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Lecture 19 Discrete-Time Transfer Functions

- 7 Transfer Function of a Discrete-Time Systems (2

lectures) Impulse sampler, Laplace transform of

impulse sequence, z transform. Properties of the

z transform. Examples. Difference equations and

differential equations. Digital filters. - Specific objectives for today
- Properties of the z-transform
- z-transform transform equations
- Solving difference equations

Lecture 19 Resources

- Core material
- SaS, OW, C10
- Related Material
- MIT lecture 22 23
- The discrete time transfer function

(z-transform), closely mirrors the Laplace

transform (continuous time transfer function) and

is the z-transform of the impulse response of the

difference equation.

Introduction to Discrete Time Transfer Fns

- A discrete-time LTI system can be represented as

a (first order) difference equation of the form - This is analogous to a sampled CT differential

equation - This is hard to solve analytically, and wed like

to be able to perform some form of analogous

manipulation like continuous time transfer

functions, i.e. - Y(s) H(s)X(s)

Linearity of the z-Transform

- If
- and
- Then
- This follows directly from the definition of the

z-transform (as the summation operator is linear,

see Example 3). It is easily extended to a

linear combination of an arbitrary number of

signals

ROCR1

ROCR2

ROC R1R2

Time Shifting z-Transforms

- If
- Then
- Proof
- This is very important for producing the

z-transform transfer function of a difference

equation which uses the property

ROCR

ROCR

Example Linear Time Shift

- Consider the input signal
- We know that
- So

Discrete Time Transfer Function

- Consider a first order, LTI differential equation

such as - Then the discrete time transfer function is the

z-transform of the impulse response, H(z) - As Zdn 1, taking the z-transform of both

sides of the equation (linearity we get), for the

impulse response

Discrete Time Transfer Function

- The discrete-time transfer function of an LTI

system is a rational polynomial in z. (This is

equivalent to the transfer function of a

continuous time differential system which is a

rational polynomial in s) - As usual the z-transform transfer function can

computed by either - If the difference equation is known, take the

z-transform of each sides when the input signal

is an impulse dn - If the discrete-time impulse response signal is

known, calculate the z-transform of the signal

hn. - In either case, the same result will be obtained.

Convolution using z-Transforms

- The z-transform also has the multiplication

property, i.e. - Proof is identical to the Fourier/Laplace

transform convolution and follows from

eigensystem property - Note that pole-zero cancellation may occur

between H(z) and X(z) which extends the ROC - While this is true for any two signals, it is

particularly important as H(z) represents the

transfer function of discrete-time LTI system

Example 1 First Order Difference Equation

- Calculate the output of a first order difference

equation of a input signal xn 0.5nun - System transfer function (z-transform of the

impulse response) - The (z-transform of the) output is therefore

ROC zgt0.8

Example 2 2nd Order Difference Equation

- Consider the discrete time step input signal
- to the 2nd order difference equation
- To calculate the solution, multiply and express

as partial fractions

Lecture 19 Summary

- The z-transform is linear
- There is a simple relationship for a signal

time-shift - This is fundamental for deriving the transfer

function of a difference equation which is

expressed in terms of the input-output signal

delays - The transfer function of a discrete time LTI

system is the z-transform of the systems impulse

response - It is a rational polynomial in the complex number

z. - Convolution is expressed as multiplication
- and this can be solved for particular signals and

systems

Lecture 19 Exercises

- Theory
- SaS, OW 10.20
- Matlab
- Use the inverse z-transform in the symbolic

Matlab toolbox to verify Examples 1 and 2 for a

first and second order system. See Lecture 18

for the ztrans() and iztrans() commands - Try changing the coefficients associated with

Examples 1 and 2 and verify their behaviour.

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