# Distortion Less Transmission

Transmission is said to be distortion-less if the input and output have identical wave shapes. i.e., in distortion-less transmission, the input x(t) and output y(t) satisfy the condition:

y (t) = Kx(t - td)

Where td = delay time and

k = constant.

Take Fourier transform on both sides

FT[ y (t)] = FT[Kx(t - td)]

= K FT[x(t - td)]

According to time shifting property,

= KX(w)$e^{-j \omega t_d}$

$\therefore Y(w) = KX(w)e^{-j \omega t_d}$

Thus, distortionless transmission of a signal x(t) through a system with impulse response h(t) is achieved when

$|H(\omega)| = K \,\, \text{and} \,\,\,\,$ (amplitude response)

$\Phi (\omega) = -\omega t_d = -2\pi f t_d \,\,\,$ (phase response)

A physical transmission system may have amplitude and phase responses as shown below: