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 12 1 2 1 ) , ( nn 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
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2D Z Transform ff ¦ ¦ ±f ±f 12 2 1 2 1 2 1 2 1 ) , ( ) , ( nn n n z z n n x z z X ¾ makes sense when f ² ) , ( 2 1 z z X ^` f ² ³ ) , ( which for and of values ROC 2 1 2 1 z z X z z ¦¦ ± ± 2 1 2 1 2 1 ) , ( n n z z n n n N converges finite sum ± ± o  ³ 2 1 2 1 2 1 2 1 2 1 2 1 ) , ( ) , ( ) , ( ) , ( n n z z n n d z z D z z 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 X e z e z X j j ² stable system Transfer function: ^` ¦¦ ³ ³ ³ ³ 2 1 12 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( n n nn 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: ³ ³ ³ ³ ) , ( ) , ( ) , ( ) , ( 2 2 1 1 2 1 2 2 1 1 2 1 mm m n m n x m m a m n m n y m m b EEE 508 - Lecture 3
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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) 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 Im{ z 2 } 1 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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Sampling Sampling in one domain l
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eee508_lect3 - EEE 508 - Digital Image Processing and...

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