Introduction to Digital Systems

Saint-Petersburg State UniversityFaculty of Applied Mathematics – Control Processes

Lections 13 ─ 16

prof. Evgeny I. Veremey

Part 3. Analog System Discretization

Analog Systems Discretization 1

1. The signals sampling

Sampling Period: Sampling frequency:

0 2 4 6 8 10 12 14 16 18 20-10

-5

0

5

10

n

f[n] Example:

Analog Systems Discretization 2

Uncertainties of the analog signal

restoration

Zero order hold and first order hold

restoration

Analog Systems Discretization 3

0 1 2 3 4 5 6 7 8

-1

0

1

a) Initial continuous-time function fa(t)=sin t

0 1 2 3 4 5 6 7 8

-1

0

1

b) Discrete-time function f[n]=sin 0.75n

0 1 2 3 4 5 6 7 8

-1

0

1

c) Inverse transition to continuous time

t (s)

The sampling of harmonic continues-time functions

Analog Systems Discretization 4

The sampling of harmonic continues-time functions

0 2 4 6 8 10 12 14 16 18 20

-1

-0.5

0

0.5

1

t (s)

0 2 4 6 8 10 12 14 16 18 20

-1

-0.5

0

0.5

1

t (s)

Frequencies are not comparable:

Frequencies are comparable:

Analog Systems Discretization 5

The aliasing effect

-1

0

1a) f(t)=sin0.2t

-1

0

1

b) fd[]=sin0.2

-1

0

1c) function f(t)=sin2.2t discretization

0 1 2 3 4 5 6 7 8 9 10-1

0

1d) function f(t)=-sin1.8t discretization

t (c)

Analog Systems Discretization 6

2. The base conceptions of analog systems discretization

Analog system:

Direct Euler method:

Approximate calculation of the definite integral

Inverse Euler method:

Trapezium method:

Analog Systems Discretization 7

Direct Euler method:

Inverse Euler method:

3. Discretization methods for the analog LTI systems

Cauchy formula:

Analog Systems Discretization 8

Cauchy formula:

Matrix exponent:

Analog Discrete )(tu )(tcy ]}[{ nu ]}[{ ny

H Hd

ApproximateADM

PrecisePDM

Analog Systems Discretization 9

Analog-digital matrices correspondence:

Nonsingular matrix A:

Analog Systems Discretization 10

Two ways for approximate discretization:

1. Taylor approximation

2. Pade approximation

Direct Euler method:

Analog Systems Discretization 11

Inverse Euler method

(Pade m=0, l=1):

Trapezium method

(Pade m=1, l=1):

Analog Systems Discretization 12

4. Тhe precise discretization taking into account input signal waveform

]}[{ nu )(tu )(tcy ]}[{ ny

dH

ZOH H S LTI discretization with preliminary ZOH:

Case 1. δ-functions sequence:

1

11 )()()( )()(1

n

n

nnn

t

t

tnc

ttnc detet Buxx AA

Analog Systems Discretization 13

4. Тhe precise discretization taking into account input signal waveform

]}[{ nu )(tu )(tcy ]}[{ ny

dH

ZOH H S LTI discretization with preliminary ZOH:

Case 1: δ-functions sequence:

1

11 )()()( )()(1

n

n

nnn

t

t

tnc

ttnc detet Buxx AA

Analog Systems Discretization 14

Case 2: piecewise constant input signal :

Nonsingular matrix A:

MATLAB implementation:

Analog Systems Discretization 15Case 3: piecewise exponential input signal :

Case 4: sequence of rectangular impulses :

Analog Systems Discretization 16

5. Тhe precise discretization of LTI systems with tf-models

LTI discretization with preliminary ZOH:

]}[{ nu )(tu )(tcy ]}[{ ny

dH (z)

ZOH H(s) S ]}[{ nu ]}[{ ny

Hd(z)

ZOH

Analog Systems Discretization 17