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# P15_Analog_Filters_P1.pdf

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# P15_Analog_Filters_P1.pdf

P15_Analog_Filters_P1

P15_Analog_Filters_P1

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## Weitere Verwandte Inhalte

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### P15_Analog_Filters_P1.pdf

1. 1. 1 Analog Filters Analog Filters (I) Introduction Yogananda Isukapalli
2. 2. Analog Filter Specifications w’ p w’ s w’ Analog Frequency A m p l I t u d e ds 0 1+dp 1 1-dp | ) ( | w¢ j H Pass band Stop band , for 1 ) ( 1 p p p j H w w d w d ¢ £ ¢ + £ ¢ £ - , for ) ( ¥ £ ¢ £ ¢ £ ¢ w w d w s s j H , ) ( log 20 , ) 1 ( log 20 10 10 dB dB s s p p d a d a - = - - = Where ap and as are the peak pass band ripple and minimum stop band attenuation in dB respectively 2
3. 3. 3 Analog Filter x(t) y(t) Review : å å = = + = M k k k k N k k k k dt t y d b dt t x d a t y 0 0 ) ( ) ( ) ( Example : a1= 1 and all other “a’s” and “b’s” are zero. dt t dx t y ) ( ) ( = Ideal Differentiator Example : Second order Analog Filter 2 2 2 1 1 0 ) ( ) ( dt y d b dt dy b dt dx a t x a t y + + + =
4. 4. 4 Transfer Function : å å = = - = M i i i N i i i s b s a s H 0 0 1 ) ( 2 1 1 . 2 1 ) ( 2 1 1 ) ( 2 ) ( ) ( : + + = + + = - + = s s s H s s s H dt dy dt dx t x t y Example Pole: s = -1/2 Zero: s = -1 Gain = 1/2 Im(s) Re(s) S- plane pole zero plot
5. 5. 5 Impulse Response : 0 0 t 1 0 1 1 1 1 0 0 ) ( 1 ) ( : ) ( ) ( ³ < - = - = = t t b e b a t h s b a s H Example s H L t h Stable Analog Filter : Define h(t) ® 0 as t ® ¥ Implies that all poles must lie to the left of the imaginary axis : Im(s) Re(s) S- plane Unstable Re(s)> 0 Stable Re(s)< 0 | | | | | | | | | | | | | | | | | | | | | | / / / / / / / / / / / / / / / / / / /
6. 6. 6 Stability and Pole Locations : Analog Filters with M poles: t p m t p t p m m M N m e g e g e g t h p s g p s g p s g s H p s p s p s z s z s z s G s H + + + = - + + - + - = - - - - - - = ......... ) ( ....... ) ( ) .....( )......... )( ( ) .....( )......... )( ( ) ( 2 1 2 1 2 2 1 1 2 1 2 1 Partial Fraction (no repeated poles) Since stable means h(t) ® 0 as t ® ¥ It follows that : pi < 0 If pi is a complex pole : 1 | | 0 ) Re( ) ( ) Im( ) Im( ) Re( = < = t p j i t p j t p i i i i i e p e e g t h Follows:
7. 7. 7 Analog filters are stable when their poles are in the left half of the s-plane, where as digital filters are stable when their poles are confined within the unit circle. z- plane Im(s) Re(s) S- plane Im(z) Re(z) Stable Stable Analog Filter : Magnitude Function : Magnitude Square Function : ) ( ) ( | ) ( ) ( w w w w ¢ < ¢ = = ¢ ¢ = j H j H s H j H j s w w ¢ = - = ¢ j s s H s H j H | ) ( ) ( ) ( 2
8. 8. 8 6 2 2 2 / 1 2 2 2 2 2 1 4 | ) ( | : ) 1 ( 1 | ) ( | 1 1 | ) ( | 1 1 ) ( ) ( 1 1 ) ( ) ( : 1 1 ) ( 1 1 ) ( : w w w w w w w w w ¢ + ¢ + = ¢ ¢ + = ¢ ¢ + = ¢ ¢ + = - - = - + - = - + = ¢ = j H j H j H s H s H s s H s H Follows s s H s s H j s Example Example Find H(s) or H(-s)
9. 9. 9 ) 1 )( 1 ( ) 2 ( ) ( ) 1 )( 1 ( ) 2 ( ) ( ) 1 )( 1 ( ) 1 ( ) 1 )( 1 ( ) 1 ( ) 1 )( 1 ( ) 2 )( 2 ( 1 4 | ) ( | ) ( ) ( : 2 2 2 3 2 3 3 3 6 2 2 2 2 s s s s s H s s s s s H s s s s s s s s s s s s s s s j H s H s H Follows + - - - = - + + + + = + - + = + + + - = - + - + - = - - = - = ¢ ¢ = - w w H(s) : poles in the left - half of the s - plane H(-s) : poles in the right - half of the s - plane