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Chapter 2 Discrete System Analysis – Discrete Signals. Continuous-time analog signal. Sampling of Continuous-time Signals. sampler. Output of sampler. T. How to treat the sampling process mathematically ?.

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Chapter 2 Discrete System Analysis – Discrete Signals

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Chapter 2Discrete System Analysis – Discrete Signals


Continuous-time analog signal

Sampling of Continuous-time Signals

sampler

Output of sampler

T

How to treat the sampling process mathematically ?

For convenience, uniform-rate sampler(1/T) with finite sampling duration (p) is assumed.

p

1

time

T

where p(t) is a carrier signal (unit pulse train)


carrier signal p(t)

PAM

This procedure is called a pulse amplitude modulation (PAM)

The unit pulse train is written as

By Fourier series

or

magnitude

phase


-4s

-3s

-2s

-s

0

s

2s

3s

4s


|F(j)|

1

c : Cutoff frequency

-s /2

-c

0

c

s/2


0

s

-s

frequency folding

0

-s

s


Theorem : Shannon’s Sampling Theorem

To recover a signal from its sampling, you must sample at least twice the highest frequency in the signal.

Remarks:

i) A practical difficulty is that real signals do not have Fourier

transforms that vanish outside a given frequency band.

To avoid the frequency folding (aliasing) problem,

it is necessary to filter the analog signal before sampling.

Note: Claude Shannon (1917 - 2001)


ii) Many controlled systems have low-pass filter characteristics.

iii) Sampling rates > 10 ~ 30 times of the BW of the system.

iv) For the train of unit impulses

-2s

-s

s

0


|X(j)|

|X*(j)|

|Y(j)|

Ideal filterG(j)

1

1/T

1/T

-c

0

c

-c

0

c

-c

0

c

reconstruction

Remarks :

  • Impulse response of ideal low-pass filter (non-causality)

  • Ideal low-pass filter is not realizable in a physical system.

  •  How to realize it in a physical system ? ZOH or FOH

T

sampling


ii)(Aliasing) It is not possible to reconstruct exactly a continuous-time signal in a practical control system once it is sampled.


iii)(Hidden oscillation)

If the continuous-time signal involves a frequency component equal to n times the sampling frequency

(where n is an integer), then that component may not appear in the sampled signal.


Signal Reconstruction

How to reconstruct (approximate) the original signal from the sampled signal?

-ZOH (zero-order hold)

- FOH (first-order hold)


ZOH (Zero-order Hold)

zero-order hold reconstruction

k-1

k

k+1

time

T


phase lag


Ideal low-pass filter


Remarks:

i) The ZOH behaves essentially as a low-pass filter.

ii) The accuracy of the ZOH as an extrapolator depends greatly

on the sampling frequency, .

  • iii) In general, the filtering property of the ZOH is used almost

  • exclusively in practice.


k-1

k

k+1

T

FOH (First-order Hold)


When k =0,


2

1

0

T

2T

3T

time

-1


first

zero

zero

first

Large lag(delay) in high frequency makes a system unstable


  • Remark:

  • At low frequencies, the phase lag produced by the ZOH exceeds that of FOH, but as the frequencies become higher, the opposite is true


T

Z-transform


  • Because R(z) is a power series in z-1 , the theory of power series may be applied to determine the convergence of

  • the z-transform.

ii) The series in z-1 has a radius of convergence  such that the

series converges absolutely when | z-1 |<

iii) If   0, the sequence {rk} is said to be z-transformable.


Remark: unit impulse =

Z-transform of Elementary Functions

i) Unit pulse function

ii) Unit step function


iii) Ramp function


iv) Polynomial function

v) Exponential function


vi) Sinusoidal function


Remark: Refer Table 2-1 in pp.29-30 (Ogata)

Also, refer Appendix B.2 Table in pp. 702-703(Franklin)


Correspondence with Continuous Signals

z=esT

s-plane

z-plane


-1

0

1

ImZ

ImZ

0

1

ReZ

ReZ

j

0

j

-2

1

0


ImZ

ImZ

1

ReZ

ReZ

j

0

0

fixed

j

0


Important Properties and Theorems of the z-transform

1. Linearity

2. Time Shifting


3. Convolution

4. Scaling

5. Initial Value Theorem


6. Final Value Theorem


Remark: Refer Table 2-2 in p. 38.(Ogata)

Also, refer Appendix B.1 Table in p.701(Franklin)


Discrete-time

domain

Continuous-time

domain

S-plane

Z-plane


Inverse z-transform

  • Power Series Method (Direct Division)

  • ii) Computational Method :

  • - MATLAB Approach - Difference Equation Approach

  • iii) Partial Fraction Expansion Method

  • iv) Inversion Integral Method


Example 1) Power Series Method


Example 2) Computational Method


Example 3) Partial Fraction Expansion Method

Remark:


Example 4) Inverse Integral Method :

where cis a circle with its center at the origin of the z plane such that all poles of F(z)zk-1 are inside it.


Case 1) simple pole

Case 2) m multiple poles


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