1. 教育部網路通訊人才培育先導型計畫 Probability, Random Processes and Noise 1 Example 5-15 Random binary wave (1/4) Consider a random binary wave, a sample function is shown in following figure. Assume these two symbols (with duration T and amplitude ) are equally likely, and the presence in any one interval is independent of all other intervals. (a) Find the autocorrelation function of the random binary wave. 5.2

2. 教育部網路通訊人才培育先導型計畫 Probability, Random Processes and Noise 2 Example 5-15 Random binary wave (2/4) The starting time is equally likely to lie anywhere between zero and T seconds. So is the sample value of a uniformly distributed random variable, with Let and be random variables obtain by observing at times and, respectively. (1) When, the random variables and occur in different pulse intervals and therefore independent. We thus have 5.2 【 Sol. 】

3. 教育部網路通訊人才培育先導型計畫 Probability, Random Processes and Noise 3 Example 5-15 Random binary wave (3/4) (2) When, with and, the random variables and occur in the same pulse interval if and only if the delay satisfies the condition. We thus obtain the conditional expectation: Averaging this result over all possible values of, we get 5.2

4. 教育部網路通訊人才培育先導型計畫 Probability, Random Processes and Noise 4 Example 5-15 Random binary wave (4/4) By similar reasoning for any other values of, we conclude that the autocorrelation function is only a function of the time difference, as shown by 5.2