# Discrete-time system transfer function

I am attempting to calculate the discrete-time system transfer function G_HP(Z) from G(s).

The continuous transfer function, $$G(s)=\frac{5e^{-\tau s}}{s+5}$$ Where $$\tau =1$$ and $$e ^{-\tau s}$$ is the pure time delay in the system.

A given hint, $$Z[e^{-\tau s}G(s)]=z^{-k}Z[(G(s)]$$ Where $$k=\frac{\tau}{T}$$ I have already calculated the value of T, using bode plots, finding the cut off frequency.

I understand that I need to apply the z-transform to the input and output signals, using the given hint,

$$Z[e^{-\tau s}G(s)] = z^{-k}Z[G(s)]$$

Finding the Z-transform of G(s) $$Z[G(s)] = Z\left[\frac{5}{s+5}\right] = \frac{5Z}{Z-1} \frac{1}{1-(-e^{-T})} = \frac{5Z}{(Z-1)(1+e^{-T})}$$

Substituting this into the equation for $$Z[e^{-\tau s} G(s)]$$ $$Z[e^{-\tau s} G(s)] = Z^{-k} Z[G(s)] = Z^{-k} \frac{5Z}{(Z-1)(1+e^{-T})}$$

Substituting $$k=\frac{\tau}{T}$$ and simplifying: $$G_{HP}(Z) = Z[e^{-\tau s} G(s)] = Z^{-\frac{\tau}{T}} \frac{5Z}{(Z-1)(1+e^{-T})} = \frac{5Z{1-\frac{\tau}{T}}}{(Z-1)(1+e{-T})}$$

So the discrete-time system transfer function is: $$G_{HP}(Z) = \frac{5Z{1-\frac{\tau}{T}}}{(Z-1)(1+e{-T})}$$

Am I on the right page? Cheers

• Your first formula seems unusual. Commented Apr 13, 2023 at 9:27
• It's academic, however thermal systems can take this form. Commented May 10, 2023 at 5:44