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8/13/2019 3 Kundur DiffEq Implement Handouts
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Difference Equations and Implementation 2.4 Difference Equations
System RealizationThere is a practical and computationally efficient means of implementing all FIR and a family of IIR systems that makes use of . . .
. . . difference equations.
All LTI systems
All systems
Professor Deepa Kundur (University of Toronto )Difference Equations and Implementation 5 / 23
Difference Equations and Implementation 2.4 Difference Equations
System RealizationThere is a practical and computationally efficient means of implementing all FIR and a family of IIR systems that makes use of . . .
. . . difference equations.
All LTI FIR systems All LTI IIR systems
All LTI systems
All systems
Professor Deepa Kundur (University of Toronto )Difference Equations and Implementation 6 / 23
Difference Equations and Implementation 2.4 Difference Equations
System RealizationThere is a practical and computationally efficient means of implementing all FIR and a family of IIR systems that makes use of . . .
. . . difference equations.
All LTI FIR systems All LTI IIR systems
All LTI systems
SystemsDescribedby LCCDEs
All systems
Professor Deepa Kundur (University of Toronto )Difference Equations and Implementation 7 / 23
Difference Equations and Implementation 2.4 Difference Equations
System Realization
General expression for N th-order LCCDE:
N
k =0
ak y (n k ) =M
k =0
b k x (n k ) a0 1
Initial conditions: y ( 1), y ( 2), y ( 3), . . . , y ( N ).
Need: (1) constant scale, (2) addition, (3) delay elements.
Professor Deepa Kundur (University of Toronto )Difference Equations and Implementation 8 / 23
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Difference Equations and Implementation 2.5 Implementation of Discrete-time Systems
Building Block Elements
Constant multiplier:
Signal multiplier: +
Unit delay:
Unit advance:
Adder: +
Professor Deepa Kundur (University of Toronto )Difference Equations and Implementation 9 / 23
Difference Equations and Implementation 2.5 Implementation of Discrete-time Systems
FIR System Realization
Finite Impulse Response Systems and Nonrecursive Implementation
Professor Deepa Kundur (University of Toronto )Difference Equations and Implementation 10 / 23
Difference Equations and Implementation 2.5 Implementation of Discrete-time Systems
FIR System Realization: Example Consider a 5-point local averager:
y (n) = 15
n
k = n 4
x (k ) n = 0 , 1, 2, . . .
The impulse response is given by:
h(n) = 1
5
n
k = n 4
(k )
= 1
5 (n 4) +
15
(n 3) + 15
(n 2) +
15
(n 1) + 15
(n)
Professor Deepa Kundur (University of Toronto )Difference Equations and Implementation 11 / 23
Difference Equations and Implementation 2.5 Implementation of Discrete-time Systems
FIR System Realization: Example Consider a 5-point local averager:
y (n) = 15
n
k = n 4
x (k ) n = 0 , 1, 2, . . .
Indeed FIR!
Professor Deepa Kundur (University of Toronto )Difference Equations and Implementation 12 / 23
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Difference Equations and Implementation 2.5 Implementation of Discrete-time Systems
FIR System Realization: Example Consider a 5-point local averager:
y (n) = 15
n
k = n 4
x (k ) n = 0 , 1, 2, . . .
Memory requirements stay constant; only need to store 5 values(4 last + present).
xed number of adders required
Professor Deepa Kundur (University of Toronto )Difference Equations and Implementation 13 / 23
Difference Equations and Implementation 2.5 Implementation of Discrete-time Systems
FIR System Realization: Example
y (n) = 1
5
n
k = n 4
x (k ) =n
k = n 4
15
x (k )
y (n) = 1
5x (n 4) +
15
x (n 3) + 15
x (n 2) +
1
5x (n 1) +
1
5x (n)
+ + + +
1/5 1/5 1/5 1/5 1/5
Professor Deepa Kundur (University of Toronto )Difference Equations and Implementation 14 / 23
Difference Equations and Implementation 2.5 Implementation of Discrete-time Systems
FIR System Realization: General
y (n) =M 1
k =0
b k x (n k )
+ + + + +
...
...Requires:
M multiplications M 1 additions M 1 memory elements
Professor Deepa Kundur (University of Toronto )Difference Equations and Implementation 15 / 23
Difference Equations and Implementation 2.5 Implementation of Discrete-time Systems
FIR System Realization
Innite Impulse Response Systems and Recursive Implementation
Professor Deepa Kundur (University of Toronto )Difference Equations and Implementation 16 / 23
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Difference Equations and Implementation 2.5 Implementation of Discrete-time Systems
IIR System Realization: Example Consider an accumulator:
y (n) =n
k =0
x (k ) n = 0 , 1, 2, . . . for y ( 1) = 0.
The impulse response is given by:
h(n) =n
k =0
(k ) = (n) + (n 1) + (n 2) +
= 1 n 0
0 n < 0h[n]
n
1
0 1 2 3 4
. . .
Professor Deepa Kundur (University of Toronto )Difference Equations and Implementation 17 / 23
Difference Equations and Implementation 2.5 Implementation of Discrete-time Systems
IIR System Realization: Example Consider an accumulator:
y (n) =n
k =0
x (k ) n = 0 , 1, 2, . . . for y ( 1) = 0.
IIR memory requirements seem to grow with increasing n!
Professor Deepa Kundur (University of Toronto )Difference Equations and Implementation 18 / 23
Difference Equations and Implementation 2.5 Implementation of Discrete-time Systems
IIR System Realization: Example
y (n) =n
k =0
x (k )
=n 1
k =0x (k ) + x (n)
= y (n 1) + x (n) y (n) = y (n 1) + x (n)
+
recursive implementation
Professor Deepa Kundur (University of Toronto )Difference Equations and Implementation 19 / 23
Difference Equations and Implementation 2.5 Implementation of Discrete-time Systems
Direct Form I vs. Direct Form II Realizations
y (n) = N
k =1
ak y (n k ) +M
k =0
b k x (n k )
is equivalent to the cascade of the following systems:
v (n)
output 1=
M
k =0
b k x (n k )
input 1nonrecursive
y (n)
output 2=
N
k =1
ak y (n k ) + v (n)
input 2recursive
Professor Deepa Kundur (University of Toronto )Difference Equations and Implementation 20 / 23
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