Correlator () cos(o. k Threshold sin(o k) ∑() Channel Correlator FIGURE 73.2 Envelope detector for a noncoherent signal in additive white Gaussian noise. signal ag(hence it is"matched"to the signal), and announcing the signals presence if the filter output exceeds the decision threshold at any time Detection of random signals In some applications, particularly in remote sensing applications such as sonar and radio astronomy, it is appropriate to consider the signal sequence S,, S2,.,Sn itself to be a random sequence, statistically independent of the noise. In such cases, the likelihood ratio formula of (73.6) is still valid with the parameter vector 0 simply taken to be the signal itself. However, for long measurement records (i.e, large n),(73.6)is not a very practical formula except in some specific cases, the most important of which is the case in which the signal is Gaussian. In particular, if the signal is Gaussian with zero-mean and autocorrelation sequence n eE[,I, then the likelihood ratio is a monotonically increasing function of the quantity ∑∑ (73.13) with ak the element in the kth row and hh column of the positive-definite matrix QAl-(I+R/0) (73.14) where I denotes the n X n identity matrix, and R is the covariance matrix of the signal, i.e., it is the n X n matrix with elements ru Note that(73. 13)is a quadratic function of the measurements; thus, a detector based on the comparison of this quantity to a threshold is known as a quadratic detector. The simplest form of this detector results from the situation in which the signal samples are, like the noise samples, i i d. In this case, the quadratic function (73. 13)reduces to a positive constant multiple of the quantity Yk2 e 2000 by CRC Press LLC© 2000 by CRC Press LLC signal {ak} (hence it is “matched’’ to the signal), and announcing the signal’s presence if the filter output exceeds the decision threshold at any time. Detection of Random Signals In some applications, particularly in remote sensing applications such as sonar and radio astronomy, it is appropriate to consider the signal sequence S1 , S2 , . . ., Sn itself to be a random sequence, statistically independent of the noise. In such cases, the likelihood ratio formula of (73.6) is still valid with the parameter vector u simply taken to be the signal itself. However, for long measurement records (i.e., large n), (73.6) is not a very practical formula except in some specific cases, the most important of which is the case in which the signal is Gaussian. In particular, if the signal is Gaussian with zero-mean and autocorrelation sequence rk,l = D E{SkSl}, then the likelihood ratio is a monotonically increasing function of the quantity (73.13) with qk,l the element in the kth row and lth column of the positive-definite matrix (73.14) where I denotes the n 3 n identity matrix, and R is the covariance matrix of the signal, i.e., it is the n 3 n matrix with elements rk,l . Note that (73.13) is a quadratic function of the measurements; thus, a detector based on the comparison of this quantity to a threshold is known as a quadratic detector. The simplest form of this detector results from the situation in which the signal samples are, like the noise samples, i.i.d. In this case, the quadratic function (73.13) reduces to a positive constant multiple of the quantity (73.15) FIGURE 73.2 Envelope detector for a noncoherent signal in additive white Gaussian noise. qk l YkYl l n k n , = = Â Â 1 1 Q D I - + I R - ( /s ) 2 1 Yk k n 2 =1 Â