The $\delta$ function and orthogonal bases

The discrete and periodic time function and spectrum can be written as, respectively

$$x_T(t)=\sum_{m=-\infty}^{\infty} x[m]\delta(t-mt_0)$$

$$X_F(f)=\sum_{n=-\infty}^{\infty} X[n]\delta(f-nf_0)$$

The $\delta$ function used above satisfies the following:

1. 
   
   $$\delta(t-\tau)=\left\{ \begin{array}{ll} 0 & t\ne \tau \\ \infty & t=\tau \end{array} \right.$$
   
   

2. 
   
   $$\int_{-\infty}^{\infty} \delta(t)dt=1$$
   
   

3. 
   
   $$\int_{-\infty}^{\infty} x(t)\delta(t-\tau)dt=x(\tau)$$