# z

```z Transform System Analysis
Block Diagrams and Transfer
Functions
Just as with continuous-time systems, discrete-time systems are
conveniently described by block diagrams and transfer functions
can be determined from them. For example, from this discretetime system block diagram the difference equation can be
determined.
y [ n ] = 2 x [ n ] − x [ n − 1] − (1 / 2 ) y [ n − 1]
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Block Diagrams and Transfer
Functions
From a z-domain block diagram the transfer function can
be determined.
Y ( z ) = 2 X ( z ) − z −1 X ( z ) − (1 / 2 ) z −1 Y ( z )
Y( z )
2 − z −1
2z − 1
H (z) =
=
=
−1
X ( z ) 1 + (1 / 2 ) z
z +1/ 2
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System Stability
A system is stable if its impulse response is
absolutely summable. That requirement
translates into the z-domain requirement that
all the poles of the transfer function must lie
in the open interior of the unit circle.
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System Interconnections
Parallel
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System Interconnections
Feedback
Y( z )
H1 ( z )
H1 ( z )
H (z) =
=
=
X ( z ) 1 + H1 ( z ) H 2 ( z ) 1 + T ( z )
T ( z ) = H1 ( z ) H 2 ( z )
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Responses to Standard Signals
N (z)
If the system transfer function is H ( z ) =
the z transform
D(z)
z N (z)
of the unit-sequence response is H −1 ( z ) =
which
z − 1 D(z)
can be written in partial-fraction form as
N1 ( z )
z
Y( z ) = z
+ H (1)
D(z)
z −1
N1 ( z )
If the system is stable the transient term z
dies out
D(z)
z
and the forced response is H (1)
.
z −1
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Responses to Standard Signals
Kz
Let the system transfer function be H ( z ) =
.
z− p
z Kz
K ⎛ z
pz ⎞
Then Y ( z ) =
−
−
⎜
z − 1 z − p 1 − p ⎝ z − 1 z − p ⎟⎠
(
)
K
and y [ n ] =
1 − p n+1 u [ n ].
1− p
(
)
Let the constant K be 1 − p. Then y [ n ] = 1 − p n+1 u [ n ].
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Responses to Standard Signals
Unit Sequence Response
One-Pole System
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Responses to Standard Signals
Unit Sequence Response
Two-Pole System
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Responses to Standard Signals
N (z)
If the system transfer function is H ( z ) =
the z transform
D(z)
of the response to a cosine applied at time n = 0 is
N ( z ) z ⎡⎣ z − cos ( Ω0 ) ⎤⎦
Y( z ) =
D ( z ) z 2 − 2z cos ( Ω0 ) + 1
Let p1 = e jΩ0 . Then the system response can be written as
⎛ N1 ( z ) ⎞
y[n] = Z ⎜ z
+ H ( p1 ) cos Ω0 n + ∠ H ( p1 ) u [ n ]
⎟
⎝ D(z) ⎠
(
−1
)
and, if the system is stable, the forced response is
(
)
H ( p1 ) cos Ω0 n +  H ( p1 ) u [ n ]
a sinusoid with, generally, different magnitude and phase.
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z Transform - Laplace Transform
Relationships
Let a signal x(t) be sampled to form
x [ n ] = x ( nTs )
and impulse sampled to form
xδ ( t ) = x ( t )δ Ts ( t )
These two signals are equivalent in the sense that their
impulse strengths are the same at corresponding times and
the correspondence between times is t = nTs .
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z Transform - Laplace Transform
Relationships
Let a discrete-time system have the impulse response h [ n ] and let a
continuous-time system have the impulse response
If x [ n ] is applied to the
discrete-time system and
hδ ( t ) = ∑ n=0 h [ n ]δ ( t − nTs ) .
∞
xδ ( t ) is applied to the
continuous-time system,
their responses will be
equivalent in the sense
that the impulse strengths
are the same.
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z Transform - Laplace Transform
Relationships
The transfer function of the discrete-time system is
∞
H ( z) = ∑ h[n] z−n
n=0
and the transfer function of the continuous-time system is
∞
Hδ ( s ) = ∑ h [ n ] e− nTs s
n=0
The equivalence between them can be seen in the transformation
Hδ ( s ) = H ( z ) z→esTs
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z Transform - Laplace Transform
Relationships
The relationship z = e sTs maps points in the s plane into
corresponding points in the z plane.
Different contours in the s plane map into the same contour in
the z plane.
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z Transform - Laplace Transform
Relationships
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Simulating Continuous-Time
Systems with Discrete-time Systems
The ideal simulation of a continuous-time system by a discrete-time
system would have the discrete-time system’s excitation and response
be samples from the continuous-time system’s excitation and
response. But that design goal is never achieved exactly in real
systems at finite sampling rates.
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Simulating continuous-time Systems
with discrete-time Systems
One approach to simulation is to make the impulse response of the
discrete-time system be a sampled version of the impulse response
of the continuous-time system.
h [ n ] = h ( nTs )
With this choice, the response of the discrete-time system to a discretetime unit impulse consists of samples of the response of the continuoustime system to a continuous-time unit impulse. This technique is called
impulse - invariant design.
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Simulating continuous-time Systems
with discrete-time Systems
When h [ n ] = h ( nTs ) the impulse response of the discrete-time
system is a sampled version of the impulse response of the
continuous-time system but the unit discrete-time impulse is not
a sampled version of the unit continuous-time impulse.
A continuous-time impulse cannot be sampled. First, as a practical
matter the probability of taking a sample at exactly the time of
occurrence of the impulse is zero. Second, even if the impulse were
sampled at its time of occurrence what would the sample value be?
The functional value of the impulse is not defined at its time of
occurrence because the impulse is not an ordinary function.
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Simulating continuous-time Systems
with discrete-time Systems
In impulse-invariant design, even though the impulse response is a
sampled version of the continuous-time system’s impulse response
that does not mean that the response to samples from any arbitrary
excitation will be a sampled version of the continuous-time
system’s response to that excitation.
All design methods for simulating continuous-time systems with
discrete-time systems are approximations and whether or not the
approximation is a good one depends on the design goals.
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Sampled-Data Systems
Real simulations of continuous-time systems by discrete-time
systems usually sample the excitation with an ADC, process
the samples and then produce a continuous-time signal with a
DAC.
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Sampled-Data Systems
An ADC simply samples a signal and produces numbers. A
common way of modeling the action of a DAC is to imagine
the discrete-time impulses in the discrete-time signal which
drive the DAC are instead continuous-time impulses of the
same strength and that the DAC has the impulse response of a
zero-order hold.
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Sampled-Data Systems
The desired equivalence between a continuous-time and a discrete-time system is illustrated below.
The design goal is to make y d ( t ) look as much like y c ( t ) as
possible by choosing h [ n ] appropriately.
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Sampled-Data Systems
Consider the response of the continuous-time system not to the
actual signal x(t) but rather to an impulse-sampled version of it
xδ ( t ) =
∞
∑ x ( nT )δ (t − nT ) = x (t )δ (t )
s
s
Ts
n=−∞
The response is
y ( t ) = h ( t ) ∗ xδ ( t ) = h ( t ) ∗
∞
∞
∑ x [ m ]δ (t − mT ) = ∑ x [ m ] h (t − mT )
s
m=−∞
s
m=−∞
where x [ n ] = x ( nTs ) and the response a the nth multiple of Ts
is y ( nTs ) =
∞
∑ x [ m ] h (( n − m ) T ) .
s
m=−∞
The response of a discrete-time system with h [ n ] = h ( nTs ) to the
excitation x [ n ] = x ( nTs ) is y [ n ] = x [ n ] ∗ h [ n ] =
∞
∑ x[ m]h[n − m] .
m=−∞
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Sampled-Data Systems
The two responses are equivalent in the sense that the values
at corresponding discrete-time and continuous-time times are
the same. 12/29/10
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Sampled-Data Systems
Modify the continuous-time system to reflect the last analysis.
Then multiply the impulse responses of both systems by Ts
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Sampled-Data Systems
In the modified continuous-time system
∞
⎡ ∞
⎤
y ( t ) = xδ ( t ) ∗Ts h ( t ) = ⎢ ∑ x ( nTs )δ ( t − nTs ) ⎥ ∗ h ( t ) Ts = ∑ x ( nTs ) h ( t − nTs ) Ts
⎣ n=−∞
⎦
n=−∞
In the modified discrete-time system
y[n] =
∞
∞
∑ x [ m ] h [ n − m ] = ∑ x [ m ] T h (( n − m ) T )
s
m=−∞
s
m=−∞
where h [ n ] = Ts h ( nTs ) and h ( t ) still represents the impulse
response of the original continuous-time system. Now let Ts
approach zero.
lim y ( t ) = lim
Ts →0
Ts →0
∞
∑ x ( nT ) h (t − nT )T
s
n=−∞
s
s
∞
=
∫ x (τ ) h (t − τ ) dτ
−∞
This is the response y c ( t ) of the original continuous-time system.
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Sampled-Data Systems
Summarizing, if the impulse response of the discrete-time
system is chosen to be Ts h ( nTs ) then, in the limit as the
sampling rate approaches infinity, the response of the
discrete-time system is exactly the same as the response
of the continuous-time system.
Of course the sampling rate can never be infinite in practice.
Therefore this design is an approximation which gets better
as the sampling rate is increased.
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Sampled-Data Systems
A continuous-time system is characterized by the transfer function
1
Hs s = 2
.
s + 40s + 300
Design a discrete-time system to approximate this continuous-time
system. Use two different sampling rates f s = 10 and f s = 100 and
()
compare step responses.
The impulse response of the continuous-time system is
1 −10t
h t =
e − e−30t u t .
20
()
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) ()
29
Sampled-Data Systems
The discrete-time impulse response is
Ts −10nTs
−30nTs
h ⎡⎣ n ⎤⎦ =
e
−e
u ⎡⎣ n ⎤⎦
20
and the transfer function is its z transform
(
)
Ts ⎛
⎞
z
z
Hz z = ⎜
−
−10Ts
−30T
20 ⎝ z − e
z − e s ⎟⎠
()
The step response of the continuous-time system is
2 − 3e−10t + e−30t
yc t =
u t
600
and the response of the discrete-time system to a unit sequence is
()
()
⎤
−10T
−30T
−10T
−30T
Ts ⎡⎢
e s −e s
e s −10nTs
e s −30nTs ⎥
y ⎡⎣ n ⎤⎦ =
+ −10T
e
− −30T
e
u ⎡⎣ n ⎤⎦
−10Ts
−30Ts
s
s
⎥
20 ⎢ 1− e
e
−1
e
−1
1− e
⎣
⎦
(
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)(
)
30
Sampled-Data Systems
The response of the DAC is
⎛ t − Ts ( n + 1 / 2 ) ⎞
y d ( t ) = ∑ y [ n ] rect ⎜
.
⎟
T
⎝
⎠
∞
n=0
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Standard Realizations
•  Realization of a discrete-time system closely
parallels the realization of a continuous-time system
•  The basic forms, Direct Form II, cascade and
parallel have the same structure
•  A continuous-time system can be realized with
integrators, summing junctions and multipliers
•  A discrete-time system can be realized with delays,
summing junctions and multipliers
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Standard Realizations
Direct Form II
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