There are two basic types of digital communications applications in which the problem(73. 18)arises. One is in M-ary data transmission, in which a symbol alphabet with M elements is used to transmit data, and a decision among these M symbols must be made in each symbol interval [Proakis, 1983]. The other type of application in which(73. 18)arises is that in which data symbols are correlated in some way because of intersymbol interference, coding, or multiuser transmission. In such cases, each of the M possible signals represents a frame of data symbols, and a joint decision must be made about the entire frame since individual symbol decisions cannot be decoupled. Within this latter framework, the problem(73. 18)is known as sequence detection. The basic distinction between M-ary transmission and sequence detection is one of degree. In typical M-ary transmission, the number of elements in the signaling alphabet is typically a small power of 2(say 8 or 32), whereas the number of symbols in a frame of data could be on the order of thousands. Thus, solution of (73. 18)by exhaustive search is prohibitive for sequence detection, and less complex algorithms must be used. Typical digital communications applications in which sequence detection is necessary admit dynamic program ming solutions to(73. 18)(see, e.g., Verdu[1993]) Detection of Signals in More General Noise Processes In the foregoing paragraphs, we have described three basic detection procedures: correlation detection of signals that are completely known, envelope detection of signals that are known except for a random phase, and quadratic detection for Gaussian random signals. These detectors were all derived under an assumption of white Gaussian noise. This assumption provides an accurate model for the dominant noise arising in many communication channels. For example, the thermal noise generated in signal processing elect quately described as being white and Gaussian. However, there are also many channels in which the statistical behavior of the noise is not well described in this way, particularly when the dominant noise is produce the physical channel rather than in the receiver electronics One type of noise that often ise that is Gaussian but not white. In this case, the detection (73. 1)-(73.2)can be converted to an equivalent problem with white noise by applying a linear filtering known as prewhitening to the measurements. In particular, on denoting the noise covariance matrix can write ∑=CCr (73.19) where C is an n X n invertible, lower-triangular matrix and where the superscript T denotes matrix transpo- sition. The representation(73. 19)is known as the Cholesky decomposition. On multiplying the measurement vector YA(Y, Y2,..., Y)satisfying(73. 1)-(73. 2)with noise covariance 2, by C-i, we produce an equivalent (in terms of information content)measurement vector that satistifies the model(73. 1)-(73. 2)with white Gaussian noise and with the signal conformally transformed. This model can then be treated using the methods evisu In other channels, the noise can be modeled as being i.i. d. but with an amplitude distribution that is not Gaussian. This type of model arises, for example, in channels dominated by impulsive phenomena, such as urban radio channels. In the non-Gaussian case the procedures discussed previously lose their optimality as defined in terms of the error probabilities. These procedures can still be used, and they will work well under many conditions; however, there will be a resulting performance penalty with respect to op timum pi dures based on the likelihood ratio. Generally speaking, likelihood-ratio-based procedures for non-Gaussia hannels involve more complex nonlinear processing of the measurements than is required in the detectors, although the retention of the i.i. d. assumption greatly simplifies this problem. A treatment of for such channels can be found in Kassam [1988] When the noise is both non-Gaussian and dependent, the methodology is less well developed, although some techniques are available in these cases. An overview can be found in Poor and Thomas [1993] Robust and Nonparametric Detection All of the procedures outlined in the preceding subsection are based on the assumption of a known(possibly up to a set of unknown parameters)statistical model for signals and noise. In many practical situations it e 2000 by CRC Press LLC© 2000 by CRC Press LLC There are two basic types of digital communications applications in which the problem (73.18) arises. One is in M-ary data transmission, in which a symbol alphabet with M elements is used to transmit data, and a decision among these M symbols must be made in each symbol interval [Proakis, 1983]. The other type of application in which (73.18) arises is that in which data symbols are correlated in some way because of intersymbol interference, coding, or multiuser transmission. In such cases, each of the M possible signals represents a frame of data symbols, and a joint decision must be made about the entire frame since individual symbol decisions cannot be decoupled. Within this latter framework, the problem (73.18) is known as sequence detection. The basic distinction between M-ary transmission and sequence detection is one of degree. In typical M-ary transmission, the number of elements in the signaling alphabet is typically a small power of 2 (say 8 or 32), whereas the number of symbols in a frame of data could be on the order of thousands. Thus, solution of (73.18) by exhaustive search is prohibitive for sequence detection, and less complex algorithms must be used. Typical digital communications applications in which sequence detection is necessary admit dynamic program￾ming solutions to (73.18) (see, e.g., Verdú [1993]). Detection of Signals in More General Noise Processes In the foregoing paragraphs, we have described three basic detection procedures: correlation detection of signals that are completely known, envelope detection of signals that are known except for a random phase, and quadratic detection for Gaussian random signals. These detectors were all derived under an assumption of white Gaussian noise. This assumption provides an accurate model for the dominant noise arising in many communication channels. For example, the thermal noise generated in signal processing electronics is ade￾quately described as being white and Gaussian. However, there are also many channels in which the statistical behavior of the noise is not well described in this way, particularly when the dominant noise is produced in the physical channel rather than in the receiver electronics. One type of noise that often arises is noise that is Gaussian but not white. In this case, the detection problem (73.1)–(73.2) can be converted to an equivalent problem with white noise by applying a linear filtering process known as prewhitening to the measurements. In particular, on denoting the noise covariance matrix by (, we can write ( = CCT (73.19) where C is an n 3 n invertible, lower-triangular matrix and where the superscript T denotes matrix transpo￾sition. The representation (73.19) is known as the Cholesky decomposition. On multiplying the measurement vector Y = D (Y1 , Y2 , . . ., Yn )T satisfying (73.1)–(73.2) with noise covariance (, by C–1, we produce an equivalent (in terms of information content) measurement vector that satistifies the model (73.1)–(73.2) with white Gaussian noise and with the signal conformally transformed. This model can then be treated using the methods described previously. In other channels, the noise can be modeled as being i.i.d. but with an amplitude distribution that is not Gaussian. This type of model arises, for example, in channels dominated by impulsive phenomena, such as urban radio channels. In the non-Gaussian case the procedures discussed previously lose their optimality as defined in terms of the error probabilities. These procedures can still be used, and they will work well under many conditions; however, there will be a resulting performance penalty with respect to optimum procedures based on the likelihood ratio. Generally speaking, likelihood-ratio-based procedures for non-Gaussian noise channels involve more complex nonlinear processing of the measurements than is required in the standard detectors, although the retention of the i.i.d. assumption greatly simplifies this problem. A treatment of methods for such channels can be found in Kassam [1988]. When the noise is both non-Gaussian and dependent, the methodology is less well developed, although some techniques are available in these cases. An overview can be found in Poor and Thomas [1993]. Robust and Nonparametric Detection All of the procedures outlined in the preceding subsection are based on the assumption of a known (possibly up to a set of unknown parameters) statistical model for signals and noise. In many practical situations it is