2. Discrete-Time Transfer Functions
Chapter 5
Discrete-Time Process Models
Discrete-Time Transfer Functions
Now let us calculate the transient response of a combined discrete-time and continuous-time system, as shown below.
The input to the continuous-time system G(s) is the signal:
The system response is given by the convolution integral:

3. Discrete-Time Transfer Functions
Chapter 5
Discrete-Time Process Models
Discrete-Time Transfer Functions
With
For 0 ≤ τ ≤ t,
We assume that the output sampler is ideally synchronized with the input sampler.
The output sampler gives the signal y*(t) whose values are the same as y(t) in every sampling instant t = jTs.
Applying the Z-transform yields:

4. Discrete-Time Transfer Functions
Chapter 5
Discrete-Time Process Models
Discrete-Time Transfer Functions
Taking i = j – k, then:
For zero initial conditions, g(iTs) = 0, i < 0, thus:
where
Discrete-time Transfer Function
The Z-transform of Continuous-time Transfer Function G(s)
The Z-transform of Input Signal u(t)

5. Discrete-Time Transfer Functions
Chapter 5
Discrete-Time Process Models
Discrete-Time Transfer Functions
When the sample-and-hold device is assumed to be a zero-order hold, then the relation between G(s) and G(z) is

6. Chapter 5
Discrete-Time Process Models
Example
Find the discrete-time transfer function of a continuous system given by:
where

7. Approximation of Z-Transform
Chapter 5
Discrete-Time Process Models
Approximation of Z-Transform
Previous example shows how the Z-transform of a function written in s-Domain can be so complicated and tedious.
Now, several methods that can be used to approximate the Z-transform will be presented.
Consider the integrator block as shown below:
The integration result for one sampling period of Ts is:

8. Approximation of Z-Transform
Chapter 5
Discrete-Time Process Models
Approximation of Z-Transform
Forward Difference Approximation (Euler Approximation)
The exact integration operation presented before will now be approximated using Forward Difference Approximation.
This method follows the equation given as:
Taking the Z-transform of the above equation:
while
Thus, the Forward Difference Approximation is done by taking or

9. Approximation of Z-Transform
Chapter 5
Discrete-Time Process Models
Approximation of Z-Transform
Backward Difference Approximation
The exact integration operation will now be approximated using Backward Difference Approximation.
This method follows the equation given as:
Taking the Z-transform of the above equation:
while
Thus, the Forward Difference Approximation is done by taking or

10. Approximation of Z-Transform
Chapter 5
Discrete-Time Process Models
Approximation of Z-Transform
Trapezoidal Approximation (Tustin Approximation, Bilinear Approximation)
The exact integration operation will now be approximated using Backward Difference Approximation.
This method follows the equation given as:
Taking the Z-transform,
while
Thus, the Forward Difference Approximation is done by taking or

11. Example Find the discrete-time transfer function of
Chapter 5
Discrete-Time Process Models
Example
Find the discrete-time transfer function of
for the sampling time of Ts = 0.5 s, by using (a) ZOH, (b) FDA, (c) BDA, (d) TA.
(a) ZOH
(b) FDA

12. Chapter 5
Discrete-Time Process Models
Example
(c) BDA
(d) TA

13. Input-Output Discrete-Time Models
Chapter 5
Discrete-Time Process Models
Input-Output Discrete-Time Models
A general discrete-time linear model can be written in time domain as:
where m and n are the order of numerator and denominator, k denotes the time instant, and d is the time delay.
Defining a shift operator q–1, where:
Then, the first equation can be rewritten as: or or

14. Input-Output Discrete-Time Models
Chapter 5
Discrete-Time Process Models
Input-Output Discrete-Time Models
The polynomials A(q–1) and B(q–1) are in descending order q–1, completely written as follows:

15. Chapter 5
Discrete-Time Process Models
Homework 6
(a) Find the discrete-time transfer functions of the following continuous-time transfer function, for Ts = 0.25 s and Ts = 1 s. Use the Forward Difference Approximation
(b) Calculate the step response of both transfer functions for 0 ≤ t ≤ 5 s.
(c) Compare the step response of both transfer functions with the step response of the continuous-time transfer function G(s) in one plot.