What is the transfer function for this system?

[Dustinsfl]

In a long-distance telephone communication, an echo is sometimes encountered due to the transmitted signal being reflected at the receiver, sent back down the line, reflected again at the transmitter, and returned to the receiver. The impulse response for a system that modles this effect is shown in figure, where we have assumed tat only one echo is received. The parameter \(T\) corresponds to the one-way travel time along the communication channel, and the parameter \(\alpha\) represents the attenuation in amplitude between transmitter and receiver.
View attachment 2099
Determine the system function \(H(s)\) and associated region of convergence for the system.

How can I determine a transfer function from this?

 

[chisigma]

In a long-distance telephone communication, an echo is sometimes encountered due to the transmitted signal being reflected at the receiver, sent back down the line, reflected again at the transmitter, and returned to the receiver. The impulse response for a system that modles this effect is shown in figure, where we have assumed tat only one echo is received. The parameter \(T\) corresponds to the one-way travel time along the communication channel, and the parameter \(\alpha\) represents the attenuation in amplitude between transmitter and receiver.
View attachment 2099
Determine the system function \(H(s)\) and associated region of convergence for the system.

How can I determine a transfer function from this?

Is...

$\displaystyle h(t) = \alpha\ \delta(t - T) + \alpha^{3}\ \delta (t - 3\ T)\ (1)$

... so that...

$\displaystyle H(s) = \alpha\ e^{- s\ T} + \alpha^{3}\ e^{- 3\ s\ T}\ (2)$

Kind regards

$\chi$ $\sigma$

 

[Dustinsfl]

Is...

$\displaystyle h(t) = \alpha\ \delta(t - T) + \alpha^{3}\ \delta (t - 3\ T)\ (1)$

... so that...

$\displaystyle H(s) = \alpha\ e^{- s\ T} + \alpha^{3}\ e^{- 3\ s\ T}\ (2)$

Kind regards

$\chi$ $\sigma$

We know that exponential are never zero. How can we represent \(H\) in terms of zeros and poles? The only time I see that \(H\) can be zero is if \(\alpha = 0, \pm i\) but is that right?

 

[chisigma]

We know that exponential are never zero. How can we represent \(H\) in terms of zeros and poles? The only time I see that \(H\) can be zero is if \(\alpha = 0, \pm i\) but is that right?

The answer is very simple: in this case H(s) doesn't have neither zeros neither poles but only 'pure delays'...

Kind regards

$\chi$ $\sigma$

 

[CellCoree]

The transfer function for this system can be determined by taking the Laplace transform of the impulse response shown in the figure. This will give us the system function \(H(s)\) in the s-domain, where s is the complex frequency variable.

\(H(s) = \frac{\alpha e^{-Ts}}{1+\alpha e^{-Ts}}\)

The region of convergence for this system will depend on the values of T and \(\alpha\). Generally, the region of convergence will be the entire s-plane except for the poles of the transfer function, which are at \(s = -\frac{1}{T}\) and \(s = \infty\).

To determine the transfer function, we can also use the fact that the transfer function is the ratio of the output signal to the input signal in the frequency domain. In this case, the output signal is the received signal and the input signal is the transmitted signal. Therefore, the transfer function can be written as:

\(H(s) = \frac{\text{Received signal}}{\text{Transmitted signal}}\)

We can also represent this system as a block diagram, with the transmitted signal as the input and the received signal as the output. The transfer function can then be determined by taking the Laplace transform of the output signal and dividing it by the Laplace transform of the input signal.

In summary, the transfer function for this system is given by \(H(s) = \frac{\alpha e^{-Ts}}{1+\alpha e^{-Ts}}\) and the region of convergence is the entire s-plane except for the poles at \(s = -\frac{1}{T}\) and \(s = \infty\). This transfer function can be used to analyze and design the system for optimal performance in long-distance telephone communication.