Using Fourier Transform

• It is possible to obtain the response of a continuous LTI system via Fourier transform
• Should only be used when:
  • Convolution integral is not manageable
  • Step responses are unknown (cannot leverage homogeneity/additivity)
  • System is not a sum of sinusoids
• For a system described by impulse response $h(t)$ and input $x(t)$:
  1. $x(t) \xrightarrow{\mathscr{F}}X(j\omega)$
  2. $h(t) \xrightarrow{\mathscr{F}}H(j\omega)$
  3. $X(j\omega) \cdot H(j\omega) = Y(j\omega)$
  4. $Y(j\omega) \xrightarrow{\mathscr{F}^{-1}}y(t)$
• This is possible due to convolution property of Fourier transform

Using Sum of Sinusoids

• See Frequency Response

References

• System response (course handout)