Question

1-5. Reexpress each sum in terms of the sawtooth function t and/or the staircase function t. (Since t - t = t, there are more than one e

Answer 1

To reexpress each sum in terms of the sawtooth function t and/or the staircase function t, we use the definitions of these functions. The sawtooth function s(t) increases linearly from 0 to 1 within each period. The staircase function u(t) increases by 1 each time the input t goes from negative to non-negative. We can express each sum by manipulating these functions accordingly.

Step-by-step explanation:

To reexpress each sum in terms of the sawtooth function t and/or the staircase function t, we need to understand what these functions represent. The sawtooth function t, denoted as s(t), is a periodic function that increases linearly from 0 to 1 within each period and then jumps back to 0. The staircase function t, denoted as u(t), is a function that increases by 1 each time the input t goes from negative to non-negative. With this understanding, we can express each sum in terms of these functions.

1. s(t) + s(t) = 2s(t)
2. s(t) + u(t) = s(t) + (u(0) - u(t-1)) = s(t) + 1 - u(t-1)
3. u(t) + u(t) = 2u(t)
4. u(t) + s(t) = u(t) + (s(t) - s(t-1)) = u(t) + s(t) - s(t-1)
5. s(t) + t = s(t) + t - t + u(t) - u(t) = s(t) + t - s(t-1) - u(t-1)