Z -transform Modeling Systems and Processes (11MSP) Bohumil Kov a - - PowerPoint PPT Presentation

Matehematical tools Z-transform

Z-transform

Modeling Systems and Processes (11MSP) Bohumil Kov´ aˇ r, Jan Pˇ rikryl, Miroslav Vlˇ cek

Department of Applied Mahematics CTU in Prague, Faculty of Transportation Sciences

7th lecture 11MSP Tuesday, April 16th 2019

verze: 2019-04-15 13:29

Matehematical tools Z-transform

Table of content

1 Matehematical tools

Motivation Usage

2 Z-transform

Matehematical tools Z-transform

Mathematical tools

Motivation

We want to analyze the behavior of some system, or to design a system that has exactly specified parameters. We use

• physical model, based on physical laws
• black-box model, based on observation, identification

Analysis of the real system behavior is a complex process (the model represents one or more differential or difference equations of higher order ) ⇒ numerical solution. How to analysis simplify?

Matehematical tools Z-transform

Mathematicla tools

Usage

System design or analysis in time domain (inputs and outputs are function of time) are very laborious. Conversion to the frequency domain (inputs and outputs are a function of a complex variable called angular frequency) provides us

• a fundamentally different tool to understand the function of

the system

• often drastically reduces the complexity of mathematical

calculations needed for system analysis.

Matehematical tools Z-transform

Table of content

1 Matehematical tools 2 Z-transform

About the origin of discrete transformation Definition Properties Z-transform tables

Matehematical tools Z-transform

About the origin of discrete transformation

How a continuous-time system becomes discrete

f (t) t

Matehematical tools Z-transform

About the origin of discrete transformation

How a continuous-time system becomes discrete

f (t) t

Matehematical tools Z-transform

About the origin of discrete transformation

How a continuous-time system becomes discrete

f [n] n

Matehematical tools Z-transform

About the origin of discrete transformation

Signal sampling

The relation between the continuous function f (t) and the ideally sampled function f ∗(t) can be formally written as f ∗(t) =

∞

• n=0

f (t) ∗ δ(t − nT) =

∞

• n=0

∞

−∞

f (τ) · δ(t − nT − τ) dτ = f (nT)δ(t − nT) ≡ {fn}∞

n=0 ,

where T is the signal sampling period. From a continuous function f : R → R becomes the sequence of real values fn : N → R. In this sequence, it is customary not to provide a signal sampling

• period. We denote it f [n] and

f [n] = {f (nT)}∞

n=0.

Matehematical tools Z-transform

About the origin of discrete transformation

Signal sampling

If we are looking for the Laplace transform of f ∗(t), we get L {f ∗(t)} = ∞ f ∗(t) e−ptdt = ∞

∞

• n=0

f (t) δ(t − nT) e−ptdt =

∞

• n=0

∞ δ(t − nT) f (t) e−ptdt =

∞

• n=0

f (nT)e−pnT ≡

∞

• n=0

f [n]z−n, where we introduced a complex variable z = epT and f [n] denotes the n-th sample of the corresponding continuous function sampled with period T.

Matehematical tools Z-transform

Z-transform

Definition

Definition (Z-transform) Z-transform of the sequence f [n] it is defined by an infinite series F(z) =

∞

• n=0

f [n]z−n, which we often refer to F(z) = Z {f [n]}. The inverse-transformation has the form of an integral along the closed curve C, which contains all the singular points of the F(z). For all n = 0, 1, . . . , ∞ f [n] = 1 2πi

• C

F(z)zn−1 dz ≡ Z−1 {F(z)} .

Matehematical tools Z-transform

Properties of the Z-transform

Linearity

Theorem (Linearity) Z-transform is linear: Z

• k

akfk[n]

• =
• k

akZ {fk[n]} Z−1

• m

bmFm(z)

• =
• m

bmZ−1 {Fm(z)}

Matehematical tools Z-transform

Properties of the Z-transform

Scaling in the z-domain

Theorem (Scaling in the z-domain) For F(z) = Z {f [n]} is a−nf [n] = Z−1 {F(az)} F(a−1z) = Z {anf [n]}

Matehematical tools Z-transform

Properties of the Z-transform

Time delay

The time delay (or advance) theorems are very important for transforming difference equations into algebraic equations in the Z-plane similarly to derivatives theorem of Laplace transform for continuous-time systems. Z {f [n − m]} = z−mZ {f [n]} = z−mF(z) |∀n−m<0: f [n−m]=0 Z {f [n + m]} = zm

• Z {f [n]} −

m−1

• ν=0

f [ν]z−ν

• = zm
• F(z) −

m−1

• ν=0

f [ν]z−ν

Matehematical tools Z-transform

Properties of the Z-transform

Partial sum transformation

Theorem (Partial sum transformation) Partial sum of sequence f [n] can be transformed as Z n

• ν=0

f [ν]

• =

z z − 1F(z) Z n−1

• ν=0

f [ν]

• =

1 z − 1F(z)

Matehematical tools Z-transform

Properties of the Z-transform

Transformation of forward diference

For m = 1, 2, . . . , ∞ a diference of m-th order is ∆0f [n] = f [n], ∆1f [n] = f [n + 1] − f [n], ∆2f [n] = ∆1 ∆1f [n]

• = f [n + 2] − 2f [n + 1] + f [n]

∆mf [n] = ∆1 ∆m−1f [n]

• The Z- transformations of first two difererences are:

Z

• ∆1f [n]
• = (z − 1)F(z) − f [0]z

Z

• ∆2f [n]
• = (z − 1)2F(z) − f [0] z(z − 1) + ∆1f [0]z

Matehematical tools Z-transform

Properties of the Z-transform

Convolution

Theorem (Convolution) If F(z) = Z {f [n]} and G(z) = Z {g[n]} then, for the discrete convolution of both sequences Z {f [n] ∗ g[n]} = Z ∞

• m=0

f [n − m] · g[m]

• = F(z) · G(z)

Matehematical tools Z-transform

Properties of the Z-transform

Differentiation

Theorem (Properties of the Z-transform) Simple differentiation of F(z) will take effect on the source function f [n] as multiplication by variable n: Z {nf [n]} = −z dF(z) dz

Matehematical tools Z-transform

Z-transform tables

f [n] = Z−1 {F(z)} F(z) = Z {f [n]} δ[n] 1 1[n] 1 1 − z−1 an 1 1 − a · z−1 n z−1 (1 − z−1)2 n · an−1 z−1 (1 − az−1)2