Digital System Frequency Response and Modeling 1 Lecture Outline Sampling Theorem Frequency Response ADC Model DAC Model Combined Models 2 Sampling Theorem Sampling is necessary for the processing of analog data using digital elements ID: 674692
Download Presentation The PPT/PDF document "Digital and Non-Linear Control" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Slide1
Digital and Non-Linear Control
Digital System Frequency Response and Modeling
1Slide2
Lecture Outline
Sampling Theorem
Frequency Response
ADC Model
DAC ModelCombined Models
2Slide3
Sampling Theorem
Sampling is necessary for the processing of analog data using digital elements.
Successful digital data processing requires that the samples reflect the nature of the analog signal and that analog signals be recoverable from a sequence of samples.
3Slide4
Sampling Theorem
Following figure shows two distinct waveforms with identical samples.
Obviously, faster sampling of the two waveforms would produce distinguishable sequences.
4Slide5
Sampling Theorem
The band limited signal with
can be reconstructed from the discrete-time waveform
if and only if the sampling angular frequency
satisfies (so sampling frequency needs to double the required reconstruction frequency)
5
Slide6
Selection of Sampling Frequency
A general signal often has a finite “effective bandwidth” beyond which its spectral components are negligible.
This allows us to treat physical signals as band limited and choose a suitable sampling rate for them based on the sampling theorem.
In practice, the sampling rate chosen is often larger than the lower bound specified in the sampling theorem.
A rule of thumb is to choose
as
6
Slide7
Frequency Response
Consider a transfer function
of an LTI system. For the discrete input
, the steady state output is
.
In other words, setting
in
gives magnitude and phase change. Refer to page 40-41 in the textbook.
7Slide8
Example 1
Compute
the steady state response for the following system
due to the sampled sinusoidal
Set
, so
the steady state response is
8Slide9
Digital Control System Modeling
A common configuration of digital control system is shown in following figure.
9Slide10
ADC Model
Assume that
ADC outputs are exactly equal in magnitude to their inputs (i.e., quantization errors are negligible)
The ADC yields a digital output instantaneously
Sampling is perfectly uniform (i.e., occur at a fixed rate)The ADC can be modeled as an ideal sampler with sampling period
T.
10
T
t
u
*
(
t
)
0
t
u
(
t
)
0Slide11
0
t
u
*
(
t
)
t
u
(
t
)
0
T
t
δ
T
(
t
)
0
×
=
modulating
pulse(carrier)
modulated
wave
Modulation
signal
u
(
t
)
u*
(
t
)
Sampling ProcessSlide12
DAC Model
Assume that
DAC outputs are exactly equal in magnitude to their inputs.
The DAC yields an analog output instantaneously.
DAC outputs are constant over each sampling period (ZOH).
The input-output relationship of the DAC is given by
12
u
(k)
u
(
t
)
u
h
(
t
)Slide13
DAC Model
Unit impulse response of ZOH
13Slide14
DAC Model
As shown in figure the impulse response is a unit pulse of width T.
A pulse can be represented as a positive step at time zero followed by a negative step at time
T
.
Using the Laplace transform of a unit step and the time delay theorem for Laplace transforms,
14
Slide15
DAC Model
Thus, the transfer function of the ZOH is
15
Slide16
DAC, Analog Subsystem, and ADC Combination
Transfer Function
The cascade of a DAC, analog subsystem, and ADC is shown in following figure. The time difference between the negative step and positive step is T.
16Slide17
DAC and
Analog Subsystem
Using the DAC model, and assuming that the transfer function of the analog subsystem is
G
(s), the transfer function of the DAC and analog subsystem cascade is
17
Slide18
DAC and
Analog Subsystem
The corresponding impulse response is
The impulse response is the analog system step response minus a second step response delayed by one sampling period.
18
Slide19
DAC and Analog Subsystem
19
Slide20
DAC and
Analog Subsystem
Inverse Laplace yields
Where
20
Slide21
DAC, Analog Subsystem, and ADC Combination
Transfer Function
The analog response is sampled to give the sampled impulse response
(ADC
part)
By
z
-transform, we can obtain the
z
-transfer function of the DAC (zero-order hold), analog subsystem, and ADC (ideal sampler) cascade.
21
Slide22
DAC, Analog Subsystem, and ADC Combination
Transfer Function
Z-Transform is given as
The * in above equation is to emphasize that sampling of a time function is necessary before
z
-transformation.
Having made this point, the equation can be rewritten more conveniently as
22
Slide23
Example 2
Find G
ZAS
(
z) for the transfer function of the system given as
Rewrite transfer function in standard form
23
Solution
Slide24
Example 2
Where
and
Now we know
Therefore,
The corresponding partial fraction expansion is
24
Slide25
Example 2
Using the
z
-transform table, the desired
z-domain transfer function is
25
Slide26
Example 3
Find G
ZAS
(
z) for the transfer function of the system given as
Rewrite transfer function in standard form
26
Solution
Slide27
Example 3
Where
and
Now we know
Therefore,
The corresponding partial fraction expansion is
27
Slide28
Example 3
The desired
z
-domain transfer function can be obtained as
28