Impulse response and transfer function significance

Impulse response

The impulse response $h(t)$ of an LTI system is the output produced when the input is a unit impulse $\delta(t)$, with the system initially at rest:

$$\delta(t) \;\to\; \boxed{\text{LTI system}} \;\to\; h(t)$$

Because any input can be written as a sum of shifted, scaled impulses, $x(t) = \int x(\lambda)\delta(t-\lambda)d\lambda$, linearity and time invariance give the output as a convolution:

$$y(t) = x(t) * h(t) = \int_{-\infty}^{\infty} x(\lambda)\,h(t-\lambda)\,d\lambda$$

Transfer function

The transfer function $H(f)$ is the Fourier transform of the impulse response, and equals the ratio of output to input spectra:

$$H(f) = \int_{-\infty}^{\infty} h(t)e^{-j2\pi ft}\,dt = \frac{Y(f)}{X(f)} = |H(f)|\,e^{j\theta(f)}$$

$|H(f)|$ is the amplitude (magnitude) response and $\theta(f)$ the phase response. For an input $e^{j2\pi f_0t}$ the output is $H(f_0)e^{j2\pi f_0t}$.

Example: an RC low-pass filter has $h(t) = \frac{1}{RC}e^{-t/RC}u(t)$ and $H(f) = \frac{1}{1 + j2\pi fRC}$.

Significance

• Either $h(t)$ or $H(f)$ completely characterizes an LTI system; knowing one, the output for any input can be found.
• Convolution in time becomes multiplication in frequency, $Y(f) = H(f)X(f)$, which simplifies analysis of filters, channels and cascaded systems ($H = H_1H_2$).
• $|H(f)|$ and $\theta(f)$ show bandwidth and whether the system causes amplitude or phase distortion (distortionless needs $|H| =$ constant, $\theta$ linear).
• $h(t)$ tells whether a system is causal ($h(t) = 0$ for $t<0$) and stable ($\int|h(t)|dt < \infty$).
• Used to design filters, equalizers and matched filters, and to find output PSD: $S_y(f) = |H(f)|^2S_x(f)$.