Constant Delay or Time Delay

fig: Constant Delay

Delay might occur during transmission through channel such as wires and wireless medium.  
Let \(V_1\) is applied as input that introduces a delay of \(D\) then the output \(V_2\) will be delayed replica of input.  
The delayed relationship is given as

\[
V_2(t) = V_1(t-D)
\]

From Fourier analysis we know that any signal can be decomposed into infinite summation of sinusoidal. Let such signals be

\[
V_1 = A \sin (\omega t + \Phi) \quad \dots 1
\]

Since the input and output signal is differed by a phase angle \(\Phi = \omega D\), we can assume that all frequency components of the input signals are delayed by same quantity \(D\) and without change in its amplitude. Then the output will be delayed replica of input.  

i.e.,

\[
V_2 = A \sin (\omega t - \omega D + \Phi) \quad \dots 2
\]

where \(V_1\) is input signal and \(V_2\) is output signal that is differed by phase angle \(\theta = - \omega D \quad \dots 3\)

In phase notation:

\[
V_2 = V_1(t - D)
\]

Taking Laplace Transform,

\[
V_2(s) = V_1(s) \cdot e^{-sD}
\]

For normalized value of delay (\(D = 1\)), we can assume that \(D = 1\)

\[
V_2(s) = V_1(s) \cdot e^{-s}
\]

\[
\frac{V_2(s)}{V_1(s)} = e^{-s} = e^{-j\omega}
\]

This is the required transfer function for constant delay but it’s not possible to realize \(e^{-s}\) by lumped circuit elements. The best method is to use Polynomial.

i.e.,

\[
T(s) = \frac{N(s)}{D(s)}
\]

For \(s = j \omega\)

\[
T(j \omega) = \frac{N(j \omega)}{D(j \omega)}
\]

\(T(j \omega)\) can be represented by its real and imaginary terms:

\[
T(j \omega) = R(\omega) + j X(\omega)
\]

Where,

\[
\theta = \tan^{-1} \left( \frac{X(\omega)}{R(\omega)} \right)
\]

From equation 3, we have

\[
\theta = - \omega D \quad \text{(which is known as Phase delay)}
\]

\[
D = - \frac{\theta}{\omega} = -1 \cdot \tan^{-1} \left( \frac{X(\omega)}{R(\omega)} \right)
\]

For small values of delay \(D\),

\[
D = - \frac{d\theta}{d\omega}
\]

\[
D = - \frac{1}{1 + \left( \frac{X}{R} \right)^2} \cdot \frac{d}{d\omega} \left( \frac{X}{R} \right) 
= - \frac{R^2 X^2 + R^2 \cdot \left[ R \frac{dX}{d\omega} - X \frac{dR}{d\omega} \right]}{R^2}
\]

\[
D = - \frac{R \frac{dX}{d\omega} - X \frac{dR}{d\omega}}{R^2 + X^2}
\]

If the phase is linear with negative slope, the magnitude is constant then delay will be constant. The obtained signal is distortion free.  

fig: Phase Delay

Here:

\[
\theta = - \omega D \quad \Rightarrow \text{Phase Delay}
\]

\[
D = - \frac{d\theta}{d\omega} \quad \Rightarrow \text{signal delay or group delay or Envelope delay}
\]

For the ideal transmission system, the magnitude of its transfer function is constant, being equal to unity for all \(\omega\), while phase delay is a linear function of \(\omega\), i.e., delay is constant as per figure above.