Impulse response usefulness; impulse response of LPF

Impulse response

The impulse response $h(t)$ of a system is the output obtained when a unit impulse $\delta(t)$ is applied at the input, with the system initially at rest. For an LTI system it fully describes the system.

Why it is useful in communication system design

• Output for any input: $y(t) = x(t) * h(t)$, so one measurement or calculation of $h(t)$ predicts the response to any signal (speech, pulses, modulated waves).
• Frequency response: its Fourier transform is the transfer function $H(f)$, which gives the bandwidth, amplitude and phase response; this tells whether a channel or filter will distort the signal and whether an equalizer is needed.
• Pulse spreading and ISI: $h(t)$ of a channel shows how much a transmitted pulse spreads, which sets the maximum symbol rate; pulse-shaping (raised-cosine) filters are designed from it.
• Matched filter design: the optimum receiver filter has $h(t) = s(T - t)$.
• Causality and stability checks: realizable only if $h(t) = 0$ for $t < 0$; stable if $\int|h(t)|dt < \infty$.
• Noise analysis: output PSD $S_y(f) = |H(f)|^2S_x(f)$.

Impulse response of a low-pass filter

For an ideal LPF of bandwidth $B$ (gain 1, delay $t_d$), $h(t) = 2B\,\text{sinc}[2B(t - t_d)]$, a sinc pulse centred at $t_d$ (non-causal, so not realizable). For a practical RC low-pass filter, $h(t) = \frac{1}{RC}e^{-t/RC}u(t)$.