Distortionless transmission conditions

Distortionless transmission means the output is an exact copy of the input in shape; it may only be scaled in amplitude and delayed in time.

Time-domain condition:

$$y(t) = K\,x(t - t_0)$$

where $K$ is a constant gain and $t_0 \ge 0$ is a constant delay.

Impulse response: put $x(t) = \delta(t)$:

$$h(t) = K\,\delta(t - t_0)$$

Frequency response: take the Fourier transform of $y(t) = K x(t-t_0)$ using the time-shift property:

$$\begin{aligned} Y(\omega) &= K\,X(\omega)\,e^{-j\omega t_0} \\ H(\omega) &= \frac{Y(\omega)}{X(\omega)} = K\,e^{-j\omega t_0} \end{aligned}$$

Conditions:

1. Constant magnitude: $|H(\omega)| = K$ for all frequencies (at least over the band of the signal). Every frequency component is amplified equally, so there is no amplitude distortion.
2. Linear phase: $\theta(\omega) = \angle H(\omega) = -\omega t_0$ (a straight line through the origin with slope $-t_0$). Then every frequency component is delayed by the same time, so there is no phase distortion.

The delay is constant because the group delay is

$$t_g = -\frac{d\theta(\omega)}{d\omega} = t_0$$

 |H(w)|                 angle H(w)
  K  +--------------      |\
     |                    |  \  slope = -t0
     |                    |    \
   --+-------------> w  --+------\----> w

In practice, it is enough for these conditions to hold over the bandwidth of the input signal.