What is the correct range of tau for case (4) in the Convolution 2 picture?

[aruwin]

Hi! Can someone show me how to get the range of tau for case (4) in the picture? Because when I draw it, the overlapped range I get is from t-T/2 to T, but in the solution, it says t-T to T/2.

View attachment 3916

 

[jvicens]

Sure, I'd be happy to help! Let's first define the variables in case (4) for reference:
- t = starting time
- T = period of the wave
- tau = time delay

To determine the range of tau for case (4), we need to consider the following conditions:
- The wave starts at time t and ends at time t+T
- The time delay, tau, is the time difference between when the wave starts and when it reaches its maximum amplitude

Based on these conditions, we can see that the range of tau can vary depending on the value of t. Let's break it down into two cases:

Case 1: When t < T/2
In this case, the wave starts at time t and ends at time t+T. The time delay, tau, is the time difference between when the wave starts (t) and when it reaches its maximum amplitude (t+tau). Since the maximum amplitude occurs at t+T/2, the time delay would be T/2. Therefore, the range of tau would be from 0 to T/2.

Case 2: When t > T/2
In this case, the wave starts at time t and ends at time t+T. However, the maximum amplitude occurs at t-T/2. This means that the time delay, tau, is the time difference between when the wave starts (t) and when it reaches its maximum amplitude (t-T/2). Therefore, the range of tau would be from T/2 to T.

In conclusion, the range of tau for case (4) is from 0 to T/2 when t < T/2, and from T/2 to T when t > T/2. I hope this helps clarify any confusion! Let me know if you have any further questions.