1. Lecture 4: Sampling and Reconstruction
Data Reconstruction (Hold)
Reading: Chapter 3 of the textbook
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2. A Typical Digital Control System
Sampling
Reconstruction
continuous-time
signal
discrete-time system
continuous-time system 𝑇
𝑟(𝑡)
A/D
Computer
D/A
Data
Hold
Plant
𝑦(𝑡) _
𝑒(𝑡)
discrete-time
signal
digital signal
discrete-time
signal
continuous-time
signal
Sensor
Sampling and data reconstruction needed to interface the digital computer with the physical world
Sampling
Continuous-time Signals
Discrete-time Signals
Data
reconstruction

3. Sampler
A sampler obtains from a continuous-time signal a discrete-time signal by sampling every 𝑇 seconds
Input is a continuous-time signal 𝑒 𝑡 , 𝑡≥0
Output is a discrete-time signal 𝑒(𝑘𝑇), 𝑘=0,1,…
Both signals have identical values at sampling moments
Not a lossless procedure
Not a time-invariant procedure
𝑒(𝑡)
𝑒(𝑘𝑇) 𝑇
𝑒(𝑡)
𝑒(𝑘𝑇) ⋯ 𝑇 2𝑇 3𝑇 4𝑇 𝑡

4. A/D Converter (ADC)
(From Wikipedia: 4-channel stereo multiplexed analog-to-digital converter WM8775SEDS made by Wolfson Microelectronics placed on a X-Fi Fatal1ty) Pro sound card.

5. Reconstruction: Zero-Order Hold
Data reconstruction needed to convert digital controller output into a continuous-time signal to drive the plant
Ideally both signals should agree at the sampling moment: 0, 𝑇, 2𝑇, …
The simplest data reconstruction is zero-order hold
Given a discrete time signal 𝑒(𝑘𝑇), 𝑘=0,1,…
A continuous-time signal 𝑒 𝑡 , 𝑡≥0, is reconstructed whose value at any time 𝑡 is equal to the value of 𝑒(𝑘𝑇) in the closest sampling moment preceding 𝑡:

6. D/A Converter (DAC) AK4396 DAC for Headphone Output
(from
An 8-Digit DAC implemented by R/2R resistor network
From Wikipedia: 8-channel digital-to-analog converter Cirrus Logic CS4382 as used in a soundcard.

7. Sampled-Data Control System
𝑒(𝑘𝑇)
𝑢(𝑘𝑇)
𝑢(𝑡)
𝑒(𝑡)
Digital controller
Plant
𝑦(𝑡)
Sampler
Data hold _
Does there exist a transfer function from 𝑒(𝑡) to 𝑢(𝑡)?
What about from 𝑒(𝑘𝑇) to 𝑢(𝑘𝑇)?
Let us start from the simplest case: digital controller does nothing
𝑒(𝑡)
𝑒 (𝑡)
𝑒(𝑘𝑇)
Sampler
Data hold
sampler and hold

8. Sampler and Zero-Order Hold
What is the effect of sampler and hold in frequency domain?
𝑒(𝑡)
𝑒 (𝑡)
𝑒(𝑘𝑇)
Sampler
Data hold
𝐸(𝑠)
𝐸(𝑧)
𝐸 (𝑠)
𝑒 (𝑡)
𝑒(𝑡) ⋯ 𝑇 2𝑇 3𝑇 4𝑇

9. Sampler and Zero-Order Hold
Laplace transform of 𝑒(𝑡):
𝐸 𝑠 =ℒ[𝑒 𝑡 ]
𝑧-transform of 𝑒(𝑘𝑇):
𝐸 𝑧 = 𝑘=0 ∞ 𝑒 𝑘𝑇 𝑧 −𝑘
Laplace transform of 𝑒 𝑡 :
𝑒 𝑡 =𝑒 0 1 𝑡 −1 𝑡−𝑇 +𝑒 𝑇 1 𝑡−𝑇 −1 1−2𝑇 +⋯
= 𝑘=0 ∞ 𝑒 𝑘𝑇 𝑒 −𝑘𝑇𝑠 ⋅ 1− 𝑒 −𝑇𝑠 𝑠
Independent of signal 𝑒(𝑡)
𝐸 ∗ 𝑠 =𝐸 𝑧 ​ 𝑧= 𝑒 𝑇𝑠
𝑒 (𝑡)
𝑒(𝑘𝑇) ⋯ 𝑇 2𝑇 3𝑇 4𝑇

10. Sampler and Zero-Order Hold (cont.)
Sampler and zero-order hold can be broken up in two steps:
𝐸 𝑠 𝑇
𝐸 ∗ 𝑠
𝐸 𝑠
𝐺 ℎ0 𝑠 = 1− 𝑒 −𝑇𝑠 𝑠
Ideal sampler
0-th order data hold
𝐸 ∗ 𝑠 =𝑒 0 +𝑒 𝑇 𝑒 −𝑇𝑠 +𝑒 2𝑇 𝑒 −2𝑇𝑠 +⋯
In time domain: 𝑇
𝑒 𝑡
𝑒 ∗ 𝑡
𝑒 𝑡 𝑇
where 𝑒 ∗ 𝑡 is a string of impulses whose amplitudes are modulated by 𝑒(𝑡):
𝑒 ∗ 𝑡 =𝑒 0 𝛿 𝑡 +𝑒 𝑇 𝛿 𝑡−𝑇 +𝑒 2𝑇 𝛿 𝑡−2𝑇 +⋯

11. Ideal Sampler 𝑒 ∗ 𝑡 =𝑒 0 𝛿 𝑡 +𝑒 𝑇 𝛿 𝑡−𝑇 +𝑒 2𝑇 𝛿 𝑡−2𝑇 +⋯
𝑒 𝑡
𝑒 ∗ 𝑡
𝑒 ∗ 𝑡 =𝑒 0 𝛿 𝑡 +𝑒 𝑇 𝛿 𝑡−𝑇 +𝑒 2𝑇 𝛿 𝑡−2𝑇 +⋯
=𝑒 𝑡 ⋅[𝛿 𝑡 +𝛿 𝑡−𝑇 +𝛿 𝑡−2𝑇 +⋯]
𝛿 𝑇 𝑡
𝛿 𝑇 𝑡
𝛿 𝑇 𝑡 is a string of unit impulses T seconds apart:
𝛿 𝑇 𝑡 𝑇 2𝑇 3𝑇 4𝑇
𝑒 𝑡
𝑒 ∗ 𝑡
Impulse
Modulator

12. Remarks on Ideal Sampler
Not a physical entity (introduced for theoretical convenience)
Linear but not time-invariant, cannot be modeled by a transfer function
Can map different 𝑒(𝑡) to the same 𝑒 ∗ (𝑡)
If 𝑒(𝑡) is discontinuous at a sampling time 𝑘𝑇, choose
If 𝑒(𝑡) has impulses at sampling times, then 𝑒 ∗ 𝑡 is not well defined
𝑒 ∗ 𝑡 =𝑒 𝛿 𝑡 +𝑒 𝑇 + 𝛿 𝑡−𝑇 +⋯

13. Sampler and Hold: First Perspective
In the time domain: 𝑇
𝑒(𝑡)
𝑒(𝑘𝑇)
𝑒 (𝑡)
Zero-Order Hold
Sampler
In the frequency domain (a mixture of z- and Laplace transforms): 𝑇
𝐸 𝑠
𝐸 𝑧
𝐸 𝑠
Zero-Order Hold
Sampler
𝐸 𝑧 = 𝑘=0 ∞ 𝑒 𝑘𝑇 𝑧 −𝑘
is called the z-transform of 𝐸(𝑠)

14. Sampler and Hold: Second Perspective
In the time domain: 𝑇
𝑒(𝑡)
𝑒 ∗ (𝑡)
𝑒 (𝑡)
Ideal sampler
Data hold
In the frequency domain: 𝑇
𝐸 𝑠
𝐸 ∗ 𝑠
𝐸 𝑠
𝐺 ℎ0 𝑠 = 1− 𝑒 −𝑇𝑠 𝑠
𝐸 ∗ 𝑠 = 𝑘=0 ∞ 𝑒 𝑘𝑇 𝑒 −𝑘𝑇𝑠
is called the star-transform of 𝐸(𝑠)

15. Relating the Two Perspectives
𝐸(𝑧)
sampler
z-transform
𝐸 ∗ 𝑠 =𝐸 𝑧 ​ 𝑧= 𝑒 𝑇𝑠
𝐸(𝑠)
𝐸 (𝑠)
Ideal sampler
Zero-order hold
star transform
𝐺 ℎ0 𝑠 = 1− 𝑒 −𝑇𝑠 𝑠
𝐸 ∗ (𝑠)

16. Finding Star-Transform of 𝐸(𝑠)
Given E(s), find
Approach I: If 𝐸 𝑠 is a rational function (and …), then
where the summation is over all poles of E(¸)
Approach II: (see Appendix III)
where 𝜔 𝑠 = 2𝜋 𝑇 is called radian sampling frequency
𝐸 ∗ 𝑠 = 𝑘=0 ∞ 𝑒 𝑘𝑇 𝑒 −𝑘𝑇𝑠
𝐸 ∗ 𝑠 = [𝑟𝑒𝑠𝑖𝑑𝑢𝑒 𝑜𝑓 𝐸 𝜆 1 1− 𝑒 −𝑇(𝑠−𝜆) ]
𝐸 ∗ 𝑠 = 1 𝑇 𝑘=−∞ ∞ 𝐸 𝑠+𝑗𝑘 𝜔 𝑠 + 𝑒(0) 2

17. Finding 𝑧-Transform of 𝐸(𝑠)
Given E(s), find
Approach I: If 𝐸 𝑠 is a rational function (and …), then
Approach II: find 𝐸 ∗ (𝑠) first, then
Approach III: consult the 𝑧-transform table on pp. 676
𝐸 𝑧 = 𝑘=0 ∞ 𝑒 𝑘𝑇 𝑧 −𝑘
𝐸 𝑧 = [𝑟𝑒𝑠𝑖𝑑𝑢𝑒 𝑜𝑓 𝐸 𝜆 1 1− 𝑧 −1 𝑒 𝑇𝜆 ]
𝐸 𝑧 = 𝐸 ∗ 𝑠 ​ 𝑒 𝑇𝑠 →𝑧

18. Examples
𝐸 𝑠 = 1 𝑠
𝐸 𝑠 = 1 𝑠+2

19. Example 𝐸 𝑠 = 1 𝑠+𝑎 2 𝐸 𝑧 = 𝑇𝑧 𝑒 −𝑎𝑇 𝑧− 𝑒 −𝑎𝑇 2 𝐸 𝑠 = 1 (𝑠+1)(𝑠+2)
𝐸 𝑠 = 1 𝑠+𝑎 2
𝐸 𝑧 = 𝑇𝑧 𝑒 −𝑎𝑇 𝑧− 𝑒 −𝑎𝑇 2
𝐸 𝑠 = 1 (𝑠+1)(𝑠+2)
𝐸 𝑠 = 𝜔 2 𝑠 2 + 𝜔 2
𝐸 ∗ 𝑠 = 𝑒 −𝑇𝑠 sin 𝜔𝑇 1−2 𝑒 −𝑇𝑠 cos 𝜔𝑇 + 𝑒 −2𝑇𝑠

20. Effect of Time Delay on Star-Transform
If 𝑡0=𝑛𝑇 is an integer multiple of the sampling period 𝑇, then
For general 𝑡0, the above is not true
𝐸 ∗ 𝑠 = 𝑒 −𝑛𝑇𝑠 𝐸 1 ∗ 𝑠

21. Periodicity Property of 𝐸 ∗ (𝑠)
𝐸 ∗ (𝑠) is periodic in 𝑠 with period 𝑗 𝜔 𝑠 =𝑗 2𝜋 𝑇
𝐸 ∗ 𝑠+𝑗𝑚 𝜔 𝑠 = 𝐸 ∗ (𝑠)
for integer 𝑚

22. Strings of Poles of 𝐸 ∗ (𝑠)
If 𝐸(𝑠) has a pole at 𝑝, then 𝐸 ∗ (𝑠) has poles at 𝑝±𝑗𝑚 𝜔 𝑠
𝐸 ∗ (𝑠)
𝐸(𝑠)
𝑗 𝜔 𝑠 2 × × ×
Primary strip
− 𝑗 𝜔 𝑠 2 ×

23. Data Reconstruction

24. Sampler and Hold: Second Perspective
In the time domain: 𝑇
𝑒(𝑡)
𝑒 ∗ (𝑡)
𝑒 (𝑡)
Ideal sampler
Data hold
In the frequency domain: 𝑇
𝐸 𝑠
𝐸 ∗ 𝑠
𝐸 𝑠
𝐺 ℎ0 𝑠 = 1− 𝑒 −𝑇𝑠 𝑠

25. Data Reconstruction in Frequency Domain
𝑒(𝑡)
𝑒 ∗ (𝑡)
𝑒 (𝑡)
𝐸(𝑗𝜔)
𝐸 ∗ (𝑗𝜔)
𝐸 (𝑗𝜔)
Ideal sampling
Data reconstruction
𝐸 ∗ 𝑗𝜔 = 1 𝑇 𝑘=−∞ ∞ 𝐸 𝑗𝜔+𝑗𝑘 𝜔 𝑠 + 𝑒(0) 2
𝐸 𝑗𝜔 = 𝐺 ℎ0 𝑗𝜔 𝐸 ∗ (𝑗𝜔)
(periodic extension in freq. domain)
(data hold transfer function)
Shannon’s Sampling Theorem: 𝑒(𝑡) can be uniquely reconstructed from 𝑒 ∗ (𝑡) if 𝐸(𝑗𝜔) has no frequency component greater than 𝜔 𝑠 2 = 𝜋 𝑇
Need to sample at least twice as fast as the highest frequency in 𝐸(𝑗𝜔)
Perfect reconstruction by the ideal low pass filter (physically infeasible)

26. Zero-Order Hold
A physically feasible (causal) data hold transfer function is
Frequency (magnitude) response
𝐺 ℎ0 𝑠 = 1− 𝑒 −𝑇𝑠 𝑠 1
with impulse response 𝑇
|𝐺 ℎ0 𝑗𝜔 |= 1− 𝑒 −𝑗𝜔𝑇 𝑗𝜔 =𝑇 sin 𝜋𝜔 𝜔 𝑠 𝜋𝜔 𝜔 𝑠

27. First-Order Hold
Idea: Given the sampled data 𝑒 𝑘𝑇 , 𝑘=0,1,…, reconstruct the signal between sampling times by extrapolating two previous data
𝑒 𝑡 =𝑒 𝑘𝑇 + 𝑒 𝑘𝑇 −𝑒 𝑘−1 𝑇 𝑇 𝑡−𝑘𝑇 , 𝑘𝑇≤𝑡< 𝑘+1 𝑇
First-order hold is causal

28. Transfer Function of First-Order Hold
Impulse response 2 1 2𝑇 𝑇 −1
ℎ 𝑡 =1 𝑡 + 𝑡 𝑇 ⋅1 𝑡 −2⋅1 𝑡−𝑇 − 2 𝑡−𝑇 𝑇 1 𝑡−𝑇
+1 𝑡−2𝑇 + 𝑡−2𝑇 𝑇 ⋅1 𝑡−2𝑇
⇒ 𝐺 ℎ1 𝑠 = 1+𝑇𝑠 𝑇 1− 𝑒 −𝑇𝑠 𝑠 2

29. Fractional-Order Holds
𝑒 𝑡 =𝑒 𝑘𝑇 +𝜂 𝑒 𝑘𝑇 −𝑒 𝑘−1 𝑇 𝑇 𝑡−𝑘𝑇 , 𝑘𝑇≤𝑡< 𝑘+1 𝑇
where the parameter 𝜂∈[0,1]
Impulse response 2 1 2𝑇 𝑇 −1
𝐺 ℎ𝜂 𝑠 = 1−𝜂 𝑒 −𝑇𝑠 1− 𝑒 −𝑇𝑠 𝑠 + 𝜂 𝑇 𝑠 − 𝑒 −𝑇𝑠 2