Energy detection of a signal with random amplitude

I INTRODUCTION • In the classic paper, Urkowitz [1] discussed the detection of a deterministic signal of unknown structure in the presence of flat, band-limited Gaussian noise of known power density. The decision statistic is shown to have a noncentral chi-square distribution. As a result of atmosphere turbulence, multichannel wave propagation, and other reasons, received communication signals often have a random amplitude. In this case the distribution of energy decision statistic is not chi-square distribution [2]. • This paper is aimed at determined the probability of energy detection of a received communication signal with a various random amplitude. II DECISION STATIISTIC DISTRIBUTION • As in the classical paper [1], we assume that the carrier frequency f 0 , the spectrum width ∆f of the detected signal, and the spectral power density N 0 of the noise are known. The processed random signal ()()() xtstnt ι =+, (1) where ι = 0, 1, is a mixture of the detected signal () { } 0 ()Re()exp2 stAUtift πϕ =+   (2) and white noise n(t) or only white noise n(t) in accordance with the hypotheses H 1 or H 0 , respectively. In (2) A is the random amplitude, U(t) is the normalized deterministic complex envelope, and ϕ is the random initial phase. The a priori unknown binary parameter ι that is used in (1) takes the values 0 or 1, which coincide with the hypothesis subscript. • If the signal detected is present (hypothesis H 1) the distribution of decision statistic depends on distribution of signal amplitude A. It isn't difficult to show that in the case of deterministic amplitude the decision statistic has noncentral chi-square distribution [1]. In the case of random amplitude the decision-statistic distribution can differ from the noncentral chi-square distribution. But the noncentral chi-square distribution may be used as a conditional (under fixed A) distribution of the output signal from the energy receiver. • Consequently, the characteristic function Θ(jη) of the decision statistic Ξ can be written in the form () 22 /2 0 (12)exp() 12 jAq jjWAdA j µ η ηη η ∞ −   Θ=−  −   ∫ , (3) where µ = 2(T∆f + 1) is a number of degrees of freedom, 2 2 0 0 2 () T qUtdt N = ∫ , (4) and W(A) is a probability density function (pdf) of the random amplitude A. III RALEIGH …