Short note: Distortionless transmission

Distortionless transmission means the output of a system (channel, filter or amplifier) has exactly the same shape as the input; only its amplitude may be scaled by a constant and it may be delayed by a fixed time:

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

where $K$ is a constant gain and $t_d$ a constant delay.

Transfer function: taking the Fourier transform,

$$Y(f) = K X(f)\,e^{-j2\pi f t_d} \;\Rightarrow\; H(f) = K\,e^{-j2\pi f t_d}$$

and the impulse response is $h(t) = K\,\delta(t - t_d)$.

Conditions (over the band occupied by the signal):

1. Constant amplitude response: $|H(f)| = K$. All frequency components are amplified or attenuated equally.
2. Linear phase response: $\theta(f) = -2\pi f t_d$ (a straight line through the origin, or through $\pm n\pi$). All components are delayed by the same time $t_d$; the group delay $-\frac{1}{2\pi}\frac{d\theta}{df} = t_d$ is constant.

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

Types of distortion when conditions fail:

• Amplitude distortion: $|H(f)|$ not constant, so some frequencies are boosted or cut more than others.
• Phase (delay) distortion: phase not linear, so components arrive at different times; serious for video and data pulses (causes ISI).
• Non-linear distortion: the system is non-linear, creating harmonics and intermodulation products.

In practice exact distortionless transmission is impossible over all frequencies, but it is enough for $|H(f)|$ to be flat and the phase linear over the signal bandwidth. Equalizers are used to correct channel amplitude and phase response.