By Robert B. Dingle

ISBN-10: 0122165500

ISBN-13: 9780122165504

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**Additional resources for Asymptotic Expansions: Their Derivation and Interpretation**

**Sample text**

In the case of Gaussian signals, we have tractable results for some important cases. To illustrate this point, let us consider the following example. Example 9 We are interested in modeling the output of a squaring device when the input is a real, zero mean, WSS Gaussian signal. Assume that Rx (τ ) = E {x (t) x (t + τ )} describes the correlation function of the input signal. Let y (t) = x 2 (t) denote the output process. Find the correlation function of a vector of size 2 that is obtained by sampling y (t) at regular interval of Ts seconds.

20 The eye-diagram for a 6-level raised-cosine waveform p (t) = sin c t T cos πβt T 1− 4β 2 t 2 T2 where sin c (x) = sin (πx) πx These types of plots which are overlaps of all possible waveforms generated by different values of an = an is known as the eye-diagram. Note that the pulse shape is not confined to T seconds, and hence there are self-interference generated for the adjacent symbol intervals. That is, an impacts the signal during the intervals of time associated with an−1 , an−2 , …and an+1 , an+2 ,….

To examine this problem further, it is important to identify various ergodicity principles, which in simple words allow replacing ensemble averages with their time-average counterparts. In principle, a signal may be ergodic in mean, power, correlation, or distribution. This implies that 50 1 Non-Biological Signals one may obtain mean, power, correlation, or distribution of a random signal by merely observing sample paths of the random signal. 103) n=−N It becomes immediately obvious that for a mean ergodic process the ensemble average N must be constant.

### Asymptotic Expansions: Their Derivation and Interpretation by Robert B. Dingle

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