Impulse response; conditions for distortionless transmission

Impulse response

The impulse response $h(t)$ of a linear time-invariant (LTI) system is its output when the input is a unit impulse $\delta(t)$ (zero initial conditions). It completely describes the system: for any input $x(t)$, the output is the convolution $y(t) = x(t) * h(t)$, and its Fourier transform is the transfer function $H(f)$.

Conditions for distortionless transmission

Transmission is distortionless if the output is an exact copy of the input, differing only by a constant gain $K$ and a constant delay $t_d$:

$$y(t) = K\,x(t - t_d) \;\Rightarrow\; H(f) = K\,e^{-j2\pi f t_d}, \quad h(t) = K\,\delta(t - t_d)$$

So, over the whole band of the signal:

1. Flat (constant) amplitude response: $|H(f)| = K$. Every frequency component gets the same gain. Otherwise amplitude distortion occurs, e.g. high frequencies cut more than low ones, making sharp edges rounded.
2. Linear phase response: $\theta(f) = -2\pi f t_d$ (or $-2\pi f t_d \pm n\pi$). The phase shift is proportional to frequency, so every component has the same delay $t_d$ (constant group delay $\tau_g = -\frac{1}{2\pi}\frac{d\theta}{df} = t_d$). Otherwise phase (delay) distortion occurs, which spreads pulses and causes ISI in data transmission.
3. The system must be linear; a non-linear system creates harmonics and intermodulation (non-linear distortion).

 Input x(t)           Output y(t) = K x(t - td)
   |  __                 |        __
   | /  \                |       /  \   (K times)
   |/    \               |      /    \
 --+------\---> t      --+-----/------\---> t
                         |<-td->|

 |H(f)|                 theta(f)
   |                       |
 K +---------------        |\
   |                       | \  slope -2*pi*td
   |                     --+--\-----------> f
 --+---------------> f     |   \
  -B       0      B        |    \

The required band is only the signal bandwidth $|f| \le B$; outside it, $H(f)$ may be anything. Real channels are not ideal, so equalizers are used to make the overall response flat in amplitude and linear in phase.