频谱，功率谱密度

信号时域(x(t))，频域(X(omega)).
在((omega_0, omega_0 + Delta omega))区间内的信号时域表达式为
[ x_{(omega_0, omega_0 + Delta omega)}(t) = int_{omega_0}^{omega_0 + Delta omega} X(omega) e^{jomega t} d omega]
$ x_{(omega_0, omega_0 + Delta omega)}(t) $ 的功率为 $ x_{(omega_0, omega_0 + Delta omega)}^2(t) $

• 则信号在(omega_0)处的的功率谱密度为
  [ lim_{Delta omega to 0 } frac{ x^2_{(omega_0, omega_0 + Delta omega)}(t) }{Delta omega}= lim_{Delta omega to 0 } frac{ left{ int_{omega_0}^{omega_0 + Delta omega} X(omega) e^{jomega t} d omega right}^2 }{Delta omega } ]

• 信号$ x_{(omega_0, omega_0 + Delta omega)}(t) $ 在((-T,T))时间内的功率为
  [ int_{-T}^{T} x_{(omega_0, omega_0 + Delta omega)}^2(t) d t ]
  则信号在(omega_0)处的的功率谱密度为
  [ lim_{substack{Delta omega to 0 \ T to infty} } frac{ int_{-T}^{T} x_{(omega_0, omega_0 + Delta omega)}^2(t) d t }{ 2T Delta omega} ]

[ begin{aligned} sqrt{37} & = sqrt{frac{73^2-1}{12^2}} \ & = sqrt{frac{73^2}{12^2}cdotfrac{73^2-1}{73^2}} \ & = sqrt{frac{73^2}{12^2}}sqrt{frac{73^2-1}{73^2}} \ & = frac{73}{12}sqrt{1 - frac{1}{73^2}} \ & approx frac{73}{12}left(1 - frac{1}{2cdot73^2}right) end{aligned} ]