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eee508_lect3

# eee508_lect3 - EEE 508 Digital Image Processing and...

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EEE 508 - Digital Image Processing and Compression 2D DSP Basics: Systems Stability, 2D Sampling EEE 508 - Lecture 3

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Stability System is stable if a bounded input always results in a bounded output (BIBO) ¾ For LSI system, a sufficient condition for stability: f f f ± ¦ ¦ ²f ²f 1 2 1 2 1 ) , ( n n S n n h ¾ 2D FIR filters always stable ¾ Stability test is difficult for IIR filters ¾ Z-transform is used to study stability EEE 508 - Lecture 3
2D Z Transform ¦ ¦ f f ² ² 2 1 ) ( ) ( n n X ²f ²f 1 2 2 1 2 1 2 1 , , n n z z n n x z z ¾ makes sense when f ± ) , ( 2 1 z z X ^ ` f ± ) , ( which for and of values ROC 2 1 2 1 z z X z z ¦¦ ² ² 2 1 2 1 2 1 ) , ( ) ( n n z z n n n z z N converges finite sum ¦¦ ² ² o 1 2 2 1 1 2 2 1 2 1 2 1 2 1 2 1 ) , ( ) , ( , ) , ( n n n n n n z z n n d z z D z z X EEE 508 - Lecture 3 finite sum

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2D Z Transform Note: only if unit circle ROS ) , ( ) , ( 2 1 2 1 2 1 Z Z Z Z X e z e z X j j stable system Transfer function: ¦¦ ^ ` ¦¦ ² ² ² ² 2 1 1 2 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( n n n n n n z z n n b z z n n a z z X z z Y z z H n n h Z Linear constant-coefficients difference equation to implement IIR system: 1 2 n n ¦¦ ¦¦ ² ² ² ² 1 2 1 2 ) , ( ) , ( ) , ( ) , ( 2 2 1 1 2 1 2 2 1 1 2 1 m m m m m n m n x m m a m n m n y m m b EEE 508 - Lecture 3
2D Z Transform Analogous to 1D z-transform pole-zero plot is the Root Map in 2D : ¾ H ( z 1 , z 2 ) : 9 hold z 1 const. on the Unit (e.g. z 1 =1) and find the roots in z 2 (e.g., H (1, z 2 )=0) ( , ) 0) 9 repeat for all z 1 on the Unit Circle 9 Repeat the same thing with z 2 on Unit Circle and find roots in z 1 9 We get a pair of plots: Root Map consists of two parts I { } I { } R { } Im{ z 2 1 R { } Im{ z 1 1 9 Stable if root maps stay in Unit Circle Re{ z 2 Re{ z 1 EEE 508 - Lecture 3 Stable

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eee508_lect3 - EEE 508 Digital Image Processing and...

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