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Example
Suppose a pair of rabbits may born a pair of little rabbits every month, The pair of little rabbits are capability of procreation after a month. For only a pair of little rabbits in the first month, then to determine the number of pairs of rabbits in the n month.
Example 4
Consider a LTI discrete - system described by the difference equation : y (n 2) 3 y (n 1) 2 y (n) x(n 1) 2 x(n) where x(n) 2 u (n),
n
y zi (0) 0,
y zi (1) 1
determine its entire response y (n) ?
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Solution : (1) y zi ( n) ? y ( n 2) 3 y ( n 1) 2 y ( n) 0
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Therefore, the number of pairs of rabbits in n month is : y (n) 2 y (n 2) [ y (n 1) y (n 2)] y (n) y (n 1) y (n 2) 0
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(2)
y zs (n) ? Define x(n) (n) y (n) h(n)
h(n 2) 3h(n 1) 2h(n) (n 1) 2 (n) n 2 h(0) 0 n 1 h(1) 1 n 0 h(2) 3 2 1 n 1 h(3) 3 2 1 n 2 h ( 4) 3 2 1 .......... ..... h(n) u (n 1)
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Solution : 1 y (n) y (n 1) x(n) 3 1 1 y (0) y (1) (0) 0 1 1 3 3 1 1 1 y (1) y (0) (1) 1 0 3 3 3 1 1 1 1 2 y (2) y (1) (2) 0 ( ) 3 3 3 3 .......... ... 1 n y ( n) ( ) u ( n) 3
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Example 2
To find the solution of this difference equation : y (n) 2 y (n 1) x(n) x(n 1) where x(n) n 2 , y (1) 1.
a a
m n n
m
[u (m N )a ( n m ) u (n m)]
m 0 n 1 1 a n n
m N
n
a m a N a n 1 1 a
1
1 a
u ( n) a n
u (n N )
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This is a difference equation
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8.3 Solution of Difference Equation
① According to the difference equation, determine the numerical value of y(n) with initial value . ② Define y(n)=y1(n)+y2(n)，where y1(n) is the solution of the equation which right-side is zero, y2(n) is a special solution of the difference equation. ③ Define y(n)=yzi(n)+yzs(n)，yzi(n) is the zero-input response of the system， yzs(n) is the zero-state response of the system and yzs(n)=x(n)*h(n). ④ Determine the solution of the difference equation with Ztransform.
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Solution : (1) Consider the difference equation : y (n) 2 y (n 1) 0 Define y1 (n) A n then A n 2 A n 1 0 α 2 0 α 2 y1 (n) A(2) n
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(3) The entire solution : 2 1 y (n) y1 (n) y 2 (n) A(2) n 3 9 8 y (1) 1 A 9 8 2 1 n y (n) (2) n 9 3 9
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(2) Consider the right - side of difference equation : x(n) x(n 1) n 2 (n 1) 2 2n 1 Define the special solution : y 2 (n) B1n B2 B1n B2 2 B1 (n 1) 2 B2 2n 1 2 1 B1 , B2 3 9 2 1 y 2 ( n) n 3 9
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Properties of LTI System
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8.2 Difference Equation
A discrete system can be represented with a difference equation A difference equation is composed with sequences as …… , x(n+2), x(n+1), x(n), x(n1), x(n-2), x(n-3), ……
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Define y (n) represent the number of pairs of rabbits in n month, then : y (0) 0 y (4) 3 y (n 2) y (1) 1 y (2) 1 y (3) 2 y (5) 5 .......... y (n 1) y ( n)
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y zs ( n) x(n) h(n) [2 n u (n)] [u (n 1)] (2 n 1)u (n 1)
(3)
y ( n) ? y ( n) y zi (n) y zs ( n) ( 1 2 )u ( n) (2 1)u (n 1) 2(2 1)u (n)
2 3 2 0 1 1, 2 2
y zi ( n) A1 (1) n A2 ( 2) n y zi (0) 0 A1 1 y zi (1) 1 A2 1 y zi ( n) ( 1 2 n )u ( n)
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Solution : y ( n) x ( n) h( n)
m
[u (m) u (m N )] a ( n m ) u (n m) [u (m)a ( n m ) u (n m)] a m a n
in n month, there are y (n-2) pairs of rabbits are capability of procreatio n the number of pairs of these rabbits is y (n 2) 2 y (n 2). at the same time, there are [ y (n 1) y (n 2) ] pairs of rabbits are not capability of procreatio n the number of pairs of these rabbits is only [ y (n 1) y (n 2)].
Chapter 8
Analyzing DT System In TIME Domain
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8.1 Mathematic Modes of DT System
Mathematic Description: A discrete system can be represented with a difference equation. Function: Complete the calculation from x(n) to y(n).
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Example 1
To find the solution of this difference equation : 1 y (n) y (n 1) x(n) 3 where x(n) (n), y (1) 0
n
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Example 3
Consider a discrete - system, If its unit - impulse response h(n) a n u (n), where 0 a 1, Suppose input x(n) u (n) u (n N ), determine its zero state response y (n).
