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# 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 - t_{d})

Where t_{d} = delay time and

k = constant.

Take Fourier transform on both sides

FT[ y (t)] = FT[Kx(t - t_{d})]

= K FT[x(t - t_{d})]

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: