Laplace transform

The Fourier transform is not suitable for causal systems because it requires the whole signal from $-\infty < t < +\infty$. What we need is a transform which works with continuous causal signals. This is the Laplace transform:

  $${\mathrm{\bf L\!T}({h(t)})} = H(s) = \int_{0}^{\infty} h(t) e^{-st} dt$$ (77)

The Laplace transform has a couple of very useful properties:

• Integration:
    $$\int f(\tau) d\tau \Leftrightarrow \frac{1}{s} F(s)$$ (78)
• Differentiation:
    $$\frac{d}{dt} f(t) \Leftrightarrow s F(s)$$ (79)
• Shift:
    $$f(t - T) \Leftrightarrow e^{-Ts} F(s)$$ (80)
  Proof:
  

  $${\mathrm{\bf L\!T}({h(t - T)})} = \int_{0}^{\infty} h\underbrace{(t - T)}_{causal} e^{-st} dt$$ (81)

  $$= \int_{0}^{\infty} h(t) e^{-s(t + T)} dt$$ (82)

  $$= \int_{0}^{\infty} h(t) e^{-st} e^{-sT} dt$$ (83)

  $$= e^{-sT} \underbrace{\int_{0}^{\infty} h(t) e^{-st} dt}_{H(s)}$$ (84)

  $$= e^{-sT} H(s)$$ (85)

  
  

• Convolution:
    $$f(t) * g(t) \Leftrightarrow F(s) G(s)$$ (86)

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