There are several basic signal detection structures that can be derived from Eqs.(73. 1)to(73.3)under the assumption that the noise sequence consists of a set of independent and identically distributed (i.i.d. )Gaussian random variables with zero means Such a sequence is known as discrete-time white Gaussian noise. Thus, until further notice, we will make this assumption about the noise. It should be noted that this assumption is physically justifiable in many applications Detection of Known Signals If the signal sequence S,,S2,..,Sn is known to be given by a specific sequence, say SI, s2,..,s,(a situation known as coherent detection), then the likelihood ratio(73.3)is given in the white Gaussian noise case by exp (73.4 where o is the variance of the noise samples. The only part of (73. 4 )that depends on the measurements is the erm EkalskYk and the likelihood ratio is a monotonically increasing function of this quantity. Thus, optimum detection of a coherent signal can be accomplished via a correlation detector, which operates by comparing the quantity kYk (73.5) to a threshold, announcing signal presence when the threshold is exceeded. Note that this detector works on the principle that the signal will correlate well with itself, yielding a value of (73.5)when present, whereas the random noise will tend to average out in the sum(73.5), yiel relatively small value when the signal is absent. This detector is illustrated in Fig. 73.1 ∑() Comparison k=1 FIGURE 73. 1 Correlation detector for a coherent signal in additive white Gaussian noise. Detection of Parametrized Signals The correlation detector cannot usually be used directly unless the signal is known exactly. If, alternatively, the ignal is known up to a short vector e of random parameters(such as frequencies or phases)that are independent of the noise, then an optimum test can be implemented by threshold comparison of the quantity 5(6)y_1 ∑ (6)|/}p()de where we have written Sk=S(0)to indicate the functional dependence of the signal on the parameters, and where A and p(0)denote the range and probability density function, respectively, of the parameters The most important example of such a parametrized signal is that in which the signal is a modulated sinusoid with random phase; i.e e 2000 by CRC Press LLC© 2000 by CRC Press LLC There are several basic signal detection structures that can be derived from Eqs. (73.1) to (73.3) under the assumption that the noise sequence consists of a set of independent and identically distributed (i.i.d.) Gaussian random variables with zero means. Such a sequence is known as discrete-time white Gaussian noise. Thus, until further notice, we will make this assumption about the noise. It should be noted that this assumption is physically justifiable in many applications. Detection of Known Signals If the signal sequence S1, S2, . . ., Sn is known to be given by a specific sequence, say s1, s2 , . . ., sn (a situation known as coherent detection), then the likelihood ratio (73.3) is given in the white Gaussian noise case by (73.4) where s2 is the variance of the noise samples. The only part of (73.4) that depends on the measurements is the term S n k =1skYk and the likelihood ratio is a monotonically increasing function of this quantity. Thus, optimum detection of a coherent signal can be accomplished via a correlation detector, which operates by comparing the quantity (73.5) to a threshold, announcing signal presence when the threshold is exceeded. Note that this detector works on the principle that the signal will correlate well with itself, yielding a large value of (73.5) when present, whereas the random noise will tend to average out in the sum (73.5), yielding a relatively small value when the signal is absent. This detector is illustrated in Fig. 73.1. Detection of Parametrized Signals The correlation detector cannot usually be used directly unless the signal is known exactly. If, alternatively, the signal is known up to a short vector u of random parameters (such as frequencies or phases) that are independent of the noise, then an optimum test can be implemented by threshold comparison of the quantity (73.6) where we have written Sk = sk (u) to indicate the functional dependence of the signal on the parameters, and where L and p(u) denote the range and probability density function, respectively, of the parameters. The most important example of such a parametrized signal is that in which the signal is a modulated sinusoid with random phase; i.e., FIGURE 73.1 Correlation detector for a coherent signal in additive white Gaussian noise. exp s Y s / k k k k n k n - Ê Ë Á ˆ ¯ ˜ Ï Ì Ô Ó Ô ¸ ˝ Ô ˛ = = Ô Â Â 1 2 2 1 1 2 s s Yk k k n = Â 1 exp s Y ( ) s ( ) / p( )d k k k k n k n q - [ ] q q q Ê Ë Á ˆ ¯ ˜ Ï Ì Ô Ó Ô ¸ ˝ Ô ˛ Â= Â= Ô Ú 1 2 2 1 1 2 s L