Continuous-time Systems

• Represented by:
  • Differential equation
  • Impulse response (convolution integral)
  • Frequency response
  • Transfer function
• Impulse response, $h(t)$, is the system response to $\delta$
• LTI system is BIBO stable if its impulse response is absolutely integrable

Frequency Response

• Describes what the system does to sinusoidal inputs
• Broken down into magnitude and phase components

• Determined using the Fourier transform of $h(t)$

Analysis of $H(j\omega)$

Let $H(j\omega)$ be a general frequency response function of the form $H(jω)= \frac{N(j\omega)}{D(j\omega)}$. Then:

• $|H(j\omega)| = \frac{|N(j\omega)|}{|D(j\omega)|}$
• $H(j\omega) = \angle N(j\omega) - \angle D(j\omega)$

Magnitude Calculation

• If $M(j\omega) = A e^{j\phi(\omega)}$, $|M(j\omega)| = A$
• If $M(j\omega) = a + j b(\omega)$, $|M(j\omega)| = \sqrt{a^2 + b(\omega)^2}$

Phase Calculation

• If $M(j\omega) = A e^{j\phi(\omega)}$, $\angle M(j\omega) = \phi(\omega)$
• If $M(j\omega) = a + j b(\omega)$, $\angle M(j\omega) = \tan^{-1}\left(\frac{b(\omega)}{a}\right)$

Sinusoidal Response

• Response to sinusoids for an LTI system is a sinusoid of the same frequency with a shifted amplitude (gain) and phase, dependent on $H(j\omega)$
• Frequencies of the sinusoidal components cannot change and it must be two-sided
• Input format $x(t)$:

• Output format $y(t)$:

References

• Frequency response (course handout)