Slecture rep comb ECE438 Xiaozhe - Rhea

     1.1 Definition of Rep operator
      
         1.1.1 Introduction to Rep operator

         1.1.2 Definition of Rep operator

         1.1.3 Two representations of Rep operator

         1.1.4 Relationship between two representation

     1.2 Definition of Comb operator

         1.1.1 Introduction
     
         1.1.2 Definition of Comb operator

         1.1.3 Two representations of Comb operator

         1.1.4 Relationship between two representation

1.1.1 Introduction to Rep operator

In order to understand Rep operator clearly, a graphical method is introduced as follows:

In the figure above, there is a little pulse with compact supports over a definite interval. When repeating it at capital T, We can get the following figure.

This process which is denoted by Rep operator have the same meaning as taking original function x(t) and shifting it by KT in which K∈N and -∞<K<+∞.

1.1.2 Definition of Rep operator

Rep operator denotes a kind of process which can periodically replicate a function with some specific period T. Where the function has a finite domain of argument and the minimum repeating period T has to > a+b(a denotes left boundary of the graph and b denotes right boundary of the graph ).

1.1.3 Two representations of Rep operator

$ 1.rep_{T}(x[t])=\sum_{k=-\infty}^{\infty}x(t-KT) $

$ 2.rep_{T}(x[t])=x(t) \ast P_{T}(T) $

1.1.4 Relationship between two representation

$ rep_{T}(x[t])=x(t) \ast P_{T}(T) \qquad \qquad \qquad Where \quad P_{T}(T)=\sum_{k=-\infty}^{\infty}\delta(t-KT) $

            $ =x(t) \ast \sum_{k=-\infty}^{\infty}\delta(t-KT) $
            $ =\int_{-\infty}^{\infty}x(\tau)\sum_{k=-\infty}^{\infty}\delta(t-KT-\tau)d\tau $
            $ =\sum_{k=-\infty}^{\infty}\int_{-\infty}^{\infty}x(\tau)\delta(t-KT-\tau)d\tau $

$ Since \quad x(t)=\int_{-\infty}^{\infty}\delta(t-\tau)d\tau $

$ So \quad rep_{T}(x[t])=\sum_{k=-\infty}^{\infty}x(t-KT) $

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