Time-shifting a signal.

• If a signal can be expressed in the form
  $$x(t) = s(t-t_1),$$
  we say $x(t)$ is a time-shifted version of $s(t)$.

• Consider the simple function
  $$s(t) = t \qquad \qquad 0\leq t\leq 1.$$

  Figure 6: Time-shifting a signal.

  

• Shifting the function by $t_1 = 2$ seconds yields
  

  $$x(t) = s(t-2) = t-2 \qquad 0\leq t-2\leq 1$$

  $$= t-2 \qquad 2\leq t \leq 3,$$

  
  which is simply $s(t)$ with its origin shifted to the right, or delayed, by 2 seconds.
• Shifting the function by $t_1 = -1$ seconds yields
  

  $$y(t) = s(t+1) = t+1 \qquad 0\leq t+1\leq 1$$

  $$= t+1 \qquad -1\leq t \leq 0,$$

  
  which is simply $s(t)$ with its origin shifted to the left, or advanced in time, by 1 seconds.

• Summary:
  • a positive phase indicates a shift to the left on the time axis (advance in time),
  • a negative phase indicates a shift to the right (a delay in time).