Filters

Figure 14: General idea of a filter and how to describe it: either with its impulse response or with it's Laplace transforms.

Fig. 14 presents a causal filter as a black box in continuous time. We send in a signal and and we get a signal out of it. The operation of the filter is that of a convolution of the input signal or a multiplication in the Laplace space:

  $$g(t) = h(t) * x(t) \Leftrightarrow Y(s) = H(s) \cdot X(s)$$ (87)

How to characterise filters?

1. Impulse Response
   

   $$x(t) = \delta(t) \qquad \leftarrow \mbox{delta pulse}$$ (88)

   $$h(t) = y(t) \qquad \leftarrow \mbox{impulse response}$$ (89)

   
   
   The filter is fully characterised by its impulse response $h(t)$

2. Transfer function The Laplace transform of the impulse response is called Transfer Function. With the argument $j\omega$ we get the frequency response of the filter. What does the frequency response tell us about the filter? The absolute value of the
     $$\vert H(i \omega)\vert$$ (90)
   gives us the amplitude or magnitude for every frequency (compare the Fourier transform). The angle of the term $H(j\omega)$ gives us the phase shift:
     $$\phi = \arg\left(H(i \omega) \right)$$ (91)
   of the filter. In this context the group delay can be defined as:
     $$\tau_{\omega} = - \frac{d \phi (\omega)}{d\omega}$$ (92)
   which is delay for a certain frequency $\omega$. In many applications this should be kept constant for all frequencies.

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