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Therefore, <i>\u03c1<\/i><sub><i>ED<\/i><\/sub> can be expressed as follows:Both  and  are exponentially distributed, and their PDFs can be expressed as follows:AndThe following equation can be obtained:Thus, the following equation can be derived:Because  and  are mutually independent random variables, the variance of the numerator of (44) can be derived using Bienaym\u00e9's identity [40\u201342] as follows:On the basis of (), the following equation can be derived:","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Appendix A: Derivation of (<!--InternalRef removed-->18) and (<!--InternalRef removed-->19)"}},{"value":"A total of <i>L<\/i> samples of the second preamble are included in the moving sum of autocorrelation, as displayed in Fig. . Therefore, <i>\u03c1<\/i><sub><i>AD<\/i><\/sub> can be expressed as follows:where 1\u2266 <i>L<\/i>\u2266 <i>N<\/i>.The background noise and IN in  and  are mutually independent random variables.Therefore, the following equation can be obtained:On the basis of (), the following equation can be derived:The variance of the numerator of () is divided into two parts. The first part comprises  and , which are mutually independent complex random variables whose real and imaginary parts are also mutually independent random variables that have a Gaussian distribution with zero mean and variance [+ (1\u2009+\u2009<i>P<\/i><sub><i>impulse<\/i><\/sub><i>\u03bc<\/i>)]\/2 and (1\u2009+\u2009<i>P<\/i><sub><i>impulse<\/i><\/sub><i>\u03bc<\/i>)\/2, respectively. Let  and .The second part:The following equation can be derived in a similar manner as ():On the basis of (), the following equation can be obtained:Combining () and () can yield the following equation:","order":4,"name":"Ethics","group":{"name":"EthicsHeading","label":"Appendix B: Derivation of (<!--InternalRef removed-->38)"}}]}}