These relations describe a mapping between the variables z and s t h a k skt s

These relations describe a mapping between the

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These relations describe a mapping between the variables z and s. t h a ~  k skt s s a s e kt h t s H s H k k s d z k h z H  s skt e s st e z s t z s ln
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10 All right reserved. Copyright © 2013. Sharifah Saon Since s = σ + , where ω = continuous frequency, the complex variable z can be expressed as where = digital frequency in radians/sample. j t t j t t j e e e e e z s s s s F S f t s 2 2
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11 All right reserved. Copyright © 2013. Sharifah Saon Response Matching The idea behind response matching to match the time- domain analog and digital response for a given input, typically the impulse response or step response. Given the analog filter H ( s ) and invariance input x ( t ), find the analog response y ( t ) as the inverse transform of H ( s ) X ( s ). sampled x ( t ) and y ( t ) at intervals t s to obtain their sampled versions x [ n ] and y [ n ]. computed H ( z ) = Y ( z )/ X ( z ) to obtain the digital filter.
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12 All right reserved. Copyright © 2013. Sharifah Saon Summary - The response y ( t ) of H ( s ) matches the response y [ n ] of H ( z ) at the sampling instants t = nt s .
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13 All right reserved. Copyright © 2013. Sharifah Saon Example 6.1 Convert to H ( z ) by using: a) Impulse invariance transformation, b) Step invariance transformation.  2 1 4 s s s H
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Solution (Impulse Invariance Transformation) 14 All right reserved. Copyright © 2013. Sharifah Saon  z Y z X z Y z H z X e z z e z z z Y n n x n u e n u e n y t u e t u e t y s s s s s H s X s Y s X t t x s s s s t t nt nt t t 1 4 4 4 4 4 4 2 4 1 4 2 1 4 1 2 2 2
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Solution (Unit Invariance Transformation) 15 All right reserved. Copyright © 2013. Sharifah Saon  s s s s s s t t t t nt nt t t e z z e z z z X z Y z H z z z X e z z e z z z z z Y n u n x n u e n u e n u n y t u e t u e t u t y s s s s s s s H s X s Y S s X t u t x 2 2 2 2 1 2 1 4 2 1 2 4 1 2 2 4 2 2 4 2 2 2 1 4 2 2 1 4 1
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16 All right reserved. Copyright © 2013. Sharifah Saon The Bilinear Transformation defined by or where and t s = sampling intervals By letting σ = 0, the complex variable z is obtained in the form 1 1 z z C s s t s t z s s 2 2 s C s C z s t C 2 C j e j C j C z 1 tan 2
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17 All right reserved. Copyright © 2013. Sharifah Saon Since z = e j Ω , where Ω = 2 F is the digital frequency, then This is a non-linear relation between the analog frequency ω and the digital frequency Ω. When ω 0, Ω 0 ω , Ω . it is a one-to-one mapping that nonlinearly compresses the analog frequency range - < f < to the digital frequency range - < Ω < .
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