Sampling

1. AE491
Special Topics:
Digital Techniques in
Aerospace
Engineering
Semester 231
Dr. Syed Saad Azhar Ali
75-110
syed.ali@kfupm.edu.sa

2. SAMPLING, AD, DA, ZOH
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3. Analog to digital conversion A/D
• Essentially a sample and hold followed by encoding to a
digital signal.
• Sampling frequency in Hz is fs=1/T
• Sampling frequency in rad/sec is s=2f
• Result is discrete amplitude, discrete time signal

4. r*(t)
t
• Need to discretize analog
signals (in time and amplitude)
to handle with digital
computation.
r(t)
An analog signal r(t)
ANALOG TO DIGITAL CONVERSION A/D
PHYSICAL EXPLANATION
r(t) is sampled every T sec, by a
switch that is closed for a small
period.
r(t) r*(t)
T
r(t) r*(t)
r*(t) is the sampled signal
T
T

5. RECALL ON DELAYED OR SHIFTED SIGNAL
𝞭 (t-T)
T
T
T
𝞭 (t)
t

6. DELAYED IMPULSES -- PULSE TRAIN
𝛿 𝑡
𝛿 𝑡 − 𝑇
𝛿 𝑡 − 2𝑇
𝛿 𝑡 − 3𝑇
⋮
+
⟹ 𝛿 𝑡 + 𝛿 𝑡 − 𝑇 + 𝛿 𝑡 − 2𝑇 + 𝛿 𝑡 − 3𝑇 + ⋯
Impulse Function, e.g
amplitude 10 units
Delayed by T seconds
Delayed by 2T seconds
Delayed by 3T seconds
(Adding the above signals)
Also called a pulse train
𝑝 𝑡 =
𝐾=0
∞
𝛿 𝑡 − 𝐾𝑇

7. Need to discretize analog signals (in time and amplitude) to handle with
digital computation.
t
Sampled signal, r*(t) = r(t) x p(t)
r*(t)
The sampled-data signal
resulting from multiplying r(t)
by p(t)
ANALOG TO DIGITAL CONVERSION A/D
MATHEMATICAL EXPLANATION
× =
t
r(t)
t
p(t)

8. 






k
k
kT
t
kT
r
t
r
0
)
(
)
(
)
( 
ANALOG TO DIGITAL CONVERSION A/D
t
r(t)
An analog signal r(t)
t
å
¥
=
=
*
-
=
k
k
kT
t
kT
r
t
r
0
)
(
)
(
)
( d
å
¥
=
=
*
-
=
k
k
kT
t
kT
r
t
r
0
)
(
)
(
)
( d
The amplitude at the kTth instant is

9. Digital to Analog conversion D/A
• Essentially a sample and hold circuit. Usually uses a
zero-order-hold (ZOH) to keep output fixed at latest
digital value until the next sample time arrives
• The ZOH introduces a very significant time delay unless
the sampling frequency is very high
f(kT) m(t)
⟹ results in an analog signal
(continuous time-discrete value)

10. Zero Order Hold (ZOH)
• A ZOH is a device that holds the value it receives from the
sampler. It captures and remembers that value for the following T
seconds, where T is the sampling period (=1/sampling rate).
• We need to find the impact of adding a transfer function

11. Determine the closed-loop transfer function, Y(s)/R(s).
a. The sampled-data control system of a process in its closed-loop form is as
shown.
)
(
1
)
(
1
s
s
s
G


ZOH G(s) Y(s)
R(s)
T=1 sec
+
-
Example (this is from a later slide, so don’t worry at the moment)

12. Zero Order Hold (ZOH)
• A ZOH is a device that holds the value it receives from the
sampler. It captures and remembers that value for the following T
seconds, where T is the sampling period (=1/sampling rate).
• We need to find the impact of adding a transfer function
• When we apply an impulse to a system, the Laplace transform of
the resulting output is the system’s transfer function.
• We use this concept to derive the transfer function for a ZOH.
• The sampler and the resulting output appears as follows:

13. Zero Order Hold
r(t) r*(t)
T
r(t) r*(t)
r(t) r*(t)
T
r(t) r*(t)
m(t)
𝑟 𝑡 = 𝛿 𝑡
m(t)
Sample (record the value) and holding it
Lets understand this box or the hold function

14. The hold function
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u(t-3) or in this case with T=3 u(t-T)
u(t)
m(t) = u(t) - u(t-T)
The hold function can be expressed as,
𝑚 𝑡 = 𝑢 𝑡 − 𝑢 𝑡 − 𝑇
1
𝑠
−
1
𝑠
𝑒−𝑠𝑇
𝑀(𝑠) =
1 − 𝑒−𝑠𝑇
𝑠
𝑀 𝑠 =
• Any delayed function
results in 𝑒−𝑠𝑇
in the
Laplace domain.
• 𝑓 𝑡 ⇌ 𝐹 𝑠
• 𝑓 𝑡 − 𝑇 ⇌ 𝐹 𝑠 𝑒−𝑠𝑇
• Check Laplace transform
properties

15. Zero-Order Hold
The output, m(t), is expressed as,
the transfer function of ZOH, call it , GZOH (s), is:
r(t) r*(t)
T
r(t) r*(t)
r(t) r*(t)
T
r(t) r*(t)
m(t)
𝑚 𝑡 = 𝑢 𝑡 − 𝑢 𝑡 − 𝑇
𝑀 𝑠 =
1
𝑠
−
1
𝑠
𝑒−𝑠𝑇
𝑀(𝑠) =
1 − 𝑒−𝑠𝑇
𝑠
For an impulse input, [𝑟 𝑡 = 𝛿 𝑡 ⟺ 𝑅 𝑠 = 1],
𝐺𝑍𝑂𝐻 =
𝑀(𝑠)
𝑅(𝑠)
=
1 − 𝑒−𝑠𝑇
𝑠
1
=
1 − 𝑒−𝑠𝑇
𝑠

16. Why do we need digitalization… ?
• Suggest any aerospace system
• Present a general model/block diagram/dynamic equation
• Discuss
• Suggest reasons, why is there a need to digitalize the system
• Submit to BlackBoard
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17. Why Digital Signal and Digital Systems…?
• Accuracy
• Design Flexibility
• Speed
• Cost
• Storage
• Implementation errors

18. In This lecture we discussed…
• What and why of digital system
• Continuous and discrete time system/signal
• Basic block diagram of digital system
• Some basic signals
• Introduction to A/D and D/A
• Zero Order Hold
• Physical and mathematical representation
• Finally, conclusion on the advantages of using digital systems