Ch9(2)Handouts_3e
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1 1 Copyright © 2005, S. K. Mitra Spectral Transformations of Spectral Transformations of IIR Digital Filters IIR Digital Filters • Objective - Transform a given lowpass digital transfer function to another digital transfer function that could be a lowpass, highpass, bandpass or bandstop filter • has been used to denote the unit delay in the prototype lowpass filter and to denote the unit delay in the transformed filter to avoid confusion ) ( z G L ) ˆ ( z G D 1 ˆ − z 1 − z ) ( z G L ) ˆ ( z G D 2 Copyright © 2005, S. K. Mitra Spectral Transformations of Spectral Transformations of IIR Digital Filters IIR Digital Filters • Unit circles in z- and -planes defined by , • Transformation from z-domain to -domain given by • Then z ˆ z ˆ ω = j e z ω = ˆ ˆ j e z ) ˆ ( z F z = )} ˆ ( { ) ˆ ( z F G z G L D = 3 Copyright © 2005, S. K. Mitra Spectral Transformations of Spectral Transformations of IIR Digital Filters IIR Digital Filters • From , thus , hence • Recall that a stable allpass function A(z) satisfies the condition ) ˆ ( z F z = ) ˆ ( z F z = ⎪ ⎩ ⎪ ⎨ ⎧ < < = = > > 1 if , 1 1 if , 1 1 if , 1 ) ˆ ( z z z z F 4 Copyright © 2005, S. K. Mitra Spectral Transformations of Spectral Transformations of IIR Digital Filters IIR Digital Filters • Therefore must be a stable allpass function whose general form is ⎪ ⎩ ⎪ ⎨ ⎧ < > = = > < 1 if , 1 1 if , 1 1 if , 1 ) ( z z z z A ) ˆ ( / 1 z F 1 , ˆ ˆ 1 ) ˆ ( 1 1 * < α ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ α − α − ± = ∏ = l l l l L z z z F 5 Copyright © 2005, S. K. Mitra Lowpass Lowpass- to to- Lowpass Lowpass Spectral Transformation Spectral Transformation • To transform a lowpass filter with a cutoff frequency to another lowpass filter with a cutoff frequency , the transformation is where α is a function of the two specified cutoff frequencies ) ( z G L ) ˆ ( z G D c ω c ω ˆ α − α − = = − z z z F z ˆ ˆ 1 ) ˆ ( 1 1 6 Copyright © 2005, S. K. Mitra Lowpass Lowpass- to to- Lowpass Lowpass Spectral Transformation Spectral Transformation • On the unit circle we have • From the above we get • Taking the ratios of the above two expressions ω − ω − ω − α − α − = ˆ ˆ 1 j j j e e e ) 2 / ˆ tan( 1 1 ) 2 / tan( ω ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ α − α + = ω ω − ω − ω − ω − ω − α − ⋅ α ± = α − α − = ˆ ˆ ˆ ˆ 1 1 ) 1 ( 1 1 1 j j j j j e e e e e m m m
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Transcript of Ch9(2)Handouts_3e
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11Copyright 2005, S. K. Mitra
Spectral Transformations of Spectral Transformations of IIR Digital FiltersIIR Digital Filters
Objective - Transform a given lowpass digital transfer function to another digital transfer function that could be a lowpass, highpass, bandpass or bandstop filter
has been used to denote the unit delay in the prototype lowpass filter and to denote the unit delay in the transformed filter to avoid confusion
)(zGL)(zGD
1z1z
)(zGL
)(zGD2
Copyright 2005, S. K. Mitra
Spectral Transformations of Spectral Transformations of IIR Digital FiltersIIR Digital Filters
Unit circles in z- and -planes defined by ,
Transformation from z-domain to -domain given by
Then
z
z
= jez = jez
)(zFz =
)}({)( zFGzG LD =
3Copyright 2005, S. K. Mitra
Spectral Transformations of Spectral Transformations of IIR Digital FiltersIIR Digital Filters
From , thus , hence
Recall that a stable allpass function A(z) satisfies the condition
)(zFz = )(zFz =
>
1if,11if,11if,1
)(zzz
zF
4Copyright 2005, S. K. Mitra
Spectral Transformations of Spectral Transformations of IIR Digital FiltersIIR Digital Filters
Therefore must be a stable allpassfunction whose general form is
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