Energy vs Power Signals
1. Energy of a Signal
The energy of a continuous-time signal is defined as:
E = ∫ |x(t)|² dt (from -∞ to ∞)
A signal is called an energy signal if:
0 < E < ∞
Example: Time-scaled signal
g(t) = f(2t)
Energy becomes:
Eg = E / 2
General rule:
If x(at) → Energy = E / |a|
These signals usually exist for a finite duration or decay with time.
2. Power of a Signal
The average power of a signal is defined as:
P = lim (T→∞) 1/(2T) ∫ |x(t)|² dt
A signal is called a power signal if:
0 < P < ∞, E = ∞
These signals typically exist for infinite time (e.g., periodic signals).
3. Example: Power of an AM Signal
An AM signal is given by:
s(t) = Ac [1 + m cos(ωm t)] cos(ωc t)
Total transmitted power:
Ptotal = Pc (1 + m²/2)
Where:
- Pc = Ac² / (2R)
- m = modulation index
Sideband power:
PSB = Pc × (m² / 2)
This shows that an AM signal has finite power but infinite energy, hence it is a power signal.
4. Key Differences
| Property | Energy Signal | Power Signal |
|---|---|---|
| Energy | Finite | Infinite |
| Power | Zero | Finite |
| Duration | Finite | Infinite |
| Examples | Pulse, Decaying signals | Sinusoid, AM signal |
5. Summary
- Energy signals → finite energy, zero average power
- Power signals → finite power, infinite energy
- Time scaling affects energy as: E / |a|
- AM signal is a power signal