Frequency response of a CT LTI system

The frequency response $H(j\omega)$ of a continuous-time LTI system is the Fourier transform of its impulse response. It tells how the system changes the amplitude and phase of each sinusoidal (complex exponential) component of the input.

$$H(j\omega) = \int_{-\infty}^{\infty} h(t)e^{-j\omega t}\,dt$$

Eigenfunction property

If $x(t) = e^{j\omega t}$, the output is

$$\begin{aligned} y(t) &= \int_{-\infty}^{\infty} h(\tau)e^{j\omega(t-\tau)}\,d\tau \\ &= e^{j\omega t}\int_{-\infty}^{\infty} h(\tau)e^{-j\omega\tau}\,d\tau = H(j\omega)\,e^{j\omega t} \end{aligned}$$

Complex exponentials pass through unchanged in form, only multiplied by the complex number $H(j\omega)$.

Transfer function

From the convolution property, $y(t) = x(t)*h(t) \Rightarrow Y(j\omega) = X(j\omega)H(j\omega)$, so

$$H(j\omega) = \frac{Y(j\omega)}{X(j\omega)}$$

For a system described by $\sum_k a_k \frac{d^k y}{dt^k} = \sum_k b_k \frac{d^k x}{dt^k}$:

$$H(j\omega) = \frac{\sum_k b_k (j\omega)^k}{\sum_k a_k (j\omega)^k}$$

Magnitude and phase

$H(j\omega) = |H(j\omega)|\,e^{j\angle H(j\omega)}$

• $|H(j\omega)|$: gain (magnitude response), often plotted in dB on a Bode plot.
• $\angle H(j\omega)$: phase shift; $-d\angle H/d\omega$ is the group delay.
• For real $h(t)$: $|H|$ is even and $\angle H$ is odd in $\omega$.
• A sinusoid $\cos\omega_0 t$ gives output $|H(j\omega_0)|\cos(\omega_0 t + \angle H(j\omega_0))$.

Example: RC low-pass filter, $H(j\omega) = \dfrac{1}{1 + j\omega RC}$: $|H| = 1/\sqrt{1 + (\omega RC)^2}$, $\angle H = -\tan^{-1}\omega RC$. Low frequencies pass, high frequencies are attenuated.