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Distortionless Transmission through a System
A distortion is defined as the change of the shape of the signal when it is transmitted through the system. Therefore, the transmission of a signal through a system is said to be distortion-less when the output of the system is an exact replica of the input signal. This replica, i.e., the output of the system may have different magnitude and also it may have different time delay.
A constant change in the magnitude and a constant time delay in the output signal is not considered as distortion. Only the change in the shape of the signal is considered the distortion.
Mathematically, the transmission of a signal 𝑥(𝑡) is said to be distortion less transmission if the output of the system is
$$\mathrm{y(t)=kx(t-t_{d})\; \; \cdot \cdot \cdot (1)}$$
Where,
The constant k represents the change in the magnitude, either amplification or attenuation.
𝑡𝑑 is the delay time.
The block diagram of a distortion less system and typical input and output waveforms are shown in Figure-1.
Now, by taking the Fourier transform on both sides of the equation (1) and using the shifting property of Fourier transform, we obtain,
$$\mathrm{Y(\omega )=ke^{-j\omega t_{d}}X\left ( \omega \right )}$$
Hence, the transfer function of the system for the distortion less transmission is given by,
$$\mathrm{H(\omega )=\frac{Y\left ( \omega \right )}{X\left ( \omega \right )}=ke^{-j\omega t_{d}}\; \; \cdot \cdot \cdot (2)}$$
Now, by taking the inverse Fourier transform of the equation (2), the corresponding impulse response of the system is given by,
$$\mathrm{h(t)=k\delta (t-t_{d})\: \: \cdot \cdot \cdot (3)}$$
The magnitude of the transfer function is,
$$\mathrm{\left | H(\omega ) \right |=k}$$
And it is constant for all values of 𝜔.
The phase shift of the transfer function is,
$$\mathrm{\theta \left ( \omega \right )=\angle H(\omega )=-\omega t_{d}}$$
In general, the phase shift is,
$$\mathrm{\theta \left ( \omega \right )=n\pi -\omega t_{d};\; \; \left ( n\: is\:an\:integer \right )}$$
And it varies linearly with frequency. Figure-2 shows the magnitude and phase characteristics of a distortion less transmission system.
Therefore, for the distortionless transmission of a signal through a system, the magnitude of the transfer function of the system |𝐻(𝜔)| should be a constant, i.e., the input signal must undergo the same amount of amplification or attenuation for all the frequency components. Thus, for the distortionless transmission of the system, the bandwidth of the system is infinite and the phase spectrum should be proportional to the frequency. However, in practice, no system exists that have infinite bandwidth and hence the conditions of the distortion less transmission are never met exactly.
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