Chapter 2 Discrete System Analysis – Discrete Signals

Sampling of Continuous-time Signals sampler Continuous-time analog signal 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)

This procedure is called a pulse amplitude modulation (PAM) carrier signal p(t) PAM The unit pulse train is written as By Fourier series or magnitude phase

s 2s 3s 4s -s -2s -3s -4s

c s/2 -c -s /2 1 |F(j)| c : Cutoff frequency

-s s frequency folding

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

Impulse response of ideal low-pass filter (non-causality) 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 Ideal filter G(j) T c -c 1 |X(j)| 1/T |X*(j)| |Y(j)| reconstruction 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.

FOH (First-order Hold) k-1 k k+1 T

When k =0,

2 1 T 2T 3T time -1

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

Z-transform T

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.

Z-transform of Elementary Functions i) Unit pulse function Remark: unit impulse = 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

j ReZ ImZ ① ② ③ ④ ⑤ 1 -1 ③ ② ①  ④ ⑤ j ReZ ImZ 1 -2 1 

j ReZ ImZ  fixed j ReZ ImZ 1 

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)

Continuous-time domain Discrete-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 c is 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