1 1 SIGNALS AND SYSTEMS 16 December 2006 2 Transforms are regularly used to make calculation or mathematical analysis simpler. • Examples include: – Logarithms (used to simplify multiplication) slide-rule – Laplace transform. (widely used in linear analysis, circuits & systems including control systems) – Fourier transforms. (Move us from the time to frequency domain) – The Z transform is the primary mathematical tool for the analysis and synthesis of digital filters. • The system function of a digital filter is defined as the z transform of its unit-sample response. TRANSFORMS
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SIGNALS AND SYSTEMS
16 December 2006
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Transforms are regularly used to make calculation or
mathematical analysis simpler.
• Examples include:
– Logarithms (used to simplify multiplication) slide-rule
– Laplace transform. (widely used in linear analysis, circuits &
systems including control systems)
– Fourier transforms. (Move us from the time to frequency domain)
– The Z transform is the primary mathematical tool for the analysis
and synthesis of digital filters.
• The system function of a digital filter is defined as the z transform
of its unit-sample response.
TRANSFORMS
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Z-transforms
For discrete-time systems, z-transforms play the same role of Laplace transforms do in continuous-time systems
[ ]∑∞
−∞=
−=k
kzkhzH ][
Bilateral Forward z-transform
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Region of Convergence
Region of the complex z-plane for which forward z-transform converges
Im{z}
Re{z}Entire plane
Im{z}
Re{z}Complement of a disk
Im{z}
Re{z}Disk
Four possibilities (z=0 is a special case and may or may not be included)
Im{z}
Re{z}
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Z complex……….. φjnnn erz =
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z-transform infinite sum , does not converge for all zExample
x[n] = (0.5)nu[n]
X(z) = 1/(1-0.5z-1) for |z|>0.5
Region of Convergence
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ROC Properties:1)The ROC of X(z) consist of a ring in the z-plane centered about the origin.
2) The ROC does not contains any poles
3) If x[n] is of finite duration, then ROC is the entire z-plane , except possibly z=0 and/or z=infinity
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EXAMPLE:
∑∞
−∞=
− =⇔n
nznn 1][][ δδ
With an ROC consisting of the entire z plane including zero and infinity.
Consider ]1[ −nδ1]1[]1[ −
∞
−∞=
− =−⇔− ∑ zznnn
nδδ
This z transform is well defined except at z=0;
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4) If x[n] is a right sided sequence then ROC will be outside the outside excluding 0 but including infinity.
5) If x[n] is a left sided sequence then ROC will be a circle including 0 and excluding infinity.
Im{z}
Re{z}Complement of a disk
Im{z}
Re{z}Disk
ROC is BLUE
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6) If x[n] is two sided sequence then ROC will consist of a ring in z-plane
Im{z}
Re{z}
7) If the z-transform X(z) of x[n] is rational, then its ROC is bounded by poles or extends to infinity.
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x[n] = anu[n] X(z) =1/1-az-1 for |z| > |a|.
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What happens if a < 0?x[n] = anu[n] X(z) =1/1- az-1 for |z| > |a|.
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||||1
1]1[][ 1 αα <
−⇔−−−= − zfor
aznunx n
Anti causal
(left sided signals)
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))(()(
][][ 1
1||
αααα
α−−
−⇔= −
−
zzz
nunx nTwo Sided Sequence
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Inverse z-transform1) Using Contour integration……. A difficult
procedure.
[ ] [ ] dzzzFj
kf kjc
jc
1
21 −
∞+
∞−∫=
π
2) Using long Division………..A relatively easier method
3) Decomposition using partial fractions and then applying inverse transform using z transform properties
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12][
2
2
+−
++=
zz
zzzX
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21
21
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1
21][
−−
−−
+−
++=
zz
zzzX
Ratio of polynomial z-domain functions
•Divide through by the highest power of z
EXAMPLE
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Factor denominator into first-order factors
( )11
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121
1
21][
−−
−−
−
−
++=
zz
zzzX
12
1
10 1
211
][ −− −
+−
+=z
A
z
ABzX
Use partial fraction decomposition to get first-order terms
Analysis and Characterization of LTI systems using z transform1) Causality:
A discrete time LTI system is causal if and only if ROC of its system function is the exterior of a circle, including infinity
A DT LTI system with rational transfer function is causal if and only if:
a) The ROC is exterior of a circle outside the outermost pole.
b) With H(z) expressed as a ratio of polynomials in z, the order of the numerator can not be greater than the order of the denominator
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Stability:An LTI system is stable if and only if the ROC of its system function H(z) includes the unit circle. |z| = 1
EXAMPLE:
]1[2][)2/1(][ −−−= nununh nn
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Causal and Stable
A causal LTI system with rational system function H(z) is stable if and only if all of its poles lie inside the unit circle I.e. they must all have magnitude less than 1.
Example:11/1)( −−= azzH
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Causality
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Stability
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LTI system characterized by LCCD equations
Find the impulse response of the system followed by following difference equation
y[n] – 1/2y[n-1] = x[n] + 1/3 x[n-1]
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System functions and Block Diagram representations