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4.4 Energy and Power
Definition 4.27. For a signal g(t), the instantaneous power p(t) dissipated in the 1-Ω resister is pg(t) = |g(t)|^2 regardless of whether g(t) represents a voltage or a current. To emphasize the fact that this power is based upon unity resistance, it is often referred to as the normalized power.
Definition 4.28. The total (normalized) energy of a signal g(t) is given by
Eg =
−∞
pg(t)dt =
−∞
|g(t)|^2 dt = lim T →∞
∫ T
−T
|g(t)|^2 dt.
4.29. By the Parseval’s theorem discussed in 2.39, we have
Eg =
−∞
|g(t)|^2 dt =
−∞
|G(f )|^2 df.
Definition 4.30. The average (normalized) power of a signal g(t) is given by
Pg = lim T →∞
T
∫^ T^ /
−T / 2
|g (t)|^2 dt = lim T →∞
2 T
∫ T
−T
|g(t)|^2 dt.
Definition 4.31. To simplify the notation, there are two operators that used angle brackets to define two frequently-used integrals:
(a) The “time-average” operator:
〈g〉 ≡ 〈g (t)〉 ≡ lim T →∞
T
∫ T / 2
−T / 2
g (t)dt = lim T →∞
2 T
∫ T
−T
g (t)dt (46)
(b) The inner-product operator:
〈g 1 , g 2 〉 ≡ 〈g 1 (t) , g 2 (t)〉 =
−∞
g 1 (t)g∗ 2 (t)dt (47)
4.32. Using the above definition, we may write
- Eg = 〈g, g〉 = 〈G, G〉 where G = F {g}
- Pg =
|g|^2
- Parseval’s theorem: 〈g 1 , g 2 〉 = 〈G 1 , G 2 〉 where G 1 = F {g 1 } and G 2 = F {g 2 } 4.33. Time-Averaging over Periodic Signal: For periodic signal g(t) with period T 0 , the time-average operation in (46) can be simplified to
〈g〉 =
T 0
T 0
g (t)dt
where the integration is performed over a period of g.
Example 4.34. 〈cos (2πf 0 t + θ)〉 =
Similarly, 〈sin (2πf 0 t + θ)〉 =
Example 4.35.
cos^2 (2πf 0 t + θ)
Example 4.36.
ej(2πf^0 t+θ)
= 〈cos (2πf 0 t + θ) + j sin (2πf 0 t + θ)〉
Example 4.37. Suppose g(t) = cej^2 πf^0 t^ for some (possibly complex-valued) constant c and (real-valued) frequency f 0. Find Pg.
4.38. When the signal g(t) can be expressed in the form g(t) =
k
ckej^2 πfkt
and the fk are distinct, then its (average) power can be calculated from
Pg =
k
|ck|^2
Example 4.44. Suppose g (t) = 2 cos
2 π
3 t
2 π
5 t
. Find Pg.
4.45. For periodic signal g(t) with period T 0 , there is also no need to carry out the limiting operation to find its (average) power Pg. We only need to find an average carried out over a single period:
Pg =
T 0
T 0
|g (t)|^2 dt.
(a) When the corresponding Fourier series expansion g(t) =
n=−∞
cnejnω^0 t is known,
Pg =
∑^ ∞
k=−∞
|ck|^2
(b) When the signal g(t) is real-valued and its (compact) trigonometric Fourier series expansion g(t) = c 0 + 2
k=
|ck| cos (kω 0 t + ∠ck) is known,
Pg = c^20 + 2
∑^ ∞
k=
|ck|^2
Definition 4.46. Based on Definitions 4.28 and 4.30, we can define three distinct classes of signals:
(a) If Eg is finite and nonzero, g is referred to as an energy signal.
(b) If Pg is finite and nonzero, g is referred to as a power signal. (c) Some signals^17 are neither energy nor power signals.
- Note that the power signal has infinite energy and an energy signal has zero average power; thus the two categories are mutually exclusive.
Example 4.47. Rectangular pulse
(^17) Consider g(t) = t− 1 / (^41) [t 0 ,∞)(t), with t 0 > 0.
Example 4.48. Sinc pulse
Example 4.49. For α > 0, g(t) = Ae−αt (^1) [0,∞)(t) is an energy signal with Eg = |A|^2 / 2 α.
Example 4.50. The rotating phasor signal g(t) = Aej(2πf^0 t+θ)^ is a power signal with Pg = |A|^2.
Example 4.51. The sinusoidal signal g(t) = A cos(2πf 0 t + θ) is a power signal with Pg = |A|^2 /2.
4.52. Consider the transmitted signal x(t) = m(t) cos(2πfct + θ)
in DSB-SC modulation. Suppose M (f − fc) and M (f + fc) do not overlap (in the frequency domain).
(a) If m(t) is a power signal with power Pm, then the average transmitted power is Px =
Pm.
(b) If m(t) is an energy signal with energy Em, then the transmitted energy is Ex =
Em.
- Q: Why is the power (or energy) reduced?
- Remark: When x(t) =
2 m(t) cos(2πfct + θ) (with no overlapping between M (f − fc) and M (f + fc)), we have Px = Pm.
