Specifically, we show that for every 'moment detector' there exists an SNR below which detection becomes impossible in the presence of noise uncertainty. In the neighborhood of that SNR wall, we show how the sample complexity of detection approaches infinity. We also show that if our radio has a finite dynamic range (upper and lower limits to the voltages we can quantize), then at low enough SNR, any detector can be rendered useless even under moderate noise uncertainty. <s> BIB002 </s> A review on spectrum sensing for cognitive radio: challenges and solutions <s> Noise Power Uncertainty and Estimation <s> Sensing (signal detection) is a fundamental problem in cognitive radio. In this paper, a new method is proposed based on the eigenvalues of the covariance matrix of the received signal. It is shown that the ratio of the maximum eigenvalue to the minimum eigenvalue can be used to detect the signal existence. Based on some latest random matrix theories (RMT), we can quantize the ratio and find the threshold. The probability of false alarm is also found by using the RMT. The proposed method overcomes the noise uncertainty difficulty while keeps the advantages of the energy detection. The method can be used for various sensing applications without knowledge of the signal, the channel and noise power. Simulations based on randomly generated signals and captured ATSC DTV signals are presented to verify the methods. <s> BIB003 </s> A review on spectrum sensing for cognitive radio: challenges and solutions <s> Noise Power Uncertainty and Estimation <s> Sensing (signal detection) is a fundamental problem in cognitive radio. The statistical covariances of signal and noise are usually different. In this paper, this property is used to differentiate signal from noise. The sample covariance matrix of the received signal is computed and transformed based on the receiving filter. Then two detection methods are proposed based on the transformed sample covariance matrix. One is the covariance absolute value (CAV) detection and the other is the covariance Frobenius norm (CFN) detection. Theoretical analysis and threshold setting for the algorithms are discussed. Both methods do not need any information of the signal, the channel and noise power as a priori. Simulations based on captured ATSC DTV signals are presented to verify the methods. <s> BIB004 </s> A review on spectrum sensing for cognitive radio: challenges and solutions <s> Noise Power Uncertainty and Estimation <s> Spectrum sensing is a fundamental component in a cognitive radio. In this paper, we propose new sensing methods based on the eigenvalues of the covariance matrix of signals received at the secondary users. In particular, two sensing algorithms are suggested, one is based on the ratio of the maximum eigenvalue to minimum eigenvalue; the other is based on the ratio of the average eigenvalue to minimum eigenvalue. Using some latest random matrix theories (RMT), we quantify the distributions of these ratios and derive the probabilities of false alarm and probabilities of detection for the proposed algorithms. We also find the thresholds of the methods for a given probability of false alarm. The proposed methods overcome the noise uncertainty problem, and can even perform better than the ideal energy detection when the signals to be detected are highly correlated. The methods can be used for various signal detection applications without requiring the knowledge of signal, channel and noise power. Simulations based on randomly generated signals, wireless microphone signals and captured ATSC DTV signals are presented to verify the effectiveness of the proposed methods. <s> BIB005
For many detection methods, the receiver noise power is assumed to be known a priori, in order to form the test statistic and/or set the test threshold. However, the noise power level may change over time, thus yielding the socalled noise uncertainty problem. There are two types of noise uncertainty: receiver device noise uncertainty and environment noise uncertainty. The receiver device noise uncertainty comes from BIB001  BIB002 : (a) nonlinearity of receiver components and (b) time-varying thermal noise in these components. The environment noise uncertainty is caused by transmissions of other users, either unintentionally or intentionally. Because of the noise uncertainty, in practice, it is very difficult to obtain the accurate noise power. Let the estimated noise power be σ 2 η = ασ 2 η , where α is called the noise uncertainty factor. The upper bound on α (in dB scale) is then defined as B = sup 10 log 10 α , where B is called the noise uncertainty bound. It is usually assumed that α in dB scale, that is, 10 log 10 α, is uniformly distributed in the interval [−B, B]  . In practice, the noise uncertainty bound of a receiving device is normally below 2 dB   , while the environment/interference noise uncertainty can be much larger  . When there is noise uncertainty, it is known that the energy detection is not effective BIB001  BIB002  . To resolve the noise uncertainty problem, we need to estimate the noise power in real time. For the multi-antenna case, if we know that the number of active primary signals, K, is smaller than M, the minimum eigenvalue of the sample covariance matrix can be a reasonable estimate of the noise power. If we further assume to know the difference M − K, the average of the M − K smallest eigenvalues can be used as a better estimate of the noise power. Accordingly, instead of comparing the test statistics with an assumed noise power, we can compare them with the estimated noise power from the sample covariance matrix. For example, we can compare T BCED and T ED with the minimum eigenvalue of the sample covariance matrix, resulting in the maximum to minimum eigenvalue (MME) detection and energy to minimum eigenvalue (EME) detection, respectively BIB003 BIB005 . These methods can also be used for the single-antenna case if signal samples are time-correlated BIB005 . Figures 1 and 2 show the Receiver Operating Characteristics (ROC) curves (P d versus P f a ) at SNR = −15 dB, N = 5000, M = 4, and K = 1. In Figure 1 , the source signal is i.i.d and the flat-fading channel is assumed, while in Figure 2 , the source signal is the wireless microphone signal   and the multipath fading channel (with eight independent taps of equal power) is assumed. For Figure 2 , in order to exploit the correlation of signal samples in both space and time, the received signal samples are stacked as in BIB004 . In both figures, "ED-x dB" means the energy detection with x-dB noise uncertainty. Note that both BCED and ED use the true noise power to set the test threshold, while MME and EME only use the estimated noise power as the minimum eigenvalue of the sample covariance matrix. It is observed that for both cases of i.i.d source (Figure 1 ) and correlated source (Figure 2) , BCED performs better than ED, and so does MME than EME. Comparing Figures 1 and 2 , we see that BCED and MME work better for correlated source signals, while the reverse is true for ED and EME. It is also observed that the performance of ED degrades dramatically when there is noise power uncertainty.