Properties of delta $\delta(t)$

$$\int_{-\infty}^{\infty}\delta(t-t_0)\phi(t)dt=\phi(t_0) \mbox{\qquad \qquad sifting}$$

$$\int_{-\infty}^{\infty}\delta(t)dt=1$$

$$\delta(t)=0 \qquad \vert t\vert>0 \quad \delta(0)=\infty$$

$$\delta(at)=\frac{1}{\vert a\vert}\delta(t)$$

$$\delta(t-t_0)\ast h(t)=h(t-t_0)$$

$$u(t)=\int_{-\infty}^{t}\delta(\tau)d\tau \Leftrightarrow \delta(t)=\frac{d}{dt}u_{-1}(t)$$

• Systems: (Continuous - but also applies to discrete)

  Define transformations as
  

  y(t)=T[x(t)]
  
  

• Linearity

  if
  

  y₁(t)=T[x₁(t)]
  
  
  
  
  y₂(t)=T[x₂(t)]
  
  
  and a and b are constant, then if

  
  

  T[ax₁(t)+bx₂(t)]=ay₁(t)+by₂(t)
  
  

  the system is linear (superposition and scaling)

• Time invariance

  
  

  $$T[x(t-t_0)]=y[t-t_0] \mbox{\qquad} \forall x(t),t_0$$
  
  

  LTI $\rightarrow$ linear, time invariant (has an impulse response)

  Causality

  $y(t_0)=\left.T[x(t)]\right\vert _{t=t_0}$ depends only on $x(t),t\le t_0$