2. Agenda
Mathematical representation of impulse sampling.
The convolution integral method for obtaining the z-transform.
Examples
Properties
Inverse Z-Transform
Long Division
Partial Fraction
Solution of Difference Equation
Mapping between s-plane to z-plane
2
3. Impulse Sampling
Fictitious Sampler.
The output of the sampler is a train of impulses.
Let’s define the train of impulses
𝛿 𝑇 𝑡 =
𝑘=0
∞
𝑥 𝑘𝑇 𝛿 𝑡 − 𝑘𝑇
3
5. Impulse Sampling
The Laplace transform of 𝑥∗
𝑡
𝑋∗ 𝑠 = 𝑥 0 𝛿 𝑡 + 𝑥 𝑇 𝑒−𝑠𝑇 + ⋯ + 𝑥 𝑘𝑇 𝑒−𝑠𝑘𝑇 + ⋯
𝑋∗ 𝑠 =
𝑘=0
∞
𝑥 𝑘𝑇 𝑒−𝑠𝑘𝑇
If we define 𝑧 = 𝑒 𝑠𝑇
⟹ 𝑠 =
1
𝑇
ln 𝑧
𝑋∗ 𝑠
𝑠=
1
𝑇
ln 𝑧
= 𝑋 𝑧 =
𝑘=0
∞
𝑥 𝑘𝑇 𝑧−𝑘
The Laplace transform of sampled signal 𝑥∗ 𝑡 has been shown to be the
same as z-transform of the signal 𝑥 𝑡 if 𝑒 𝑠𝑇
is defined as z.
5
6. Data Hold Circuits
Data hold is a process of generating a continuous-time signal ℎ(𝑡) from
a discrete time sequence 𝑥∗ 𝑡 .
A hold circuit approximately reproduces the signal applied to the
sampler.
ℎ 𝑘𝑇 + 𝑡 = 𝑎 𝑛 𝑡 𝑛 + 𝑎 𝑛−1 𝑡 𝑛−1 + ⋯ + 𝑎1 𝑡 + 𝑎0
Note that the signal ℎ 𝑘𝑇 must equal 𝑥 𝑘𝑇 , hence
ℎ 𝑘𝑇 + 𝑡 = 𝑎 𝑛 𝑡 𝑛 + 𝑎 𝑛−1 𝑡 𝑛−1 + ⋯ + 𝑎1 𝑡 + 𝑥 𝑘𝑇
The simplest data-hold is obtained when 𝑛 = 0 [Zero-Order Hold (ZOH)]
ℎ 𝑘𝑇 + 𝑡 = 𝑥 𝑘𝑇
When 𝑛 = 1 [First-Order Hold (FOH)]
ℎ 𝑘𝑇 + 𝑡 = 𝑎1 𝑡 + 𝑥 𝑘𝑇
6
9. Zero-Order Hold (ZOH)
Since, ℒ ℎ1 𝑡 = 𝐻1 𝑠 = ℒ ℎ2 𝑡 = 𝐻2 𝑠
𝐻2 𝑠 =
1 − 𝑒−𝑠𝑇
𝑠
𝑘=0
∞
𝑥 𝑘𝑇 𝑒−𝑘𝑠𝑇
=
1 − 𝑒−𝑠𝑇
𝑠
𝑋∗
(𝑠) = 𝐺ℎ0 𝑠 𝑋∗
(𝑠)
Then the transfer function of the ZOH is
𝐺ℎ0 𝑠 =
1 − 𝑒−𝑠𝑇
𝑠
Thus , the real sampler and zero-order hold can be replaced by a
mathematically equivalent continuous time system that consists of an
impulse sampler and a transfer function
1−𝑒−𝑠𝑇
𝑠
.
9
10. First Order Hold (FOH)
The equation of the first order hold is ℎ 𝑘𝑇 + 𝑡 = 𝑎1 𝑡 + 𝑥 𝑘𝑇 𝑓𝑜𝑟 0 ≤ 𝑡 ≤ 𝑇
By applying the condition ℎ (𝑘 − 1)𝑇 = 𝑥 (𝑘 − 1)𝑇
The constant 𝑎1 can be determined as follows:
ℎ (𝑘 − 1)𝑇 = −𝑎1 𝑇 + 𝑥 𝑘𝑇 = 𝑥 (𝑘 − 1)𝑇
𝑎1 =
𝑥 𝑘𝑇 − 𝑥 (𝑘 − 1)𝑇
𝑇
10
11. First Order Hold (FOH)
Hence,
ℎ (𝑘 − 1)𝑇 = 𝑥 𝑘𝑇 +
𝑥 𝑘𝑇 − 𝑥 (𝑘 − 1)𝑇
𝑇
Suppose that the input 𝑥 𝑡 is unit- step function
ℎ 𝑡 = 1 +
𝑡
𝑇
1 𝑡 −
𝑡 − 𝑇
𝑇
1 𝑡 − 𝑇 − 1(𝑡 − 𝑇)
11
12. First Order Hold (FOH)
The Laplace transform of the last equation
𝐻 𝑠 =
1
𝑠
+
1
𝑇𝑠2
−
1
𝑇𝑠2
𝑒−𝑠𝑇 −
1
𝑠
𝑒−𝑠𝑇 =
1 − 𝑒−𝑠𝑇
𝑠
+
1 − 𝑒−𝑠𝑇
𝑇𝑠2
𝐻 𝑠 = 1 − 𝑒−𝑠𝑇
𝑇𝑠 + 1
𝑇𝑠2
The Laplace transform of the input 𝑥∗
𝑡 is
𝑋∗
𝑠 =
𝑘=0
∞
1 𝑘𝑇 𝑒−𝑘𝑠𝑇
=
1
1 − 𝑒−𝑠𝑇
12
13. First Order Hold (FOH)
Since, 𝐻 𝑠 = 1 − 𝑒−𝑠𝑇 𝑇𝑠+1
𝑇𝑠2 = 𝐺ℎ1 𝑠 𝑋∗ 𝑠
Hence, the transfer function of the FOH is
𝐺ℎ1 𝑠 =
𝐻 𝑠
𝑋∗ 𝑠
= 1 − 𝑒−𝑠𝑇 2
𝑇𝑠 + 1
𝑇𝑠2
𝑮 𝒉𝟏 𝒔 =
𝟏 − 𝒆−𝒔𝑻
𝒔
𝟐
𝑻𝒔 + 𝟏
𝑻
13
14. Obtaining the z-Transform by the Convolution Integral Method
Calculating 𝑋∗
𝑠 from the original
signal 𝑋 𝑠
By substituting 𝒛 for 𝒆 𝒔𝑻
to obtain 𝑋(𝑧) from the sampled signal 𝑋∗
𝑠
For simple pole
For multiple pole of order n
14
17. Reconstructing Original Signals from Sampled Signals
Sampling Theorem
If the sampling frequency is sufficiently high compared with the highest-
frequency component involved in the continuous-time signal, the amplitude
characteristics of the continuous-time signal may be preserved in the
envelope of the sampled signal.
To reconstruct the original signal from a sampled signal, there is a certain
minimum frequency that the sampling operation must satisfy.
We assume that 𝑥(𝑡) does not contain any frequency components above 𝜔1
rad/sec.
17
18. Reconstructing Original Signals from Sampled Signals
Sampling Theorem
If 𝜔𝑠 , defied as 2𝜋/𝑇 is greater than 2 𝜔1 , where 𝜔1 is the highest-
frequency component present in the continuous-time signal 𝑥(𝑡), then the
signal 𝑥(𝑡) can be reconstructed completely from the sampled signal 𝑥(𝑡).
The frequency spectrum:
18
20. Reconstructing Original Signals from Sampled Signals
Ideal Low-pass filter
The ideal filter attenuates all complementary components to zero and will
pass only the primary component.
If the sampling frequency is less than twice the highest-frequency
component of the original continuous-time signal, even the ideal filter
cannot reconstruct the original continuous-time signal.
20
21. Reconstructing Original Signals from Sampled Signals
Ideal Low-pass filter is NOT physically realizable
For the ideal filter an output is required prior to the application of the input
to the filter
21
22. Reconstructing Original Signals from Sampled Signals
Frequency response of the Zero-Order Hold.
The transfer function of the ZOH
22
24. Reconstructing Original Signals from Sampled Signals
Frequency response characteristics of the Zero-Order Hold.
The comparison of the ideal filter and the ZOH.
ZOH is a Low-pass filter, although its function is not quite good.
The accuracy of the ZOH as an extrapolator depends on the sampling
frequency.
24
25. Reconstructing Original Signals from Sampled Signals
Folding
The phenomenon of the overlap in the frequency spectra.
The folding frequency (Nyquist frequency): 𝜔 𝑁
𝜔 𝑁 =
1
2
𝜔𝑠 =
𝜋
𝑇
In practice, signals in control systems have high-frequency components, and
some folding effect will almost always exist.
25
26. Reconstructing Original Signals from Sampled Signals
Aliasing
The phenomenon that frequency component n 𝜔𝑠 ± 𝜔2 shows up at
frequency 𝜔2 when the signal 𝑥(𝑡) is sampled.
To avoid aliasing, we must either choose the sampling frequency high
enough or use a prefilter ahead of the sampler to reshape the frequency
spectrum of the signal before the signal is sampled.
26
40. Obtaining Response Between Consecutive Sampling Instants
The z-transform analysis will not give information on the reponse
between two consecutive sampling instants.
Three methods for providing a response between consecutive sampling
instants are commonly available:
1) Laplace transform method
2) Modified z-Transform method
3) State-Space method.
40
42. Realization of Digital Controllers and Digital Filters
A digital filter is a computational algorithm that converts an input
sequence of numbers into an output sequence in such a way that the
characteristics of the signal are changed in some prescribed fashion.
A digital filter processes a digital signal by passing desirable frequency
components of the digital input signal and rejecting undesirable ones.
In general, a digital controller is a form of digital filter.
In general, “Realization” means determining the physical layout for the
appropriate combination of arithmetic and storage operation.
Realization techniques are
Direct realization.
Standard realization.
Series realization.
Parallel realization.
Lader realization. 42
44. Standard Programming
In direct programming, the numerator uses a set of 𝑚 delay elements
and the denominator uses a different set of 𝑛 delay elements. Thus the
total number of delay elements used in direct programming is (𝑛 + 𝑚).
The standard programming uses a minimum number of delay elements
(𝑛).
44
47. Note:
In realizing digital controllers or digital filters, it is important to have a
good level of accuracy . Basically, three sources of errors affect the
accuracy:
1) The quantization error due to the quantization of the input signal into a
finite number of discrete levels. The quantization noise may be considered
white noise; the variance of the noise is 𝜎2
= 𝑄2
/12.
2) The error due to the accumulation of round-off errors in the arithmetic
operations in the digital system.
3) The error due to quantization of the coefficients 𝑎𝑖 and 𝑏𝑖 of the pulse
transfer function. This error may become large as the order of the pulse
transfer function is increased.
• That is, in a higher–order digital filter in direct structure, small errors in the
coefficients cause large errors in the locations of the poles and zeros of the
filter.
47
48. Decomposition Techniques
Note that the third type of error listed may be reduced by
mathematically decomposing a higher-order pulse transfer function into
a combination of lower-order pulse transfer functions.
For decomposing higher-order pulse transfer functions in order to avoid
the coefficient sensitivity problem, the following three approaches are
commonly used.
1) Series Programming.
2) Parallel Programming
3) Lader Programming
48
49. Series Programming
The 𝐺(𝑧) may be decomposed as follows:
Then the block diagram for the digital filter 𝐺(𝑧) is a series connection
of p component digital filters as shown.
49
50. Parallel Programming
The 𝐺(𝑧) is expanded using the partial
fractions as follows:
Then the block diagram for the digital
filter 𝐺(𝑧) is a parallel connection of
p component digital filters as shown.
50
51. Lader Programming
The 𝐺(𝑧) is decomposed into a continued-fractions form as follows:
The G(z) may be written as follows:
51
52. Lader Programming (Cont.)
Then the block diagram for the digital filter 𝐺(𝑧) is a Lader connection
of p component digital filters as shown.
52