Addition of u(t-1) and u(t-3)

To add u(t-1) and u(t-3), first understand what each term represents.

The unit step function is defined as:

u(t-a) =

0,    t < a
1,    t ≥ a
        

Step 1: Understand Each Function

1. u(t-1)

u(t-1) =

0,    t < 1
1,    t ≥ 1
          

2. u(t-3)

u(t-3) =

0,    t < 3
1,    t ≥ 3
          

Step 2: Add Them

y(t) = u(t-1) + u(t-3)
        

Case 1: t < 1

Both are zero.

y(t) = 0 + 0 = 0
          

Case 2: 1 ≤ t < 3

u(t-1) = 1 and u(t-3) = 0

y(t) = 1 + 0 = 1
          

Case 3: t ≥ 3

Both are 1.

y(t) = 1 + 1 = 2
          

Final Answer (Piecewise Form)

y(t) =

0,    t < 1
1,    1 ≤ t < 3
2,    t ≥ 3
        

Graphical Interpretation

• At t = 1 → signal jumps from 0 to 1
• At t = 3 → signal jumps from 1 to 2

The result is a two-step staircase signal.