Energy of a Time-Scaled Signal

If a signal f(t) has energy E, then the energy of f(2t) will be:

E / 2

Explanation

The energy of a continuous-time signal is defined as:

E = ∫ |f(t)|² dt   (from -∞ to ∞)

Now consider the time-scaled signal:

g(t) = f(2t)

Energy of g(t):

Eg = ∫ |f(2t)|² dt

Let:

x = 2t
dt = dx / 2

Substituting:

Eg = ∫ |f(x)|² (dx / 2)

Eg = (1/2) ∫ |f(x)|² dx

Eg = E / 2

General Rule

If a signal is time-scaled as f(at), then its energy becomes:

Energy = E / |a|

For a = 2:

Energy = E / 2

Final Answer

Energy of f(2t) = E / 2

Time compression (like f(2t)) reduces the signal energy by the scaling factor.