I am given the following sampling signal function, where $\delta$ is the Dirac delta function, t is time, and Ts is the sampling period.
First, I am asked to plot the signal. I do not understand the meaning of the summation from 0 to $\infty$ when plotting the function. What does the summation mean in this context?
Second, I am asked to find the "integral" of the function above. I know I can move the integration to the inside of the summation, so I thought I would just be integrating $\delta(t-kT)$ – but the textbook uses a notation that I don't understand where the variable $t$ is replaced by $\tau$ inside the integral, while keeping $t$ in the integral limits. What is the meaning of $\tau$ here? Is this some sort of standard notation? How do I solve the integral if both $t$ and $\tau$ are involved?
Sorry – I know there are many questions. Thanks in advance for helping clarify this for me.
Best Answer
This is indeed an impulse train. T is the gap between impulses. The sum is there because each term of the sum adds just one impulse to the train. As you count through k = 0,1,2,3... you get deltas with arguments t,t-T,t-2T,t-3T... -- in other words the impulses at t=0,T,2T,3T...
The integral of an impulse train is a "staircase function". It will be 0 for t<0, then increase by 1 at every integer multiple of T. You'll have to make 0,T,2T,3T... the horizontal "tick marks" of your graph.
Tau is just a variable of integration over time. If you are going to integrate a function over TIME up to the current TIME, there are two different times you are talking about. In this example, tau is being used to represent any time in the past, and you're integrating over the entire past up to the current time t.