Understanding the Concept:
Distortionless transmission implies that the output signal $y(t)$ is an exact replica of the input signal $x(t)$, scaled in amplitude and delayed in time, without any alterations to its spectral wave shape. Mathematically, it is expressed as:
\[
y(t) = K \cdot x(t - t_d)
\]
where $K$ represents a constant amplification/attenuation scaling factor, and $t_d$ is a uniform time propagation delay.
Step 1: Finding the required Frequency Domain Transfer Function.
To establish the constraints on the system, let us evaluate the continuous-time Fourier Transform (CTFT) of the ideal distortionless output relationship:
\[
\mathcal{F}\{y(t)\} = \mathcal{F}\{K \cdot x(t - t_d)\}
\]
Applying the linear time-shifting property of the Fourier Transform:
\[
Y(\omega) = K \cdot X(\omega) \cdot e^{-j\omega t_d}
\]
The system transfer function $H(\omega)$ is defined as the ratio of output to input spectra:
\[
H(\omega) = \frac{Y(\omega)}{X(\omega)} = K \cdot e^{-j\omega t_d}
\]
Step 2: Extracting Magnitude and Phase constraints.
Expressing $H(\omega)$ in its standard polar configuration, we write:
\[
H(\omega) = |H(\omega)| \cdot e^{j\angle H(\omega)}
\]
Equating this with our derived relation $K \cdot e^{-j\omega t_d}$, we find:
• Magnitude Response Requirement:
\[
|H(\omega)| = K \quad (\text{Constant for all operational frequencies})
\]
If the magnitude varies, different frequency components of the signal are scaled unequally, causing amplitude distortion.
• Phase Response Requirement:
\[
\angle H(\omega) = -\omega t_d \quad (\text{Strictly linear function of frequency }\omega)
\]
If the phase is non-linear, different frequencies undergo different time delays, causing phase/delay distortion.
Therefore, both constant magnitude response and linear phase response are simultaneously required.